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Volume 14, Issue 3
Modeling the Lid Driven Flow: Theory and Computation

Makram Hamouda, Roger Temam & Le Zhang

Int. J. Numer. Anal. Mod., 14 (2017), pp. 313-341.

Published online: 2017-06

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  • Abstract

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 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.

  • AMS Subject Headings

76D07, 35D30, 76D03

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-14-313, author = {Makram Hamouda, Roger Temam and Le Zhang}, title = {Modeling the Lid Driven Flow: Theory and Computation}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2017}, volume = {14}, number = {3}, pages = {313--341}, abstract = {

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 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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10010.html} }
TY - JOUR T1 - Modeling the Lid Driven Flow: Theory and Computation AU - Makram Hamouda, Roger Temam & Le Zhang 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 -

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 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.

Makram Hamouda, Roger Temam and Le Zhang. (2017). Modeling the Lid Driven Flow: Theory and Computation. International Journal of Numerical Analysis and Modeling. 14 (3). 313-341. doi:
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