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): Sweta Sharma, Sunil, Poonam Sharma

Email(s): Email ID Not Available

Address: Sweta Sharma1, Sunil1, Poonam Sharma2 1Department of Mathematics and Scientific Computing, National Institute of Technology Hamirpur, Hamirpur, (H.P.) – 177005, India. 2Department of Mathematics, NSCBM Govt. College Hamirpur, Hamirpur, (H.P.) –177005, 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:




Global stability for thermal convection in a Navier-Stokes-Voigt fluid

 

Sweta Sharma1, Sunil1, Poonam Sharma2

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

Hamirpur, (H.P.) – 177005, India.

2Department of Mathematics, NSCBM Govt. College Hamirpur, Hamirpur, (H.P.) –177005, India.

*Corresponding Author E-mail:

 

ABSTRACT:

In the present work, we show that the global stability threshold for thermal convection in a Navier-Stokes-Voigt fluid layer with free-free, free-rigid and rigid-rigid boundaries is exactly the same as the linear instability boundary. This optimal result is important because linearized instability theory has captured completely the physics of the onset of convection. The eigen value problem is formed by using normal mode analysis for linear analysis and using energy method for nonlinear analysis. The critical Rayleigh number is found to be exactly same using both linear and nonlinear analysis for different boundary cases. It ensures the absence of any sub-critical instability. An important role of a Kelvin-Voigt parameter has been seen in an energy decay for nonlinear stability whereas it doesn’t affect the value of Rayleigh number.

 

KEY WORDS: thermal convection, Navier-Stokes-Voigt fluid, stability analysis, energy method,  normal mode analysis method.

 

1.     INTRODUCTION:

The work on Navier-Stokes-Voigt (NSV) fluid emerges from the Russian literature which is highly focused on a class of materials having viscoelastic property. Oskolkov [1] and Oskolkov [2] presented the model for Kelvin-Voigt fluid. This model was used for analysing the motion of fluids possessing very weak viscoelastic properties. The Navier-Stokes-Voigt fluid is a subclass of thermal Kelvin-Voigt fluids. However, it is certain that the series of Oskolkov’s works played a major role in the study of the NSV equations and stimulated further research in this direction. Sviridyuk [3] established the solvability of the weakly compressible NSV equations. The local-in-time unique solvability of the system of Navier-Stokes-Voigt equations is proved by Sviridyuk  [4]. Various slip problems are studied in the papers of Ladyzhenskaya [5]. Kaya and Celebi [6] gave the solid result for g-Kelvin-Voight equations by proving the existence and uniqueness of its weak solutions. Korpusov and Sveshnikov [7]  investigated the blowup of solutions to the NSV equations with a cubic source. The thermal effects for Kelvin-Voigt models were studied by Sukacheva and Matveeva  [8]. Matveeva and Sukacheva  [9] has been worked on a class of Sobolev type equations. The Navier-Stokes-Voigt model has the same solutions in the steady-state as the Navier-Stokes equations analysed by Layton and Rebholz [10]. The response of stress to changes in the velocity gradient is instantaneous for a Navier-Stokes fluid. This particular behaviour is not recognized by all real fluids, namely various types of viscoelastic and complex fluids. These fluids have a stress which does not react instantly, instead they recall the velocity gradient history. The Navier-Stokes equations express the motion of viscous fluid substance which may help with the draft of aircraft, cars, power stations, model of ocean currents, the analysis of pollution and is also used to model the weather, airflow around the wing, flow of water in a pipe.

 

The use of NSV model for image inpainting has been discussed by Ebrahimi [11]. In the latest work of Straughan [12] on the thermosolutal convection with a NSV fluid, at higher thermal Rayleigh and salt Rayleigh numbers, the transition to oscillatory convection occurs as the Kelvin-Voigt parameter increases which is very useful for solar pond. The aforementioned paper discusses the linear stability and nonlinear stability for free-free boundary surfaces. Straughan [13] gave the results on the stability for convection in a Kelvin-Voigt type of variable order at free-free boundary and also gave the result for Navier-Stokes-Voigt fluid at free-free boundary surfaces.

 

This paper presents the work on the linear and nonlinear analysis for the thermal convection of viscoelastic Navier-Stokes-Voigt fluid employing the model of Straughan [12] by using different boundary surfaces such as free-free, rigid-free, rigid-rigid. Recent study has been focused on results of linear instability and energy stability for all bounding surfaces. The outcomes of the study give the same eigen value problem for both linear and nonlinear analysis. This gives the global stability for thermal convection in a Navier-Stokes-Voigt fluid. Thus, no sub-critical instabilities occur for Navier-Stokes-Voigt fluid. Also, the critical Rayleigh number is calculated for different boundary surfaces such as free-free, rigid-free, rigid-rigid. The Kelvin-Voigt parameter shows an important role in energy decay. As far as we know, no attempts have been made to perform the aforementioned analysis by considering all bounding surfaces.

 

2.     Mathematical formulation of the problem:

Here, we consider an infinite, horizontal Navier-Stokes-Voigt fluid layer of thickness , heated from below. Within cartesian coordinates, the z-axis is aligned vertically upwards with the gravitational force . This fluid is assumed to occupy the layer . The assumption for temperature  is  at  and



REFERENCES:

1.     A. P.  Oskolkov, Initial-boundary value problems for the equations of motion of Kelvin–Voigt fluids and Oldroyd fluids, Trudy Math. Inst. Steklov, 179, 126-164, 1989.

2.     A. P. Oskolkov, Nonlocal problems for the equations of motion of Kelvin-Voigt fluids, Journal of Mathematical Sciences, 75, 2058-2078, 1995.

3.     G. A. Sviridyuk, On a model of the dynamics of a weakly compressible viscoelastic fluid, Russian Mathematics, 38, 59-68, 1994.

4.     G. A. Sviridyuk and T. G. Sukacheva, On the solvability of a nonstationary problem describing the dynamics of an incompressible viscoelastic fluid, Mathematical Notes, 63, 388-395, 1998.

5.     O. A. Ladyzhenskaya, On the global unique solvability of some two-dimensional problems for the water solutions of polymers, Journal of Mathematical Sciences, 99, 888-897, 2000.

6.     M. Kaya and A. O. Celebi, Existence of weak solutions of the g-Kelvin-Voight equation, Mathematical and Computer Modelling, 49, 497-504, 2009.

7.     M. O. Korpusov and  A. G. Sveshnikov, Blow-up of Oskolkov’s system of equations,  Sbornik: Mathematics, 200, 549-572, 2009.

8.     O. P. Matveeva and T. G. Sukacheva, On a homogeneous model of the non-compressible viscoelastic Kelvin-Voigt fluid of the non-zero order, Journal of Samara State Technical University, 5, 33-41, 2010.

9.     T. G. Sukacheva  and A. O. Kondyukov, On a class of Sobolev type equations, Bulletin of the South Ural State University. Series Mathematical Modelling, Programming and Computer Software, 7, 5-21, 2014.

10. W. J. Layton and L. G. Rebholz, On relaxation times in the Navier-Stokes-Voigt model, International Journal of Computational Fluid Dynamics, 27, 184-187, 2013.

11. M. A. Ebrahimi and M. Holst, The Navier–Stokes–Voight model for image inpainting, IMA Journal of Applied Mathematics, 78, 869-894, 2013.

12. B. Straughan, Thermosolutal Convection with a Navier-Stokes-Voigt fluid,  Applied Mathematics & Optimization, 84, 2587-2599, 2021.

13. B. Straughan, Instability thresholds for thermal convection in a Kelvin–Voigt fluid of variable order, Rendiconti del Circolo Matematico di Palermo Series 2, 71, 187-206, 2022.

14. Sunil and A. Mahajan, A nonlinear stability analysis for magnetized ferrofluid heated from below, Proceedings of the Royal Society A, 464, 83-98, 2008.

15. Sunil and R. Devi, Global stability for thermal convection in a couple stress fluid saturating a porous medium with temperature pressure dependent viscosity: Galerkin method, International Journal of Engineering, 25, 221-229, 2012.

 



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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