Macroscopic and microscopic stabilization mechanisms of unstable interface with interfacial mass flux

Ilyin, D. V.; Abarzhi, S. I.

doi:10.1063/5.0040842

Unstable interfaces are omnipresent in plasma processes in nature and technology at astrophysical and at molecular scales. This work investigates the interface dynamics with interfacial mass flux and focuses on the interplay of macroscopic and microscopic stabilization mechanisms, due to the inertial effect and the surface tension, respectively, with the destabilizing acceleration. We derive solutions for the interfacial dynamics conserving mass, momentum, and energy and find the critical values of the acceleration, density ratio, and surface tension separating the stable and unstable regimes. While the surface tension influences only the interface, its presence leads to the formation of vortical structures in the bulk. The vortical structures are energetic in nature, and the velocity field is shear free at the interface. We find that the conservative dynamics is unstable only when it is accelerated and when the acceleration value exceeds a threshold combining the contributions of macroscopic and microscopic mechanisms. In the unstable regime, the interface dynamics corresponds to the standing wave with the growing amplitude and has the growing interface velocity. For strong accelerations and weak surface tensions typical for high energy density plasmas, the unstable conservative dynamics is the fastest when compared to other instabilities; it has finite values of the initial perturbation wavelength at which the interface is stabilized and at which its growth is the fastest. We elaborate extensive theory benchmarks for experiments and simulations and outline its outcomes for application problems in nature and technology.

I. INTRODUCTION

Non-equilibrium transport, interfaces, and mixing are omnipresent in plasma processes in nature and technology, at astrophysical and at molecular scales, in high and in low energy density regimes.¹ Examples include thermonuclear flashes in type-Ia supernova, coronal mass ejections in the solar flares, downdrafts in planetary magneto-convection, plasma instabilities in the Earth ionosphere, unstable laser ablated plasmas in inertial confinement fusion, and plasma thrusters.^2–12,30,35 While plasma processes are necessarily electro-magnetic with charged particles and magnetic fields, the better understanding of non-equilibrium dynamics of interfaces and mixing in neutral plasmas (fluids) is required for the grip and control of realistic plasmas at macroscopic (hydrodynamic) scales.^1–16,35 This work investigates the interface dynamics with the interfacial mass flux and focuses on the interplay of the macroscopic and microscopic stabilization mechanisms with the destabilizing acceleration.

In plasmas, the interface is a boundary separating the phases of matter with distinct thermodynamics properties and having the interfacial mass flux, such as hot and cold plasmas in laser fusion.^1–19 The dynamics of interfaces with interfacial mass flux is a long-standing problem in science, mathematics, and engineering; it is challenging to study in theory, experiments, and simulations.^1–19 To tackle these frontiers, the theory of interface dynamics was recently developed.^17–19 It elaborated the general framework for the problem of interface stability, directly linked the microscopic interfacial transport to macroscopic flow fields, and identified the inertial mechanism of the interface stabilization and the new fluid instability.^17–19,30 The theory^17–19,30 considered the ideal dynamics free from microscopic stabilization mechanisms caused by interactions of particles at molecular scales.¹⁶ These microscopic mechanisms may be present in realistic plasmas and may include radiation transport, dissipation, diffusion, and other effects.^{3–12,20–24} Their influences on the interface dynamics call for systematic investigations.^1,17,30

The dynamics of neutral plasmas (ideal fluids) is governed at continuous scales by the laws of conservation of mass, momentum, and energy, with the closure equation of state.¹⁷ For realistic plasmas, the set of the governing equations is more complicated and may include the entropy equation and the magneto-hydrodynamics.¹⁶ Among a broad range of microscopic processes that can stabilize the dynamics, there exists the process that influences only the boundary conditions at the interface and keeps the governing equations in the bulk unaffected. This is the surface tension understood as a tension at the phase boundary.^16,17

Physically, surface tension is always present in a multiphase matter, because at microscopic scales it is caused by the anisotropy of interactions between the particles near the interface which results in energy consumption with the increasing interface area.^16,17 For instance, in laser ablated plasmas in fusion relevant conditions with negligible effects of dissipation and diffusion, the fluid phases (the hot and cold plasmas) are formed by the sharply and rapidly changing flow fields, and the interface between these phases has the surface energy.^4–7 In experiments on pulsed laser–matter interactions with solid targets, the surface tension is known to play a significant role, especially when the density changes strongly, e.g., at the end of the pulse and at the onset of solidification.¹¹ The surface tension can be essential for plasma discharges formed in and interfacing with liquids—an emerging topic of plasma physics research.¹² Besides, surface tension is often an important stabilizing factor in Lagrangian methods of numerical simulations, which are valuable tools for the investigations and modeling of the multiphase matter at the extremes.^13–15

For the interface dynamics with the interfacial mass flux, the theory^17–19,30 found that the conservative dynamics (i.e., the dynamics conserving mass, momentum, and energy in the bulk and at the interface) has potential velocity fields in the bulk, is shear free at the interface, and is stabilized by the macroscopic inertial mechanism. The theory^17–19 also found the extreme sensitivity of the flow fields in the bulk to the boundary conditions at the interface and revealed that the formation of vortical structures in bulk is associated with the energy imbalance at the interface.

Questions thus appear: What may the qualitative and quantitative influence be of microscopic stabilization mechanisms on the interface stability and the structure of flow fields? What may the nature of the vortical field be, and is it produced “energetically” or “dynamically?” How may the macroscopic and microscopic stabilizations interplay with the destabilizing acceleration, and what are the conditions for the macroscopic mechanism to dominate the microscopic stabilization? These issues are critical to address in order to better understand non-equilibrium dynamics of interfaces and mixing in high energy density settings in nature, and elaborate the means of control of plasma processes in technology.

In this work, we investigate the interface dynamics with the interfacial mass flux and we focus on the interplay of macroscopic and microscopic stabilization mechanisms due to the inertial effect and surface tension, respectively, with the destabilizing acceleration. We choose the surface tension, since this microscopic process impacts only the interface.

We find that even the surface tension influences only the interface, it leads to the formation of vortical structures in the bulk. The resultant dynamics couples the interface perturbations with potential and vortical components of the velocity field in the bulk. The volumetric vortical field is energetic in nature, since it is produced by the excess of surface energy; and, yet, the velocity field is shear free at the interface. The conservative dynamics is unstable only when it is accelerated and when the acceleration value exceeds a threshold value combining contributions of the macroscopic inertial and the microscopic surface tension mechanisms. The interface dynamics is dominated by the interplay of the macroscopic mechanisms—the destabilizing acceleration and the stabilizing inertial effect, whereas vortical structures are created in the bulk to balance the energy excess produced by surface tension at microscopic scales at the interface. The instability of the conservative dynamics has unique properties unambiguously differentiating it from other instabilities.^25–27 Particularly, for strong acceleration and vanishing surface tension values typical for high energy density plasmas, it has the fastest growth and the largest perturbation wavelength at which this growth is achieved. Based on the obtained results, we identify the theory benchmarks for future experiments and simulations and outline its outcomes for application problems in nature and technology.^{1–15,20–24}

II. METHOD

A. Governing equations

In the inertial frame of reference, the dynamics of neutral plasmas (ideal fluid) is governed by the conservation of mass, momentum, and energy as

\frac{\partial ρ}{\partial t} + \frac{\partial ρ v_{i}}{\partial x_{i}} = 0, \frac{\partial ρ v_{i}}{\partial t} + \frac{\partial ρ v_{i} v_{j}}{\partial x_{j}} + \frac{\partial P}{\partial x_{i}} = 0, \frac{\partial E}{\partial t} + \frac{\partial (E + P) v_{i}}{\partial x_{i}} = 0.

(1)

Here, $x_{i}$ are the spatial coordinates, $(x_{1}, x_{2}, x_{3}) = (x, y, z)$ ⁠, $t$ is time, $(ρ, v, P, E)$ are the fields of density $ρ$ ⁠, velocity $v$ ⁠, pressure $P$ and energy density $E = ρ (e + v^{2} / 2)$ ⁠, and $e$ is the specific internal energy.^16–19 The inertial frame of reference is referred to the frame of reference moving with constant velocity ${\tilde{V}}_{0}$ ⁠; for definiteness ${\tilde{V}}_{0} = (0, 0, {\tilde{V}}_{0})$ ⁠.^17,18 The governing equations (1) are augmented with the closure equation—the equation of state associating the internal energy and pressure.¹⁷ For plasmas, one usually employs the equation of state $P = (2 / 3) ρ E$ ⁠.^16,34,35

We mark the fields of the heavy (light) fluid as ${(ρ, v, P, E)}_{h (l)}$ ⁠, and we introduce a continuous local scalar function $θ (x, y, z, t)$ to describe the fluid interface. The function value is $θ = 0$ at the interface and it is $θ > 0$ (⁠ $θ < 0$ ⁠) in the heavy (light) fluid.^{21–23,28,29} With the Heaviside step-function $H (θ)$ ⁠, the flow fields are $(ρ, v, P, E) = {(ρ, v, P, E)}_{h} H (θ) + {(ρ, v, P, E)}_{l} H (- θ)$ ⁠.^17–19

At the interface, the balance of fluxes of mass and normal and tangential components of momentum and energy obey the boundary conditions

[\tilde{j} \cdot n] = 0, [(P + \frac{{(\tilde{j} \cdot n)}^{2}}{ρ}) n] = 0, [\frac{(\tilde{j} \cdot n) (\tilde{j} \cdot τ)}{ρ} τ] = 0, [(\tilde{j} \cdot n) (W + \frac{{\tilde{j}}^{2}}{2 ρ^{2}})] = 0,

(2)

where the jump of functions across the interface is denoted with $[\dots]$ ⁠; the unit vectors normal and tangential at the interface are $n$ and $τ$ with $n = \nabla θ / | \nabla θ |$ and $(n \cdot τ) = 0$ ⁠; the mass flux across the moving interface is $\tilde{j} = ρ (n \dot{θ} / | \nabla θ | + v)$ ⁠; and the specific enthalpy is $W = e + P / ρ$ in physics definition.^15–19

The interfacial boundary conditions Eq. (2) are derived from the governing equations (1) independently of the equation of state and hold true for ideal fluids with any equation of state.^16–19 Particularly, the boundary conditions in Eq. (2) are valid for compressible and incompressible ideal fluids, for two- and three-dimensional flows, for arbitrary positioning of the interface relative the mass flux, and for small and large perturbations.^17–19 The interfacial boundary conditions Eq. (2) are exact since they are derived from the conservation laws Eq. (1) in the inertial frame of reference and are independent of the velocity of the inertial frame of reference. This general formulation allows us to stay free from the postulate of constancy of the interface velocity and to examine and quantify the sensitivity of the dynamics to the boundary conditions at the interface, including the structure of the flow fields and the interface stability.^16–19,30

We consider the spatially extended flow, which is unbounded in the $z$ direction and is periodic in the $(x, y)$ plane. The heavy (light) fluid is located in the lower (upper) sub-domain. The boundary conditions at the outside boundaries of the domain are

{v_{h} |}_{z \to - \infty} = V_{h} = (0, 0, V_{h}), {v_{l} |}_{z \to + \infty} = V_{l} = (0, 0, V_{l})

(3)

with the constant velocity magnitude(s) $V_{h (l)}$ ⁠, Fig. 1. The boundary conditions Eq. (3) hold true for both compressible and incompressible fluids and for any equation of state.^16–19,30 Figure 1 illustrates the schematics of the dynamics.

FIG. 1.

View large Download slide

Schematics of the dynamics in a far field (not to scale). Blue color marks the planar (dashed line) interface and the perturbed (solid line) interface.

The inertial frame of reference moves with constant velocity ${\tilde{V}}_{0}$ relative the laboratory frame of reference. The interface velocity in the laboratory frame of reference is $\tilde{V}$ ⁠. For a steady planar interface normal to the mass flux, the interface velocity is constant. The velocity of the inertial frame of reference can be chosen equal to the velocity of the steady planar interface in the laboratory frame of reference, ${\tilde{V}}_{0} = \tilde{V}$ ⁠. For a non-steady non-planar interface arbitrary positioned relative to the mass flux, the velocity of the inertial frame of reference ${\tilde{V}}_{0}$ and the interface velocity $\tilde{V}$ are distinct quantities, $\tilde{V} \neq {\tilde{V}}_{0}$ ⁠.^17,18 In this general case, the interface velocity $\tilde{V}$ obeys the relation

\tilde{V} n = - {vn |}_{θ \to 0^{+}} = - {(\tilde{j} / ρ) n |}_{θ \to 0^{+}} .

(4)

As per the usual convention,¹⁶ there is an important particular case, in which the heavy fluid is at rest in the laboratory frame of reference. In this case, the velocity of the inertial frame of reference ${\tilde{V}}_{0}$ is the same as the velocity of the steady planar interface. This leads to ${\tilde{V}}_{0} = - V_{h}$ with $V_{0} = | V_{0} | = V_{h}$ ⁠.^17–19,30 This situation often applies for dynamics of ideal fluids free from thermal heat flux Eq. (1). It may occur in steady flames with uniform flow fields, when in the laboratory frame of reference the heavy fluid (unburned gas) is at rest and the light fluid (burned gas) and the interface both move, whereas in the moving frame of reference, the steady planar interface is at rest and the heavy and light fluids both move. In the existing models of laser ablated plasmas with thermal heat flux the interface velocity (i.e., the ablation velocity) is a free parameter and its magnitude is adjusted by the comparison with data.^31–37

The initial conditions are the initial perturbations of the interface and the flow fields. For ideal fluids, they define the dimensionality, the symmetry, the length-scale, and the timescale of the dynamics.

The dynamics is the subject to the acceleration and the surface tension. The destabilizing acceleration $g$ is directed along the $z$ direction from the heavy fluids to the light fluid, $g = (0, 0, g)$ ⁠. The interfacial surface tension is the tension between the phases with the coefficient $σ, σ \geq 0$ ⁠.^16,17

We consider a sample case of a two-dimensional flow periodic in the $x$ direction, free from motion in the $y$ direction and spatially extended in the $z$ direction. The interfacial function $θ$ is set as

θ = - z + z^{*} (x, t), \dot{θ} = \frac{\partial z^{*}}{\partial t}, \nabla θ = (\frac{\partial z^{*}}{\partial x}, 0, - 1), | \nabla θ | = \sqrt{1 + {(\frac{\partial z^{*}}{\partial x})}^{2}} .

(5)

B. Linearized dynamics

Perturbations: The governing equations (2)–(4) are challenging. They can be simplified by the conditions of small perturbations, mass flux directionality, and incompressibility. Indeed, the unperturbed flow fields are uniform ${\tilde{j}, v, P, W} = {J, V, P_{0}, W_{0}}$ ⁠. We slightly perturb the flow fields as $\tilde{j} = J + j$ ⁠, $v = V + u$ ⁠, $P = P_{0} + p$ ⁠, and $W = W_{0} + w$ ⁠, with $| j | ≪ | J |$ ⁠, $| u | ≪ | V |$ $| p | ≪ | P_{0} |$ and $| w | ≪ | W_{0} |$ ⁠, and the fluid density as $ρ \to ρ + δ ρ$ with $| δ ρ | ≪ | ρ |$ ⁠. The small perturbations decay away from the interface. For the unperturbed interface, the normal and tangential unit vectors are ${n, τ} = {n_{0}, τ_{0}}$ ⁠; for the slightly perturbed interface $n = n_{0} + n_{1}$ and $τ = τ_{0} + τ_{1}$ ⁠, with $| n_{1} | ≪ | n_{0} |$ ⁠, $| τ_{1} | ≪ | τ_{0} |$ ⁠, and $| \dot{θ} / | \nabla θ | | ≪ | V |$ ⁠.

Physics assumptions: We presume that the leading order mass flux $J = ρ V$ is normal to the interface, with $J \cdot τ_{0} = 0$ ⁠, and $J \cdot n_{1} = 0$ for a two-dimensional flow. The dynamics are incompressible, with the speed of sound $c$ being the largest velocity scale. This leads to $(P_{0} + {(J \cdot n_{0})}^{2} / ρ) \to P_{0}$ and $(W_{0} + J^{2} / 2 ρ^{2}) \to W_{0}$ for $(| V | / c) \to 0$ ⁠. Density variations are negligible $| δ ρ / ρ | ≪ | u | / | V |$ ⁠.^16–19 We emphasize that the density variations are negligible within the distinct fluid phase, i.e., with $| δ ρ_{h (l)} / ρ_{h (l)} | ≪ 1$ in the heavy (light) fluid, whereas the ratio of the fluid densities may vary broadly $(ρ_{h} / ρ_{l}) \in [1, \infty)$ with the Atwood number $A = (ρ_{h} - ρ_{l}) / (ρ_{h} + ρ_{l})$ varying as $A \in [0, 1]$ ⁠.³⁰

Boundary conditions: To the leading order, the boundary conditions at the interface are

[J_{n}] = 0, [P_{0} n_{0}] = 0, [J_{n} W_{0}] = 0,

(6a)

where $J_{n} = J \cdot n_{0}$ ⁠, and the uniform flow fields in the bulk are

{(ρ, v, P, W)}_{h (l)} = {(ρ, V, P_{0}, W_{0})}_{h (l)} .

(6b)

In Eq. (6a) the specific enthalpy is $W_{0} = {\bar{W}}_{0} + C_{P} Θ$ ⁠, where $C_{P}$ is the specific heat at constant pressure and $Θ$ is the temperature. In the incompressible limit, the physics enthalpy $W_{0}$ (accounting for the enthalpy of formation) is continuous at the interface $[W_{0}] = 0$ ⁠, whereas the value ${\bar{W}}_{0}$ has a jump at the interface $[{\bar{W}}_{0}] = \bar{Q}$ ⁠, where $\bar{Q} = - [C_{p} Θ]$ is the specific heat release.^16–19,30 The enthalpy ${\bar{W}}_{0}$ is often applied in engineering applications to calibrate the heat flux, since for a given system its value is well-known and tabulated.^{16,21–24,31–35}

To the first order, the boundary conditions at the interfaces are

\begin{matrix} [j \cdot n_{0}] = 0, [((p + \frac{2 (J \cdot n_{0})}{ρ}) (j \cdot n_{0})) n_{0}] = 0, \\ [(J \cdot n_{0}) \frac{(J \cdot τ_{1} + j \cdot τ_{0})}{ρ} τ_{0}] = 0, [(J \cdot n_{0}) (w + \frac{(J \cdot j)}{ρ^{2}})] = 0, \end{matrix}

(7a)

where $j_{n} = j \cdot n_{0}$ and $j_{τ} = j \cdot τ_{0}$ ⁠. The perturbed flow fields in the bulk are^16–19

\nabla \cdot u = 0, \dot{u} + (V \cdot \nabla) u + \frac{\nabla p}{ρ} = 0,

(7b)

where the fields $(ρ, V, u, p)$ are ${(ρ, V, u, p)}_{h (l)}$ in the bulk of the heavy (light) fluid. The boundary conditions for the perturbed flow fields away from the interface are

{u_{h} |}_{z \to - \infty} = 0, {u_{l} |}_{z \to + \infty} = 0.

(7c)

The perturbed velocity of the interface is $\tilde{V} = {\tilde{V}}_{0} + \tilde{v}$ ⁠, $| \tilde{v} | ≪ | {\tilde{V}}_{0} |$ ⁠. Up to the first order, it is

\tilde{V} = {\tilde{V}}_{0} + \tilde{v}, \tilde{v} n_{0} = - {(u_{h} n_{0} + \dot{θ}) |}_{θ = 0^{+}} .

(8)

C. Interfacial shear and interfacial boundary conditions

To conclude the formulation of the boundary conditions, we emphasize that, in excellent agreement with the classical results,¹⁶ for the interface dynamics with interfacial mass flux, the tangential component of the velocity is continuous at the interface, and the velocity field is shear-free at the interface Eqs. (2), (6), and (7).^17–19,30

Indeed, in Eq. (2), since $[\tilde{j} \cdot n] = 0$ and $\tilde{j} = ρ (n \dot{θ} / | \nabla θ | + v)$ ⁠, the condition of continuity of the tangential component of momentum at the interface $[(\tilde{j} \cdot n) ((\tilde{j} \cdot τ) / ρ) τ] = 0$ is equivalent to the condition of continuity of the tangential component of velocity at the interface $[v \cdot τ] = 0$ ⁠. This implies that for the dynamics of the interface with the interfacial mass, the velocity field is shear-free at the interface, in agreement with Ref. 16. To the zeroth order in Eq. (2), the condition $[J_{n} ((J \cdot τ_{0}) / ρ) τ_{0}] = 0$ is transformed to $[V \cdot τ_{0}] = 0$ [note that $V \cdot τ_{0} = 0$ is identically zero Eq. (6a)]. To the first order in Eqs. (2) and (7a), the condition $[J_{n} (J \cdot τ_{1} + j \cdot τ_{0}) / ρ] = 0$ is reduced to $[V \cdot τ_{1} + u \cdot τ_{0}] = 0$ ⁠. The latter is the first order component of the tangential velocity $v = V + u$ at the perturbed interface with the tangential vector $τ = τ_{0} + τ_{1}$ [for a two-dimensional flow with interfacial function $θ$ in Eq. (5) these are $τ_{0} = (1, 0, 0)$ and $τ_{1} = (0, 0, \partial z^{*} / \partial x)$ ]. While the tangential component of the perturbed velocity $u \cdot τ_{0}$ may have a jump at the interface, the tangential component of the velocity $v$ is continuous at the interface $[v \cdot τ] = 0$ ⁠, including the zeroth order $[V \cdot τ_{0}] = 0$ and the first order $[V \cdot τ_{1} + u \cdot τ_{0}] = 0$ ⁠. For the dynamics of the interface with the interfacial mass flux, the velocity is shear-free at the interface.

There is the important particular case in the boundary conditions Eq. (2), when the mass flux is conserved at the interface $[\tilde{j} \cdot n] = 0$ and when it is zero at the interface ${\tilde{j} \cdot n |}_{θ = 0} = 0$ ⁠. In this case, the condition for the conservation of tangential component of momentum in Eq. (2) $[(\tilde{j} \cdot n) ((\tilde{j} \cdot τ) / ρ) τ] = 0$ is reduced to $[\tilde{j} \cdot τ] = arbitrary$ ⁠, and, since $\tilde{j} = ρ (n \dot{θ} / | \nabla θ | + v)$ ⁠, to the condition $[v \cdot τ] = arbitrary$ (and the condition for the conservation of energy is reduced to $[W] = arbitrary$ ⁠). Hence, in the case of zero mass flux at the interface, the tangential component of velocity (and enthalpy) may experience arbitrary jump at the interface. This may lead to the appearance of shear and the formation of shear-driven vortical structures at the interface. The dynamics of the interface with the zero mass flux corresponds to the classical Rayleigh–Taylor and Richtmyer–Meshkov instabilities.^{16–19,30,36–39}

D. Structure of fundamental solutions

Perturbation waves: To find the structure of the perturbation waves, we consider equations for the conservation of mass and momentum in the bulk Eq. (7b). With the velocity field being a sum of potential and vortical components, $u = \nabla f + \nabla \times F$ ⁠, with $F = (0, F, 0)$ for a two-dimensional flow, we represent the perturbation waves as $(f, F, p) = (\hat{f}, \hat{F}, \hat{p}) exp (ikx - K z + Ω t)$ ⁠, where $Ω$ is a growth-rate (frequency), $k = 2 π / λ$ is the wave-vector, $λ$ is the spatial period (the wavelength), $V = | V |$ ⁠, and $K$ is the wave-vector of the perturbation wave in the direction of motion, to be found. This reduces Eq. (7b) to the linear system

(\begin{matrix} - ρ (k^{2} - K^{2}) & 0 & 0 \\ ρ (Ω - K V) & 1 & 0 \\ 0 & 0 & ρ (Ω - K V) \end{matrix}) (\begin{matrix} \hat{f} \\ \hat{p} \\ \hat{F} \end{matrix}) exp (ikx - K z + Ω t) = 0.

(9a)

The solutions for the system Eq. (9a) identify the perturbation waves as

\begin{matrix} K = - k, (\hat{f}, \hat{p}, \hat{F}) = (1, \frac{1}{ρ (Ω - k V)}, 0); \\ K = + k, (\hat{f}, \hat{p}, \hat{F}) = (1, \frac{1}{ρ (Ω + k V)}, 0); \\ K = \frac{Ω}{V}, (\hat{f}, \hat{p}, \hat{F}) = (0, 0, 1) . \end{matrix}

(9b)

By further applying the boundary conditions away from the interface Eq. (7c), we find that in Eq. (9b) the solutions with $K = \mp k$ correspond to potential components of the perturbed velocity fields and the associated pressure perturbations of the heavy fluid and the light fluid, respectively, whereas the solution $K = Ω / k$ corresponds to the vortical component of the perturbed velocity field of the light fluid. The vortical component is independent of and does not contribute to the field of pressure, indicating that the vortical field may be generated energetically (i.e., by energy excess) rather than dynamically (i.e., by pressure). It is worth noting that the de-coupling of the vortical field and pressure is held for compressible dynamics with thermal transport and heat sources, to be studied in detail in our future works.

Structure of solutions: The perturbation waves analysis in Eqs. (9a) and (9b) finds that in the sub-domain $z \in (- \infty, z^{*})$ the velocity field has the functional form $u = \nabla f$ with $f = \hat{f} exp (ikx + k z + Ω t)$ ⁠, whereas in the sub-domain $z \in (z^{*}, + \infty)$ the velocity field has the functional form $u = \nabla f + \nabla \times F$ with $F = (0, F, 0)$ ⁠, $f = \hat{f} exp (ikx - k z + Ω t)$ and $F = \hat{F} exp (ikx - (Ω / V) z + Ω t)$ ⁠. Hence, in the dynamics of Eq. (7) the perturbed velocity of the heavy fluid is potential, and the perturbed velocity of the light fluid has potential and vortical components^9,16–18,25

u_{h} = \nabla Φ_{h}, u_{l} = \nabla Φ_{l} + \nabla \times Ψ_{l} .

(9c)

In addition to the mathematical attributes in Eqs. (9a) and (9b), the solution structure in Eq. (9c) has the clear physics rationale: In the heavy fluid, the velocity field is potential in accordance with the Kelvin theorem. In the light fluid, the velocity field can be a superposition of the potential and vortical components. Recall that Kelvin's circulation theorem states that in an inviscid flow of a barotropic fluid with conservative body forces the circulation around any closed contour which encloses the same fluid elements moving with the fluid remains constant with time. Since for the heavy fluid the velocity field is potential away from the interface, it remains potential in the entire sub-domain up to the interface. This structure of the solution Eq. (9c) agrees with the observations and is established for any initial conditions.^16,17,27,30

The fluid potential and vortical fields and the interface perturbation are

\begin{matrix} Φ_{h} = Φ exp (ikx + k z + Ω t), Φ_{l} = \tilde{Φ} exp (ikx - k z + Ω t), \\ Ψ_{l} = (0, Ψ_{l}, 0), Ψ_{l} = Ψ exp (ikx - \tilde{k} z + Ω t), \\ z^{*} = Z exp (ikx + Ω t), \end{matrix}

(9d)

where $Ω$ is the growth-rate (the characteristic frequency, the eigenvalue) of the system equations (7), $k = 2 π / λ$ is the wave-vector, and $λ$ is the spatial period (the wavelength).

In consistency with Eq. (9b), the perturbed pressure in Eq. (7) is free from contributions from the perturbed vortical field.^16–19,25 This leads to

p_{h (l)} = - ρ_{h (l)} ({\dot{Φ}}_{h (l)} + V_{h (l)} \frac{\partial Φ_{h (l)}}{\partial z}), \tilde{k} = Ω / V_{l} .

(9e)

Emphasize that the wave-vector of the vortical field of the light fluid velocity $\tilde{k} = Ω / V_{l}$ is set by the growth-rate (the characteristic frequency) $Ω$ ⁠. The sign of the real part of $\tilde{k}$ is defined by the sign of $Ω$ ⁠, with $Re [\tilde{k}] > 0 (< 0)$ for $Re [Ω] > 0 (< 0)$ for the unstable (stable) dynamics. The imaginary part of $\tilde{k}$ is set by the imaginary part of $Ω$ ⁠, as $Im [\tilde{k}] = Im [Ω] / V_{l}$ ⁠.

The perturbed enthalpy has the form

w_{h (l)} = ε_{h (l)} + \frac{p_{h (l)}}{ρ_{h (l)}} = - ({\dot{Φ}}_{h (l)} + V_{h (l)} \frac{\partial Φ_{h (l)}}{\partial z}),

(9f)

where we account for that for ideal incompressible fluids the perturbations $ε$ of the internal energy $e$ are zero in both fluids, $ε_{h (l)} = 0$ ⁠.

Under the acceleration $g = (0, 0, g)$ and the surface tension, the perturbed pressure and the enthalpy are modified as^16–19

\begin{matrix} p_{h (l)} \to p_{h (l)} + ρ_{h (l)} g z, (p_{h} - p_{l}) \to (p_{h} - p_{l}) + σ \frac{\partial^{2} z^{*}}{\partial x^{2}}, \\ w_{h (l)} \to w_{h (l)} + g z, (w_{h} - w_{l}) \to (w_{h} - w_{l}) + \frac{σ}{ρ_{h}} \frac{\partial^{2} z^{*}}{\partial x^{2}} . \end{matrix}

(9g)

The perturbed dynamics are incompressible (sub-sonic) for a broad range of values of the acceleration, $ρ g / k P_{0} ≪ 1$ ⁠, and the surface tension, $σ k / P_{0} ≪ 1$ ⁠.¹⁶

The system of the equations governing the interface dynamics is thus reduced to the linear system $M r = 0$ ⁠, where vector $r$ is $r = {(Φ_{h}, Φ_{l}, V_{h} z^{*}, Ψ_{l})}^{T}$ ⁠, and the matrix $M$ is defined by the boundary conditions at the interface Eqs. (2), (6), and (7).¹⁷

Dimensionless units: For ideal incompressible fluids, the characteristic length-scale $1 / k$ and timescale $1 / k V_{h}$ of the dynamics are set by the initial conditions, and the characteristic density scale is set by the heavy fluid density $ρ_{h}$ ⁠.^16–18 We use the dimensionless values of the growth-rate $ω = Ω / k V_{h}$ ⁠, and the density ratio $R = ρ_{h} / ρ_{l}$ ⁠, $R \geq 1$ ⁠, leading to $V_{l} / V_{h} = R, \tilde{k} / k = ω / R$ ⁠. We use the dimensionless values of the gravity $G = g / k V_{h}^{2}$ ⁠, $G \geq 0$ ⁠, and the surface tension $T = (σ / ρ_{h}) (k / V_{h}^{2})$ ⁠, $T \geq 0$ ⁠. We consider a broad range of values $G \geq 0$ and $T \geq 0$ ⁠, as long as the perturbed dynamics are incompressible (sub-sonic) Eq. (9). We use dimensionless values of the flow fields, the interface, and the variables as $φ = Φ / (V_{h} / k)$ ⁠, $\tilde{φ} = \tilde{Φ} / (V_{h} / k)$ ⁠, $ψ = Ψ / (V_{h} / k)$ ⁠, $\bar{z} = k Z$ ⁠, and $k x \to x, k z \to z, k V_{h} t \to t$ ⁠. The dimensionless potentials are $φ_{h} = φ e^{i x + z + ω t}$ ⁠, $φ_{l} = \tilde{φ} e^{i x - z + ω t}$ ⁠, and $Ψ_{l} = (0, ψ_{l}, 0)$ with $ψ_{l} = ψ e^{i x - (\tilde{k} / k) z + ω t}$ ⁠. The dimensionless velocities are $u_{h} = \nabla φ_{h}, u_{l} = \nabla φ_{l} + \nabla \times Ψ_{l}$ and the interface perturbation is $z^{*} = \bar{z} e^{i x + ω t}$ ⁠.

Fundamental solutions: In the dimensionless form, the elements of the matrix $M$ are the functions of the growth-rate (the eigenvalue) $ω$ ⁠, the density ratio $R$ ⁠, the acceleration value $G$ ⁠, and the surface tension value $T$ as $M = M (ω, R, G, T)$ ⁠.^17,18 The condition $\det M (ω_{i}, R, G, T) = 0$ defines the eigenvalues $ω_{i}$ ⁠. The associated eigenvectors ${\tilde{e}}_{i}$ are derived by the reducing the matrix $M (ω_{i}, R, G, T)$ to row-echelon form.^17,18 The $4 \times 4$ matrix $M$ has four fundamental solutions $r_{i} = r_{i} (ω_{i}, {\tilde{e}}_{i})$ ⁠, $i = 1 \dots 4$ ⁠, corresponding to 4 degrees of freedom and four independent variables obeying four equations, Eqs. (7) and (9).

Solution $r$ for the system $M r = 0$ is a linear combination of the fundamental solutions $r_{i},$

r = \sum_{i = 1}^{4} C_{i} r_{i} .

(10)

Here, $C_{i}$ are the integration constants and $r_{i} = {\tilde{e}}_{i} e^{ω_{i} t}$ are the fundamental solutions with ${\tilde{e}}_{i} = {(φ e^{i x + z}, \tilde{φ} e^{i x - z}, \bar{z} e^{i x}, ψ e^{i x - (\tilde{k} / k) z})}_{i}^{T}$ and with the associated vector $e_{i} = {(φ, \tilde{φ}, \bar{z}, ψ)}_{i}^{T}$ ⁠.

E. The applicability and future developments of the theoretical framework

Our present work focuses on the effect of surface tension because this microscopic process influences only the conditions at the interface and keeps unaffected the conditions in the bulk. Our analysis is free from studying the effects of viscosity and other microscopic processes. These stabilizing microscopic processes may play an important role in high energy plasmas, and we address their detailed consideration to the future. We note that since the kinematic viscosity $ν$ may set a length-scale $ν / V_{h}$ ⁠, our analysis is valid for the wavelength of the initial perturbation, which is substantially greater than this characteristic scale $λ ≫ (ν / V_{h})$ ⁠.

In the present work, we consider the dynamics of ideal fluids free from thermal heat flux. In inertial confinement fusion and in laser ablated plasmas, the governing equations have to be augmented with the equation for the thermal heat flux, the terms describing the transport of internal energy, and the associated boundary conditions (in hydrodynamic approximation). In this case, the interface velocity is the ablation velocity. Our theoretical framework Eqs. (1)–(10) can be extended to rigorously account for the thermal heat flux and the associated effects, and to directly derive the ablation velocity in various regimes, including the regimes dominated by thermal diffusion, advection, and weak compressibility. We address these studies to the future. For the purposes of the present paper, in the absence of thermal heat flow, the ablation velocity can be set as the velocity of the steady planar interface, $V_{0} = - V_{h}$ with $V_{0} = | V_{0} | = V_{h}$ ⁠, similarly to the usual convention¹⁶ (as discussed in Sec. III I).

III. RESULTS

A. Fundamental solutions

We consider the conservative dynamics balancing the fluxes of mass, momentum, and energy at the interface, Eq. (7). For this dynamics, the matrix $M$ is

M = (\begin{matrix} - R & - 1 & - ω + R ω & i \\ 1 & - 1 & 1 - R & i ω / R \\ R - R ω & R + ω & G (R - 1) - R T & - 2 i R \\ ω & - ω & T + ω - R ω & i R \end{matrix}) .

(11a)

Its determinant is $\det M = i ((R - 1) / R) (ω - R) (ω + R) (ω^{2} (R - 1) + R (R - 1) - G (R + 1) + T R)$ ⁠, and the values $ω_{i}$ and $e_{i}$ are

\begin{matrix} ω_{1 (2)} = \pm i \sqrt{R} \sqrt{1 - \frac{G (R + 1)}{R (R - 1)} + \frac{T}{R - 1}}, \\ e_{1 (2)} = {(φ, \tilde{φ}, 1, ψ)}_{1 (2)}^{T}; \\ ω_{3} = R, e_{3} = {(0, i, 0, 1)}^{T}; ω_{4} = - R, \\ e_{4} = {(\frac{2 i}{R + 1}, - \frac{i (R - 1)}{R + 1}, 0, 1)}^{T}, \end{matrix}

(11b)

where the components ${φ, \tilde{φ}, ψ}$ of the eigenvectors are functions on $R, G, T$ ⁠. Among the solutions for the conservative dynamics Eq. (11), the fundamental solutions $r_{1} (ω_{1}, e_{1})$ and $r_{2} (ω_{2}, e_{2})$ depend on the values of the acceleration $G$ and the surface tension $T$ ⁠, and the fundamental solutions $r_{3} (ω_{3}, e_{3})$ and $r_{4} (ω_{4}, e_{4})$ depend only on the density ratio $R$ ⁠.^17,18

In regard to the fundamental solutions $r_{1 (2)}$ in Eq. (11), for some values of the acceleration, the surface tension and the density ratio, these solutions are stable, with $r_{1} = r_{2}^{*}$ and $ω_{1} = ω_{2}^{*}$ with $Re [ω_{1 (2)}] = 0$ ⁠. For some other values of the acceleration, the surface tension, and the density ratio, one of these solutions is unstable, $r_{1}$ with $Re [ω_{1}] > 0$ ⁠, whereas the other is stable, $r_{2}$ with $Re [ω_{2}] < 0$ ⁠. For these solutions, the interface perturbations are coupled with the potential and vortical components of the velocities of the fluids' bulk in the stable and the unstable regimes.

In regard to the fundamental solutions $r_{3 (4)}$ in Eq. (11), the solution $r_{3}$ is unstable, $ω_{3} = R$ and $Re [ω_{3}] > 0$ ⁠, and the solution $r_{4}$ is stable, $ω_{4} = - R$ and $Re [ω_{4}] < 0$ ⁠. Remarkably, for the formally unstable solution $r_{3}$ ⁠, the interface perturbation and the perturbed fields of the velocities and pressure are identically zero in the entire domain at any time for any integration constant $C_{3}$ ⁠, with $z^{*} = 0, u_{h (l)} = 0, p_{h (l)} = 0$ ⁠.^17,18 For the formally stable fundamental solution $r_{4}$ ⁠, we must set the integration constant $C_{4} = 0$ ⁠, in order for this solution to obey at any time the conditions ${u_{l} |}_{z \to + \infty} = 0$ ⁠. This is because the vortical component of the velocity, $\nabla \times Ψ_{l} \neq 0$ ⁠, while decaying in time, increases exponentially away from the interface. Note that for solution $r_{4}$ the vorticity value is zero, since $\nabla \times u_{l} = (0, (1 - {(\tilde{k} / k)}^{2}) ψ_{l}, 0)$ and ${(\tilde{k} / k)}^{2} = {(ω / R)}^{2} = 1$ ⁠.^17,18

The accelerated conservative dynamics with surface tension is non-degenerate, since it has four fundamental solutions. By defining the solution $r_{CDGT}$ in the stable regime as $r_{CDGT} = (r_{1} + r_{2}) / 2$ and in the unstable regime as $r_{CDGT} = r_{1}$ ⁠, we analyze the properties of this dynamics below, Table I. Sub-script stands for conservative dynamics with the gravity and the tension.

TABLE I.

Fundamental solution for the conservative dynamics with the acceleration and the surface tension.

$r = r_{CDGT}$ ⁠, $ω = ω_{CDGT}$ ⁠, $e = e_{CDGT} = {(φ, \tilde{φ}, \bar{z}, ψ)}_{CDGT}^{T}$
$ω$	$\frac{\sqrt{G + R + G R - R^{2} - R T}}{\sqrt{- 1 + R}}$
$φ$	$- \frac{1}{R + G (1 + R) - R (R^{2} + T)} (\sqrt{- 1 + R} (T \sqrt{R + G (1 + R) - R (R + T)} - G (\sqrt{- 1 + R} + \sqrt{R + G (1 + R) - R (R + T)}) - R (\sqrt{- 1 + R} + \sqrt{R + G (1 + R) - R (R + T)}) + R^{2} (\sqrt{- 1 + R} + \sqrt{R + G (1 + R) - R (R + T)})))$
$\tilde{φ}$	$\frac{1}{- G (1 + R) + R (- 1 + R^{2} + T)} (G (- 1 + R) R - G \sqrt{- 1 + R} \sqrt{R + G (1 + R) - R (R + T)} - (- 1 + R) R ((- 1 + R) R + T - \sqrt{- 1 + R} \sqrt{R + G (1 + R) - R (R + T)})$
$\bar{z}$	$1$
$ψ$	$- \frac{i (- 1 + R) R T}{R + G (1 + R) - R (R^{2} + T)}$

We emphasize that while in the foregoing our attention is focused on properties of the solution(s) $r_{CDGT}$ ⁠, and while solutions $r_{3 (4)}$ might appear unnecessary, the solutions $r_{3 (4)}$ are important to consider, since they reveal the non-degenerate character of the accelerated conservative dynamics with surface tension, which has four fundamental solutions for four independent variables obeying four governing equations and having four independent degrees of freedom.

The reader is referred to papers^17–19,30 discussing in detail the non-degeneracy of the conservative dynamics and the degeneracy of the dynamics of Landau–Darrieus and Rayleigh–Taylor instabilities and the associated physical properties. Lifting the degeneracy of Rayleigh–Taylor unstable dynamics may lead to Richtmyer–Meshkov instability, where the (sub-sonic) initial growth-rate can be set by a shock and/or an impulsive acceleration. Lifting the degeneracy of Landau–Darrieus instability (LDI) may lead to scale-invariant (power-law) rather scale-dependent (exponential) dynamics.^17–19,30

In Secs. III B–III I, Table I represents the fundamental solution $r_{CDGT}$ for the conservative dynamics with the acceleration and the surface tension. Since the eigenvector $e_{CDGT}$ defines the direction in the space of variables, its components are determined up to a constant.^16–19,30 In this work, for convenience, we consider the flow fields for given interface perturbation, and we present $e_{CDGT}$ in such form that its third element (corresponding to the interface perturbation) equals to unity. Table II represents the regions of stability and instability of the inertial conservative dynamics with the surface tension in a broad range of parameters. Table III represents the regions of stability and instability of the accelerated conservative dynamics with the surface tension in a broad range of parameters.

TABLE II.

Stability/instability regions for the inertial conservative dynamics with the surface tension.

Dynamics	Stability region	Instability region	Critical value
${r_{CDGT} \|}_{G = 0}$	$T > {{\tilde{T}}_{c r} \|}_{G = 0}$	$N / A$	${{\tilde{T}}_{c r} \|}_{G = 0} = 0$
${r_{CDGT} \|}_{G = 0}$	$R > {{\tilde{R}}_{c r} \|}_{G = 0}$	$N / A$	${{\tilde{R}}_{c r} \|}_{G = 0} = 1$

TABLE III.

Stability/instability regions for the accelerated conservative dynamics with the surface tension.

Dynamics	Stability region	Instability region	Critical values
$r_{CDGT}$	$T > {\tilde{T}}_{c r}$	$T < {\tilde{T}}_{c r}$	${\tilde{T}}_{c r} = \frac{G (R + 1) - R (R - 1)}{R}$
	$G < {\tilde{G}}_{c r}$	$G > {\tilde{G}}_{c r}$	${\tilde{G}}_{c r} = R \frac{R - 1}{R + 1} + T \frac{R}{R + 1}$
	$R > {\tilde{R}}_{c r}$	$R < {\tilde{R}}_{c r}$	${\tilde{R}}_{c r} = \frac{1 + G - T + \sqrt{{(1 + G - T)}^{2} + 4 G}}{2}$

Dynamics	Stability region	Instability region	Critical values
$r_{CDGT}$	$T > {\tilde{T}}_{c r}$	$T < {\tilde{T}}_{c r}$	${\tilde{T}}_{c r} = \frac{G (R + 1) - R (R - 1)}{R}$
	$G < {\tilde{G}}_{c r}$	$G > {\tilde{G}}_{c r}$	${\tilde{G}}_{c r} = R \frac{R - 1}{R + 1} + T \frac{R}{R + 1}$
	$R > {\tilde{R}}_{c r}$	$R < {\tilde{R}}_{c r}$	${\tilde{R}}_{c r} = \frac{1 + G - T + \sqrt{{(1 + G - T)}^{2} + 4 G}}{2}$

In the dimensional units, with $V_{0} = | V_{0} | = V_{h}$ ⁠, we obtain for the growth-rate in Eq. (11b),

Ω_{CDGT}^{2} = - {(k V_{0})}^{2} (\frac{ρ_{h}}{ρ_{l}}) - \frac{σ k^{3}}{(ρ_{h} - ρ_{l})} + g k (\frac{ρ_{h} + ρ_{l}}{ρ_{h} - ρ_{l}}),

(11c)

illustrating that the interface stability is defined by the interplay of stabilizing macroscopic inertial and microscopic surface tension mechanisms with the destabilizing acceleration.

Our results are illustrated with Figs. 2–5. Figure 2 presents the perturbed flow fields for the inertial conservative dynamics with surface tension. Figure 3 shows the perturbed flow fields for the stable accelerated conservative dynamics with surface tension. Figure 4 presents the perturbed flow fields for the unstable accelerated conservative dynamics with surface tension. Figures 2–4 show real parts of functions and values and are given in dimensionless units, as defined in Sec. II D. Figure 5 presents the growth-rate values for the accelerated conservative dynamics, the Landau–Darrieus dynamics and Rayleigh–Taylor dynamics with the surface tension in a broad range of parameters in the dimensionless units, as defined in Sec. II D.

FIG. 2.

View large Download slide

Flow fields for the inertial conservative dynamics with surface tension: the perturbed velocity (top left), the perturbed velocity streamlines (top middle), the interface perturbation (top right), the perturbed pressure (bottom left), the velocity vortical component (bottom middle), and the vorticity (bottom right). Real parts of fields and functions are shown at some instance of time and some values of the density ratio and the surface tension. Values are given in dimensionless units, as defined in Sec. II D.

FIG. 3.

View large Download slide

Flow fields for the stable accelerated conservative dynamics with surface tension: the perturbed velocity (top left), the perturbed velocity streamlines (top middle), the interface perturbation (top right), the perturbed pressure (bottom left), the velocity vortical component (bottom middle), and the vorticity (bottom right). Real parts of fields and functions are shown at some instance of time and some values of the density ratio, the acceleration, and the surface tension. Values are given in dimensionless units, as defined in Sec. II D.

FIG. 4.

View large Download slide

Flow fields for the unstable accelerated conservative dynamics with surface tension: the perturbed velocity (top left), the perturbed velocity streamlines (top middle), the interface perturbation (top right), the perturbed pressure (bottom left), the velocity vortical component (bottom middle), and the vorticity (bottom right). Real parts of fields and functions are shown at some instance of time and some values of the density ratio, the acceleration, and the surface tension. Values are given in dimensionless units, as defined in Sec. II D.

FIG. 5.

View large Download slide

Growth-rates for the accelerated conservative dynamics (purple), Landau–Darrieus dynamics (blue), and Rayleigh–Taylor dynamics (light blue) with the surface tension. Dependence of the growth-rates on the surface tension at some values of the density ratio and the acceleration (top), on the acceleration at some values of the density ratio and the surface tension (bottom). Solid (dashed) line marks real (imaginary) part; circles mark the critical values.

B. Inertial dynamics free from surface tension

The inertial conservative dynamics free from surface tension was investigated in detail in Ref. 17. Briefly, for $G = 0, T = 0$ ⁠, the solution ${r_{CDGT} |}_{G = 0, T = 0} = {r_{CDGT} (ω_{CDGT}, {\tilde{e}}_{CDGT}) |}_{G = 0, T = 0}$ is

ω_{CDGT} |_{G = 0, T = 0} = \pm i \sqrt{R}, {e_{CDGT} |}_{G = 0, T = 0} = \frac{e + e^{*}}{2}, e = {(i \frac{R - 1}{\sqrt{R} + i}, \frac{- \sqrt{R} (R - 1)}{\sqrt{R} + i}, 1, 0)}^{T} .

(12a)

The inertial conservative dynamics free from surface tension ${r_{CDGT} |}_{G = 0, T = 0}$ is stable, and it has potential flow fields in the fluids' bulk and is shear free at the interface, Table I. This dynamics is stabilized by the inertial mechanism leading to stable oscillations of the interface velocity near the constant value, $\tilde{V} = {\tilde{V}}_{0} + \tilde{v}$ ⁠, with $\tilde{v} \cdot n_{0} \sim e^{\pm i \sqrt{R} t}$ ⁠. Physically, when the interface is perturbed, the parcels of the heavy and the light fluids follow the interface perturbation thus causing the change of momentum and energy of the system. To conserve the momentum and energy, the interface as whole changes its velocity. This causes the reactive force to occur and to stabilize the dynamics.¹⁷

In the dimensional units, with $V_{0} = | V_{0} | = V_{h}$ ⁠, we obtain from Eq. (12a) the frequency

Ω_{CDGT} |_{g = 0, σ = 0} = \pm i k V_{0} \sqrt{\frac{ρ_{h}}{ρ_{l}}},

(12b)

illustrating that for the inertial conservative dynamics the interface is stable and it is stabilized by the macroscopic inertial mechanism. For a given value of $V_{0}$ ⁠, the frequency of the oscillations increases with the increase in the density ratio. The interface perturbations and interface velocity oscillate “slowly” with $| Ω_{CDGT} |_{g = 0, σ = 0} / (k V_{0}) | \to 1$ for fluids with similar densities $(ρ_{h} / ρ_{l}) \to 1^{+}$ ⁠, and oscillate “quickly” with $| Ω_{CDGT} |_{g = 0, σ = 0} / (k V_{0}) | \to \infty$ for fluids with very different densities $(ρ_{h} / ρ_{l}) \to \infty$ ⁠. Since for incompressible dynamics the speed of sound $c$ is the largest velocity scale, for validity of theoretical solution, the limiting value of the density ratio is $(ρ_{h} / ρ_{l}) \sim {(c / V_{0})}^{2}$ ⁠.

C. Inertial dynamics with surface tension

Consider now the properties of the inertial conservative dynamics with the surface tension. For $G = 0, T > 0$ ⁠, the solution is ${r_{CDGT} |}_{G = 0} = {r_{CDGT} (ω_{CDGT}, {\tilde{e}}_{CDGT}) |}_{G = 0}$ with

ω_{CDGT} |_{G = 0} = \pm i \sqrt{R} \sqrt{1 + \frac{T}{R - 1}}, {e_{CDGT} |}_{G = 0} = \frac{e + e^{*}}{2}, e = {(φ, \tilde{φ}, 1, ψ)}^{T},

(13a)

where quantities ${φ, \tilde{φ}, ψ}$ are the functions on $R, T$ ⁠, Tables I and II, and Fig. 2. For small surface tension values $T \to 0$ ⁠, in agreement with Eq. (12), the components of the eigenvector in Eq. (13a) are

T \to 0, φ \to \frac{(R - 1) (1 + i \sqrt{R})}{R + 1}, \tilde{φ} \to \frac{i \sqrt{R} (R - 1) (1 + i \sqrt{R})}{R + 1}, ψ \to \frac{i T}{R + 1} .

(13b)

For the inertial conservative dynamics with finite surface tension value, the flow field for solution ${r_{CDGT} |}_{G = 0}$ has the following structure. The velocity field is potential in the bulk of the heavy fluid, and has potential and vortical components in the bulk of the light fluid. The appearance of the vortical field is due to the contribution of the surface energy to the perturbed enthalpy, and it defines the field's strength, Eq. (9). In the limit of zero surface tension, the velocities are potential in both fluids.^17–19,30

The inertial dynamics with the surface tension ${r_{CDGT} |}_{G = 0}$ is stable for $T > {{\tilde{T}}_{c r} |}_{G = 0}$ ⁠, ${{\tilde{T}}_{c r} |}_{G = 0} = 0$ (for $R > {{\tilde{R}}_{c r} |}_{G = 0}$ ⁠, ${{\tilde{R}}_{c r} |}_{G = 0} = 1$ ⁠), Table II. The eigenvalue $ω_{CDGT} |_{G = 0}$ is imaginary, $Re [ω_{CDGT} |_{G = 0}] = 0$ ⁠. This suggests that the length-scale of the vortical field $\tilde{k} = (k / R) ω_{CDGT} |_{G = 0}$ is also imaginary, $Re [\tilde{k}] = 0$ ⁠. The dynamics ${r_{CDGT} |}_{G = 0}$ describes the standing wave stably oscillating in time. For this wave, in the bulk of the heavy fluid the velocity field is potential, and it decays away from the interface. In the bulk of the light fluid, the velocity field has the potential and vortical components. The potential component decays away from the interface. The vortical field is periodic in the $x$ direction with the period $λ = 2 π / k$ and is periodic in the $z$ direction with the period $\tilde{λ} = 2 π / | \tilde{k} |$ ⁠. The vorticity field $\nabla \times u_{l}$ is periodic in the $(x, z)$ plane. Hence, in the light fluid bulk, the dynamics ${r_{CDGT} |}_{G = 0}$ has the stably oscillating periodic vortical structure with constant amplitude, Fig. 2.

Mathematically, the appearance of the vortical field and vorticity field periodic in the $z$ direction of motion is associated with the pure imaginary character of the frequency $ω$ in solution Eq. (13) setting the purely imaginary wave-vector $\tilde{k}$ ⁠, with $Im [\tilde{k}] = (k / R) Im [ω_{CDGT} |_{G = 0}]$ ⁠. It is prescribed by the structure of the perturbation waves in Eq. (9) and the structure of the eigenvectors of the fundamental solution in Eq. (10). Physically, by comparing the solutions for the inertial conservative dynamics with surface tension and free from surface tension in Eqs. (12) and (13), we find that the vortical field and the vorticity field are energetic (rather than dynamic) in nature. These fields are decoupled from the pressure field (at least, in the linear approximation) and are produced by the excess of energy, which is caused by the contribution of surface energy to the perturbed enthalpy, Eqs. (9) and (10), Table I. Our work is the first (to the authors' knowledge) to illustrate the energetic nature of the volumetric vortical field of the velocity in the problem of interface dynamics with interfacial mass flux. Remarkably, the vortical and vorticity fields in the bulk of the light fluid are produced even when the interface is stable, in order to balance the energy excess at the interface, Fig. 2.

The boundary condition ${u_{l} |}_{z \to + \infty} = 0$ requires us to set for the dynamics ${r_{CDGT} |}_{G = 0}$ the integration constant equal zero. This leads to the zero perturbations of the interface and the flow fields and to the constancy of the interface velocity $\tilde{V} = {\tilde{V}}_{0}$ ⁠. Some slight modifications of the boundary conditions away from the interface $z \to + \infty$ by an external noise may lead to a non-zero integration constant ${C_{CDGT} |}_{G = 0}$ for the dynamics ${r_{CDGT} |}_{G = 0}$ ⁠. These modifications may include slight modulations of the uniform velocity field of the light fluid away from the interface, as $V_{l} \to V_{l} + δ V_{l}$ ⁠, and may be present in realistic systems. In this case, the interface velocity for the dynamics $r_{CDGT}$ may experience slight oscillations near the constant value, as $\tilde{V} = {\tilde{V}}_{0} + \tilde{v}$ with $\tilde{v} n_{0} = - {(u_{h} n_{0} + \dot{θ}) |}_{θ = 0} \sim e^{\pm i | ω_{CDGT} |_{G = 0} | t}$ ⁠.

In the dimensional units, with $V_{0} = | V_{0} | = V_{h}$ ⁠, we obtain for the frequency in Eq. (13a),

Ω_{CDGT} |_{g = 0} = \pm i \sqrt{{(k V_{0})}^{2} (\frac{ρ_{h}}{ρ_{l}}) + \frac{σ k^{3}}{(ρ_{h} - ρ_{l})}} .

(13c)

For the inertial conservative dynamics with surface tension, the interface is stabilized by the macroscopic inertial mechanism and by the microscopic surface tension. The inertial mechanism (the surface tension) dominates for $V_{0}^{2} (ρ_{h} / ρ_{l}) (ρ_{h} - ρ_{l}) {(σ k)}^{- 1} \to \infty (0)$ ⁠. For given values of $V_{0}, σ, k$ in Eq. (13c), the inertial term ${(k V_{0})}^{2} (ρ_{h} / ρ_{l})$ increases with the increase in the density ratio and becomes “singular” $(ρ_{h} / ρ_{l}) \to \infty$ ⁠, whereas the surface tension term $(σ k^{3}) / (ρ_{h} - ρ_{l})$ increases with the decrease in the density ratio and becomes singular for $(ρ_{h} / ρ_{l}) \to 1^{+}$ ⁠. Since for the incompressible dynamics, the speed of sound $c$ is the largest velocity scale, for the validity of the theoretical solution, the limiting values of the density ratio and the surface tension are $(ρ_{h} / ρ_{l}) \sim {(c / V_{0})}^{2}$ and $σ \sim (ρ_{h} - ρ_{l}) (c^{2} / k)$ ⁠. In realistic systems, the surface tension may depend on the density ratio and may vanish for $(ρ_{h} / ρ_{l}) \to 1^{+}$ ⁠.²⁰

D. Accelerated dynamics free from surface tension

The accelerated conservative dynamics free from surface tension is investigated in detail in Refs. 17 and 18. Briefly, for $G > 0, T = 0$ the solution ${r_{CDGT} |}_{T = 0} = {r_{CDGT} (ω_{CDGT}, {\tilde{e}}_{CDGT}) |}_{T = 0}$ is

\begin{matrix} G < G_{c r}, ω_{CDGT} |_{T = 0} = \pm i \sqrt{R} \sqrt{1 - \frac{G}{G_{c r}}}, \\ {e_{CDGT} |}_{T = 0} = \frac{e + e^{*}}{2}, e = {(φ, \tilde{φ}, 1, 0)}^{T}; \\ G > G_{c r}, ω_{CDGT} |_{T = 0} = \sqrt{R} \sqrt{\frac{G}{G_{c r}} - 1}, \\ {e_{CDGT} |}_{T = 0} = e = {(φ, \tilde{φ}, 1, 0)}^{T}, G_{c r} = \frac{R (R - 1)}{R + 1}, \end{matrix}

(14a)

where quantities ${φ, \tilde{φ}}$ are the functions on $R, G$ ⁠, and $G_{c r}$ is the critical value.

For $G > 0$ ⁠, the solution's stability is defined by the interplay of the buoyancy and the inertia, or the gravity and the reactive force. For $G < G_{c r}$ ⁠, the inertial effect dominates, and the reactive force exceeds the gravity. The dynamics is stable and describes the standing wave stably oscillating in time. For $G > G_{c r}$ ⁠, the buoyant effect dominates, and the gravity exceeds the reactive force. The dynamics is unstable; it describes the standing wave with the growing amplitude, has the potential velocity fields in the bulk, and is shear free at the interface. The dynamics is the superposition of two motions—the motion of the interface as whole with the growing velocity and the growth of the interface perturbations.¹⁷

In the dimensional units, with $V_{0} = | V_{0} | = V_{h}$ ⁠, we obtain for the growth-rate in Eq. (14a),

Ω_{CDGT} |_{σ = 0} = \sqrt{- {(k V_{0})}^{2} \frac{ρ_{h}}{ρ_{l}} + g k \frac{ρ_{h} + ρ_{l}}{ρ_{h} - ρ_{l}}} .

(14b)

This illustrates that for the accelerated conservative dynamics free from surface tension, the interface stability is determined by the interplay of the stabilizing inertial mechanism and the destabilizing acceleration, with the inertia (buoyancy) dominating for $k V_{0}^{2} (ρ_{h} / ρ_{l}) (ρ_{h} - ρ_{l}) / (ρ_{h} + ρ_{l}) > g (< g)$ ⁠, and with the threshold acceleration value $g_{c r} = k V_{0}^{2} (ρ_{h} / ρ_{l}) (ρ_{h} - ρ_{l}) / (ρ_{h} + ρ_{l})$ ⁠, where $g_{c r} \to k V_{0}^{2} (ρ_{h} / ρ_{l})$ for $(ρ_{h} / ρ_{l}) \to \infty$ and $g_{c r} \to k V_{0}^{2} (ρ_{h} - ρ_{l}) / (ρ_{h} + ρ_{l})$ for $(ρ_{h} / ρ_{l}) \to 1^{+}$ ⁠. For given values of $V_{0}, g, k$ in Eq. (14b), the inertial term ${(k V_{0})}^{2} (ρ_{h} / ρ_{l})$ becomes singular for $(ρ_{h} / ρ_{l}) \to \infty$ ⁠, and the buoyant term $g k (ρ_{h} + ρ_{l}) / (ρ_{h} - ρ_{l})$ becomes singular for $(ρ_{h} / ρ_{l}) \to 1^{+}$ ⁠. For the incompressible dynamics, the speed of sound $c$ is the largest velocity scale, and for the validity of the theoretical solution the limiting values of the density ratio and acceleration are $(ρ_{h} / ρ_{l}) \sim {(c / V_{0})}^{2}$ ⁠, $g \sim k c^{2} (ρ_{h} - ρ_{l}) / (ρ_{h} + ρ_{l})$ ⁠.

In order to accurately evaluate the limiting cases with account for the dependence of the threshold $g_{c r}$ on the density ratio $(ρ_{h} / ρ_{l})$ ⁠, we represent in Eq. (14b) the acceleration as $g = f g_{c r}$ ⁠, where $f$ is some regular function on the density ratio and $f > 1 (< 1)$ in the unstable (stable) regime

Ω_{CDGT} |_{σ = 0} = \sqrt{(\frac{ρ_{h} + ρ_{l}}{ρ_{h} - ρ_{l}}) (g_{c r} k) (f - 1)}, g_{c r} = k V_{0}^{2} (\frac{ρ_{h}}{ρ_{l}}) (\frac{ρ_{h} - ρ_{l}}{ρ_{h} + ρ_{l}}), f = \frac{g}{g_{c r}} .

(14c)

For fluids with similar densities $(ρ_{h} / ρ_{l}) \to 1^{+}$ the growth-rate approaches $Ω_{CDGT} |_{σ = 0} \to k V_{0} \sqrt{(f - 1)}$ ⁠, and for fluids with very different densities $(ρ_{h} / ρ_{l}) \to \infty$ it is $Ω_{CDGT} |_{σ = 0} \to k V_{0} \sqrt{(ρ_{h} / ρ_{l}) (f - 1)}$ ⁠.¹⁷

E. Accelerated dynamics with surface tension

Investigate now in detail the accelerated conservative dynamics with the surface tension.

Fundamental solution: For the accelerated conservative dynamics with the surface tension, the solution $r_{CDGT} = r_{CDGT} (ω_{CDGT}, {\tilde{e}}_{CDGT})$ is

\begin{matrix} \tilde{G} < G_{c r}, ω_{CDGT} = \pm i \sqrt{R} \sqrt{1 - \frac{\tilde{G}}{G_{c r}}}, \\ e_{CDGT} = \frac{e + e^{*}}{2}, e = {(φ, \tilde{φ}, 1, ψ)}^{T}; \\ \tilde{G} > G_{c r}, ω_{CDGT} = \sqrt{R} \sqrt{\frac{\tilde{G}}{G_{c r}} - 1}, \\ e_{CDGT} = e = {(φ, \tilde{φ}, 1, ψ)}^{T}; \\ \tilde{G} = G - \frac{T R}{R + 1}, G_{c r} = \frac{R (R - 1)}{R + 1} . \end{matrix}

(15a)

The quantities ${φ, \tilde{φ}, ψ}$ depend on $R, G, T$ with $ψ = - i T R (R - 1) / (G (R + 1) - R (R^{2} + T - 1))$ ⁠, and ${ψ |}_{T = 0} = 0$ ⁠, Tables I and III and Figs. 3 and 4. For small surface tension values $T \to 0$ ⁠, in agreement with Eqs. (12) and (14), the components of the eigenvector in Eq. (15b) are

\begin{matrix} T \to 0, φ \to \frac{(\sqrt{R - 1} + \sqrt{G (R + 1) - R (R - 1)}) \sqrt{R - 1}}{R + 1}, \\ \tilde{φ} \to \frac{(- R \sqrt{R - 1} + \sqrt{G (R + 1) - R (R - 1)}) \sqrt{R - 1}}{R + 1}, \\ ψ \to \frac{- iTR (R - 1)}{G (R + 1) - R (R - 1)} . \end{matrix}

(15b)

Recall that for linear systems eigenvectors define directions in the space of variables and are determined up to constant; we present the eigenvector components for given interface perturbation, by choosing its third element as unity, Eq. (15a). In the dimensional units, the growth-rate in Eq. (15a) is

\begin{matrix} \tilde{g} < g_{c r}, Ω_{CDGT} = \pm i k V_{h} \sqrt{\frac{ρ_{h}}{ρ_{l}}} \sqrt{1 - \frac{\tilde{g}}{g_{c r}}}; \\ \tilde{g} > g_{c r}, Ω_{CDGT} = k V_{h} \sqrt{\frac{ρ_{h}}{ρ_{l}}} \sqrt{\frac{\tilde{g}}{g_{c r}} - 1}; \\ \tilde{g} = g - \frac{σ k}{(ρ_{h} + ρ_{l}) V_{h}^{2}}, g_{c r} = k V_{h}^{2} (\frac{ρ_{h}}{ρ_{l}}) (\frac{ρ_{h} - ρ_{l}}{ρ_{h} + ρ_{l}}) . \end{matrix}

(15c)

Stability and instability of the fundamental solution: For $G > 0$ ⁠, stability of the dynamics $r_{CDGT}$ is defined by the interplay of the buoyancy, the inertia and the surface tension, or—the gravity, the reactive force and the tension force. The stability surface in the parameter space $(R, G, T)$ is defined by the condition $\tilde{G} = G_{c r}$ balancing the buoyancy (the gravity) with the combined contributions of the inertial stabilization mechanism (the reactive force) and the surface tension (the tension force). For $\tilde{G} < G_{c r}$ ⁠, the buoyancy is dominated and the reactive and tension forces exceed the gravity. The dynamics is stable and describes the standing wave stably oscillating in time. For $\tilde{G} > G_{c r}$ ⁠, the buoyant effect dominates and the gravity exceeds the reactive and the tension forces. The dynamics is unstable and describes the standing wave with the growing amplitude.

For given $R, T$ values, the dynamics is stable (unstable) for $G < (>) {\tilde{G}}_{c r}$ ⁠, where the threshold acceleration value is ${\tilde{G}}_{c r} = R (R - 1 + T) / (R + 1)$ ⁠, with ${\tilde{G}}_{c r} \to G_{c r}$ for $T \to 0$ ⁠. For given $R, G$ values, the dynamics is stable (unstable) for $T > (<) {\tilde{T}}_{c r}$ ⁠, where the critical surface tension value is ${\tilde{T}}_{c r} = (G (R + 1) - R (R - 1)) / R$ ⁠. For given $G, T$ values, the dynamics is stable (unstable) for $R > (<) {\tilde{R}}_{c r}$ with ${\tilde{R}}_{c r} = (1 + G - T + \sqrt{{(1 + G - T)}^{2} + 4 G}) / 2$ ⁠, Table III.

Structure of the flow fields: Consider the structure of the flow fields for the dynamics $r_{CDGT}$ for given interface perturbation. In the limit of zero surface tension, $T \to 0$ ⁠, the vortical component vanishes, $ψ \to 0$ ⁠, and the accelerated conservative dynamics free from surface tension has potential velocity fields in the fluids' bulks.^17,18 For a finite value of the surface tension, $T > 0$ ⁠, in the dynamics $r_{CDGT}$ ⁠, the vortical component is finite, $ψ \neq 0$ ⁠. This accelerated conservative dynamics with surface tension has potential velocity field in the bulk of the heavy fluid, and the velocity field combining the potential and vortical components in the bulk of the light fluid. The appearance of the vortical field in the light fluid bulk is due to the surface energy contribution to the enthalpy jump at the interface $[w]$ ⁠, which is $\sim \bar{z} T$ in the dimensionless units, Eq. (9f), and which defines the strength of the vortical field, Table I and Figs. 3 and 4.

Stable dynamics: For $G < {\tilde{G}}_{c r}$ (⁠ $T > {\tilde{T}}_{c r}$ ⁠) (⁠ $R > {\tilde{R}}_{c r}$ ⁠), the accelerated conservative dynamics with surface tension $r_{CDGT}$ is stable: The buoyancy (the gravity) is dominated by the combined effects of the inertial stabilization mechanism and the surface tension (the reactive force and the tension force). In this regime, the eigenvalue $ω_{CDGT}$ is purely imaginary, $Re [ω_{CDGT}] = 0$ ⁠. The length-scale of the vortical field $\tilde{k} = (k / R) ω_{CDGT}$ is also purely imaginary, $Re [\tilde{k}] = 0$ ⁠. The solution $r_{CDGT}$ is the standing wave stably oscillating in time. In the heavy fluid bulk, the velocity field is potential; it is periodic in the $x$ direction and decays away from the interface $z \to - \infty$ ⁠. In the bulk of the light fluid, the velocity field combines the potential and the vortical components. The potential component is periodic in the $x$ and decays away from the interface $z \to + \infty$ ⁠. The vortical component and the vorticity field are periodic in the $x$ direction with the period $λ = 2 π / k$ and are also periodic in the $z$ direction with period $\tilde{λ} = 2 π / \tilde{k}$ ⁠, Tables I and III and Fig. 4.

Similarly to inertial conservative dynamics with surface tension, for the stable accelerated conservative dynamics with surface tension, the vortical and vorticity fields are periodic in the $z$ direction of motion. Mathematically, this periodicity is associated with the pure imaginary character of the frequency $ω$ in solution Eq. (15) in the stable regime. Physically, the appearance of the vortical and vorticity fields follows from the energetic nature of these fields. Vortical structures are created in the bulk to balance the excess of energy caused by the surface tension, Eq. (9), Tables I and III and Fig. 4.

For the stable dynamics $r_{CDGT}$ ⁠, the boundary condition ${u_{l} |}_{z \to + \infty} = 0$ requires us to set its integration constant $C_{CDGT}$ equal zero. For zero integration constant the perturbations fields are also zero and the interface velocity is constant, $\tilde{V} = {\tilde{V}}_{0}$ ⁠. Some slight modifications of the boundary condition away from the interface may lead to a non-zero integration constant $C_{CDGT}$ for solution $r_{CDGT}$ ⁠. These modifications may include the slight modulations of the uniform velocity field of the light fluid by some external noise, $V_{l} \to V_{l} + δ V_{l}$ ⁠, which may be present in realistic systems. In this case, the interface velocity for the dynamics $r_{CDGT}$ may experience slight oscillations near the constant value, $\tilde{V} = {\tilde{V}}_{0} + \tilde{v}$ with $\tilde{v} n_{0} = - {(u_{h} n_{0} + \dot{θ}) |}_{θ = 0} \sim e^{\pm i | ω_{CDGT} | t}$ ⁠.

Unstable dynamics: The accelerated conservative dynamics with surface tension $r_{CDGT}$ is unstable for $G > {\tilde{G}}_{c r}$ (⁠ $T < {\tilde{T}}_{c r}$ ⁠) (⁠ $R < {\tilde{R}}_{c r}$ ⁠): The buoyancy (the gravity) dominates the combined effects of the inertial stabilization mechanism and the surface tension (the reactive and the tension forces). In this regime, the eigenvalue $ω_{CDGT}$ is real and positive, $Re [ω_{CDGT}] > 0$ and $Im [ω_{CDGT}] = 0$ ⁠. The dynamics $r_{CDGT}$ couples the interface perturbation with the vortical and potential components of the velocity fields $\nabla φ_{h}, \nabla φ_{l}, \nabla \times Ψ_{l}$ ⁠, achieving their extreme values near the interface, and, while increasing in time, decaying away from the interface, Tables I and III and Fig. 4.

For the unstable dynamics $r_{CDGT}$ with $G > {\tilde{G}}_{c r}$ (⁠ $T < {\tilde{T}}_{c r}$ ⁠) (⁠ $R < {\tilde{R}}_{c r}$ ⁠), for given interface perturbation, the structure of the vortical field differs from that in the stable regime: This field is no longer periodic in the $z$ direction of motion, and its wave-vector of the vortical field is $\tilde{k} = (k / R) ω_{CDGT}$ ⁠. When surface tension value decreases, $T \to 0$ ⁠, the vortical field strength decreases, leading to potential velocity fields in the bulks.^17,18

For the unstable dynamics $r_{CDGT}$ with $G > {\tilde{G}}_{c r}$ (⁠ $T < {\tilde{T}}_{c r}$ ⁠; $R < {\tilde{R}}_{c r}$ ⁠), the interface velocity increases with time, $\tilde{V} = {\tilde{V}}_{0} + \tilde{v}$ with $\tilde{v} n_{0} \sim e^{| ω_{CDGT} | t}$ ⁠. The resultant dynamics is the superposition of two motions—the motion of the interface as whole with the growing interface velocity and the growth of the interface perturbations.

Summary: The accelerated conservative dynamics with surface tension can be stable or unstable depending on the values of the acceleration, the surface tension, and the density ratio. In the stable regime, the resultant dynamics has the stable unperturbed flow fields and the constant interface velocity, when the boundary conditions away from the interface are strictly obeyed. It corresponds to the stably oscillating standing wave, has the stable periodic vortical structure in the bulk, and has the interface velocity slightly oscillating near the constant value, for noisy boundary conditions away from the interface, Fig. 3.

In the unstable regime, the interface perturbations grow and so does the interface velocity. The dynamics couples the interface perturbation with the potential velocity field in the heavy fluid bulk and the potential and vortical components of the velocity field in the light fluid bulk, and is shear-free at the interface. The strength of the vortical field in the light fluid bulk depends on the surface tension, Fig. 4.

F. Landau–Darrieus and Rayleigh–Taylor instabilities and conservative dynamics

Our theoretical framework can be applied to study other interfacial dynamics influenced by the acceleration and the surface tension, Fig. 5.^{17–19,25–27,30} We give a brief outline here and provide details in the future.

Landau–Darrieus and Rayleigh–Taylor instabilities: For the classical Landau's dynamics that corresponds to Landau–Darrieus instability and that conserves mass and momentum and imposes special condition for the perturbed mass flux at the interface, the dynamics $r_{LDGT}$ has growth-rate $ω_{LDGT}$ ⁠,

ω_{LDGT} = \frac{1}{R + 1} (- R + \sqrt{R^{3} + R^{2} - R + (G - \frac{T R}{R - 1}) (R^{2} - 1)}) .

(16a)

It is unstable for $T < {\bar{T}}_{c r}$ ⁠, ${\bar{T}}_{c r} = (G + R) (R - 1) / R$ ⁠, for $G > {\bar{G}}_{c r}$ ⁠, ${\bar{G}}_{c r} = - R + T R / (R - 1)$ ⁠, and for $R > {\bar{R}}_{c r}$ ⁠, ${\bar{R}}_{c r} = (1 + G - T + \sqrt{{(1 + G - T)}^{2} - 4 G}) / 2$ ⁠, and it is stable otherwise.^20,25

In the dimensional units, with $V_{0} = | V_{0} | = V_{h}$ ⁠, we obtain from Eq. (16a),

Ω_{LDGT} = (- k V_{0} \frac{ρ_{h}}{ρ_{h} + ρ_{l}} + \sqrt{{(k V_{0})}^{2} (\frac{ρ_{h}}{ρ_{l}}) \frac{(ρ_{h}^{2} + ρ_{h} ρ_{l} - ρ_{l}^{2})}{{(ρ_{h} + ρ_{l})}^{2}} + (g k) (\frac{ρ_{h} - ρ_{l}}{ρ_{h} + ρ_{l}}) - \frac{σ k^{3}}{ρ_{h} + ρ_{l}}}) .

(16b)

In Eq. (16a), the acceleration and the surface tension terms are “regular” for any density ratio $(ρ_{h} / ρ_{l})$ ⁠. The growth-rate becomes singular $Ω_{LDGT} / (k V_{0}) \to \sqrt{ρ_{h} / ρ_{l}} \to \infty$ for $(ρ_{h} / ρ_{l}) \to \infty$ ⁠.

Landau–Darrieus instability (LDI) is sometimes considered in theoretical physics as third prospect of Landau.^16,27 The resolutions of two other prospects of (the 1962 Noble Laureate) Landau—self-similar character of fluctuations in phase transitions and structure of superconductors—were recognized with 1982 and 2003 Noble prizes.^16,41 While predicted by Landau in 1994, Landau–Darrieus instability was never been directly observed in experiments in plasmas; it was attempted to observe in experiments and simulations in reactive fluids under conditions departing from those in Landau's theory (i.e., for ideal incompressible fluids separated by a sharp interface).^19,27,30 Recent studies^17–19,30 found that the classical Landau's solution and the postulate of the constancy of the interface velocity for Landau–Darrieus instability lead to the degeneracy of the dynamics and are incompatible with the condition of the conservation of energy at the interface.^16–19,30

For Rayleigh–Taylor instability (RTI), the boundary conditions at the interface differ from those for the conservative dynamics and LDI. In RTI, there is no mass flux across the interface; the normal component of velocity and pressure are continuous at the interface, and the tangential component of velocity and enthalpy are discontinuous at the interface. Rayleigh–Taylor dynamics $r_{RTGT}$ has the growth-rate $ω_{RTGT}$ ⁠,

ω_{RTGT} = \sqrt{\frac{R - 1}{R + 1}} \sqrt{G - \frac{T R}{R - 1}} .

(16c)

It is unstable for $T < {\hat{T}}_{c r}$ ⁠, ${\hat{T}}_{c r} = G (R - 1) / R$ ⁠, for $G > {\hat{G}}_{c r}$ ⁠, ${\hat{G}}_{c r} = T R / (R - 1)$ ⁠, and for $R > {\hat{R}}_{c r}$ with ${\hat{R}}_{c r} = G / (G - T)$ ⁠, and it is stable otherwise.^20,26,27

In the dimensional units, with $V_{0} = | V_{0} | = V_{h}$ ⁠, we obtain from Eq. (16c) for the growth-rate

Ω_{RTGT} = \sqrt{g k \frac{ρ_{h} - ρ_{l}}{ρ_{h} + ρ_{l}} - \frac{σ k^{3}}{ρ_{h} + ρ_{l}}} .

(16d)

In Eq. (16d), the acceleration and the surface tension terms are regular for any density ratio $(ρ_{h} / ρ_{l})$ ⁠.

In the presence of the acceleration and the surface tension, the conservative dynamics and Landau–Darrieus and Rayleigh–Taylor dynamics are driven by distinct boundary conditions at the interface. Hence, these dynamics may have distinct qualitative and quantitative properties.^17–19,30

Comparison of the conservative dynamics with Landau–Darrieus and Rayleigh–Taylor instabilities: We briefly compare the growth-rates of the conservative dynamics, Landau–Darrieus, Rayleigh–Taylor instabilities for fluids with very similar and very different densities. The reader is referred to Refs. 17–19 for detailed comparisons of properties of inertial and accelerated conservative dynamics free from surface tension with those of Landau–Darrieus and Rayleigh–Taylor instabilities.

For the inertial dynamics with surface tension $G = 0, T = const$ ⁠, the growth-rates are

\begin{matrix} ω_{CDGT} = \pm i \sqrt{R} \sqrt{1 + \frac{T}{R - 1}}, \\ ω_{LDGT} = \frac{1}{R + 1} (- R + \sqrt{R^{3} + R^{2} - R - T R (R + 1)}), \\ ω_{RTGT} = \pm i \sqrt{\frac{R - 1}{R + 1}} \sqrt{\frac{T R}{R - 1}} . \end{matrix}

(17a)

At $T = const$ and for fluids with very different densities $R \to \infty$ the growth-rates Eq. (17a) approach

ω_{CDGT} \to \pm i \sqrt{R}, ω_{LDGT} \to \sqrt{R}, ω_{RTGT} \to \pm i \sqrt{T}

(17b)

with the stable conservative and Rayleigh–Taylor dynamics, and the unstable Landau–Darrieus dynamics. At $T = const$ and for fluids with similar densities $R \to 1^{+}$ the growth-rates Eq. (17a) are

ω_{CDGT} \to \pm i \sqrt{1 + \frac{T}{R - 1}} ω_{LDGT} \to \frac{- 1 + \sqrt{1 - 2 T}}{2}, ω_{RTGT} \to \pm i \sqrt{\frac{T}{2}}

(17c)

with the stable conservative and Rayleigh–Taylor dynamics, and with the stability of Landau–Darrieus dynamics depending on the surface tension strength.¹⁶ Note that the magnitude of $ω_{CDGT}$ in Eq. (17c) increases for $R \to 1^{+}$ at $T = const$ ⁠. This increase has a clear physical interpretation: For fluids with very similar densities, $R \to 1^{+}$ ⁠, a two-fluid system may become a single-fluid system and fast stable oscillations homogenize the dynamics. Note also that in realistic systems the surface tension may depend on the density ratio, with $T \to 0$ for $R \to 1^{+}$ ⁠.¹⁶

For the accelerated dynamics free from surface tension $G = const, T = 0$ ⁠, the growth-rates are

\begin{matrix} ω_{CDGT} = \pm i \sqrt{R} \sqrt{1 - \frac{G}{G_{c r}}}, G_{c r} = \frac{R (R - 1)}{R + 1}, \\ ω_{LDGT} = \frac{1}{R + 1} (- R + \sqrt{R^{3} + R^{2} - R + G (R^{2} - 1)}); \\ ω_{RTGT} = \sqrt{G \frac{R - 1}{R + 1}} . \end{matrix}

(17d)

The unstable conservative dynamics in Eq. (17d) has the largest growth-rate, when compared to Rayleigh–Taylor and Landau–Darrieus instabilities for $G > G^{*}, G^{*} = (R^{2} - 1) / 4, G^{*} > G_{c r}$ ⁠, and, in dimensional units, $g > g^{*}, g^{*} > g_{c r}, g^{*} = k V_{h}^{2} ({(ρ_{h} / ρ_{l})}^{2} - 1) / 4, g_{c r} = k V_{h}^{2} (ρ_{h} / ρ_{l}) (ρ_{h} - ρ_{l}) / (ρ_{h} + ρ_{l})$ ⁠.^17–19

At $G = const$ and for fluids with very different densities $R \to \infty$ ⁠, the growth-rates approach

ω_{CDGT} \to \sqrt{R} \sqrt{\frac{G}{G_{c r}} - 1}, G_{c r} \to R; ω_{LDGT} \to \sqrt{G + R}; ω_{RTGT} = \sqrt{G}

(17.5)

with the unstable Landau–Darrieus and Rayleigh–Taylor dynamics, and with the stability of the conservative dynamics depending on whether the acceleration magnitude is smaller or larger than the threshold value $G_{c r}$ ⁠. In the stable regime, $G < G_{c r}$ with $G ≪ G_{c r}$ ⁠, the frequency approaches $ω_{CDGT} \to \pm i \sqrt{R}$ ⁠.^17–19 In the unstable regime, $G > G_{c r}$ ⁠, the conservative dynamics instability has the largest growth-rate for $G > G^{*}$ ⁠, $G^{*} \to R^{2} / 4$ ⁠.^17–19

At $G = const$ and for fluids with similar densities $R \to 1^{+}$ the growth-rates approach

ω_{CDGT} \to \pm i \sqrt{\frac{G}{G_{c r}} - 1}, G_{c r} \to \frac{(R - 1)}{2}; ω_{LDGT} \to \sqrt{\frac{G (R - 1)}{2}}; ω_{RTGT} \to \sqrt{\frac{G (R - 1)}{2}}

(17.6)

with the unstable Landau–Darrieus and Rayleigh–Taylor dynamics, and with the stability of the conservative dynamics depending on whether the acceleration strength is smaller or larger than the threshold value $G_{c r}$ ⁠. In the stable regime, $G < G_{c r}$ with $G ≪ G_{c r}$ ⁠, the frequency approaches $ω_{CDGT} \to \pm i$ ⁠.^17–19 In the unstable regime, $G > G_{c r}$ ⁠, the unstable conservative dynamics has the largest growth-rate for $G > G^{*}$ ⁠, $G^{*} \to (R - 1) / 2$ ⁠.^17–19

G. Mechanisms of stabilization and destabilization and structure of flow fields

Based on the foregoing results, we analyze the mechanisms of stabilization and destabilization of the interface dynamics and their link to the structure of flow fields, Tables I and III and Figs. 3–5.

Acceleration: Since the acceleration is directed from the heavy fluid to the light fluid, its qualitative role is to destabilize the interface dynamics. For $G \to \infty, T \to 0$ the dynamics ${r_{CDGT}, r_{LDGT}, r_{RTGT}}$ are unstable, and their growth-rates are

ω_{CDGT} \to \sqrt{G} \sqrt{\frac{R + 1}{R - 1}}; ω_{LDGT} \to \sqrt{G} \sqrt{\frac{1}{R + 1}}; ω_{RTGT} \to \sqrt{G} \sqrt{\frac{R - 1}{R + 1}} .

(18)

For strong accelerations and weak surface tension, the unstable conservative dynamics has the largest growth-rate when compared to Landau–Darrieus and Rayleigh–Taylor instabilities, Fig. 5.

Surface tension: The qualitative role of the surface tension is to stabilize the interface dynamics, Fig. 5. For $G < R (R - 1) / 2$ ⁠, the conservative dynamics is stabilized by the smallest surface tension value, when compared to Landau–Darrieus and Rayleigh–Taylor instabilities, ${\tilde{T}}_{c r} < {\hat{T}}_{c r} < {\bar{T}}_{c r}$ ⁠. For $G > R (R - 1)$ ⁠, the conservative dynamics requires the largest surface tension value for the stabilization, and ${\hat{T}}_{c r} < {\bar{T}}_{c r} < {\tilde{T}}_{c r}$ ⁠.

Density ratio: For fluids with contrasting densities, $R \to \infty$ ⁠, and with finite $G, T$ ⁠, the growth-rates are

ω_{CDGT} \to i \sqrt{R}; ω_{LDGT} \to \sqrt{R}; ω_{RTGT} \to \sqrt{G - T},

(18.2)

suggesting that the conservative dynamics is stable, Landau–Darrieus dynamics is unstable, and that stability of Rayleigh–Taylor dynamics is set by the interplay of the acceleration and the surface tension. This dramatic qualitative difference in the behaviors is due to the inertial stabilization mechanism.

Inertial mechanism: The inertial mechanism is the essential property of the conservative dynamics of the interface with the interfacial mass flux.^17,18 This mechanism is absent in Landau–Darrieus and Rayleigh–Taylor instabilities.^25–27 For weak accelerations, it effectively reduces the surface tension values required for the interface stabilization. For strong accelerations, it leads to the largest stabilizing surface tension and the largest growth-rate of the conservative dynamics, when compared to Landau–Darrieus and Rayleigh–Taylor instabilities, Fig. 5. We find that the macroscopic inertial mechanism is the primary stabilization mechanism for the interface dynamics with the interfacial mass flux.

Structure of flow fields: The inertial mechanism is the macroscopic stabilization mechanism of the conservative dynamics of the interface with the interfacial mass flux; it is due to the conservation of mass, momentum and energy. The surface tension is the microscopic stabilization mechanism; it is caused by anisotropy of interactions between the particles near the interface and results in energy consumption with the increasing interface area. Since the surface tension influences only the interface, one might expect that it keeps unaffected the structure of the flow fields and influences only the growth of the interface. Our theory reveals that the structure of flow fields in the bulk is tightly connected to the microscopic transport at the interface: The bulk “adjusts” to the interface. For the conservative dynamics, the surface tension produces the excess of energy at the interface. This leads to the formation of vortical structures in the bulk, which are energetic in nature, and couples the potential and vortical components of the velocity with the interface perturbations, Figs. 2–4.

H. Characteristic length-scales

In case of the accelerated conservative dynamics with surface tension, in the unstable regime in Eq. (15c), the value of the growth-rate in the dimensional units is

\tilde{g} > g_{c r}, Ω_{CDGT} = k V_{h} \sqrt{\frac{ρ_{h}}{ρ_{l}}} \sqrt{\frac{\tilde{g}}{g_{c r}} - 1}; \tilde{g} = g - \frac{σ k}{(ρ_{h} + ρ_{l}) V_{h}^{2}}, g_{c r} = k V_{h}^{2} (\frac{ρ_{h}}{ρ_{l}}) (\frac{ρ_{h} - ρ_{l}}{ρ_{h} + ρ_{l}}) .

(19a)

The values of gravity $g$ ⁠, the velocity $V_{h}$ ⁠, the surface tension $σ$ ⁠, and the fluid densities $ρ_{h (l)}$ define the characteristic length-scales and timescales of the dynamics of ideal fluids. These include the critical wave-vector value, at which the interface is stabilized, and the maximum wave-vector value, at which the maximum growth-rate is achieved of the interface perturbations, and the associated length- and timescales. For the dynamics $r_{CDGT}$ with the growth-rate $Ω = Ω_{CDGT}$ Eq. (15), the critical and the maximum wave-vector values are, respectively,

\begin{matrix} k_{c r} = \frac{1}{2 σ} [- V_{h}^{2} (\frac{ρ_{h}}{ρ_{l}}) (ρ_{h} - ρ_{l}) + \sqrt{4 σ g (ρ_{h} + ρ_{l}) + {(V_{h}^{2} (\frac{ρ_{h}}{ρ_{l}}) (ρ_{h} - ρ_{l}))}^{2}}], \\ k_{max} = \frac{1}{6 σ} [- 2 V_{h}^{2} (\frac{ρ_{h}}{ρ_{l}}) (ρ_{h} - ρ_{l}) + \sqrt{12 σ g (ρ_{h} + ρ_{l}) + {(2 V_{h}^{2} (\frac{ρ_{h}}{ρ_{l}}) (ρ_{h} - ρ_{l}))}^{2}}] . \end{matrix}

(19b)

For vanishing surface tension values, due to the presence of the inertial mechanism, the critical and maximum wave-vector values remain finite, $k_{c r (max)} / (g / V_{h}^{2}) \sim O (1)$ ⁠, and their ratio approaches $2$ ⁠,

| \frac{σ (ρ_{h} + ρ_{l})}{V_{h}^{2} {(ρ_{h} - ρ_{l})}^{2}} \frac{g}{V_{h}^{2}} {(\frac{ρ_{h}}{ρ_{l}})}^{2} | \to 0; k_{c r} \to \frac{g}{V_{h}^{2}} \frac{ρ_{l}}{ρ_{h}} (\frac{ρ_{h} + ρ_{l}}{ρ_{h} - ρ_{l}}); k_{max} \to \frac{1}{2} \frac{g}{V_{h}^{2}} \frac{ρ_{l}}{ρ_{h}} (\frac{ρ_{h} + ρ_{l}}{ρ_{h} - ρ_{l}}); \frac{k_{c r}}{k_{max}} \to 2.

For very large surface tension values, the critical and maximum wave-vector approach zero, $k_{c r (max)} / (g / V_{h}^{2}) \to 0$ ⁠, and their ratio approaches $\sqrt{3} \approx 1.73$ ⁠,

| \frac{σ (ρ_{h} + ρ_{l})}{V_{h}^{2} {(ρ_{h} - ρ_{l})}^{2}} \frac{g}{V_{h}^{2}} {(\frac{ρ_{h}}{ρ_{l}})}^{2} | \to \infty; k_{c r} \to \sqrt{\frac{g (ρ_{h} + ρ_{l})}{σ}}; k_{max} \to \frac{1}{\sqrt{3}} \sqrt{\frac{g (ρ_{h} + ρ_{l})}{σ}}; \frac{k_{c r}}{k_{max}} \to \sqrt{3} .

Note that while for given values of $V_{h}, g, ρ_{h}, ρ_{l}$ ⁠, the wave-vector values $k_{c r (max)}$ depend strongly on the surface tension $σ$ ⁠, their ratio $(k_{c r} / k_{max})$ varies only slightly with the surface tension.

I. Outcome for experiments and simulations

Outcome for experiments: Our results clearly indicate that for the conservative dynamics, the accelerated interface can be stable even for ideal fluids with vanishing surface tension, when the acceleration value is smaller than a threshold, similarly to ablative Rayleigh–Taylor instabilities in fusion plasmas.^{4–7,20–24} We further find that microscopic stabilizations can be dominated by the macroscopic inertial stabilization. Moreover, according to our results, for the accelerated conservative dynamics the growth of the interface perturbations is also accompanied by the growth of the interface velocity. The latter can be applied to qualitatively explain the quick extinction of the hot spot in inertial confinement fusion, which is observed in the experiments at the National Ignition Facility.^{4–7,20–24}

According to our results, in the extreme regime of strong accelerations typical for high energy density plasmas,^4–7 the instability of the conservative dynamics is the fastest. For given values of the parameters $V_{h}, ρ_{h (l)}, σ$ ⁠, one can unambiguously observe this instability by the increasing the acceleration strength $g$ ⁠. In some experiments in high energy density plasmas, the parameters $V_{h}, g, σ, ρ_{h}, ρ_{l}$ may be a challenge to vary systematically.^{4–7,11,23,24} To address this challenge for given values $V_{h}, ρ_{h (l)}, σ, g$ ⁠, one may vary the wavelength of the initial perturbation $λ$ ⁠, and may observe the interface stabilization at the wave-vector $k = k_{c r}$ and the fastest growth-rate of the unstable interface at the wave-vector $k = k_{max}$ with the associated length- and timescales $λ_{c r (max)} = 2 π / k_{c r (max)}$ and $τ_{c r (max)} = {(k_{c r (max)} V_{h})}^{- 1}$ ⁠. These results can be applied for the experimental design.^{4–7,11,23,24}

Our theory elaborates for experiments and simulations the new extensive benchmarks. For instance, the existing observations usually measure the growth and growth-rate of the perturbation amplitude.^3–14,23,24 We derive the growth and the growth-rate of the perturbation amplitude, and we also provide the detailed information on the structure of flow fields. Our analysis reveals that by measuring the flow fields at macroscopic scales far from the interface, one can capture the transport properties at microscopic scales at the interface. This information is especially important for systems where data are challenging to obtain, including plasma fusion, supernovae, and plasma thrusters.^2–12

Outcome for simulations: Numerical simulations are a powerful tool for investigations of the multiphase matter at the extremes.¹ Tremendous progress was recently achieved in numerical modeling of hydrodynamic instabilities and mixing in high energy density settings.^4–7,35 Existing simulations are now applied in a broad parameter regime, including compressibility, realistic equations of state, radiation transport, thermal heat flow, and others effect.^4–7,35 The dynamics of interfaces and mixing in realistic plasmas is, however, a problem even more challenging than the Millennium problem of Navier–Stokes equation: In addition to solving governing equations in the bulk, it requires one to solve also the boundary value problem at a freely evolving unstable interface, to solve the ill-posed initial value problem, and to account for finite-time singularities, non-locality, and statistical unsteadiness.¹ Even the linearized perturbation wave analysis, similar to that in Eq. (9), is very complex in the case of compressible dynamics with thermal heat flow. To accurately model hydrodynamic instabilities and interfacial mixing in high energy density settings, numerical simulations are required to track unstable interfaces, to capture small scales dissipative processes, to span substantial range of spatial and temporal scales, to use highly accurate numerical methods and massive computations, and to extensively verify and validate the numerical results.^{1,4–7,13–15} Theoretical benchmarks can help to further advance the methods of numerical modeling of complex processes in high energy density plasmas.^{1,4–7,13–15}

Our analysis elaborates the extensive benchmarks for the interface dynamics, which were not diagnosed in detail before. These include the growth and the growth-rate of the interface perturbations; the dependence of the flow fields on the density ratio, the acceleration strength and the surface tension; the unsteadiness of the interface velocity, and others. Our qualitative and quantitative analytical results—the new fluid instability, the interplay of macroscopic and microscopic stabilization mechanisms, and the structure of flow fields—can serve for further advancements of Lagrangian and Eulerian methods of numerical modeling of multiphase matter with sharply changing flow fields in high energy density settings.^1,13–15

Comparison with observations and other models: In order to directly compare our extensive theoretical benchmarks with experiments, a scrupulous analysis of unprocessed raw data is required.^{4–7,23,24,34,35} Since the raw data are challenging to access,^34,35 we briefly compare here the growth-rate values for the conservative dynamics with those proposed by the models^{20,21,31–33,36,37} of ablative Rayleigh–Taylor and Richtmyer–Meshkov instabilities. We focus on highly contrasting fluid densities typical for high energy density plasmas, $R \to \infty$ ⁠, omit for simplicity the surface tension effect, $T \to 0$ ⁠, and we use the dimensional units $Ω, k, V_{h}$ and $R = ρ_{h} / ρ_{l}$ ⁠. In the absence of thermal heat flow, we set the interface velocity (i.e., the “ablation velocity”) in Eq. (8) as $V_{0} = - V_{h}$ with $V_{0} = | V_{0} | = V_{h}$ ⁠, as per the usual convention,¹⁶ and we associate this velocity magnitude with the rate of mass ablation $\dot{m}$ as $\dot{m} = ρ_{h} V_{h}$ ⁠.^{20–24,31–35} In addition to the growth-rate, we identify the quantities that fully characterize the dispersion curve.

Per our theory, for the conservative dynamics free from surface tension and thermal heat flow and for fluids with very different densities, $ρ_{h} / ρ_{l} \to \infty$ ⁠, the growth-rate $Ω_{C D} = {Ω_{CDGT} |}_{σ = 0, (ρ_{h} / ρ_{l}) \to \infty}$ ⁠, the critical $k_{c r}$ wave-vector at which the dynamics is stabilized $Ω_{C D_{c r}} = {Ω_{C D} |}_{k = k_{c r}} = 0$ ⁠, the maximum $k_{max}$ wave-vector at which the dynamics has the fastest growth, the maximum growth-rate value of the accelerated dynamics $Ω_{C D_{max}} = {Ω_{C D} |}_{k = k_{max}}$ ⁠, and frequency of the inertial dynamics $Ω_{C D_{i n}} = {Ω_{C D} |}_{g = 0}$ are

\begin{matrix} Ω_{C D} = \sqrt{g k - {(k V_{0})}^{2} (\frac{ρ_{h}}{ρ_{l}})}; Ω_{C D max} = \frac{1}{2} \frac{g}{V_{0}} \sqrt{\frac{ρ_{l}}{ρ_{h}}}, \\ Ω_{C D_{i n}} = \pm i k V_{0} \sqrt{\frac{ρ_{h}}{ρ_{l}}}; \\ k_{c r} = \frac{g}{V_{0}^{2}} (\frac{ρ_{l}}{ρ_{h}}), k_{max} = \frac{1}{2} \frac{g}{V_{0}^{2}} (\frac{ρ_{l}}{ρ_{h}}), \frac{k_{c r}}{k_{max}} = 2. \end{matrix}

(20a)

The models^{20,21,31–33} of ablative Rayleigh–Taylor and Richtmyer–Meshkov instabilities apply the plasma's equation of state and account for that for fluids (plasmas) with highly contrasting densities $(ρ_{h} / ρ_{l}) \to \infty$ ⁠, the energy flux $Q = J \bar{Q}$ is large, $\bar{Q} = [{\bar{W}}_{0}]$ ⁠, the thermal conductivity $κ$ of the light fluid is small, and the thermal transport in the heavy fluid is negligible.^{20–23,31–33} By further connecting the perturbed thermal heat flux $δ Q$ to the energy flux $Q$ and the interface perturbations, the model³¹ proposed for ablative Rayleigh–Taylor instability the growth-rate which we denote as $Ω_{PSI}$ ⁠. To fully describe the dispersion curve for the growth-rate $Ω_{PSI}$ ⁠,³¹ we evaluate here the values of the critical $k_{c r}$ and maximum $k_{max}$ wave-vectors, the maximum growth-rate of the accelerated dynamics $Ω_{P S I_{max}} = {Ω_{PSI} |}_{k = k_{max}}$ ⁠, and frequency of the inertial dynamics $Ω_{P S I_{i n}} = {Ω_{PSI} |}_{g = 0}$ corresponding to ablative Richtmyer–Meshkov instability. These values are

\begin{matrix} Ω_{PSI} = - 2 f k V_{0} + \sqrt{g k - {(k V_{0})}^{2} (\frac{ρ_{h}}{ρ_{l}}) + {(2 f k V_{0})}^{2}}; \\ Ω_{PSI max} \approx \frac{1}{2} \frac{g}{V_{0}} \sqrt{\frac{ρ_{l}}{ρ_{h}}}, Ω_{P S I_{i n}} \approx \pm i k V_{0} \sqrt{\frac{ρ_{h}}{ρ_{l}}}; \\ k_{c r} \approx \frac{g}{V_{0}^{2}} (\frac{ρ_{l}}{ρ_{h}}), k_{max} \approx \frac{1}{2} \frac{g}{V_{0}^{2}} (\frac{ρ_{l}}{ρ_{h}}) . \end{matrix}

(20b)

In Eq. (20b), the value of the factor $f$ is on the order of unity, $f \sim 1$ ⁠, and the approximate equalities account for that $(ρ_{l} / ρ_{h}) \to 0$ in high energy density plasmas. The model³¹ reported excellent agreement with simulations.²¹

Model²³ proposed a slightly different growth-rate formula in order to describe, with account for results,^32,33 ablative Rayleigh–Taylor and ablative Richtmyer–Meshkov instabilities for fluids with highly contrasting densities, which we denote as $Ω_{AVG}$ ⁠. To fully describe the dispersion curve for the growth-rate $Ω_{AVG}$ ⁠,^23,32,33 we evaluate here the values of the critical $k_{c r}$ and maximum $k_{max}$ wave-vectors, the maximum growth-rate value of the accelerated dynamics $Ω_{A V G_{max}} = {Ω_{AVG} |}_{k = k_{max}}$ and the frequency $Ω_{A V G_{i n}} = {Ω_{AVG} |}_{g = 0}$ of the inertial dynamics corresponding to ablative Richtmyer–Meshkov instability. These values are

\begin{matrix} Ω_{AVG} = - 2 k V_{0} + \sqrt{g k - {(k V_{0})}^{2} (\frac{ρ_{h}}{ρ_{l}})}; \\ Ω_{AVG max} \approx \frac{1}{2} \frac{g}{V_{0}} \sqrt{\frac{ρ_{l}}{ρ_{h}}}, Ω_{A V G_{i n}} \approx \pm i k V_{0} \sqrt{\frac{ρ_{h}}{ρ_{l}}}; \\ k_{c r} \approx \frac{g}{V_{0}^{2}} (\frac{ρ_{l}}{ρ_{h}}), k_{max} \approx \frac{1}{2} \frac{g}{V_{0}^{2}} (\frac{ρ_{l}}{ρ_{h}}), \end{matrix}

(20c)

where approximate equalities account for that $(ρ_{l} / ρ_{h}) \to 0$ ⁠. The models^23,32,33 reported excellent agreement with experiments, which observed high frequency oscillations (slowly decaying with time) in ablative Richtmyer–Meshkov instability with highly contrasting densities.

The pioneering model of ablative Rayleigh–Taylor instability^20,21 proposed the growth-rates, which we denote as $Ω_{BAK}$ ⁠, with $Ω_{BAK g} = {Ω_{BAK} |}_{(g / k V_{0}^{2}) \to \infty}$ for strong accelerations

Ω_{BAK} = - f k V_{0} + \sqrt{g k}; Ω_{BAK g} \approx \sqrt{g k},

(20d)

where the factor $f$ value is $f \sim O (1)$ ⁠.

We note that what is presently called the “ablative Richtmyer–Meshkov instability” was first investigated in the works.^36,37 These works identified significant differences between the ablative and classical Richtmyer–Meshkov instabilities, in excellent agreement with our theory and with experiments and simulations.^36–38 Particularly, while both in ablative and classical Richtmyer–Meshkov instabilities the growth-rate may oscillate in time, in classical Richtmyer–Meshkov instability the oscillations occur near a finite growth-rate value,³⁸ whereas in ablative Richtmyer–Meshkov instability the growth-rate contains only an oscillatory part, and the perturbation amplitude saturates.^36,37 This leads to significant differences in the increase in the ablation surface deformation in the ablative case when compared to the classical case.^36,37 The model of ablative Richtmyer–Meshkov^36,37 applied the Rankine–Hugoniot conditions at the shock front and the Chapman–Jouguet deflagration conditions at the laser ablation surface, and found for the growth-rate, which we denote as $Ω_{N I}$ ⁠,

Re [Ω_{N I}] = 0; Im [Ω_{N I}] \neq 0.

(20e)

By comparing expressions Eq. (20), we find that our theory Eq. (20a) accurately describes the dispersion curves Eqs. (20a)–(20e), including the critical and maximum wave-vector values, the maximum growth-rates of ablative RTI, the oscillations of ablative RMI, as well as the strong acceleration limit.^{20–23,31–33} It is thus agrees with available experiments and simulations.

Our theory obtains the new important results: the inertial stabilization mechanism (being the dominant macroscopic mechanism), the unsteadiness of the interface velocity (oscillating near the constant value in the stable regime, and increasing with time in the instable regime), and the structure of the flow fields (being free from the interfacial shear, and having energetic in nature vortical field in the presence of microscopic stabilization); it identifies the extensive benchmarks, which were not diagnosed before, and thus opens new avenues for experimental studies in high energy density plasmas.^1–7

Our theory strictly obeys the conservation laws and provides the rigorous consideration of the problem of the interface dynamics with the interfacial mass flux. This work presents the detailed study of the interplay of macroscopic inertial and microscopic surface tension stabilization mechanisms with the destabilizing acceleration. Our theory can be further developed to rigorously account for the thermal transport, the compressibility, the equation of state, the magnetic field, and other effects characterizing realistic plasma environments. It can systematically evaluate the influence of these effects on the interface velocity, stability, and growth, and on the structure of flow fields. It can thus elaborate a unified theory framework for the studies of interfacial dynamics in a broad range of processes, including extreme conditions of high energy density, to be done in the future.

Experiments and theory: In our theory, the acceleration is a body force, and the acceleration magnitude is set constant in order to simplify the analysis Eqs. (1)–(20). In experiments in laser-ablated plasmas, the acceleration is an effective acceleration, and it is time-dependent. To set a model experiment with a (quasi-) constant acceleration is a challenging task; it is the target, laser-drive, and experiment specific.⁶

For instance, in order to set a quasi-constant acceleration of a planar foil target, one needs first to irradiate the target with constant laser intensity.⁶ Once the initial shock breaks out the back and the rarefaction comes back to the front side of the foil, the foil accelerates as whole unit. This acceleration slowly decreases with time, because the critical density position (where the laser energy is deposited) moves away from the ablation front (as plasma is ablated off the foil), where the ablation pressure is generated. One needs next to slightly increase the laser intensity vs time incident of the target to create a constant acceleration. One needs then to iterate. After a few iterations, one may approach a quasi-constant acceleration, which is tuned to the target, the laser drive, and the experiment. In compound rippled targets, the process is even more complex, and the experiments are the state-of-the-art.^4–7

Our analysis finds that in the unstable regime of the conservative dynamics, the interface velocity is a complex function of time even for a constant acceleration. Physically, this is caused by the need to conserve mass, momentum, and energy in the perturbed system. The time-dependence of the interface velocity can be exponential in the linear regime and a power-law in the non-linear regime. In a frame of reference moving with the interface velocity, the system may experience an effective acceleration, which is a complex function of time and is also dependent on the perturbation amplitude. While this opens new and exciting perspectives for theory, it also indicates that the problems of ablative Rayleigh–Taylor and Richtmyer–Meshkov instabilities are even more challenging than they may appear.

One may need to elaborate a new experimental approach to the problem, such that, on the one hand, it would closely mimic transport processes in laser-ablated plasmas by employing a striking similarity of non-equilibrium dynamics of interfaces and mixing in the vastly different physical regimes, and, on the other hand, it would be affordable and repeatable in a broad range of parameters, setups and conditions. In classical plasmas, this approach was applied in the large plasma device (LAPD); it enabled, through immense statistics of high quality data, the discoveries of the spiky structures in flux ropes and the new topologies in magnetic reconnection.³⁹ In classical Rayleigh–Taylor instabilities, the approach was employed in jelly experiments and enabled the direct observation of the order in interfacial mixing at very high Reynolds numbers.⁴⁰

We refer to the 1924 Noble Lecture of Robert A. Milliken emphasized: “The fact that Science walks forward on two feet, namely, theory and experiment… Sometimes it is one foot which is put forward first, sometimes the other, but continuous progress is only made by the use of both—by theorizing and then testing, or by finding new relations in the process of experimenting and then bringing the theoretical foot up and pushing it on beyond, and so on in unending alternations.”

Summary: It is traditionally believed that the interface dynamics can be stabilized by factors depending on the microscopic properties of matter, occurring due to interactions of constituting particles.^2–12,16 Our analysis suggests that while these factors indeed may play a stabilizing role and may influence the structure of the flow fields, the conservative dynamics of the interface with the interfacial mass flux is stabilized primarily by the inertial mechanism, which is enabled by the macroscopic motion of the interface as whole. This mechanism is absent in other instabilities.^17,25–27 This mechanism can be applied for the better understanding of a broad range of phenomena in nature, including thermonuclear flashes in type-Ia supernova, coronal mass ejections in the Solar flares, and plasma instabilities in the Earth ionosphere, and for the grip and control of high energy density plasmas in technology, including fusion, nano-fabrication, and medicine.^1–12,23,24

The scrupulous analyses of raw and processed data, the experimental approaches enabling high repeatability in a broad range of parameters, setups and conditions, and the vast and ample statistics of crispy experimental data are now in demand in order to grasp the non-equilibrium dynamics of interfaces and mixing in plasmas in high and in low energy density regime.^1–41

IV. DISCUSSION AND CONCLUSION

Non-equilibrium dynamics, interfaces, and mixing ubiquitously occur in high and in low energy density plasmas, at astrophysical and at molecular scales. In this work, we theoretically studied the interfacial dynamics with interfacial mass flux, and we scrupulously investigated the interplay of the destabilizing acceleration with the macroscopic and microscopic stabilizations mechanisms due to the inertial effect and the surface tension, respectively, Eqs. (1)–(19), Tables I–III and Figs. 1–5.¹⁷

Our theory finds: The macroscopic inertial mechanism is the primary mechanism of the interface stabilization. The interface velocity is unsteady—it oscillates near the constant value in the stable regime, and increases with time in the unstable regime. The fluid velocity is shear-free at the interface, and the vortical and vorticity fields are energetic in nature—the vortical structures are produced to balance the energy excess at the interface, caused by, e.g., microscopic stabilization. Our results agree qualitatively with available observations, elaborate new extensive benchmarks for future experiments and simulations, and serve for better understanding of plasma processes in nature and technology.^1–12

We identified in the broad parameter regime the regions of stability and instability for the interface dynamics conserving mass, momentum, and energy Eqs. (1)–(18), Fig. 1. We found that even the surface tension influences only the interface, the bulk adjusts, and the presence of the microscopic stabilization may lead to the formation of volumetric vortical structures, Eqs. (11)–(15), Figs. 2–4.

The instability of the conservative dynamics can develop only in the presence of the acceleration and only when its magnitude exceeds a threshold, Eqs. (11)–(15), Table III. This threshold value reflects the contributions of the macroscopic inertial stabilization mechanism and the microscopic surface tension mechanism and is finite for zero surface tension, Table III. In the unstable regime, the dynamics couples the interface perturbations with the potential and vortical components of the velocity fields in the fluids' bulk and is shear-free at the interface, Fig. 4. The resultant dynamics describes the standing wave with the growing amplitude and has the growing interface velocity. In the extreme regime of strong acceleration and weak surface tension typical for high energy density plasmas, the interface dynamics is set primarily by the interplay of macroscopic mechanisms—the inertial stabilization and the destabilizing acceleration, whereas the microscopic surface tension influences the structure of the flow fields in the bulk, Eq. (15). These properties unambiguously differentiate the unstable conservative dynamics from Landau–Darrieus and Rayleigh–Taylor instabilities, Eqs. (15)–(18), and open new opportunities for control of plasma processes in nature and technology.^1–27

Our theory agrees qualitatively with available observations. For instance, it can be applied to explain the quick extinction of the hot spot in inertial confinement fusion observed in experiments at the National Ignition Facility.^4–7 For experimental and numerical studies of unstable interfaces,^{1,4–7,10–15,23,24} our analysis provides the growth-rate of the interface perturbations and the growth of the interface velocity, defines the regions of the stable and unstable dynamics, identifies the structure of flow fields, and finds the values of the wave-vector of the initial perturbation at which the dynamics are stabilized and at which it has the largest growth, Eqs. (1)–(19). According to our results, by measuring the flow fields in the bulk at macroscopic scales, one can capture the transport properties at microscopic scales at the interface. These theoretical results indicate a strong need in further advancement of methods of numerical modeling and experimental diagnostics in plasmas in high and in low energy density regimes.^1–24

Our theoretical framework can be further connected to realistic plasma environments, where the dynamics are usually accompanied by magnetic fields, radiation transport, compressibility, dissipation, diffusion, and non-local forces.^2–12 Our general approach can be extended to systematically incorporate these microscopic stabilization mechanisms, to analyze the interplay of the interface stability with the structure of flow fields, and to elaborate a unified theory framework for the studies of interfacial dynamics in a broad range of processes including ablative Rayleigh–Taylor instabilities in fusion plasmas, nuclear synthesis in type-Ia supernova, plasma discharge interfacing with liquids, dynamics of supercritical fluids, and D'yakov–Kontorovich instability of the shock waves, to be done in the future.¹

AUTHORS' CONTRIBUTIONS

The authors contributed to the work as follows: S.I.A. designed the research; D.V.I. and S.I.A. performed the research; D.V.I. and S.I.A. analyzed the data; D.V.I. and S.I.A. discussed the results; and D.V.I. and S.I.A. wrote the paper.

ACKNOWLEDGMENTS

The authors thank the University of Western Australia, AUS (Project Grant No. 10101047), and the National Science Foundation, USA (Award No. 1404449) for the support. The authors express their gratitude to Dr. Bruce A. Remington for inspiring discussions on high energy density plasma experiments.

DATA AVAILABILITY

The methods, the results, and the data presented in this work are freely available to the readers in the paper and on the request from the authors.

References

1.

S. I.

Abarzhi

and

W. A.

Goddard

, “

Interfaces and mixing: Non-equilibrium transport across the scales

,”

Proc. Natl. Acad. Sci.

116

,

18171

(

2019

).

https://doi.org/10.1073/pnas.1818855116

Google Scholar

Crossref

2.

D.

Arnett

,

Supernovae and Nucleosynthesis

(

Princeton University Press

,

1996

).

Google Scholar

Crossref

3.

J. B.

Bell

,

M. S.

Day

,

C. A.

Rendleman

,

S. E.

Woosley

, and

M. A.

Zingale

, “

Direct numerical simulations of type Ia supernovae flames. I. The Landau-Darrieus instability

,”

Astrophs. J.

606

,

1029

(

2004

);

https://doi.org/10.1086/383023

Crossref

Google Scholar

J. B.

Bell

,

M. S.

Day

,

C. A.

Rendleman

,

S. E.

Woosley

, and

M. A.

Zingale

“

Direct numerical simulations of type Ia supernovae flames. II. The Rayleigh-Taylor instability

,”

Astrophs. J.

608

,

883

(

2004

).

https://doi.org/10.1086/420841

Crossref

Google Scholar

4.

O. A.

Hurricane

,

D. A.

Callahan

,

D. T.

Casey

,

E. L.

Dewald

,

T. R.

Dittrich

,

T.

Döppner

,

M. B.

Garcia

,

D. E.

Hinkel

,

L. F. B.

Hopkins

,

P.

Kervin

et al, “

The high-foot implosion campaign on the National Ignition Facility

,”

Phys. Plasmas

21

,

056314

(

2014

).

https://doi.org/10.1063/1.4874330

Google Scholar

Crossref

5.

S. W.

Haan

,

J. D.

Lindl

,

D. A.

Callahn

,

D. S.

Clark

,

J. D.

Salmonson

,

B. A.

Hammel

,

L. J.

Atherton

,

R. C.

Cook

,

M. J.

Edwards

,

S.

Glenzer

,

A. V.

Hamza

,

S. P.

Hatchett

,

M. C.

Herrmann

et al, “

Point design targets, specifications, and requirements for the 2010 ignition campaign on the National Ignition Facility

,”

Phys. Plasmas

18

,

051001

(

2011

).

https://doi.org/10.1063/1.3592169

Google Scholar

Crossref

6.

B. A.

Remington

,

H.-S.

Park

,

D. T.

Casey

,

R. M.

Cavallo

,

D. S.

Clark

,

C. M.

Huntington

,

C. C.

Kuranz

,

A. R.

Miles

,

S. R.

Nagel

,

K. S.

Raman

, and

V. A.

Smalyuk

, “

Rayleigh–Taylor instabilities in high-energy density settings on the National Ignition Facility

,”

Proc. Natl. Acad. Sci. U.S.A.

116

,

18233

(

2019

).

https://doi.org/10.1073/pnas.1717236115

Google Scholar

Crossref

PubMed

7.

R. P.

Drake

, “

Perspectives on high-energy-density physics

,”

Phys. Plasmas

16

,

055501

(

2009

).

https://doi.org/10.1063/1.3078101

Google Scholar

Crossref

8.

A.

Mahalov

, “

Multiscale modeling and nested simulations of three-dimensional ionospheric plasmas

,”

Phys. Scr.

89

,

098001

(

2014

).

https://doi.org/10.1088/0031-8949/89/9/098001

Google Scholar

Crossref

9.

Y. B.

Zeldovich

and

Y. P.

Raizer

,

Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena

, 2nd ed. (

Dover

,

New York

,

2002

).

Google Scholar

10.

T. C.

Underwood

,

K. T.

Loebner

,

V. A.

Miller

, and

M. A.

Cappelli

, “

Dynamic formation of stable current-driven plasma jets

,”

Sci. Rep.

9

,

2588

(

2019

).

https://doi.org/10.1038/s41598-019-39827-6

Google Scholar

Crossref

PubMed

11.

S.

Lugomer

, “

Laser-generated Richtmyer-Meshkov and Rayleigh-Taylor instabilities in a semiconfined configuration: Bubble dynamics in the central region of the Gaussian spot

,”

Phys. Scr.

94

,

015001

(

2019

).

https://doi.org/10.1088/1402-4896/aae71e

Google Scholar

Crossref

12.

T.

Kaneko

,

K.

Baba

, and

R.

Hatakeyama

, “

Static gas-liquid interfacial direct current discharge plasmas using ionic liquid cathode

,”

J. Appl. Phys.

105

,

103306

(

2009

).

https://doi.org/10.1063/1.3133213

Google Scholar

Crossref

13.

V. V.

Zhakhovsky

,

A. P.

Kryukov

,

V. Y.

Levashov

,

I. N.

Shishkova

, and

S. I.

Anisimov

, “

Mass and heat transfer between evaporation and condensation surfaces

,”

Proc. Natl. Acad. Sci. U.S.A.

116

(

37

),

18209

(

2019

).

https://doi.org/10.1073/pnas.1714503115

Google Scholar

Crossref

PubMed

14.

K.

Kadau

,

J. L.

Barber

,

T. C.

Germann

,

B. L.

Holian

, and

B. J.

Alder

, “

Atomistic methods in fluid simulation

,”

Philos. Trans. R. Soc. A

368

,

1547

(

2010

).

https://doi.org/10.1098/rsta.2009.0218

Google Scholar

Crossref

15.

Z.

Dell

,

A.

Pandian

,

A. K.

Bhowmick

,

N. C.

Swisher

,

M.

Stanic

,

R. F.

Stellingwerf

, and

S. I.

Abarzhi

, “

Maximum initial growth-rate of strong-shock-driven Richtmyer-Meshkov instability

,”

Phys. Plasmas

24

,

090702

(

2017

).

https://doi.org/10.1063/1.4986903

Google Scholar

Crossref

16.

L. D.

Landau

and

E. M.

Lifshitz

,

Theory Course I-X

(

Pergamon Press

,

New York

,

1987

).

Google Scholar

17.

S. I.

Abarzhi

,

D. V.

Ilyin

,

W. A.

Goddard

III, and

S.

Anisimov

, “

Interface dynamics: New mechanisms of stabilization and destabilization and structure of flow fields

,”

Proc. Natl. Acad. Sci. U.S.A.

116

(

37

),

18218

(

2019

).

https://doi.org/10.1073/pnas.1714500115

Google Scholar

Crossref

PubMed

18.

D. V.

Ilyin

,

Y.

Fukumoto

,

W. A.

Goddard

, and

S. I.

Abarzhi

, “

Analysis of dynamics, stability, and flow fields' structure of an accelerated hydrodynamic discontinuity with interfacial mass flux by a general matrix method

,”

Phys. Plasmas

25

,

112105

(

2018

).

https://doi.org/10.1063/1.5008648

Google Scholar

Crossref

19.

S. I.

Abarzhi

,

Y.

Fukumoto

, and

L. P.

Kadanoff

, “

Stability of a hydrodynamic discontinuity

,”

Phys. Scr.

90

,

018002

(

2015

).

https://doi.org/10.1088/0031-8949/90/1/018002

Google Scholar

Crossref

20.

S. E.

Bodner

, “

Rayleigh–Taylor instability and laser-pellet fusion

,”

Phys. Rev. Lett.

33

,

761

–

764

(

1974

).

https://doi.org/10.1103/PhysRevLett.33.761

Google Scholar

Crossref

21.

H. J.

Kull

and

S. I.

Anisimov

, “

Ablative stabilization in the incompressible Rayleigh–Taylor instability

,”

Phys. Fluids

29

,

2067

(

1986

).

https://doi.org/10.1063/1.865593

Google Scholar

Crossref

22.

J.

Sanz

, “

Self-consistent analytical model of the Rayleigh-Taylor instability in inertial confinement fusion

,”

Phys. Rev. Lett.

73

,

2700

–

2703

(

1994

).

https://doi.org/10.1103/PhysRevLett.73.2700

Google Scholar

Crossref

PubMed

23.

Y.

Aglitskiy

,

A. L.

Velikovich

,

M.

Karasik

,

N.

Metzler

,

S. T.

Zalesak

,

A. J.

Schmitt

,

L.

Phillips

,

J. H.

Gardner

,

V.

Serlin

,

J. L.

Weaver

, and

S. P.

Obenschain

, “

Basic hydrodynamics of Richtmyer–Meshkov-type growth

,”

Philos. Trans. R. Soc., Ser. A

368

,

1739

–

1768

(

2010

).

https://doi.org/10.1098/rsta.2009.0131

Google Scholar

Crossref

24.

H.

Azechi

,

T.

Sakaiya

,

S.

Fujioka

,

Y.

Tamari

,

K.

Otani

,

K.

Shigemori

,

M.

Nakai

,

H.

Shiraga

,

N.

Miyanaga

, and

K.

Mima

, “

Comprehensive diagnosis of growth rates of the ablative Rayleigh-Taylor instability

,”

Phys. Rev. Lett.

98

,

045002

(

2007

).

https://doi.org/10.1103/PhysRevLett.98.045002

Google Scholar

Crossref

PubMed

25.

R.

Lord

, “

Investigations of the character of the equilibrium of an incompressible heavy fluid of variable density

,”

Proc. London Math. Soc.

s1-14

,

170

–

177

(

1882

).

https://doi.org/10.1112/plms/s1-14.1.170

Google Scholar

Crossref

26.

G. I.

Taylor

, “

The instability of liquid surfaces when accelerated in a direction perpendicular to their planes

,”

Proc. R. Soc. London, Ser. A

201

,

192

–

196

(

1950

).

Google Scholar

27.

L. D.

Landau

, “

On the theory of slow combustion

,”

Acta Physicochim URSS

19

,

77

–

85

(

1944

).

Google Scholar

28.

S. P.

D'yakov

, “

Ob ustoichivosti udarnyh voln

,”

Z. Exp. Teor. Fiz.

27

,

288

–

295

(

1954

) (in Russian).

Google Scholar

29.

V. M.

Kontorovich

, “

Concerning the stability of shock waves

,”

Sov. Phys. JETP

6

,

1179

–

1180

(

1958

).

Google Scholar

30.

D. V.

Ilyin

,

W. A.

Goddard

III, and

S. I.

Abarzhi

, “

Inertial dynamics of an interface with interfacial mass flux: Stability and flow fields' structure, inertial stabilization mechanism, degeneracy of Landau's solution, effect of energy fluctuations, and chemistry-induced instabilities

,”

Phys. Fluids

32

,

082105

(

2020

).

https://doi.org/10.1063/5.0013165

Google Scholar

Crossref

31.

A. R.

Piriz

,

J.

Sanz

, and

L. F.

Ibañez

, “

Rayleigh–Taylor instability of steady ablation fronts: The discontinuity model revisited

,”

Phys. Plasmas

4

,

1117

(

1997

).

https://doi.org/10.1063/1.872200

Google Scholar

Crossref

32.

V. N.

Goncharov

, “

Theory of the ablative Richtmyer–Meshkov instability

,”

Phys. Rev. Lett.

82

,

2091

–

2094

(

1999

).

https://doi.org/10.1103/PhysRevLett.82.2091

Google Scholar

Crossref

33.

V. N.

Goncharov

,

R.

Betti

,

R. L.

McCrory

,

P.

Sorotokin

, and

C. P.

Verdon

, “

Self-consistent stability analysis of ablation fronts with large Froude numbers

,”

Phys. Plasmas

3

,

1402

(

1996

).

https://doi.org/10.1063/1.871730

Google Scholar

Crossref

34.

J. P.

Knauer

,

R.

Betti

,

D. K.

Bradley

,

T. R.

Boehly

,

T. J. B.

Collins

,

V. N.

Goncharov

,

P. W.

McKenty

,

D. D.

Meyerhofer

,

V. A.

Smalyuk

,

C. P.

Verdon

,

S. G.

Glendinning

,

D. H.

Kalantar

, and

R. G.

Watt

, “

Single-mode, Rayleigh-Taylor growth-rate measurements on the OMEGA laser system

,”

Phys. Plasmas

7

,

338

(

2000

).

https://doi.org/10.1063/1.873802

Google Scholar

Crossref

35.

R.

Betti

and

O. A.

Hurricane

, “

Inertial-confinement fusion with lasers

,”

Nat. Phys.

12

,

435

(

2016

).

https://doi.org/10.1038/nphys3736

Google Scholar

Crossref

36.

R.

Ishizaki

and

K.

Nishihara

,

Phys. Rev. Lett.

78

,

1920

(

1997

).

https://doi.org/10.1103/PhysRevLett.78.1920

Crossref

37.

K.

Nishihara

,

R.

Ishizaki

,

J. G.

Wouchuk

,

Y.

Fukuda

, and

Y.

Shimuta

,

Phys. Plasmas

5

,

1945

(

1998

).

https://doi.org/10.1063/1.872864

Crossref

38.

M.

Herrmann

,

P.

Moin

, and

S. I.

Abarzhi

,

J. Fluid Mech.

612

,

311

–

338

(

2008

).

https://doi.org/10.1017/S0022112008002905

Crossref

39.

W.

Gekelman

,

S. W.

Tang

,

T. D.

Haas

,

S.

Vincena

,

P.

Pribyl

, and

R.

Sydora

,

Proc. Natl. Acad. Sci. U.S.A.

116

,

18239

(

2019

).

https://doi.org/10.1073/pnas.1721343115

Crossref

PubMed

40.

E. E.

Meshkov

and

S. I.

Abarzhi

,

Fluid Dyn. Res.

51

,

065502

(

2019

).

https://doi.org/10.1088/1873-7005/ab3e83

Crossref

41.

R. A.

Milliken

, Nobel Lecture, 1924, see https://www.nobelprize.org/prizes/physics/1923/millikan/lecture/.

2021

Author(s)

Macroscopic and microscopic stabilization mechanisms of unstable interface with interfacial mass flux

I. INTRODUCTION

II. METHOD

A. Governing equations

B. Linearized dynamics

C. Interfacial shear and interfacial boundary conditions

D. Structure of fundamental solutions

E. The applicability and future developments of the theoretical framework

III. RESULTS

A. Fundamental solutions

B. Inertial dynamics free from surface tension

C. Inertial dynamics with surface tension

D. Accelerated dynamics free from surface tension

E. Accelerated dynamics with surface tension

F. Landau–Darrieus and Rayleigh–Taylor instabilities and conservative dynamics

G. Mechanisms of stabilization and destabilization and structure of flow fields

H. Characteristic length-scales

I. Outcome for experiments and simulations

IV. DISCUSSION AND CONCLUSION

AUTHORS' CONTRIBUTIONS

ACKNOWLEDGMENTS

DATA AVAILABILITY

References

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

Macroscopic and microscopic stabilization mechanisms of unstable interface with interfacial mass flux

I. INTRODUCTION

II. METHOD

A. Governing equations

B. Linearized dynamics

C. Interfacial shear and interfacial boundary conditions

D. Structure of fundamental solutions

E. The applicability and future developments of the theoretical framework

III. RESULTS

A. Fundamental solutions

B. Inertial dynamics free from surface tension

C. Inertial dynamics with surface tension

D. Accelerated dynamics free from surface tension

E. Accelerated dynamics with surface tension

F. Landau–Darrieus and Rayleigh–Taylor instabilities and conservative dynamics

G. Mechanisms of stabilization and destabilization and structure of flow fields

H. Characteristic length-scales

I. Outcome for experiments and simulations

IV. DISCUSSION AND CONCLUSION

AUTHORS' CONTRIBUTIONS

ACKNOWLEDGMENTS

DATA AVAILABILITY

References

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only