Practical Error Estimates for Reynolds' Lubrication Approximation and its Higher Order Corrections

PRACTICAL ERROR ESTIMATES FOR REYNOLDS’ LUBRICATION APPROXIMATION AND ITS HIGHER ORDER CORRECTIONS JON WILKENING Abstract. Reynolds’ lubrication approximation is used extensively to study flows between moving machine parts, in narrow channels, and in thin films. The solution of Reynolds’ equation may be thought of as the zeroth order term in an expansion of the solution of the Stokes equations in powers of the aspect ratio e of the domain. In this paper, we show how to compute the terms in this expansion to arbitrary order on a two-dimensional, x-periodic domain and derive rigorous, a-priori error bounds for the difference between the exact solution and the truncated expansion solution. Unlike previous studies of this sort, the constants in our error bounds are either independent of the function h(x) describing the geometry, or depend on h and its derivatives in an explicit, intuitive way. Specifically, if the expansion is truncated at order 2k, the error is O(e 2k+2 ) and h enters into R 1 the error bound only through its first and third inverse moments 0 h(x) −m dx, m = 1, 3 and via ∂ x h ‚ ∞ , 1 ≤ ≤ 2k + 2. We validate our estimates by comparing with finite the max norms ‚ ! h element solutions and present numerical evidence that suggests that even when h is real analytic and periodic, the expansion solution forms an asymptotic series rather than a convergent series. Key words. Incompressible flow, lubrication theory, asymptotic expansion, Stokes equations, thin domain, a-priori error estimates AMS subject classifications. 76D08, 35C20, 41A80 1. Introduction. Reynolds’ lubrication equation [22, 20, 16, 12] is used exten- sively in engineering applications to study flows between moving machine parts, e.g. in journal bearings or computer disk drives. It is also used in micro- and bio-fluid me- chanics to model creeping flows through narrow channels and in thin films. Although there is a vast literature (including several textbooks) on viscous flows in thin geome- tries, the equations are normally derived either directly from physical arguments [16], or using formal asymptotic arguments [12]. This is acceptable in most circumstances as the original equations (Stokes or Navier–Stokes) have also been derived from phys- ical considerations, and by now the lubrication equations have been used frequently enough that one can draw on experience and intuition to determine whether they will work well for a given problem. On the other hand, as soon as the geometry of interest develops (or approaches) a singularity, or if we wish to compute several terms in the asymptotic expansion of the solution in powers of the aspect ratio e, we rapidly leave the space of problems for which we can use experience as a guide; thus, it would be helpful to have a rigorous proof of convergence to serve as a guide to identify the features of the geometry that could potentially invalidate the approximation. For example, in [25], the author and A. E. Hosoi used lubrication theory to study the optimal wave shapes that an animal such as a gastropod should use as it propagates ripples along its muscular foot to crawl over a thin layer of viscous fluid. In certain limits of this constrained optimization problem, the optimal wave shape develops a kink or cusp in the vicinity of the region closest to the substrate, and there is a competing mechanism controlling the size of the modeling error (singularity formation vs. nearness to the substrate). We found that shape optimization within (zeroth order) lubrication theory drives the geometry ∗ Department of Mathematics and Lawrence Berkeley National Laboratory, University of Cali- fornia, Berkeley, CA 94720 (wilken@math.berkeley.edu). This work was supported in part by the Director, Office of Science, Advanced Scientific Computing Research, U.S. Department of Energy under Contract No. DE-AC02-05CH11231.