# Inf-sup estimates for the Stokes problem in a periodic channel Page: 1 of 18

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INF-SUP ESTIMATES FOR THE STOKES PROBLEM IN A

PERIODIC CHANNEL

JON WILKENING *

Abstract. We derive estimates of the Babuska-Brezzi inf-sup constant /3 for two-dimensional

incompressible flow in a periodic channel with one flat boundary and the other given by a periodic,

Lipschitz continuous function h. If h is a constant function (so the domain is rectangular), we show

that periodicity in one direction but not the other leads to an interesting connection between /3

and the unitary operator mapping the Fourier sine coefficients of a function to its Fourier cosine

coefficients. We exploit this connection to determine the dependence of /3 on the aspect ratio of the

rectangle. We then show how to transfer this result to the case that h is C1,1 or even C0,1 by a

change of variables. We avoid non-constructive theorems of functional analysis in order to explicitly

exhibit the dependence of /3 on features of the geometry such as the aspect ratio, the maximum slope,

and the minimum gap thickness (if h passes near the substrate). We give an example to show that

our estimates are optimal in their dependence on the minimum gap thickness in the C1,1 case, and

nearly optimal in the Lipschitz case.

Key words. Incompressible flow, Stokes equations, Babuska-Brezzi inf-sup condition, gradient,

divergence, Sobolev space, dual space

AMS subject classifications. 76D03, 46E35, 42A16

1. Introduction. Many problems of industrial and biological importance in-

volve fluid flow in narrow channels with moving boundaries [7, 11]. Examples include

the flow of oil in journal bearings or between moving machine parts, the flow of air be-

tween disk drive platters and read-write heads, or the flow of mucus under a crawling

gastropod [13]. A primary objective in all these problems is to solve for the pressure

required to maintain incompressibility. Indeed, it is the pressure that determines the

load sustainable by a journal bearing, and that provides propulsion against viscous

drag forces in peristaltic locomotion. However, only the gradient of pressure enters

directly into the Stokes or Navier-Stokes equations; thus, regardless of the method

used to solve the equations, the pressure must be determined via its gradient.

The fundamental fact that makes it possible to extract p from Vp is that the

gradient is an isomorphism from L2(Q), the space of mean-zero square integrable

functions, onto the subspace of linear functionals in H-1(Q)2 that annihilate the di-

vergence free vector fields u E HJ()2; see Section 2 below. The inf-sup constant

0 (or rather, its inverse) gives a bound on the norm of the inverse of this operator.

Thus the magnitude of p (and our ability to estimate errors in p) depends to a large

extent on the size of )--1. However, to the author's knowledge, every existing proof

(e.g. [4, 8]) that )-1 is finite relies on Rellich's compactness theorem to extract a sub-

sequence whose lower order derivatives converge, making it impossible to determine

how large )-1 might be or how it depends on Q. The proof in [4] also uses the closed

graph theorem, which, like Rellich's theorem, leads to constants that depend on Q in

an uncontrollable way. These proofs are appropriate for pathological domains with

bulbous regions connected by thin, circuitous pathways; however, for "nice domains",

it should be possible to obtain better estimates of the constants existing theorems

are of limited practical use.

*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.

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Wilkening, Jon. Inf-sup estimates for the Stokes problem in a periodic channel, article, December 10, 2008; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc900046/m1/1/: accessed April 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.