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Shear-induced Casimir forces in liquid layers

In stationary nonequilibrium states a coupling between hydrodynamic modes causes thermal fluctuations to become long ranged inducing nonequilibrium Casimir forces or pressures. Here we consider nonequilibrium Casimir pressures induced in liquids by a velocity gradient. Specifically, we have obtained explicit expressions for the magnitude of the shear-induced pressure enhancement in a liquid layer between two horizontal plates that complete and correct results previously presented in the literature. In contrast to nonequiibrium Casimir pressures induced by a temperature gradient, kinetic theory shows that nonequilibrium contributions from short-range fluctuations are no longer negligible. In addition, it is noted that computer simulations of model fluids in shear observe effects from molecular correlations at nanoscales that have a different physical origin. The idea that such computer simulations probe shear-induced pressures resulting from coupling of long-wavelength hydrodynamic modes is erroneous.

In this letter we consider Casimir forces due to longrange velocity fluctuations in liquids in shear [6][7][8]. For the case of a liquid subjected to planar laminar Couette flow, we have obtained explicit expressions for the shearinduced pressure enhancement δp, which complete and correct results obtained by previous authors [9][10][11][12]. We provide quantitative estimates for the magnitude of these shear-induced Casimir pressures. We clarify an essential difference between the Casimir pressures resulting from macroscopic long-range fluctuations and forces resulting from the effect of fluctuations at nanoscales which are the ones observed in computer simulations [13][14][15][16][17][18][19].
To maintain consistency with a previous analysis of velocity fluctuations in planar Couette flow by two of us [20,21], we continue to use a coordinate system which the x coordinate is in the stream-wise direction, the y coordinate in the span-wise direction, and the z coordinate in the wall-normal direction. The liquid layer is located between two horizontal plates located at z = ±L moving with constant velocities ±U in the x direction. We assume no-slip boundary conditions for the velocities at z = ±L [22]. The local fluid velocity can be decomposed as v = v 0 (z) + δv, where v 0 = v 0 (γz, 0, 0) is the average velocity dependent on the shear rate γ = U/L with a component only in the stream-wise direction x, and where δv(r, t) is a fluctuating-velocity contribution dependent on location r (x, y, z) and on time t. Just as in our previous work on a pressure enhancement induced by a temperature gradient [23], we use nonlinear fluctuating hydrodynamics. Since density fluctuations have been shown to decay faster than velocity fluctuations [7], the leading fluctuation renormalization to the pressure tensor is, see Supplemental Material: δP (r) = ρ δv(r)δv(r) NE . (1) Here ρ is the mass density and the average is taken over the stationary nonequilibrium state which is independent of the time t. The diagonal elements δp ii = ρ δv i δv i contribute to the shear-induced pressure enhancement, such that δp = 1 3 (δp xx + δp yy + δp zz ) . The off-diagonal elements all vanish except for δp xz = ρ δv x δv z , yielding a fluctuation-induced contribution to the shear viscosity η [11]. We find that all diagonal elements depend on the shear rate γ and the Reynolds number Re = γL 2 /ν as where k B is Boltzmann's constant, T the temperature, and ν the kinematic viscosity. Here ϕ (Re) is a crossover function such that lim Re→∞ ϕ ii (Re) = 1 and lim Re→0 ϕ ii (Re) = (V 0 ii /V ∞ ii )(Re) 1/2 . Specifically, the two limiting cases are We emphasize that the anomalous dependence of δp on γ 3/2 and on Lγ 2 is a consequence of a coupling between macroscopic viscous modes and not from static or dynamic correlations at molecular scales.
In previous publications we have shown how the correlation functions for the wall-normal velocity fluctuations and for the wall-normal vorticity fluctuations can be derived by solving the fluctuating Orr-Sommerfeld and Squire equations [20,21]. For large L and, hence, for large Re, we have obtained an exact solution of these fluctuating hydrodynamics equations, since in this limit the solution becomes independent of the boundary conditions. For small L and, hence, for small Re, the solution is strongly affected by the no-slip boundary condition for the velocity. In this limit it is difficult to get an exact solution [24,25] and we have settled for an approximate solution in a Galerkin approximation [20,21]. Explicit expressions for the diagonal elements of the shear-induced pressure tensor can be directly related to the solutions previously obtained for the wall-normal velocity and vorticity fluctuations as shown in the Supplemental Material and which does not involve any new physics [26]. The only additional step required is integration of the correlation functions over wave numbers so as to get the intensity of the velocity fluctuations in real space. The resulting values are V ∞ xx = +0.0847, V ∞ yy = +0.0173, V ∞ zz = +0.0106, and V 0 xx = +0.001438, V 0 yy = +0.000480, V 0 zz = +0.000392. Actually, the solution from the fluctuating hydrodynamics equations yield expressions for V 0 ii that depend on the z coordinate as a consequence of the boundary conditions at z = ±L. However, just as in the case of the Casimir pressures induced by a temperature gradient [23], mechanical equilibrium, combined with conservation of mass, causes a uniform pressure enhancement equal to the height-averaged value obtained from the fluctuating hydrodynamics equations. Hence, we only quote here the height-averaged values for V 0 ii . We thus conclude: We do not consider contributions from the sound modes here, since they are smaller by a factor of (U/c) 1/2 , where c is the speed of sound. Terms of higher order in the shear-rate have also been neglected as discussed in Section III of the Supplemental Material [26]. Attempts to determine the shear-induced pressure tensor in the absence of boundary conditions have been made by Kawasaki and Gunton [9] and by Yamada and Kawasaki [10]. While they did find that the shearinduced pressure varies with the shear rate as γ 3/2 , the numerical coefficients are substantially different from the values found by us as shown in Table I. Ernst et al. [11] determined the traceless part of the shear-induced pressure tensor using a kinetic-theory approach. Our results for the traceless part of the shearinduced pressure tensor are in perfect agreement with those obtained by Ernst et al. as shown in Table II. In this table we also see perfect agreement with their offdiagonal stress, V xz , which gives a generalized viscosity. Hence, we are confident that we have obtained correct expressions for the shear-induced pressure enhancement as given by Eqs. (5) and (6). Wada and Sasa [12] have only determined the wall-normal component of the shearinduced pressure tensor. They find V ∞ zz = +0.0106 in agreement with our result, but their value V 0 zz = 0.000553 slightly differs from V 0 zz = 0.000392 found by us. The reason is that Wada and Sasa used periodic boundary conditions which are mathematically convenient, but physically unrealistic.
We see that the shear-induced pressure enhancement is many orders of magnitude smaller than the Casimir pressures induced by the presence of a temperature gradient. One reason is that temperature fluctuations decay more slowly than velocity fluctuations and, hence, are more strongly impacted by the presence of a temperature gradient. Another reason is that the shear-induced pressure enhancement has a kinetic origin, while the pressure enhancement from a temperature gradient has a potential origin that in liquids is several orders of magnitude larger.
An important consequence is that, unlike the case of a temperature gradient [23], effects from short-range corre-  lations to the shear-induced pressure enhancement may not be negligible. To elucidate a possible contribution from short-range correlations, we note from nonequilibrium statistical mechanics that δp = κγ 2 , where κ is a nonlinear Burnett coefficient. These nonlinear Burnett coefficients are known to diverge as L → ∞ [38]. We may decompose this Burnett coefficient as the sum of a finite short-range contribution κ (0) and a long-range contribution Lκ (1) [23], yielding a short-range (SR) and a long-range (LR) contribution the shear-induced pressure enhancement: where δp SR = κ (0) γ 2 and δp LR = Lκ (1) γ 2 . Comparing with Eq. (6), we note that the shear-induced Casimir pressure arises from the same long-wavelength hydrodynamic modes that cause the nonlinear Burnett coefficient κ to diverge. A complete kinetic theory for the nonlinear Burnett coefficients of real fluids is not available, but it is possible to get an order-of-magnitude estimate by extending the theory of Enskog for the transport properties of a dense gas of hard spheres to the quadratic level [39].
Starting from an expression for the pressure tensor of a gas of hard spheres provided by Dufty [40] and retaining only the collisional transfer contribution, which is the dominant one at high densities, we obtain where σ is the hard-sphere diameter, n the number density, and χ the value of the radial distribution function at contact between the spheres. For liquid water ρ = nm = 10 3 kg m −3 , m = 3×10 −26 kg, σ = 3×10 −10 m. Estimating χ ∼ = 5 for a dense liquid, we then conclude from Eq. (8) that for water with L = 0.5 mm and U = 0.5 ms −1 On comparing Eq. (10) with Eq. (7) we see that the SR contribution to the induced-pressure enhancement is indeed smaller than the LR contribution to the shearinduced pressure enhancement, but may not be entirely negligible, even at L = 0.5 mm. The SR contribution becomes even more important at smaller values of L. From Eq. (9) it follows that, for a fixed velocity U, δp SR will increase as L −2 , while δp LR , due to the long-range velocity fluctuations, will only increase either as L −3/2 for large values of Re or even less as L −1 for small values of Re.
The first molecular dynamics (MD) simulations on a 3-dimensional sheared fluid consisting of a small number of Lennard-Jones (LJ) particles were performed by Evans [13]. He found results that seemed, especially near the triple point, to indicated a nonequilibrium (NE) pressure enhancement that was proportional to γ 3/2 , but with a coefficient that was orders of magnitude larger than the coefficient to be expected from Eq. (3). He noted a similarity with the so-called molasses tail observed in MD simulations of the equilibrium stress-tensor time correlation functions that determines the shear viscosity [41]. In turns out that in this time-dependent correlation function, again near the triple point of LJ particles or near freezing of hard-sphere particles, an apparent long-time tail proportional to 1/t 3/2 appears, but with a coefficient, again, several orders of magnitude larger than the theoretically expected long-time tail coefficient. It was subsequently realized that this molasses tail was not due to long-wave length MC effects, but was due to molecular scale MC effects related to structural relaxation in dense fluids [42][43][44][45]. This theory explains the magnitude of observed molasses tails and predicts that this 1/t 3/2 behavior will crossover to an exponential decay on a structural time scale τ s ≈ S(k 0 )/Dk 2 0 , where D is the selfdiffusion coefficient and k 0 the wave number where the static structure factor S(k) has its maximum [43]. For a review of these molecular scale MC effects, the reader is referred to a forthcoming book of Dorfman et al. [46]. Another complication is that the computer simulations use extremely large shear rates γ ≈ 10 11 − 10 12 s −1 . At such large shear rates, where γ > τ −1 s , the NE pressure is also determined by molecular-scale MC effects. The molecular-scale effects will not only depend on the intermolecular potential adopted, but, at a given density, also on the number of free paths sampled, and, hence, on the number of particles used in the simulations.
Generally, it never makes sense, except in some asymp-totic limit, to fit the shear-induced pressure enhancement in terms of a simple power law. For example, Eq. (8) suggests for sufficiently large L a fit to δp = A SR γ 2 + δp LR (11) with δp LR ∝ Lγ 2 for Re < 1 or δp LR ∝ γ 3/2 for Re > 1, and with A SR independent of L. In MD simulations that probe structural relaxation effects the appropriate fit should have an additional term δp s ∝ γ 3/2 for γ > τ −1 s and δp s ∝ γ 2 for γ < τ −1 s . That is, for γ > τ −1 s , δp s will renormalize δp LR in Eq. (11) and for γ < τ −1 s , δp s will renormalize A SR in this equation.
Almost all discussions of computer-simulation studies currently available [14][15][16][17][18][19] have ignored the effects of the molecular-scale correlations that are dominant at nanoscales. Lee and Cumming [14,15] found an enhancement ∝ γ 3/2 . But without checking the coefficient, they assumed to have found agreement with both the results of Evans [13] and with Eq. (3), which is impossible as explained above. Sadus and coworkers [16][17][18][19] have found effective exponents for the shear-rate dependence ranging from 1.5 to 2 without any theoretical analysis of the results.
The theoretical expression, Eq. (2), for the shearinduced pressure enhancement follows from a solution of the fluctuating hydrodynamics equations for the longrange velocity fluctuations. Recently, Varghese et al. [47] have tried to obtain a numerical solution of the fluctuating hydrodynamics equations for a multiparticle collision dynamics fluid, a model with an ideal-gas equation of state. They conclude that the shear-induced pressure enhancement obtained over about one decade of the shearrate appears to scale as γ 2 and therefore does not agree with Eq. (5). However, the magnitude of the enhancement seems to be indeed of the order given by our Eqs. (5) and (6). It would be of interest to pursue such calculations for a larger ranges of L and Re numbers, so as to probe a possible crossover from a behavior ∝ Lγ 2 for small Re to ∝ γ 3/2 for large Re. Varghese et al. [47] conclude their paper with the following comment: "It therefore remains for further theoretical and simulation studies to establish a unified picture of the exponent associated with the hydrodynamic pressure under shear". This Physical Review Letter attempts to address this issue.
We thank J.R. Dorfman for valuable discussions and R. Monchaux for some comments regarding Couette-flow experiments. The research at the Complutense University was supported by grant ESP2017-83544-C3-2-P of the Spanish Agencia Estatal de Investigación. The research at the University of Maryland was supported by the US National Science Foundation under Grant No. DMR-1401449.
In this supplemental material we show how the expressions for the diagonal elements δvδv NE are related to the correlation functions previous obtained for the wall-normal velocity fluctuations δv z and for the wall-normal vorticity fluctuations δω z = ∂ y δv x − ∂ x v y in shear [1,2]. We continue to use here dimensionless variables with position r in terms of L, wave vector q in terms of L −1 , and velocity v in terms of Lγ. In addition, we continue to define a dimensionless strength of the thermal noise as [1]: We specifically consider the limits of large L and small L at a fixed velocity U which correspond to large Re and small Re, respectively.

I. CALCULATION FOR LARGE Re
For large L we can neglect the boundary conditions and solve the fluctuating Orr-Sommerfeld and Squire equations by applying a 3-dimensional Fourier transformation [1,2]. We then obtain for the equal time correlation functions in momentum space: The functions C NE zz (q), W NE zz (q), and B NE zz (q) are given by as given by Eq. (39) and Eq. (43b) in [2], which are exactly the same as Eqs. (S.5) and (S.6) here, while Eq. (S.7) for the cross-correlation can be obtained following the same techniques and is first presented here. In these equations q is the magnitude of the component q of the wave vector in the x-y plane, i.e., parallel to the plates. From these equations we can obtain the correlation functions for δv x and δv y by noting that Integration of Eqs. (S.5), (S.13), and (S.14) yield the diagonal elements of δvδv NE in real space for large Re. As an example, we consider the computation of V ∞ zz . In the dimensionless units used in this Supplemental Material we have:S To evaluate the coefficient V ∞ zz , after substitution of Eq. (S.5) into Eq. (S.15), we adopt spherical coordinates for the integration over q. We first integrate over the magnitude q of the vector q, which can be done analytically and yields the prefactor (Re) 3/2 . A second integration over the polar angle can also be performed analytically taking advantage of the symmetry properties of the integral. The final double integral, over the azimuthal angle and over the parameter β, can be simplified but not performed analytically and has been evaluated numerically: which is the value quoted in the main text of the Letter and in Table I. The other coefficients, V ∞ xx and V ∞ yy , are evaluated in a similar fashion.

II. CALCULATION FOR SMALL Re
Small Re means narrow layers. Hence, we need to take the boundary conditions for the velocity fluctuations at z = ±L into account explicitly. As shown in [1,2], this is accomplished by applying a Fourier transformation only in the stream-wise and span-wise directions, while retaining the dependence of the wall-normal coordinate z. In that case, for the equal-time correlation functions we have: where the presence of boundaries breaks the translational invariance in the wall-normal direction. We can obtain the correlation functions for δv x and δv y by noting that

(S.24)
As stated in the main text, to solve the fluctuating hydrodynamics equations and following [1,2], we adopt a Galerkin approximation. Specifically, to satisfy the boundary conditions, we assume that where the coefficients A N (ω, q ) and B N (ω, q ) are determined by projection of the equations onto the basis used for the expansion (S.25) itself, i.e., (z 2 − 1) 2 z N and (z 2 − 1) z N , and solving the resulting algebraic equations. In practice, this Galerkin approach is only useful when the expansion (S.25) is truncated at some low order, and we have truncated at N = 1. As before and for illustrative purposes, we consider in detail here only V 0 zz . For the nonequilibrium contribution C NE zz (q , z) in Eq. (S.17) the first-order Galerkin method described above, after substitution of z = z ′ , explicitly gives: q 2 x q 2 (11 + q 2 ) (1089 + 411q 2 + 42q 4 + 2q 6 ) 11(63 + 12q 2 + 2q 4 ) + 4q 2 (11 + q 2 )q 2 x Re 2 (495 + 44q 2 + 2q 4 ) (S.27) and C 2 (Re, q) = 3465 128 q 2 x 11q 4 1089 + 411q 2 + 42q 4 + 2q 6 11(495 + 44q 2 + 2q 4 ) + 4q 2 (11 + q 2 )q 2 x Re 2 (63 + 12q 2 + 2q 4 ) . (S.28) Next, we can define a z-dependent component V 0 zz (z) which, in terms of the dimensionless units adopted here, will be given by the expression: Upon substitution of Eq. (S.26) into Eq. (S.29), after taking the Re → 0 limit, we obtain: The two integrals are convergent and can be performed analytically, but the result is long and not particularly informative. We prefer to display the result numerically, namely First, there are short-ranged corrections to Eq. (9) that are of O(γ 4 ). In dimensionless variables, these terms are of relative O([γσ/v th ] 2 ), with v th ∝ k B T /m a thermal velocity. For realistic laboratory shear rates these are very small corrections.
Second, the long wavelength nonlinear terms in the nonlinear fluctuating hydrodynamic equations that renormalize the pressure are, δP ij = ρv i v j + p(ρ, ǫ)δ ij − ρv i v j − p(ρ, ǫ)δ ij . (S.33) Here the over-line denotes average values and ǫ is the internal energy density. In principle all of these nonlinearities will lead to long-ranged renormalizations of the pressure. If we neglect density and internal energy density fluctuations, then we obtain Eq. (1). Taking into account density fluctuations leads to additional renormalizations of the pressure. However, Lutsko and Dufty [3] have shown that in general density fluctuations decay faster in space than velocity fluctuations. Their work suggest that density nonlinearities will lead to a γ 11/6 contribution to the pressure, with a relatively large coefficient. Compared to the γ 3/2 terms, this term is of relative [γσ/v th ] 1/3 , which is again quite small. In any case, it would be difficult to distinguish this term from all of the analytic γ 2 terms. We also note that we have considered isothermal flow, that is, possible viscous-heating effects have been neglected [4]. This condition is commonly satisfied in computer simulations by special dynamical rules keeping the temperature constant. However, in real experiments a pressure increase resulting from viscous heating may not be negligible.