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Weak Nonlinear Oscillatory Convection in a Viscoelastic Fluid-Saturated Porous Medium Under Gravity Modulation

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Abstract

A study of thermal instability driven by buoyancy force is carried out in an initially quiescent infinitely extended horizontal porous medium saturated with viscoelastic fluid. Modified Darcy’s law is used to explain characteristics of fluid motion. The time periodic gravity field has been considered, and its effect on the system has been investigated. A weak nonlinear stability analysis has been performed for the oscillatory mode of convection, and heat transport in terms of the Nusselt number, which is governed by the complex non-autonomous Ginzburg–Landau equation, is calculated. The influence of relaxation and retardation times of viscoelastic fluid on heat transfer has been discussed. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system. Finally, it is found that overstability advances the onset of convection, and hence increases heat transfer.

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Abbreviations

\(\mathbb {A}\) :

Amplitude of convection

\(a\) :

Wavenumber

\(\delta \) :

Amplitude of gravity modulation

\(d\) :

Depth of the fluid layer

\(\vec {g}\) :

Acceleration due to gravity

\(a_c\) :

Critical wavenumber

\(Nu\) :

Nusselt number

\(p\) :

Reduced pressure

\(Ra_{D}\) :

Thermal Rayleigh-Darcy number, \(Ra_{D}=\frac{\beta _{T} g \Delta T d K}{\nu \kappa _{T}}\)

\(R_0\) :

Critical Rayleigh-Darcy number

\(T\) :

Temperature

\(\Delta T\) :

Temperature difference across the porous layer

\(t\) :

Time

\((x,z)\) :

Horizontal and vertical co-ordinates

\(\alpha _T\) :

Coefficient of thermal expansion

\(\chi \) :

Perturbation parameter

\(\kappa _{T}\) :

Effective thermal diffusivity

\(K\) :

Permeability

\(\gamma \) :

Heat capacity ratio, \(\gamma =\frac{(\rho c)_m}{(\rho c)_f}\)

\(\Omega \) :

Frequency of modulation

\(\omega \) :

Dimensionless oscillatory frequency

\(\overline{\lambda }\) :

Stress relaxation time

\(\overline{\varepsilon }\) :

Strain retardation time

\(\mu \) :

Dynamic viscosity of the fluid

\(\phi \) :

Porosity

\(\nu \) :

Kinematic viscosity, \(\left( {\frac{\mu }{\rho _{0}}} \right) \)

\(\rho \) :

Fluid density

\(\psi \) :

Stream function

\(s\) :

Slow time \(s=\chi ^2 t\)

\(\hat{k}\) :

Vertical unit vector

\(\nabla _{1}^2\) :

\(\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}}\)

\(\nabla ^{2}\) :

\(\nabla _{1}^2+\frac{\partial ^{2}}{\partial z^{2}}\)

\(b\) :

Basic state

\(c\) :

Critical

\(0\) :

Reference value

\('\) :

Perturbed quantity

\(*\) :

Dimensionless quantity

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Acknowledgments

This work was done during the lien sanctioned to the first author by Banaras Hindu University, Varanasi, India to work as professor of Mathematics at Department of Applied Mathematics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar Central University, Lucknow, India. The author B.S. Bhadauria gratefully acknowledges Banaras Hindu University, Varanasi for the same. Further, the author Palle Kiran gratefully acknowledges the financial assistance from Babasaheb Bhimrao Ambedkar University, Lucknow, India as a research fellowship.

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Authors and Affiliations

  1. Department of Applied Mathematics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow , 226 025, India

    B. S. Bhadauria & Palle Kiran

  2. Department of Mathematics, Faculty of Sciences, Banaras Hindu University, Varanasi , 221005, India

    B. S. Bhadauria

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  1. B. S. Bhadauria
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Correspondence to B. S. Bhadauria.

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Bhadauria, B.S., Kiran, P. Weak Nonlinear Oscillatory Convection in a Viscoelastic Fluid-Saturated Porous Medium Under Gravity Modulation. Transp Porous Med 104, 451–467 (2014). https://doi.org/10.1007/s11242-014-0343-2

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