Stratified thin-film flow in a rheometer

The use of rheometers for measurements of stratified fluid systems has become commonplace as interest in the dynamical features of superhydrophobic surfaces has grown. For instance, cone and plate rheometers have been used to measure laminar drag reduction over traditional air-cushion superhydrophobic surfaces,¹ although the reported slip lengths are controversial.^2,3 More recently, a parallel-plate rheometer was used to measure drag reduction over a new spray-coated superhydrophobic surface with a hierarchy of surface features and enhanced slip lengths.⁴ An alternative to traditional superhydrophobic surfaces is a liquid-infused surface,^5–8 which can provide a variety of omniphobic properties and greater robustness to high pressures. Recently, the properties of these liquid-liquid systems have been investigated in a cone and plate rheometer, and drag reduction up to 16% compared to solid substrates was reported.⁹ 

The behavior of single layer liquid films on rotating disks has a lengthy literature from the classic problem of spin-coating a fixed volume of fluid¹⁰ to the wavy instabilities produced on continuously filled rotating disks.¹¹ The analysis of the spin-coating problem has been reconsidered more recently to treat the effects of surface tension and the dynamics of spreading,¹² contact line movement,¹³ and the behavior of complex fluids.^14,15 However, the two-layer rotational problem has only received limited treatment, e.g., in the realm of geophysical flows with density-stratified but miscible layers of salt-water¹⁶ and more recently for spin-coating layered immiscible fluids.¹⁷ The latter spin-coating study determined that the long-time thinning behavior of both fluids was surprisingly insensitive to the viscosity and density ratios of the fluids but did not explore boundary conditions relevant to rheometry.

We consider the steady-state interfacial behavior of liquid-coated smooth and rough surfaces in parallel-plate and cone and plate rheometers and explore the implications of interfacial shape changes on two-phase, stratified rheological measurements. We begin by analyzing a two-fluid system on a smooth substrate, and we find analytical and numerical predictions for the interface shape including predictions of the threshold rotation rate above which the lower fluid starts to dewet the substrate. We then provide scaling arguments to show how the results for a smooth substrate can be extended to a rough substrate in which the lower fluid is initially infused within the roughness. We end with calculations of the expected change in the measured torque due to interfacial deformation.

In the rheometer system of interest (Figure 1), the lower, solid base is coated with a thin layer of lubricating oil of initial height h₀ that wets the substrate perfectly. The upper cone or plate of the rheometer is then filled by an immiscible working fluid, which wets the upper plate or cone, creating two distinct fluid layers within the gap of the rheometer.

At the outer edge of the rheometer, where the two fluids contact the air, the geometric configuration of the fluids is complex. Previous studies on the behavior of small droplets moving on liquid-infused surfaces⁸ or droplets in static equilibrium with multiple fluid phases¹⁸ have shown that surface chemistry and fluid properties can produce an entire family of different geometries, including (1) a case where one fluid entirely encapsulates the other, (2) a case where a thin film of one fluid wraps around the other forming a microscopic “cloak,” and (3) a case where one fluid forms a “collar” or small ridge-like meniscus around the base of the other, without cloaking it. For most applications of liquid-covered or liquid-infused surfaces, it is important to avoid any sort of wrapping or encapsulation, as this behavior could eventually lead to lubricant loss.⁸ Thus, the most likely practical scenario when studying these surfaces is the formation at the outer edge of a liquid ridge of the lower film about the circumference of the gap containing the upper fluid, as illustrated in Figure 1. For convenience of notation but without loss of generality, we assume in the usual case that the upper working fluid is aqueous and denote it with subscript “w” and the lower fluid is oil, denoted with subscript “o.”

We assume a cylindrical coordinate system (r, θ, z) with its origin centered at the base of the rheometer. The upper plate, with outer radius R, rotates at angular velocity Ω and is a distance H(r) away from the solid lower plate. For the parallel-plate rheometer sketched in Figure 1(a),

whereas for the cone and plate rheometer sketched in Figure 1(b),

and in all cases of practical interest H_R/R ≪ 1.

In addition, we assume that the capillary (Ca) and Reynolds (Re) numbers are small so that the thin film of oil wetting the base of the rheometer has an axisymmetric steady shape z = h(r). We define the pressures in the two phases as p_w(r, z) and p_o(r, z). The densities and viscosities of the two phases are, respectively, ρ_w, ρ_o and μ_w, μ_o and the interfacial tension between them is γ.

In order to model the effect of interfacial deformation on rheometer measurements, we must consider the magnitude of possible deformations in the central region of fluid within the rheometer gap, where the gradients in interface height are assumed small, versus deformations in the edge region surrounding the free surface, where the interface shape changes rapidly. We first consider the static configuration of the fluids and estimate the radial extent of the edge region, before estimating the relationship between dynamic interfacial deformations in the two regions.

The ridge near r = R results from the need to satisfy the Neumann triangle relation at the contact point between oil, water, and air. Moving away from the contact point, the deformation decays exponentially under the influence of gravity, with a radial length scale corresponding to the capillary length for the oil phase, defined as $ℓ c = γ / | ρ o − ρ w | g$ (similar to the problem of the meniscus against a planar wall¹⁹). Therefore, we divide the film into two regions: (1) a central region of the film, 0 < r ≲ R − ℓ_c, with negligible interfacial curvature in the static case and (2) an edge region, R − ℓ_c ≲ r ≲ R + ℓ_c, with strong interfacial curvature.

Now, once the rheometer begins spinning, centripetal forces will cause the interface to deform. Within the edge region, the two dominant length scales are H_R and ℓ_c, which for most configurations are of a similar order of magnitude as long as we assume that the density difference |ρ_o − ρ_w| ≫ γ/(gR²), which is true for many pairs of fluids. If the interface in the edge region deforms by a magnitude δ_edge, it will result in changes of interfacial curvature on the order of $δ edge / H R 2$ or $δ edge / ℓ c 2$⁠.

However, in the central region, the dominant lateral length scale is R. Interfacial deformations δ_central in this region will produce changes in curvatures of order δ_central/R². In order for the central and edge regions to be in pressure equilibrium, the two changes in curvature must be of comparable magnitude.

Equating changes in curvature in the central region with changes in curvature in the edge region, we arrive at δ_central/δ_edge ∼ (R/H_R)² ≫ 1 or δ_central/δ_edge ∼ (R/ℓ_c)² ≫ 1. Thus, interfacial deformations due to dynamic forcing will be of a much greater magnitude in the central region of the rheometer than in the edge region. We therefore confine our analysis to the central region, r ≲ R − ℓ_c, where deformations are expected to be of a much greater magnitude and therefore produce a greater effect on measurements. To further simplify the notation, we assume that ℓ_c ≪ R and thus neglect the capillary length from the remaining analysis.

In order to get a sense for the expected interfacial deformation in the central region of the stratified system, we write the order of magnitude of a force balance between the centripetal force driving the deformation and the capillary force sustaining the interface. The total pressure drop across the interface, due to centripetal acceleration, can be approximated as Δp_centripetal ≈ Ω²R²Δρ, where Δρ = ρ_w − ρ_o. The corresponding capillary pressure due to the interfacial curvature can be approximated as $Δ p capillary ≈ Δ h R 2 γ$⁠, where Δh expresses the maximum magnitude of h₀ − h(r). At equilibrium, the balance between these two stresses yields

which indicates that the magnitude of the deformation Δh compared to the undeformed film height h₀ can potentially be quite large. This ratio represents a Bond number describing the balance between centripetal and capillary forces. The detailed calculation that follows provides a more complete description of how the steady-state interface shape depends on all of the fluid and flow parameters of the problem.

It is well known that for standard rheometry, the Reynolds number is low, the leading-order velocity is purely azimuthal, and the pressure is uniform.²⁰ As we are interested in possible changes to the interface shape, we must determine higher-order velocity and pressure corrections to this leading-order base flow.

Within the rheometer gap, we assume that the velocity field is also axisymmetric and steady, so that u_θo,w(r, z), u_ro,w(r, z), and u_zo,w(r, z) are, respectively, the azimuthal, radial, and vertical velocities in the oil and water phases. The radial and azimuthal velocities can be non-dimensionalized by ΩR, and the distances can be non-dimensionalized by R in the radial direction and H_R in the vertical direction, to yield the experimentally relevant²¹ Reynolds number for the aqueous phase, $Re= ρ w Ω H R 2 μ w$⁠. The corresponding capillary number is $Ca= μ w Ω R / H R γ / H R$⁠. We then write a perturbation expansion²⁰ of the non-dimensional momentum equations in cylindrical coordinates, where velocity and pressure terms are expanded in orders of Reynolds number, as $u= u ( 0 ) + Re 1 u ( 1 ) + Re 2 u ( 2 ) +O ( Re 3 )$ and $P= P ( 0 ) + Re 1 P ( 1 ) + Re 2 P ( 2 ) +O ( Re 3 )$⁠. At leading order, we have the familiar azimuthal flow common in rheometric studies (though here coupled for two fluid layers). Grouping terms in the radial direction by Reynolds number at $O ( Re 1 )$ for either phase yields a non-dimensional radial governing equation, which we write here dimensionally as

where we have applied the lubrication approximation, assuming that the radial length scale is much larger than the vertical length scale, H_R/R ≪ 1. This approximation also implies that in the vertical momentum balance, $∂ P o , w ( 1 ) ∂ z$ is negligible compared to the radial pressure gradient and that $∂ u z o , w ( 1 ) ∂ r$ is negligible compared to its gradient in the vertical direction.

The radial momentum balance in Eq. (4) depends on the azimuthal velocity at zeroth order, which can be determined by solving $∂ 2 u θ o , w ( 0 ) /∂ z 2 =0$ subject to no-slip boundary conditions at the top and bottom surfaces, and matched velocities and shear stresses at the fluid-fluid interface, z = h(r), so that

We integrate the above zeroth-order equation in z to arrive at an equation for the azimuthal velocity in each fluid. Defining the viscosity ratio $λ= μ o μ w$⁠, we have

We recall that $∂ P θ o , w ( 0 ) /∂r=0$⁠, and thus, we seek the pressure gradient at order $O ( Re 1 )$ in order to examine the effect of secondary flows on interfacial shape. As noted above, by the lubrication approximation, the ratio of out-of-plane to radial pressure gradients is $O ( h 0 / R ) 2$⁠, and therefore, the out-of-plane contribution is neglected. The radial pressure gradients in each phase can be found by integrating Eq. (4) twice and applying the boundary conditions

along with the integral constraint that there is no net radial flux of fluid in each layer,

Taking the difference between these two radial pressure gradients and defining the density ratio between the fluids as $β= ρ o ρ w$ and the ratio between the rheometer separation distance and the film height as $ξ ( r ) = H ( r ) h ( r )$⁠, we arrive at

where the non-dimensional function f is given by

At any radial position, the pressure drop across the interface is proportional to the mean curvature of the interface, which in cylindrical coordinates, for $d h ( r ) d r ≪1$⁠, is given by $κ ( r ) = 1 r d d r r d h ( r ) d r$⁠. Using the Young-Laplace equation and denoting the dimensional pressures as p_o and p_w, we have

The negative sign reflects the notion that higher pressure in the oil phase should produce a concave-down interface shape, which has negative curvature by our convention. We now differentiate (11) with respect to r and substitute (9) to obtain a differential equation for h(r),

In order to obtain some insight into the behavior of the prefactor function, $f ξ ( r ) , β , λ$⁠, we can replace ξ(r) by introducing $α= H R h 0$⁠, assume a parallel-plate configuration, and then expand Eq. (10) in a first-degree Taylor polynomial about a base state (α₀ → ∞, β₀ = 1, λ₀ = 1) to yield

The expansion shows that the density ratio (β) dependence is quite weak and that the prefactor increases linearly with α. The β dependence appears only in the third-degree Taylor approximation and is $− 9 β / 20 α 2$⁠. We can therefore employ β = 1 for illustrative purposes, even where capillary length assumptions dictate that β is, technically, greater or smaller than unity. The fact that the prefactor is non-zero for identical but immiscible fluids is due to the secondary flows in the rheometer.

Although the function f most directly represents the radial pressure gradients in each phase, it is convenient to consider instead the secondary flow properties in order to build physical intuition about the behavior of f. At finite Reynolds number, a radial pressure gradient must balance the centripetal force and this pressure gradient distorts the interface and drives secondary, recirculating flows in the radial and out-of-plane directions. Moreover, the magnitude of the out-of-plane secondary flow, $u z ( 1 )$⁠, scales linearly with the height of its fluid layer. Therefore, we expect that as α increases and the water layer gets deeper relative to the oil layer, the secondary flows in the water layer will become stronger, and thus, f should increase linearly with α. When the two fluids are not identical, then the density and viscosity of each fluid will affect the relative strength of the secondary flows as well. As λ = μ_o/μ_w increases, the viscous resistance to secondary flows in the water phase decreases and its secondary flows become stronger, and thus, f increases. In contrast, when β = ρ_o/ρ_w increases, the centripetal force in the water phase decreases relative to the oil phase, resulting in stronger secondary flows in the oil phase and a decrease in f. Therefore, physically, the prefactor f(α, β, λ) represents the magnitude of the difference between the secondary flows in the two phases: positive f > 0 means secondary flows in the water phase are dominant (the typical scenario); negative f < 0 means the secondary flows in the oil phase are dominant.

Streamlines of the secondary flows are shown in Figure 2, where $u z ( 1 )$ was calculated from $u r ( 1 )$ via continuity. Unlike the typical single-phase rheometer problem,²⁰ each phase can actually sustain more than a single eddy, depending on the particular set of parameters (α, β, λ). In fact, two eddies are shown within the oil phase in the figure, denoted by the two roots of the radial velocity profile. For fixed (β, λ), there is a finite range of α that produces two eddies; all other scenarios produce a single eddy. The width and location of this range of α are determined by the choice of (β, λ).

Despite using the linearized curvature approximation, we observe that the governing equation for steady-state interface shape (12) is a non-linear ordinary differential equation (ODE) without a straightforward solution. Equation (12) may be solved numerically (in Sec. IV), but we can also consider an asymptotic analysis assuming that the final interface shape h(r) is not too different from some initial constant interface height, h₀.

We non-dimensionalize the film thickness by the unperturbed interface height, $h ̃ =h/ h 0$⁠, the radial dimension as $r ̃ =r/R$⁠, and the curvature as $κ ̃ =κ R 2 / h 0$⁠. The non-dimensionalization yields a parameter governing the magnitude of the deformation

ODE (12) can then be rewritten as

The parameter ϵ is similar to the original scaling in (3) localized to the interface. However, the original scaling indicated that the sign of the deformation should depend on the density difference Δρ; under the formal analysis, the sign of the deformation still depends on the density difference, but the difference is expressed via the ratio $β= ρ o ρ w$ in the function f.

For boundary conditions, we require the solution to Eq. (15) to be smooth at $r ̃ =0$ so that $d h ̃ d r ̃ | 0 =0$⁠. At the outer edge of the disk, $r ̃ →1$⁠, the interfacial curvature is assumed to be zero as discussed in Sec. II A. Finally, the total volume of fluid in the lubricant phase is conserved and thus by continuity

In order to approximate a solution to this system, we assume ϵ ≪ 1 and expand both sides of (15) in an asymptotic series in ϵ according to

Substituting $h ̃ ( r ̃ )$ and expanding the prefactor function then yield

where f(α, β, λ) is a constant function of the fluid properties and system geometry, independent of $r ̃$⁠.

Rewriting Eq. (15) to $O ( ϵ )$ yields

Equation (19) can be trivially integrated and, after enforcing the boundary conditions above, yields

Because of the form of (20), which has a prefactor that is less than $O ( 1 0 − 1 )$⁠, ϵ need not be so small for (20) to approximate the deformed interfacial shape, as discussed below. Moreover, we note that even when $ϵ h ̃ 1 ∼1$⁠, the approximate curvature remains $O ( h 0 / R 2 )$⁠, where we assume h₀/R ≪ 1. Thus, Eq. (15) remains valid even when the interfacial perturbation is relatively large.

There remains an additional constraint to enforce, which is that $h ̃ ( r ̃ ) =1+ϵ h ̃ 1 ( r ̃ ) +O ( ϵ 2 ) >0$⁠, since when $h ̃ ( r ̃ ) =0$⁠, the film dewets. The function $h ̃ 1 ( r ̃ )$ has global extrema in the domain [0, 1] at $r ̃ =0$⁠, where $h ̃ 1 ( 0 ) =5f ( α , β , λ ) /96$ and at $r ̃ =1$⁠, where $h ̃ 1 ( 1 ) =−f ( α , β , λ ) /24$⁠. The relevant extremum for establishing dewetting therefore depends on the sign of f(α, β, λ). Figure 3 shows the behavior of f(α, β, λ) for a representative value of α; for large viscosity ratios λ, the prefactor f(α, β, λ) remains positive independent of the density ratio; for small viscosity ratios, however, the prefactor becomes negative for large density ratio β, although most practical fluid pairs have β < 2, so f(α, β, λ) is nearly always positive. Only a specific set of (α, β, λ), indicated by the white line in the figure, results in a zero prefactor and therefore no deformation. This means that even when the viscosity and density ratios of the two fluids are matched, there is a non-negligible deformation of the interface due to secondary flows, as discussed above.

The general constraint on dewetting to guarantee $h ̃ ( r ̃ ) >0$ can then be written at $O ( ϵ )$ as

In order to solve full system (15) numerically, we rewrite it as a coupled system of first-order ODEs. Defining $h ̃ = y ̃ 1$ and $h ̃ ′ = y ̃ 2$⁠, $h ̃ ″ = y ̃ 3$⁠, and $∫ 0 r ̃ h ̃ ( s ) s ds= y ̃ 4$⁠, we write

The integral constraint due to continuity corresponding to (16) above yields $y ̃ 4 ( 0 ) =0$ and $y ̃ 4 ( 1 ) = 1 2$⁠. Zero curvature at the outer edge requires $y ̃ 2 ( 1 ) + y ̃ 3 ( 1 ) =0$⁠. Symmetry about $r ̃ =0$ requires $y ̃ 2 ( 0 ) =0$⁠. Finally, the system is singular as $r ̃ →0$⁠. In order to avoid the singularity, we must specify the value of $y ̃ 3 ′ ( 0 )$⁠. In the limit as $r ̃ →0$⁠, $y ̃ 2 r ̃ 2 − y 3 ̃ r ̃ →0$ by application of L’Hôpital’s rule, and thus, we enforce $y ̃ 3 ′ ( 0 ) =0$ directly in the integration scheme. The system can then be solved using collocation methods like MATLAB’s “bvp4c” solver; however, these approaches are susceptible to significant numerical stability issues as ϵf(α, β, λ) becomes large, and thus, the calculation requires great care. By employing numerical continuation through iteratively updating the initial guess of the collocation solver as ϵf(α, β, λ) increases in magnitude, stable and accurate solutions can be produced reliably.

For any set of flow parameters, the full non-linear boundary value problem presented in Sec. III A can be solved numerically to obtain the critical value of ϵ, where dewetting first occurs (i.e., where $h ̃ ( r ̃ ) =0$⁠). In order to facilitate comparison to the asymptotics, the product ϵf(α, β, λ) is plotted as a function of λ in Figure 4. If the asymptotic representation were exact, then ϵf(α, β, λ) would exhibit no dependence on λ since λ is already incorporated within the prefactor f(α, β, λ) in the asymptotic expansion. The numerical calculations for positive ϵf(α, β, λ) are quite close to the asymptotic predictions in Eq. (21) as long as λ ≪ 1 and are within a factor of two for all values of λ typically observed in liquid-infused systems,⁹ ranging from 10⁻³ ≲ λ ≲ 10. As λ becomes larger, the asymptotic upper bound turns out to be somewhat optimistic. The close agreement at small λ is due to the suppression of the non-linearity inherent in $f ( α / h ̃ ( r ̃ ) , β , λ )$ from Eq. (15). For negative ϵf(α, β, λ), the numerical calculation indicates no dewetting at all, in sharp disagreement with the asymptotic prediction; the lack of dewetting for negative ϵf(α, β, λ) is due to the unusual interface shape that develops, as noted below, which eventually violates the curvature assumptions of the governing equation.

Besides confirming the dewetting limits, we can also compare the axisymmetric film profiles from the asymptotic approximation and the numerical solutions. Results for ϵf(α, β, λ) > 0 are shown in Figure 5 and ϵf(α, β, λ) < 0 are shown in Figure 6. We observe excellent agreement between the numerical and asymptotic solutions even for $|ϵf ( α , β , λ ) |=O ( 20 )$⁠; the surprising range of ϵf(α, β, λ) for which the asymptotic estimates compare well with numerical solutions is an example of the widely observed phenomenon known as “Carrier’s rule,”²² whereby asymptotic series produce good approximations even when the asymptotic parameter is not small. For ϵf(α, β, λ) < − 96/5, the film profile begins to exhibit an inflection point, which introduces an additional radius of curvature into the analysis. Thus, the film is not observed to dewet for negative values of ϵf(α, β, λ), but the governing equation no longer obeys the assumption of linearized curvature within the central rheometer region.

The numerical procedure also allows for solution of the cone and plate variation of the problem (Figure 1(b)), which cannot be calculated by asymptotic means due to the radial dependence of $H ̃ ( r ̃ )$⁠. Results for numerical calculations with identical parameters to those in Figure 5 (but introducing ζ = H_C/h₀ = 2 for the cone geometry) are presented in Figure 7. We also include asymptotic results for a constant gap height with the same α = H_R/h₀. Note that these parallel-plate asymptotic results approximate the cone and plate numerical results quite well. To zeroth order in the expansion of the cone geometry as a function of r, the two rheometer geometries can be treated the same.

Recent experiments⁹ utilized a cone and plate configuration to study drag reduction over a silicone oil film infused into a microstructured substrate (an ordered array of posts separated by gaps that are 50 μm wide and 50 μm deep) above which an aqueous glycerine-water solution flowed under the cone of the rheometer. In the experiments, the cone angle $ϕ= 2 ° + 1 60 °$⁠, R = 20 mm, H_C = 104 μm, and H_R ≈ Rtan(ϕ) ≈ 754 μm, yielding α ≈ 15 and ζ ≈ 2. A hydrophilic pinning line was used to enforce a fixed outer meniscus in the aqueous working fluid. The largest drag reduction was observed at a viscosity ratio λ = 0.0048 with β ≈ 1. The shear rates reported were $γ ̇ = [ 10 − 100 ] s − 1$⁠, where $Ω= γ ̇ ϕ$⁠.

The corresponding range of ϵf(α, β, λ) for these shear rates is ϵf(α, β, λ) = [10 − 1200], which significantly exceeds the parallel-plate and cone and plate dewetting criteria at the higher shear rates (the dewetting for the cone and plate is predicted numerically at ϵf(α = 15, β = 2, λ = 0.0048) ≈ 17 with similar qualitative dependence on λ as in the parallel-plate case and negligible dependence on ζ).

However, our analysis assumes that the oil is entirely mobile; the presence of microstructures in the substrate with radial periodicity can significantly modify the force balance between centripetal and capillary forces. By rewriting the parameter ϵ (Eq. (14)) in the form of a Bond number as the ratio of those forces, as was introduced conceptually in (3), it is clear that the straightforward non-dimensionalization assumes that the interfacial curvature scales with the radius of the rheometer, R,

If, however, periodic roughness of wavelength a in the radial direction were to interrupt the mobility of the oil film, then the curvature in the denominator of the Bond number should scale with the dimension a, which yields an effective Bond number ϵ_eff given by

The ratio of the two measures of interfacial deformation is ϵ_eff/ϵ ≈ (a/R)². The microstructure will also affect the mobility of the oil, giving the oil a higher effective viscosity,²³ μ_o, and therefore a higher effective λ. In experiments,⁹ ϵ_eff/ϵ ≈ 6 × 10⁻⁶, which means that the relevant value of ϵ ≪ 1 even in a case of sparse surface roughness.

However, as the radial wavelength of the roughness approaches R, the deformation of the interface can become significant. Indeed, assuming a mobile film under the above flow conditions, the height of the oil film at r = 0 at the onset of dewetting can be measured from the numerically determined film profile as h(0) ≈ 2H_C, which indicates that the film can even make contact with the bottom of the cone given sufficient mobility. Unfortunately, no direct experimental validation of the film profiles was possible given the optical access constraints of most standard rheometers, including those used in the recent experiments described above.

Before dewetting occurs, we can calculate the effect of the interfacial shape change on the torque measurement in both the parallel-plate and cone and plate configurations. We continue to treat only the central region of the rheometer, ignoring changes in the outer meniscus, since the meniscus shape is largely unaffected by the secondary flows,²⁴ and thus, the torque correction is expected to obey the static scaling arguments discussed above. In particular, changes in the interface deflection between the center and edge regions were shown to scale as $δ edge δ central ∼ H R R 2 ≪1$⁠, which means the change in the rotational moment, ΔM, taking into account the lever arm, should scale as $Δ M edge Δ M central ∼ H R R 2 R 3 ( R − ℓ c ) 3 − 1$ which remains small as long as we maintain the original assumption that the relevant capillary length at the edge is small, ℓ_c ≪ R.

The torque is measured on the motor that drives the upper plate, so we calculate the angular moment exerted on the upper disk or cone by integrating the shear stress at the upper surface times its lever arm over the area of the disk. The dimensional shear stress at the top surface is $τ z θ ( r , z = H ( r ) ) = μ w ∂ u θ w ( 0 ) ∂ z$⁠. From (6), we arrive at

Non-dimensionalizing by $μ w R Ω H R$ gives

The total moment exerted on the upper disk is then

and we can denote $M ̃ 0$ to represent the moment when $h ̃ ( r ̃ ) =1$⁠, i.e., the expected moment given a flat interface. The ratio of $M ̃ / M ̃ 0$ for a range of λ and ϵ but fixed α = 15 and β = 1 is displayed in Figure 8. The overall effect on the measured torques appears to be quite significant. For ϵf(α, β, λ) > 0, the film tends to be displaced inward, thereby increasing the relative thickness of the aqueous layer at larger “lever arms” and simultaneously thinning the oil layer there. If the oil layer is less viscous, λ < 1, then the moment increases; if the oil is more viscous, the moment decreases. For ϵf(α, β, λ) < 0 (at larger $β= ρ o ρ w$⁠), the film is displaced outward, and thus, if λ < 1 the moment decreases and if λ < 1 the moment increases. Finally, changing the ratio of rheometer gap size to unperturbed film height, α, can enhance or suppress the magnitude of all of these trends, as the relative significance of the film becomes more or less important to the overall flow.

It is worth emphasizing that these calculations reflect only the situation of a completely wetting oil film with perfect mobility; once dewetting occurs on the bottom plate (or alternatively, when the oil begins to wet the top plate or cone), or if roughness constrains the oil mobility, then the effect on the torque measurements may be significantly different.

Stratified immiscible flows in a rheometer exhibit both interfacial deformations and dewetting behavior due to secondary flows that vary as a function of fluid and rheometer properties. Through asymptotic analysis and numerical calculations, we show how significant interfacial deformations can affect thin films in both parallel-plate and cone and plate rheometers and how these deformations can lead to possible dewetting of the film from the lower plate or even wetting of the upper plate or cone. The effects of these deformations on torque measurements are expected to be quite significant for a wide range of operating conditions, and the wetting behavior may also be an important consideration for future experiments on liquid-infused or coated substrates.

We acknowledge the support of the Office of Naval Research through MURI Grant Nos. N00014-12-1-0875 and N00014-12-1- 0962 (Program Manager Dr. Ki-Han Kim) and thank Professor Alexander Smits for valuable discussions. We thank the anonymous referees for help in streamlining the derivations and suggestions for improving the numerics. We also thank the editor, John Hinch, who found an error in an earlier version of the manuscript, for important feedback regarding the derivations.

Using (20), the solution to $O ( ϵ 2 )$ is found to be

and the next term in the series to $O ( ϵ 3 )$ has the form

where $P i ( r ̃ )$ are polynomial functions of order 12. Also, at $O ( ϵ 4 )$⁠, the prefactor is $1 20 ⋅ 35 ⋅ 168 ⋅ ( 96 ) 4$⁠. Thus, we see that the asymptotic series converges very quickly in its first few terms even for large values of ϵ, as can occur for asymptotic series according to “Carrier’s rule.”²² However, these asymptotic series tend to exhibit a minimum error at some finite number of terms, past which the series diverges rapidly.