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An analytical study of nonlinear double-diffusive convection in a porous medium under temperature/gravity modulation

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Abstract

The article deals with nonlinear thermal instability problem of double-diffusive convection in a porous medium subjected to temperature/gravity modulation. Three types of imposed time-periodic boundary temperature (ITBT) are considered. The effect of imposed time-periodic gravity modulation (ITGM) is also studied in this problem. In the case of ITBT, the temperature gradient between the walls of the fluid layer consists of a steady part and a time-dependent periodic part. The temperature of both walls is modulated in this case. In the problem involving ITGM, the gravity field has two parts: a constant part and an externally imposed time-periodic part. Using power series expansion in terms of the amplitude of modulation, which is assumed to be small, the problem has been studied using the Ginzburg–Landau amplitude equation. The individual effects of temperature and gravity modulation on heat and mass transports have been investigated in terms of Nusselt number and Sherwood number, respectively. Further the effects of various parameters on heat and mass transports have been analyzed and depicted graphically.

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Abbreviations

A :

Amplitude of streamline perturbation

d :

Height of the fluid layer

Da :

Darcy number Da \({=\frac{K}{d^{2}} }\)

g :

Acceleration due to gravity

g m (τ):

Modulation in gravity \({g_{m}({\tau })=\epsilon^{2}\delta_{2} Cos(\omega \tau)}\)

k c :

Critical wavenumber

Le :

Lewis number, \({Le=\frac{\kappa _{T}}{\kappa _{S}}}\)

Nu :

Nusselt number

p :

Reduced pressure

Pr :

Prandtl number, \({Pr=\frac{\nu }{\kappa _{T}}}\)

Ra S :

Solutal Rayleigh number, \({Ra_{S}=\frac{\beta_{S} g\Delta Sd^{3}}{\nu \kappa _{T}}}\)

Ra T :

Thermal Rayleigh number, \({Ra_{T}=\frac{\beta_{T} g\Delta Td^{3}}{\nu \kappa _{T}}}\)

R 0c :

Critical Rayleigh number

S :

Solute concentration

ΔS :

Solute difference across the fluid layer

Sh :

Sherwood number

t :

Time

T :

Temperature

ΔT :

Temperature difference across the fluid layer

x, y, z :

Space Co-ordinates

β T :

Coefficient of thermal expansion

β S :

Coefficient of solute expansion

δ 2 :

Horizontal wave number k 2 c + π 2

δ 1 :

Amplitude of temperature modulation

δ 2 :

Amplitude of gravity modulation

\({\epsilon}\) :

Perturbation parameter

\({\gamma }\) :

Heat capacity ratio \({\frac{(\rho c_{p})_{m}}{(\rho c_{p})_{f}}}\)

\({\kappa_{T}}\) :

Effective thermal diffusivity in horizontal direction

\({\kappa_{S}}\) :

Effective thermal diffusivity in vertical direction

μ :

Effective dynamic viscosity of the fluid

ν :

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

\({\phi}\) :

Porosity

\({\Phi^{*}}\) :

Dimensionless amplitude of solutal perturbation

\({\phi}\) :

Solutal perturbation

\({\psi}\) :

Stream function

\({\psi}\) :

Dimensionless amplitude of stream function

ρ :

Fluid density

τ :

Slow time τ = ε 2 t

\({\Theta}\) :

Temperature perturbation

\({\Theta^{*}}\) :

Dimensionless amplitude of temperature perturbation

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

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

b :

Basic state

c :

Critical

0:

Reference value

′ :

Perturbed quantity

*:

Dimensionless quantity

st :

Stationary

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Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Bangalore, 560001, India

    P. G. Siddheshwar

  2. Department of Applied Mathematics and Statistics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow, 226025, India

    B. S. Bhadauria

  3. DST-Centre for Interdisciplinary Mathematical Sciences, Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi, 221005, India

    B. S. Bhadauria & Alok Srivastava

Authors
  1. P. G. Siddheshwar
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  2. B. S. Bhadauria
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  3. Alok Srivastava
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Correspondence to B. S. Bhadauria.

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Siddheshwar, P.G., Bhadauria, B.S. & Srivastava, A. An analytical study of nonlinear double-diffusive convection in a porous medium under temperature/gravity modulation. Transp Porous Med 91, 585–604 (2012). https://doi.org/10.1007/s11242-011-9861-3

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