Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T07:16:26.488Z Has data issue: false hasContentIssue false
  • English
  • Français

A Study of Multiple Solutions for the Navier-Stokes Equations by a Finite Element Method

Published online by Cambridge University Press:  28 May 2015

Huanxia Xu ,
Ping Lin  and
Xinhui Si
Huanxia Xu*
Affiliation:
Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, China
Ping Lin*
Affiliation:
Department of Mathematics, University of Dundee, Dundee DD1 4HN, UK
Xinhui Si*
Affiliation:
Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

In this paper, a finite element method is proposed to investigate multiple solutions of the Navier-Stokes equations for an unsteady, laminar, incompressible flow in a porous expanding channel. Dual or triple solutions for the fixed values of the wall suction Reynolds number R and the expansion ratio α are obtained numerically. The computed multiple solutions for the symmetric flow are validated by comparing them with approximate analytic solutions obtained by the similarity transformation and homotopy analysis method. Unlike previous works, our method deals with the Navier-Stokes equations directly and thus has no similarity and other restrictions as in previous works. Finally we use the method to study multiple solutions for three cases of the asymmetric flow (which has not been studied before using the similarity-type techniques).

Keywords

76M1076D0574H15Finite element methodNavier-Stokes equationsporous channelexpanding walls
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 1 , February 2014 , pp. 107 - 122
Copyright
Copyright © Global Science Press Limited 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] Berman, A. S., Laminar flow in channels with porous walls, J. Appl. Phys., 24 (1953), pp. 1232–1235.Google Scholar
[2] Yuan, S. W., Further investigations of laminar flow in channels with porous walls, J. Appl. Phys., 27 (1956), pp. 267–269.Google Scholar
[3] Terrill, R. M. and Shrestha, G. M., Laminar flow through parallel and uniformly porous walls of different permeability, ZAMP, 16 (1965), pp. 471–482.Google Scholar
[4] Suryaprakasarao, U., Flow in channels with porous walls in the presence of a transverse magnetic field, Appl. Sci. Res., 9 (1961), pp. 374–382.Google Scholar
[5] Terrill, R. M., Laminar flow in a uniformly porous channel, The Aeronautical Quarterly, XV (1964), pp. 299–310.Google Scholar
[6] Terrill, R. M., Laminar flow in a uniformly porous channel with large injection, The Aeronautical Quarterly, 16 (1965), pp. 323–332.Google Scholar
[7] Robinson, W. A., The existence of multiple solutions for the laminar flow in a uniformly porous channel with suction at both walls, J. Eng. Math., 10 (1976), pp. 23–40.Google Scholar
[8] Lu, C., Macjillvary, A.D. and Hastings, S. P., Asymptotic behaviour of solutions of a similarity equation for laminar flows in channels with porous walls, IMA. J. Appl. Math., 49 (1992), pp. 139–162.Google Scholar
[9] Zaturska, M. B., Drazin, P.G. and Banks, W. H. H., On the flow of a viscous fluid driven along a channel by suction at porous walls, Fluid Dyn. Res., 4 (1988), pp. 151–178.Google Scholar
[10] Majdalani, J. and Zhou, C., Moderate-to-large injection and suction driven channel flows with expanding or contracting walls, ZAMM. Z. Angew. Math. Mech., 83 (2003), pp. 181–196.Google Scholar
[11] Majdalani, J. and Zhou, C., Large injection and suction driven channel flows with expanding and contracting walls, 31st AIAA Fluid Dynamics Conference 11-14 June 2001 Anaheim, CA.Google Scholar
[12] Majdalani, J. and Zhou, C., Inner and outer solutions for the injection driven channel flows with retractable walls, 33st AIAA Fluid Dynamics Conference 23-26 June 2003 Orlando, FL.Google Scholar
[13] Majdalani, J.,Zhou, C. and Dawson, C. A., Two dimensional viscous flow between slowly expanding or contracting walls with weak permeability, J. Biomech., 35 (2002), pp. 1399–1403.CrossRefGoogle ScholarPubMed
[14] Asghar, S.,Mushtap, M. and Hayat, T., Flow in a slowing deforming channel with weak permeability: an analytical approach, Nonlinear Anal. Real., 11 (2010), pp. 555–561.CrossRefGoogle Scholar
[15] Saeed, D., Mohammad, M. R. and Ahmad, D., Analytical approximate solutions for two dimensional viscous flow through expanding or contracting gaps with permeable walls, Centr. Euro. J. Phys., 7 (2009), pp. 791–799.Google Scholar
[16] Dauenhauer, E. C. and Majdalani, J., Exact self-similarity solution of the Navier-Stokes equations for a porous channel with orthogonally moving walls, Phys. Fluids, 15 (2003), pp. 1485–1495.Google Scholar
[17] Xu, H., Lin, Z. L., Liao, S. J., Wu, J. Z. and Majdalani, J., Homotopy based solutions of the Navier-Stokes equations for a porous channel with orthogonally moving walls, Phys. Fluids, 22 (2010), 053601.Google Scholar
[18] Durlofsky, L. and Brady, J. F., The spatial stability of a class of similarity solutions, Phys. Fluids, 27 (1984), pp. 1068–1076.Google Scholar
[19] Glowinski, R., Finite Element Methods for Incompressible Viscous Flow, In Handbook of Numerical Analysis, IX, Ciarlet, P.G. and Lions, J.-L. eds., North-Holland, Amsterdam, 2003, pp. 3–1176.Google Scholar
[20] Teman, R., Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, 1997.Google Scholar
[21] Quartapelle, L., Numerical Solution of the Incompressible Navier-Stokes Equations, International Series of Numerical Mathematics, 113, Brikhauser, 1993.Google Scholar
[22] Si, X. H., Zheng, L. C., Zhang, X. X. and C., Y. Homotopy analysis solutions for the asymmetric laminar flow in a porous channel with expanding or contracting walls, Acta Mech. Sinica, 27 (2011), pp. 208–214.CrossRefGoogle Scholar
[23] Gresho, P. M., Incompressible fluid dynamics: some fundamental formulation issues, Annu. Rev. Fluid Mech., 23 (1991), pp. 413–453.Google Scholar
[24] Lin, P., A sequential regularization method for time-dependent incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 34(3) (1997), pp. 1051–1071.Google Scholar
[25] Rempfer, D., On boundary conditions for incompressible Navier-Stokes problems, Appl. Mech. Rev., 59 (2006), pp. 107–125.Google Scholar
[26] Lin, P., Chen, X. Q. and Ong, M. T., Finite element methods based on a new formulation for the non-stationary incompressible Navier-Stokes equations, Int. J. Numer. Mech. Fluids, 46 (2004), pp. 1169–1180.Google Scholar
[27] Lin, P. and Liu, C., Simulations of singularity dynamics in liquid crystal flows: a finite element approach, J. Comput. Phys., 215 (2006), pp. 348–362.CrossRefGoogle Scholar
[28] Lu, X. L., Lin, P. and Liu, J. G., Analysis of a sequential regularization method for the unsteady Navier-Stokes equations, Math. Comput., 77(263) (2008), pp. 1467–1494.CrossRefGoogle Scholar
[29] Shi, H. D., Lin, P., Li, B. T. and Zheng, L. C., A finite element method for heat transfer of power-law flow in channels with a transverse magnetic field, 2011, submitted.Google Scholar