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Inf-sup estimates for the Stokes problem in a periodic channel | | Jon Wilkening
; | | Date: |
27 Jun 2007 | | Abstract: | We derive estimates of the Babuu{s}ka-Brezzi inf-sup constant $eta$ 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 $eta$ 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 $eta$ on the aspect ratio of the
rectangle. We then show how to transfer this result to the case that $h$ is
$C^{1,1}$ or even $C^{0,1}$ by a change of variables. We avoid non-constructive
theorems of functional analysis in order to explicitly exhibit the dependence
of $eta$ 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 $C^{1,1}$ case, and nearly optimal in the
Lipschitz case. | | Source: | arXiv, arxiv.0706.4082 | | Services: | Forum | Review | PDF | Favorites | |
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