TY - JOUR
T1 - Modeling the Lid Driven Flow: Theory and Computation
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 313
EP - 341
PY - 2017
DA - 2017/06
SN - 14
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/10010.html
KW - Stokes and related (Oseen, etc.) flows, weak solutions, existence, uniqueness, regularity theory, lid driven cavity.
AB - <p style="text-align: justify;">Motivated by the study of the corner singularities in the so-called cavity flow, we
establish in the first part of this article, the existence and uniqueness of solutions in $L^2(Ω)^2$ for
the Stokes problem in a domain $Ω$, when $Ω$ is a&nbsp;smooth domain or a convex polygon. This result
is based on a new trace theorem and we show that the trace of $u$ can be arbitrary in $L^2(∂Ω)^2$ except for a standard compatibility condition recalled below. The results are also extended to the
linear evolution Stokes problem. Then in the second part, using a finite element discretization,
we present some numerical simulations of the Stokes equations in a square modeling thus the
well known lid-driven flow. The numerical solution of the lid driven cavity flow is facilitated
by a regularization of the boundary data, as in other related equations with corner singularities
([9], [10], [45], [24]). The regularization of the boundary data is justified by the trace theorem in
the first part.</p>