Extended Reynolds lubrication model for incompressible Newtonian fluid

An extended lubrication model is proposed by taking into account a larger surface-to-surface distance than that for the Reynolds lubrication theory, and the wall-normal variation of the pressure is related to the longitudinal derivative of the local velocity of the Couette-Poiseuille flow.

gives the estimation V = εU . The Reynolds lubrication theory [20] assumes a narrow gap of ε 1, and the unsteady and convective terms are neglected for a small value of ε 2 Re, where Re = ρU L/μ is the Reynolds number, and ρ and μ are the density and viscosity of the fluid, respectively.
Retaining only the O[ε 0 ] terms, as explained later, the incompressible Navier-Stokes equations are reduced to the following equations: where p is the pressure. Coupling this with Eq. (1) yields the Reynolds lubrication equation in which the dimension of the problem is reduced because the surface-normal variation of the pressure is eliminated, i.e., p(x).
In our previous work [24], the Reynolds lubrication equation was incorporated into a particulate flow solver for correcting the flow in the interparticle region, and the numerical solution was confirmed to be in good agreement with the results of the analytical and independently conducted numerical studies (by direct numerical simulation) for eccentric-bearing film flow and fluid-particle interaction problems in a low Reynolds number range with less computational time. However, our previous study also revealed the limited applicability of the Reynolds lubrication equation to model particle-induced flows due to the strong constraints of ε 1 and ε 2 Re 1; only a limited configuration satisfies this condition, and in other cases, the variation in the wall-normal direction is non-negligible.
Indeed, in a relatively wide region of ε ∼ 1, for example, the velocity components are comparable (V ∼ U ). Further, the pressure gradient in the surface-normal direction may be non-negligible, and the Reynolds lubrication equation does not describe the correct film flow.
In the present study, we introduce a relaxed geometric condition of ε 1 while ε 2 is still small enough (i.e., ε 2 1), together with approximations of small gradient and curvature for surface profile. Then, an extended lubrication model is proposed by considering the non-negligible effect of the pressure variation in the surface-normal direction. To assess the validity of the extended lubrication model, the pressure fields obtained by the present model are compared with analytical and direct-numerical solutions for lubrication flows between moving and stationary walls.

II. GOVERNING EQUATIONS
The scaled variables u * = u/U, v * = v/V, t * = L/U, x * = x/L, y * = y/H, and p * = p/P ref are introduced in the incompressible Navier-Stokes equations, as well as the equation of continuity, Eq. (1), where t is the time and P ref = ε −2 μU L −1 . Then, the scaled Navier-Stokes equations are obtained as [25] By eliminating all the terms smaller than the order of ε 2 , the above equations are reduced to Eq. (2). On the other hand, assuming Re 1 and retaining the terms of O[ε 2 ], we have the following set of equations: The first term in the right-hand side of Eq. (3a) is reevaluated as follows: where h * = h/H. This equation suggests that the term ∂ 2 u/∂x 2 is smaller than the order of ε 2 when both the gradient and curvature of the surface profile remain small. Equation (4) may be represented in an alternative form: assuming that the gap width h(x) changes only slightly such that the variation of the width, h, is much smaller than H, then suggesting that the condition h/H 1 enables elimination of the second-order x derivative of u. Therefore, in the present study, we employ the following set of equations for lubrication: in a relatively wide region (i.e., non-Reynolds lubrication condition) of ε 1, ε 2 1, and h/H 1.

III. MODELING
By solving Eqs. (1) and (6), a lubrication model is constructed to describe the pressure variations in both the longitudinal and surface-normal directions. The following two lemmas are introduced. Lemma 1. The local pressure gradient in the x direction, ∂ p/∂x, is regarded as a function of x (and is negligibly relevant to y). Lemma 2. The pressure can be separated into the base and adjusting components: where no more independent functions of x are separable from p adj . The proofs of the lemmas are given in Appendix A. By Lemma 1, we obtain the following form of u by integrating Eq. (6a) twice with respect to y: where f 1 and f 2 are the integral constants. The boundary conditions, u = U 1 at y = 0 (surface 1) and u = U 2 at y = h(x) (surface 2), identify these constants as where is obtained, where one of the boundary conditions v| y=0 = V 1 is used. The other boundary condition, v| y=h = V 2 , imposes the following relation: Substituting Eq. (9) into the above equation, an equation analogous to the Reynolds lubrication equation is derived: Equation (11) suggests that, although it considers the pressure variation in the surface-normal direction, the surface pressure still obeys the Reynolds lubrication equation. Considering Lemma 1, this can be written as follows: where p Re (x) is the pressure determined by the Reynolds lubrication equation. An alternative form of v is derived to obtain a closed system of p. By integrating Eq. (6b) with respect to y and using Lemma 2, ∂ v/∂y is related to p adj as where f 3 is an integral constant. Comparing the above equation with the y-dependent terms in −∂ u/∂x, the following relation is obtained: From the above equation, the order of magnitude of ∂ p adj /∂x is found to be μU This fact characterizes the present problem; although p adj is small in comparison to p Re , ∂ p adj /∂y exhibits a comparable order of magnitude to ∂ p Re /∂x. Equation (15) The In this equation, the terms inside the x derivative represent the velocity of the Couette-Poiseuille flow. Equation (17) suggests that p adj is the pressure adjustment due to the spatial change of the local Couette-Poiseuille flow, induced by the gradient of p Re and moving walls, in a narrow gap. Therefore, in a feasible extended lubrication model for the non-Reynolds regime, p Re is obtained from the Reynolds lubrication equation, and then, with p adj (x, y) determined by Eq. (17), the pressure is eventually given by The three-dimensional version of the extended lubrication model is presented in Appendix C.

IV. VALIDATION
To assess the validity of the proposed extended lubrication model, pressure fields in the small gaps are compared with analytical and numerical solutions.
Wannier [26] demonstrated the analytical solution of the Stokes equation for the flow between a flat moving plate and a cylinder (as schematically shown in Fig. 2), and the pressure is given as where On the other hand, the Reynolds lubrication equation gives the pressure for this flow as where the surface-to-surface distance is approximated as h h 0 + x 2 /2a, and the boundary pressures are assigned as p Re (±∞) = 0.  Figure 3(a) shows that the pressure for the case of d/a = 1.01 is well described by the Reynolds lubrication equation. For the case of d/a = 1.18, on the other hand, Fig. 3(b) suggests that the present extended lubrication model (p adj ) reasonably captures the trend of p Wa − p Re at x/a = 0.2, where the locally evaluated value ε = h(0.2a)/2a 0.1 may be around an upper limit for ε 2 1 and h/H h/h| x=0.2a 0.02a/0.2a = 0.1 is also reasonable. The deviation of the wall-normal distribution of p adj from the Wannier case becomes non-negligible at x/a = 0.4 and 0.8; in particular, the deviation at x/a = 0.4 is large as the sign of ∂ 2 p adj /∂y 2 changes (i.e., convex to concave profiles along the y axis) at around x/a = 0.5, as typically observed later in a pressure contour map (Fig. 4). The case of d/a = 1. (ε 2 = 0.125) may be even farther from the condition of ε 2 1, and the h/H value is not fully ideal to justify Eq. (6b) for the entire x range. However, the local value h/h| x=0.2a 0.02 suggests the applicability to a limited x range, and, as Fig. 3(c) shows, the extended lubrication model still works to improve towards the correct pressure distribution. Figure 4 shows the comparison of the pressure distributions obtained by Eqs. (18) and (19) for d/a = 1.5. Considering that p Re is only x dependent, the effect of the adjustment by p adj is remarkable. Both Figs. 3 and 4 suggest that the adjustment (p adj ) added to p Re improves the y-dependent trends of the pressure towards the Wannier flow even in a wide gap region.
For a second validation case, the flow induced by a curved object traveling at a constant speed is set up to highlight the noncylindrical geometric effect and is compared with the numerical solution. As schematically shown in Fig. 5, a corrugated plate of sinusoidal geometry is placed in the middle y level of the domain bounded by the stationary flat solid walls, and the object is towed at a constant speed of U 0 in the positive x direction: where H 0 is the half-channel height and δ is the nondimensional parameter between 0 and 1. The wave number k is set to 2π/L 0 . The periodic boundary condition is applied in the x direction, and the no-slip condition is imposed on the solid surfaces. Following the lubrication analysis in Ref. [27], Periodic Periodic the p Re for this problem is given as follows: For comparison, a direct numerical simulation (DNS) is carried out with a fully validated method [27]. The spatial resolution is fixed as /H 0 = 0.05 in both directions, where is the grid spacing. The Reynolds number for the DNS is set as Re = ρU 0 L 0 /μ = 1.  The third validation case demonstrates the hydrodynamic force on the center of a spherical particle generated by the lubrication in the region between a plane wall and the particle approaching at a constant speed with no angular velocity, as illustrated in Fig. 7. The pressure that satisfies the Reynolds lubrication equation is given as [28] p Re (r) = 3μaW 0 where the paraboloid approximation h(r) h 0 + r 2 /2a is assumed for the gap distance. For the present axisymmetric problem, the adjusting component of the pressure [Eq. (C1) in Appendix C] is simplified to the following form: where z represents the distance in the wall-normal direction. With the above approximation for the profile h(r), the forces at the particle center are calculated as follows: where ε = h 0 /a. Both cases indicate that the major term is ε −1 in the limit of ε → 0, which is consistent with the prediction by asymptotic analysis [12].

V. CONCLUDING REMARKS
Our extended lubrication model proposed for a relatively wide range of aspect ratios ε satisfying ε 1 and ε 2 1 enables decomposition of the pressure into two components: the contribution by the Reynolds lubrication theory (p Re ) and the adjusting component (p adj ) varying in both longitudinal and wall-normal directions. Although the contribution of the latter is small in magnitude, its gradient in the surface-normal direction is significant. The adjusting component of pressure was described with the longitudinal derivative of the local velocity of the Couette-Poiseuille flow driven by p Re and the tangential velocity of the walls. Through comparison with the analytical and numerical results, the extended lubrication model successfully predicted the pressure distribution, particularly the surface-normal distribution, and the force acting on the object center, regardless of the geometry (even for noncircular/nonspherical) of the object.
This method is easily applicable for analyzing film flow in an interparticle narrow gap; by solving the Reynolds lubrication equation with a combined Eulerian-Lagrangian formulation as demonstrated in Ref. [24], the efficient solution of a film flow under an unresolved situation is facilitated by the extended lubrication model presented herein. In future works, a method for modeling the film flow in the network of the narrow gaps between the particles will be developed to study dense particulate flows.