Conference Proceeding

Mathematics in Space and Applied Sciences (ICMSAS-2023)
ICMSAS-2023

Subject Area: Mathematics
Pages: 331
Published On: 03-Mar-2023
Online Since: 04-Mar-2023

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Author(s): Akanksha Thakur, Sunil, Reeta Devi, Shalu Choudhary

Email(s): Email ID Not Available

Address: Akanksha Thakur1, Sunil1, Reeta Devi2, Shalu Choudhary3 1Department of Mathematics and Scientific Computing, National Institute of Technology Hamirpur, (H.P.), 177005, India 2Department of Mathematics, Government Post Degree College Nagrota Bagwan, Distt – Kangra, (H.P.), 176047, India 3Department of Mathematics, Uttaranchal University, Dehradun, Uttarakhand, 248007, India *Corresponding Author

Published In:   Conference Proceeding, Mathematics in Space and Applied Sciences (ICMSAS-2023)

Year of Publication:  March, 2023

Online since:  March 04, 2023

DOI:




The Effect of Couple Stresses on Global Stability for Thermosolutal Convection in A Fluid Saturating A Porous Medium

 

Akanksha Thakur1, Sunil1, Reeta Devi2, Shalu Choudhary3

1Department of Mathematics and Scientific Computing, National Institute of Technology

Hamirpur, (H.P.), 177005, India

2Department of Mathematics, Government Post Degree College Nagrota Bagwan,

Distt – Kangra, (H.P.), 176047, India

3Department of Mathematics, Uttaranchal University, Dehradun, Uttarakhand, 248007, India

*Corresponding Author E-mail:

 

ABSTRACT:

In this paper nonlinear analysis is performed for studying the effect of couple stress forces on convective stability in a steady, viscous, incompressible fluid saturating a porous medium. The nonlinear analysis is carried out by using energy method and Galerkin technique is used to solve the obtained eigen value problem. The linear and nonlinear thresholds that encapsulate the physics of the onset of convection are found to be same. It is observed that the permeability of porous medium and magnetization parameter tends to destabilize the system whereas the couple stress and solute gradient is seen to have a stabilizing impact on the system.

 

KEYWORDS: Double diffusive convection, Couple-stress forces, Porous medium, Rigid-Rigid leading boundaries, Galerkin method. 

 

1. INTRODUCTION:

Theory of buoyancy-driven thermal convection in a layer of Newtonian viscous fluid saturating porous medium has been subject of intensive study for the last few decades due to its relevance in various applications like geothermal systems, contaminant transport in ground water, drying processes, biomedical engineering and many others. Hydrodynamic stability theory is predominantly concerned with finding the critical Rayleigh number values for depicting stability region. Linear theory provides conditions of definite instability of hydrodynamic systems and is not certain about stability. However, energy method of nonlinear theory assures stability of hydrodynamic systems under certain conditions but cannot with certainty conclude instability. Hence to investigate the finite disturbances effects in flow, the nonlinear approach becomes inevitable.

 

Energy method is initially credited to Reynolds [1], Orr [2] but was later on refined by Serrin [3] and Joseph [4-6]. This classical energy method being successful in many problems (Galdi [7] and Galdi and Straughan [8]) has been confronted in many situations. So, in recent years the classical energy theory has been improved and many authors (Straughan and Walker [9], Kaloni and Qiao [10-12], Guo and Kaloni [13], Straughan [14], Payne and Straughan [15]) analyzed stability of fluids by generalized energy method. Ferroconvection problems have been discussed using energy method by Sunil and Mahajan [16-18].

 

 

Industrial and technological applications of fluids with couple stress forces (for example pumping fluids like synovial joints fluid, synthetic fluids, liquid crystal, animal blood) and the lubrication theory have attracted researchers to study these fluids. Fluids with couple stress forces proposed by Stokes [19] have discrete characteristics like non-symmetric stress tensor, couple stresses and body couples. Stokes [20] monograph “Theories of Fluids with Microstructure-An Introduction” gives an excellent description to this theory.

Sharma and Thakur [21] and Sharma et al. [22] have analyzed the double-diffusive convection problems in couple stress fluid saturating porous medium and rotation separately. The driving force for studying double-diffusive convection problems is their importance in many physical applications.

 

Recently, Sunil et al. [23, 24] has examined the stability problem of fluid with couple stress forces using energy method. More recently Choudhary and Sunil [25] have studied nonlinear stability analysis of double-diffusive convection in layer of fluid having couple stress forces, with free boundaries, saturating a porous medium. They observed the co-incidence of the linear instability and nonlinear stability critical thermal Rayleigh numbers indicating that the subcritical instabilities are not possible. Qin and Kaloni [26] mentioned that when boundary layer effects are considered and for flows through porous materials of high porosity, Brinkman model gives better results as compared to Darcy model.

 

The goal of this article is to perform nonlinear analysis for thermosolutal convective stability in fluid layer, having couple stress forces acting on it, confined between  boundaries, saturating a porous medium of high permeability and analyze the effects of couple stress, solute concentration and permeability on convection. Here, Brinkman model is used keeping in view the work of Qin and Kaloni [27]. This paper identifies research opportunities and challenges for future research and has far not appeared in literature to the best of my knowledge.

 

2. MATHEMATICAL MODEL:

Consider an infinite horizontal layer of thickness ‘ ’ of incompressible fluid with couple stress forces, having constant viscosity, heated and soluted from below saturating a porous medium of homogeneous nature possessing porosity . The medium permeability is  and gravity field  acts in the negative z-direction.




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[9]           Straughan, B. and Walker, D.W. (1996). Two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems. Journal of computational physics, 127, 128.

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[12]        Kaloni, P. N. and Qiao, Z. (2001). Non-linear convection in a porous medium with inclined temperature Gradient and variable gravity effects. International journal of Heat Mass Transfer, 44, 1585.

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[14]        Straughan, B. (2001).  A sharp nonlinear stability threshold in rotating porous convection. Proceedings of the Royal Society of London, A, 457, 87.

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[26]        Qin, Y. and Kaloni, P. N. (1995). Nonlinear stability problem of a rotating porous layer, Quarterly of Applied Mathematics. 53, 129.



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Author/Editor Information

Dr. Sanjay Kango

Department of Mathematics, Neta Ji Subhash Chander Bose Memorial, Government Post Graduate College, Hamirpur Himachal Pradesh-177 005, INDIA