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): Jitender Kumar, Chitresh Kumari, Jyoti Prakash

Email(s): Email ID Not Available

Address: Jitender Kumar*, Chitresh Kumari, Jyoti Prakash Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, 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:




Upper Limits to the Complex Growth Rate in Multicomponent Convection with Variable Viscosity

 

Jitender Kumar*, Chitresh Kumari, Jyoti Prakash

Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, India.

*Corresponding Author E-mail: tjitender002@gmail.com

 

Abstract:

It is proved analytically that the complex growth rate  (  and  are respectively the real and imaginary parts of ) of an arbitrary oscillatory motion of growing amplitude in multicomponent convection with variable viscosity lies inside a semi-circle in right half of the -plane whose center is origin and radius , where  are the concentration Rayleigh numbers and  is the Prandtl number. Further this result is uniformly applicable for quite general nature of the boundaries.

 

Keywords: Multicomponent convection, variable viscosity, Rayleigh number, concentration Rayleigh number, Prandtl number.

 

Introduction:

Double diffusive convection is an important area of research due to its outstanding application in various fields like engineering, astrophysics, geophysics, oceanography etc. This phenomenon has been extensively studied by Turner [1], Brandt and Fernando [3], Radko [4], Sekar et al [5], Kellner and Tilgner [6], Nield and Kuznetsov [7], Schmitt [8].

 

We can have many hydrodynamical system in nature wherein density of fluid depends on more than two stratifying agencies having different diffusivity like the Earth’s core, ocean water, geothermally heated lakes, magmas (and their laboratory models). Earlier theoretical and experimental studies of such systems where the density of fluid depends on more than two stratifying agencies include the work of Griffiths [9-10], Pearlstein et al. [11], Lopez et al. [13], Turner [17]. In the later studies, Terrones [14] investigated the cross-diffusion effects on the onset of convective instability. Poulikako [15] studied the triply diffusive convection on the onset of convection in a horizontal porous layer. Terrones [16] analysed the multicomponent convection on the onset of convection. Rionero [18] investigated the multicomponent diffusive convection in porous layer. Ryzhkov [19-20] studied long wave instability of a multicomponent fluid layer with the soret effect.

 

Prakash et al [22] derived upper bounds for the complex growth rate of an unstable oscillatory perturbation of growing amplitude, in double diffusive convection problem with viscosity variation effects included. Prakash et al [23] further extended their results to more general triply diffusive convection with viscosity variation under the action of uniform vertical magnetic field. As a further step, in the present paper, we have further extended these results to more general problem with n diffusing components known as multicomponent convection with the viscosity variation effects included.

Mathematical Formulation:

A viscous finitely heat conducting Boussinesq fluid of infinite horizontal extension is statistically confined between two horizontal boundaries  and  which are respectively maintained at uniform temperatures  and  and uniform concentrations and  in the force field of gravity(as shown in Fig.1). It is assumed that the cross-diffusion effects of the stratifying agencies can be neglected.




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22.      Prakash, J., Kumar, R., Upper Limits for the Complex Growth Rate in Double Diffusive Convection with Viscosity Variations, Recent Trends in Algebra and Mechanics, Indo American Books, (2014), 257-264 (Delhi).

23.      Prakash, J., Kumar, R., Kumar, V., Upper Bounds for the Complex Growth Rate in Magnetohydrodynamic Triply Diffusive Convection with Viscosity Variations, H. P. U. Journal, 3(2), (2015), 83-90.

24.      Prakash, J., Kumar, R., Kumari, K., Linear Triply Diffusive Convection with Viscosity Variations, International Journal of Physical and Mathematical Sciences, 5(1), (2015), 186-196.

25.      Prakash, J., Manan, S., Singh, V., Linear Stability Analysis in Multicomponent Convection, International Journal of technology, 6(2), (2016), 113-117.

 

 

 

 

 



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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