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

Makram Hamouda, Roger Temam & Le Zhang

DOI:

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 L²(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 L²(∂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.

  • Keywords

Stokes and related (Oseen etc.) flows weak solutions existence uniqueness regularity theory lid driven cavity

  • AMS Subject Headings

76D07 35D30 76D03

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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 L²(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 L²(∂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 KW - etc.) flows KW - weak solutions KW - existence KW - uniqueness KW - regularity theory KW - 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 L²(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 L²(∂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 & Le Zhang. (1970). Modeling the Lid Driven Flow: Theory and Computation.. International Journal of Numerical Analysis and Modeling. 14 (3). 313-341. doi:
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