Extensions to the Navier–Stokes equations

The Navier–Stokes equations appeared for the first time in Sur les lois des mouvements des fluides, en ayant égard à l'adhésion des molecules¹ in 1822. They arose from applying the theory of elasticity for the stain–stress equilibrium equations and extending the Newton's second law to the moving state—elastic fluid motion.

After the pioneering work of Navier, numerous investigators have completed the Navier–Stokes equations, such as Stokes. In the present days, the Navier–Stokes equations have been broadly applied to describe the physics of flow phenomena of many scientific and engineering interest, such as weather forecast, the design of aircraft and cars, and many other things.

Although they are simple looking, for decades, the existence and smoothness of the Navier–Stokes equations is still an unsolved problem. Recently, Science Webinar published a special booklet of “125 questions: exploration and discovery,” one of the questions reads that “Despite the fact that they are practically useful, proof of the Navier–Stokes equations, used to model fluid flow, remains elusive. We still do not know whether these equations are applicable at all times and particularly if global classical solutions exist for all points in a 3D space.”² 

Before 1905, namely, before Einstein proposed the special relativity theory,³ scientists and engineers understood the motion based on the Newton's classical motion law with the Galilean transformation.

The incompatibility of Newtonian classical mechanics with Maxwell's equations of electromagnetism led to Einstein's development of special relativity. Today, special relativity is proven to be the most accurate model of motion for macro behavior of physical properties at any speed when the gravitational effect is negligible. The special relativity theory corrects the classical Newtonian mechanics to handle situations involving all motions.

Once Einstein revolutionized and expanded our understanding of motion—the inseparability of the time and space in the special relativity theory, we are faced with a huge task. All the classical physics (especially classical field theory) involving of motions had to be reexamined. Therefore, inspirited by the idea of specific relativity, we will reinforce the Navier–Stokes equations in this paper.

As we know, the earliest motivation to develop special relativity theory is to restore the principle of relativity in physics by maintaining that Maxwell's equations are correct in all inertial reference frames, based on the postulate that the speed of light in vacuum space is a constant in all inertial reference frames. However, special relativity is not merely a theory about light traveling, it is a theory about the indivisibility of space and time in this sense, and the principles of relativity can be extended in all cases where there are relative motions between physical objects.

Tao's recent research⁴ focuses on the comparison of the magnitude of the convection term and the viscous term. When flow speed greatly exceeds the wave number, convection terms dominate; it might be possible that the flow velocity and wave number go to infinity. It leads to a heuristic argument that the Navier–Stokes equations cease to be an accurate model for the fluid flow in some cases. This research gives some clues that there are some implicit connections between flow velocity and wave properties of the fluid flow and consequently the wave propagation speed. (Wave number is the concept of wave function.)

Dou^5,6 proposed a total mechanical energy gradient theory for the study of flow stability and turbulent transition. (The mechanical energy gradient includes the pressure gradient.) For the pressure-derived flow, he compared the energy gradient and the viscosity term. It is found that the inflection point on the velocity profile for the pressure driven flow is a singular point hidden in the Navier–Stokes equations.

It is well known that the ratio of the second derivative of velocity with respect to time and the curvature of velocity leads to a very important relationship in physics known as the linear wave equation:

where c is the wave propagation speed. It gives an indirect indication that the pressure (energy) gradient has some implicit physical coherence with the wave property. Tao's work also shown that the linear viscosity term would have magnitude of the wavenumber and flow speed of k²v, namely, pressure gradient (mechanical energy) should show some wave behavior.

Scholle et al.^7,8 have developed, analogy to the Maxwell's equations represented by scalar and vector potentials, a general first integral of a 4D Lorentz-invariant energy–momentum equation. This methodology facilitates the derivation of a 4D Lorentz-invariant first-integral formulation of the energy–momentum equations for viscous flow, consisting of a single tensor equation. The asymptotic analysis shows that it reduces to one representing the unsteady compressible viscous flow, from which the classical Galilei-invariant field equations can also be recovered. By applying this theory to the case of acoustic wave propagation, the classical expressions for sound speed were recovered. The relativistic generalization of the 4D Lorentz-invariant viscous flow in the form of the energy–momentum equations gives the hint of the indivisibility of energy and momentum (meanwhile the indivisibility of the space and time). The sound propagation speed (pressure wave speed) is an important, coherent physical parameter of the flow field.

Getting some kind of inspiration from above works, in this paper, we first show that the mass conservation and Navier–Stokes equations are Galilean invariant, but the resulted wave equation is not. This leads us to use the Lorentz transformation. The Lorentz factor is parameterized by the local Mach number; the transformed Navier–Stokes equations will have a singularity when Mach number is equal to one. The classical Navier–Stokes equations are only valid as an approximation at very low Mach number.

Provided we have a compressible, isotropic, fluid flow field, an infinitesimal control volume is attached to the fluid flow field, for the Newtonian fluid, the resulted net force on this infinitesimal control volume is as follows:

We can write the mass conservation law and the Newton's second law in terms of the force on this infinitesimal control volume; the resulted equation of motion is the Navier–Stokes equations,

To create a basis for discussion about the problem, we first discuss the Galilean invariant property of the mass conservation and Navier–Stokes equations. For simplicity, let us consider motion along a single direction, saying along the x-direction. Suppose we have coordinates used by observer O to measure the position x of a fluid element in the flow field at time t, O(x, t). Another observer O′ moves with respect to O with a constant velocity U to define his coordinates for the position of the identical fluid element x′ at time t′ as measured in his “moving frame,” O′ (x′, t′). We further assume that at time t = 0, the origin O of frame S coincides with the origin O′ of frame S′ and all axes overlapped, see Fig. 1.

Let us to investigate the behavior of a fluid element in the flow field at a position P, whose coordinates are (x,y,z) and (x′,y′,z′) in S and S′ frames, respectively. Let t and t′ be the time of occurrence of P that observer S and S′ record their clocks. Then, the Galilean transformation relating these two inertial reference frames is

The mass conservation and momentum conservation are Galilean invariant (see  Appendix A for details), and thus, in the moving reference frame, we have

It is noticed that a differential equation will be said to be invariant under a transformation if it is left unchanged by this transformation. In this sense, the mass conservation and Navier–Stokes equations are Galilean invariant. Physically, Galilean invariance means that the mass conservation and Navier–Stokes equations governing the velocity field do not change with a change in reference frames. They are valid in every inertial frame of reference.

It is of importance to note, if one inertial frame exists, then an infinite number of other inertial frames exist, too. Since any frame that is moving at a constant velocity, U, relative to the first inertial frame can be also assigned as an inertial frame, suggesting that U can be any value. If U equals to the flow velocity of the fluid element P, this implies that we write the mass and momentum conservation in the co-moving reference frame (observer S′ stays on the moving element, P, to observe the physical properties), thus

Recalling the original derivation of mass and momentum conservation in numerous studies, indeed, these conservation equations have been derived in the co-moving reference frame. That means Eqs. (7) and (8) can be assigned in the co-moving reference frame, if we use the condition of Eq. (9).

It has been well known a shock wave will appear for Mach number equal to one. In this position of the flow field, the derivatives of the physical properties with respect to space will become infinitely. Equations (7) and (8) cannot express these flow discontinuous properties. Accordingly, Galilean transformation is not suitable to describe the fluid flow. In Sec. IV, it will be seen that Lorentz transformation can reflect these flow discontinuous properties.

In the following discussion, we focus only on two reference frames: one is lab reference (observer S), and another one is the co-moving reference frame (observer S′).

Suppose the density is function of pressure, using the chain rule, the mass conservation, Eq. (4), can be rewritten as

where

$cs$ is the propagation speed of the pressure wave in the frame S, and it depends on the medium properties.

At constant temperature, the fluid does not heat up, and the speed of the pressure wave only depends on the temperature of fluid, not on the density or pressure separately. That means the speed of sound is constant.

By taking the time derivative of the mass conservation law, Eq. (10), and the divergence of the momentum conservation law, Eq. (5), we can obtain the following equation:

where

Equation (12) shows that mass conservation and Navier–Stokes equations imply wave (differential) equations in the fluid flow field characterized by a propagation speed of c_s. The right hand side of Eq. (12) is the wave source term.

With the same procedure and using Eqs. (7) and (8), we can get the wave equations for the same fluid flow field, observed in the co-moving frame S′ (observer S′ stays onto the fluid element P):

where

Here, $cs′$ is the propagation speed of the pressure wave, observed in the co-moving frame S′.

Actually, the propagation speed of the pressure wave is the fluid properties (the equation of state), which is independent of the frame of reference, see  Appendix B for details, that is

By comparing Eqs. (12)–(14), it is shown that the propagation speed of the pressure wave, $cs$⁠, is independent of the relative motion of the observers. In another word, the propagation speed of the pressure wave is invariant in reference frames S and S′.

Mass conservation and Navier–Stokes equations are proved to be valid in every inertial frame of reference. They lead to the wave equation; the wave speed are same in every inertial frame. Only Lorentz transformation can ensure that wave speed is same in all inertial frames.

The Lorentz transformation is a linear transformation from a coordinate reference frame in spacetime to another reference frame that moves at a constant velocity, u, relative to the former. This transformation ensures that wave speed is same in all inertial frames, see Fig. 2.

The speed of pressure wave is independent of the (constant) velocity, $u$⁠. Thus, in each frame, the wave velocity is same. From the invariance of $cs,$ it follows that [here the Minkowski metric signature (−1,1,1,1) is applied, see Sec. V]:

where $Δt′$ is time interval, observed in co-moving reference frame S′, it is often called as proper time; the displacement of the fluid element, $Δx′=0, x1′=x2′$⁠, in the co-moving reference frame S′.

In the x-direction, using the condition of Eq. (17), the Lorentz transformation reads

where $γ$ is the Lorentz factor, it is defined as

$β$ is the ratio of $u$ to the wave speed of $cs$⁠,

From Secs. II and III, we recognize that in a flow field, the speed of the pressure wave, produced by every fluid element in the flow field, should obey the principle of relativity like Einstein's postulates for light traveling in the vacuum space, which are:

• The principle of relativity: the laws of physics (the mass and energy–momentum conservation) are the same and can be stated in all inertial frames of reference.

• In an isotropic, isothermal flow field, the speed of mechanical wave, $cs$⁠, is independent of the relative motion of the source (actually, it is the fluid physical property). It forms the fundamental basis for special relativity.

We want to describe the law of fluid motion for different reference frames. As mentioned above, an infinite number of inertial frames exist, with this argument, we can assign a special inertial reference frame to attach onto the investigated flowing element, the so-called co-moving reference frame and labeled as S′ frame. We will label all physical quantities (such as pressure and velocities) in the S′ frame with a “prime” symbol, and the physical quantities in the S frame (or lab reference frame) without the “prime” symbol. Two inertial reference frames have set up their own spacetime coordinate systems, such that the $o−ct−x−y−z$ for the “unprimed” coordinate system, the $o′−ct′−x′−y′−z′$ for the “primed” coordinate system, as illustrated in Fig. 3. The primed reference frame is in motion relative to the unprimed reference frame with a relative velocity of $u→(t,x→)$⁠.

Considering now an infinitesimal control volume in the flow field, the primed coordinate system S′ is placed on it, co-moving with the flow element. If we take the origins of S and S′ to coincide at time $t=t′=0$⁠, then the points (0,0) and (0 $′$⁠,0 $′$⁠) are same for primed and unprimed reference frames.

In the following, all derivations will be done in the Cartesian coordinate systems for spatial components. Minkowski metric (in spacetime) will be used. The four-dimensional coordinates in the Minkowski coordinate system for S and S′ frame are, respectively, given by

Here, letters with Greek superscript indices denote coordinates on spacetime (⁠$x0=cst, x1=x, x2=y, x3=z$⁠). The proper velocity will be measured by the ratio between observer S measured displacement vector, $r→$⁠, and proper time, $t′$⁠, elapsed on the clocks of the co-moving traveling fluid element, which is defined as

The energy–momentum tensor, which describes the density and flux of energy and momentum in spacetime, can be expressed as the following form:^9–12 

Here, $ϵ0$ is the proper “total energy density” measured by co-moving observer S′ (in the instantaneous initial frame), it must be borne in mind that this total energy also includes the “wave energy density” $ρ0cs2$⁠, namely, $ϵ0=ρ0e0+ρ0cs2$⁠. Here, $e0$ is the proper “specific internal energy density” and can be regarded as a type of “background” energy density for the “wave energy density.” $ημν=ημν=diag(−1,1,1,1)$ is the metric tensor for flat spacetime. $τμν$ represents the viscous stress part. For Newtonian fluid, in the Landau frame¹⁰ for Minkowski space, it reads

The viscous stress part represents internal friction (viscosity) between adjacent fluid elements in the infinitesimal control volume (as the shear stresses result from the velocity gradients). In the non-relativistic limit, the $τμν$ reduces to the components of the three-dimensional viscous stress tensor of Eq. (3).

$Σμν$ represents other type of energies, excluding above explicitly expressed energies, e.g., heat conduction. In this paper, we do not consider this term.

The above defined energy–momentum tensor contains the energy density, momentum density, and stresses as measured by any observer and holds in arbitrary reference frame.

If the specific internal energy, pressure energy and dissipative process are neglected, we get the energy–momentum tensor for the “dust” model:

If the specific internal energy and dissipative process (isentropic) are neglected, we have the tensor for “perfect fluid” model, it reads

The local formulation of the conservation of the energy–momentum in the motion of the system reads

where $∂ν$ denotes the components of the four-dimensional divergence operator.

Lorentz invariance holds for different observers, if all the components of the above equations are written out for the S reference frame (lab reference frame) explicitly, it will be very lengthy. Interested readers can refer to Refs. 13–15. In order to reveal the connection between the classical Navier–Stokes equations (non-relativistic) and the relativistic ones, we will see the conservation law in the co-moving frame, namely, in the limit of the non-relativistic case.

In the co-moving frame (or instantaneous inertial frame S′), the energy–momentum tensor is the limit of the non-relativistic case (which implies $γ→1$ or $cs→∞$⁠, and hence, $u→/cs→0$⁠), we have $U′μ→cs,0, 0, 0$⁠. However, the velocity gradients do not equal zero, because the co-moving frame is moving in the flow field, it is a function of space and time, there are relative motions between the adjacent fluid element.

For the energy conservation, Eq. (28) gives for $μ=0$⁠:

or rather

For spatial coordinates of $μ=1, 2, and 3$⁠, it gives

or

Accordingly, substituting the “dust” energy–momentum tensor into Eq. (30), this is just the classical equation of continuity of a fluid:

In the “dust” model, there are no fluid element interactions; thus, here the “wave speed” can be considered as the photon speed.

The classical Navier–Stokes equations in the co-moving frame can be deduced using the condition for the limit of the non-relativistic case. If the proper “background” energy density is neglected or keeps constant (or the “wave energy” density is far greater than the internal energy density, $ρ0cs2≫ρ0e0$⁠). It must be borne in mind that these conditions are not fulfilled for many practical applications. Thus, it reduces to (⁠$μ=1, 2, and 3$ represent the momentum equations):

Substituting this energy–momentum tensor into Eq. (32), namely, the conservation law of the momentum, taking the limit for the non-relativistic case $uc→0$⁠, we have their simplest form (⁠$Ui→ ui$⁠),

It is clearly recognized, if the wave speed of pressure, $cs$⁠, is constant, Eqs. (33) and (35) degrade to the classical non-relativistic mass conservation and Navier–Stokes equations, obtained for co-moving frame S′.

The relativistic Navier–Stoke equations degrade to the classical ones. It is obviously that the classical Navier–Stokes equations approximate the relativistic ones applicable to the case for very low Mach numbers (or very small relative velocities).

The relativistic Navier–Stokes equations embody the discontinuity, when the flow velocity equals to the local speed of sound (pressure wave). This singularity can be clearly identified when the conservation law, Eq. (28), is written out explicitly for reference frame S. In this paper, we will use another simply way to show this discontinuity. To that, we will turn to the Lorentz transformation. For the sake of simplicity, we will still stick to the procedure by ignoring the directions y and z, which are assumed perpendicular to the direction of the fluid element motion. Extensions to higher space-dimensions are straightforward. The Lorentz transformation, the transformation between unprimed $St,x$ and primed frames $S′t′,x′$⁠, is Eq. (18). For pressure wave traveling, the Lorentz factor is defined here as

where $Mu$ represents the local flow Mach number, and it is expressed as

It is not difficult to write out any scalar field (such as the pressure field) gradient transformation using the multivariable chain rule, which is transformed from the co-coming reference frame into the lab reference frame (unprimed one):

Using Eqs. (16), (33), and (38) and the chain rule, we can get pressure gradient observed in the S frame (lab reference frame),

When the flow velocity approaches to the local speed of sound, the Lorentz factor, Eq. (35), will become infinite.

With above definitions of the Lorentz factor of Eqs. (36) and (37) and corresponding Lorentz transformation (18), we can get the following important features.

When the relative flow velocity is smaller than the traveling speed of wave, but approaches the speed of the wave, namely, as the local Mach number approaches one, the Lorentz factor [Eq. (36)] will approach infinity. Parentheses in Eq. (39) are the pressure and kinetic energy, observed in the co-coming reference frame. Suppose they are bounded in the co-moving frame, the pressure gradient $∂p∂x$ will approach infinity, from the perspective of the observer who is positioned in the lab frame.

If the relative velocities between two reference frames, $u→$⁠, approach zero, the Lorentz factor will approach one. From Eq. (39), it can be seen that $∂p∂x$ will approach to $∂p′∂x′$⁠.

If Mach number is equal to one, the Lorentz factor is not defined. Obviously, the pressure gradient has a singularity, observed in the lab frame of S. A shock wave appears, which is characterized by a discontinuous change in physical properties. Since this reason, there exists no smooth solution of the relativistic Navier–Stokes equations. It has been agreed with the experiments and fluid dynamics practices for decades.

If the flow is supersonic, that is, if the Mach number is greater than one, the square of the Lorentz factor becomes a negative number, the flow velocity has now surpassed the local speed of sound, and a Mach cone will form.

As an approximate example, we can observe the de Laval nozzle. If we ignore viscosity and any changes perpendicular to the flow direction, we have the classical equation that relates the velocity, the square of the Lorentz factor and the local Mach number in terms of the cross-sectional area of the nozzle:

In the subsonic flow (⁠$Mu<1$⁠, $γu2$ is positive), $dudx$ and $dAdx$ have opposite signs, the flow velocity will increase accompanied by decreasing the tube cross-sectional area. In the case where M = 1.0, which makes the Lorentz factor not defined, transitions from the subsonic flow to the supersonic flow will happen. In the supersonic flow region (⁠$Mu>1$⁠, $γu2$ is negative), $dudx$ has the same sign as $dAdx$⁠, meaning that as the cross-sectional area of the nozzle increases, velocity will also increase, where a pressure wave will not propagate upstream through the fluid as viewed in the lab reference frame.

The author appreciates the constructive comments from reviewers and has incorporated the reviewers' suggestions in this revised paper.

The author has no conflicts to disclose.

The data that support the findings of this study are available within the article.

Let us consider fluid element motion along a single direction, saying along the x-direction. Suppose we have coordinates used by observer O to measure the position x of a fluid element, P, in the flow field at time t, O(x, t). Another observer O′, moving with respect to O with a constant velocity U, defines his coordinates for the position of the identical fluid element x′ at time t′, as measured in his “moving frame,” O′ (x′, t′). We further assume that at time t = 0, the origin O of frame S system coincides with the origin O′ of frame S′ system and all axes overlapped, see Fig. 4.

After a period of time Δt, the fluid element moves a distance of Δx along x-direction, measured by observer O. The origin of frame S′ locates now on the right with respect to the origin of frame S with a distance of UΔt.

Let t and t′ be the time of occurrence of P that observer S and S′ record their own clocks, respectively. Then, the Galilean transformation relating the displacement of the fluid element P in the two inertial reference frames is

Recalling the velocity definition, the velocity transformation in the x-direction is described as

where $ux,t$ represents the fluid element velocity along the x-direction, observed in frame S. $u′x′,t′$ is the velocity of the same fluid element, P, observed in frame S′.

A scalar field, $φ$ (such as hydrodynamic pressure field, density field, temperature field, etc.), is a zeroth-order tensors, it is invariant under Galilean transformation,¹⁶ thus

Using the multivariable chain rule and relation (A1), we have the following transformations between frame S and frame S′:

The mass conservation equation in the frame S reads

We can transform it from frame S to frame S′. Using Eqs. (A3)–(A5), we can get

For the convective term, we have

Adding Eqs. (A6) and (A7), we have the mass conservation in frame S′:

Next, we will transform the Navier–Stokes equations from frame S to S′. Using Eq. (A4), we have

and

Adding Eqs. (A9) and (A10), we have

The last term in the LHS of (A11) is the mass conservation equation observed in frame S′. With the help of Eq. (A8), we have

The hydrodynamic pressure field is a scalar field; thus, the pressure gradient term can be transformed from frame S to S′ as follows:

Since the stress tensor [right hand side of Eq. (4)] depends only upon velocity derivatives with respective to space coordinates and contains no time derivatives, it is unaffected by the addition of a constant U to the velocity; thus, we have

Finally we get the mass conservation and momentum conservation equations, observed in frame S′,

In Eq. (A1), U is an arbitrary scalar number, namely, there are an infinite number of inertial reference frames. However, there is a special inertial reference frame, the so-called co-moving reference frame, if U is exactly equal to fluid element flow velocity u:

It is also very important to note that these Galilean invariant properties are not shared by the velocities, as well as by boundary conditions, because the velocity is not invariant under Galilean transformation, as indicated by Eq. (A2). This implies that the wave velocity is not Galilean invariant. In addition, if the mass conservation and Navier–Stokes equations contain any source terms (“source” or “sink” terms in mass conservation and external force source in Navier–Stokes equations), they are also not Galilean invariant.

The speed of sound is a thermodynamic equilibrium property. It is a property of the medium through which the sound (pressure wave) travels. The micro-scalar fluid particles simply oscillate back and forth about their individual equilibrium position and independent of their macro-scalar flow velocities. Thus, it is frame independent; this can directly be seen from its definition^17,18

where $cs$ is the speed of sound, $p$ is the pressure, $ρ$ is the density, and $s$ is the entropy. It relates merely pressure to density variations under isentropic conditions, which are given for reversible processes in the absence of heat transfer.

In a perfectly elastic (non-dissipative) homogeneous fluid, the speed of sound is given by

If temperature and pressure are taken as the independent variables. Here, $cp$ is the specific heat at constant pressure, $s$ is the entropy, and $βs$ is the isentropic compressibility.

The isentropic compressibility is defined as

Consequently, the speed of sound, $cs$⁠, is thermodynamic properties, or may be determined from the thermodynamic equation of state (EOS) of the fluid, which is reference frame independent.

In the case of an perfect gas, for which $ρ=Mp/RT$⁠, the speed of sound well-known. For an adiabatic process, the equation is given by

For an isothermal process, it is