Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T05:30:41.826Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Analysis of a Bingham Fluid in a Thin Domain with Fourier and Tresca Boundary Conditions

Published online by Cambridge University Press:  03 June 2015

M. Dilmi ,
H. Benseridi  and
A. Saadallah
M. Dilmi*
Affiliation:
Department of Mathematics, Faculty of Maths and Infs, M’sila University, 28000, Algeria
H. Benseridi*
Affiliation:
Applied Mathematics Laboratory, Department of Mathematics, Setif I-University, 19000, Algeria
A. Saadallah*
Affiliation:
Applied Mathematics Laboratory, Department of Mathematics, Setif I-University, 19000, Algeria
*
Corresponding author. Email: mouraddil@yahoo.fr
Email: m_benseridi@yahoo.fr
Email: saadmath2009@gmail.com
Get access

Abstract

In this paper we prove first the existence and uniqueness results for the weak solution, to the stationary equations for Bingham fluid in a three dimensional bounded domain with Fourier and Tresca boundary condition; then we study the asymptotic analysis when one dimension of the fluid domain tend to zero. The strong convergence of the velocity is proved, a specific Reynolds limit equation and the limit of Tresca free boundary conditions are obtained.

Keywords

35R3576F1078M35Free boundary problemsBingham fluidasymptotic approachTresca lawReynolds equation
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 6 , December 2014 , pp. 797 - 810
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Benseridi, H. and Dilmi, M., Some inequalities and asymptotic behaviour of dynamic problem of linear elasticity, Georgian Math.J., 20(1) (2013), pp. 25–41.Google Scholar
[2]Benterki, D., Benseridi, H. and Dilmi, M., Asymptotic study of a boundary value problem governed by the elasticity operator with nonlinear term, Adv. Appl. Math. Mech., 6 (2014), pp. 191–202.CrossRefGoogle Scholar
[3]Bingham, E. C., An investigation of the laws of plastic flow, U.S. Bureau of Standards Bulletin, 13 (1916), pp. 309–353.CrossRefGoogle Scholar
[4]Boukrouche, M. and Eukaszewicz, G., On a lubrication problem with Fourier and Tresca boundary conditions, Math. Mod Meth Appl. Sci., 14(6) (2004), pp. 913–941.Google Scholar
[5]Boukrouche, M. and Elmir, R., Asymptotic analysis of non-Newtonian fluid in a thin domain with Tresca law, Nonlinear Anal. Theory Methods Appl., 59 (2004), pp. 85–105.Google Scholar
[6]Bunoui, R. and Kesavan, S., Asymptotic behaviour of a Bingham fluid in thin layers, J. Math. Anal. Appl., 293(2) (2004), pp. 405–418.CrossRefGoogle Scholar
[7]Dai, P. L. and Dai, J. Z., The Galerkin finite element method for Bingham fluid, Numer. Math. J. Chinese Univ., 24(1) (2002), pp. 31–36.Google Scholar
[8]Dean, E. J, Glowinski, R. and Guidoboni, G., On the numerical simulation ofbingham viscoplastic flow: old and new results, J. Non-Newtonian Fluid Mech., 142 (2007), pp. 36–62.CrossRefGoogle Scholar
[9]De Los Reyes, J. C. and Gonz alez, S., Path following methods for steady laminar Bingham flow in cylindrical pipes, M2AN Math. Model. Numer. Anal., 43(1) (2009), pp. 81–117.CrossRefGoogle Scholar
[10]Duvant, G. and Lions, J. L., Les Inequations en Mecanique et en Physique, Dunod Paris, 1972.Google Scholar
[11]Fuchs, M., Grotowski, J. F. and Reuling, J., On variational model for quasi-static Bingham fluids, Math. Meth. Meth. Appl. Sci., 19 (1996), pp. 991–1015.Google Scholar
[12]Fuchs, M. and Seregin, G., Regularity results for the quasi-static Bingham variational inequality in dimensions two and three, Mathematische Zeitschrift, 227 (1998), pp. 525–541.Google Scholar
[13]Glowinski, R., Theorie Generale, PremieRes Applications, Paris, Dunod, 1976.Google Scholar
[14]Glowinski, R., Numerical methods for nonlinear variational problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984.Google Scholar
[15]Tabacman, E. D. and Tarzia, D. A., Sufficient and/or necessary condition for the heat transfer coefficient on Г and the heat flux on Г2 to obtain a steady-state two-phase Stefan problem, J. Diff. Eq., 77 (1989), pp. 16–37.CrossRefGoogle Scholar
[16]Temam, R. and Ekeland, I., Analyse Convexe et Problemes Variationnels, Dunod, Gauthier-Villars Paris, 1974.Google Scholar