Inf-sup estimates for the Stokes problem in a periodic channel

Wilkening, Jon

关键词: 97;    ASPECT RATIO;    FUNCTIONAL ANALYSIS;    GEOMETRY;    INCOMPRESSIBLE FLOW;    PERIODICITY;    THICKNESS;   

DOI  :  10.2172/971259
RP-ID  :  LBNL-2298E
PID  :  OSTI ID: 971259
Others  :  TRN: US201004%%435

美国|英语

来源: SciTech Connect

【 摘 要 】

We derive estimates of the Babuska-Brezzi inf-sup constant {beta} 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 {beta} 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 {beta} on the aspect ratio of the rectangle. We then show how to transfer this result to the case that h is C{sup 1,1} or even C{sup 0,1} by a change of variables. We avoid non-constructive theorems of functional analysis in order to explicitly exhibit the dependence of {beta} 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 C{sup 1,1} case, and nearly optimal in the Lipschitz case.