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Natural Convection in a Nanofluid Saturated Rotating Porous Layer: A Nonlinear Study

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Abstract

The present paper deals with linear and nonlinear analysis of thermal instability in a rotating porous layer saturated by a nanofluid. Momentum equation with Brinkman term, involving the Coriolis term and incorporating the effect of Brownian motion along with thermophoresis has been considered. Linear stability analysis is done using normal mode technique, while for nonlinear analysis, a minimal representation of the truncated Fourier series, involving only two terms, has been used. Stationary and oscillatory modes of convection have been studied. A weak nonlinear analysis is used to obtain the concentration and thermal Nusselt numbers. The behavior of the concentration and thermal Nusselt numbers is investigated by solving the finite amplitude equations using a numerical method. Obtained results have been presented graphically and discussed in details.

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Abbreviations

D B :

Brownian diffusion coefficient

D T :

Thermophoretic diffusion coefficient

Da :

Darcy number

Pr :

Pradtl number

d :

Dimensional layer depth

k T :

Effective thermal conductivity of porous medium

k m :

Thermal diffusivity of porous medium

Le :

Lewis number

N A :

Modified diffusivity ratio

N B :

Modified particle-density increment

p :

Pressure

g :

Gravitational acceleration

Ra :

Thermal Rayleigh–Darcy number

Rm :

Basic density Rayleigh number

Rn :

Concentration Rayleigh number

t :

Time

T :

Temperature

T c :

Temperature at the upper wall

T h :

Temperature t the lower wall

v :

Nanofluid velocity

v D :

Darcy velocity \({\varepsilon {\bf v}}\)

(x*, y*, z*):

Cartesian coordinates

Ta :

Taylor number

α :

Horizontal wave number

β :

Proportionality factor

\({\varepsilon }\) :

Porosity

μ :

Viscosity of the fluid

\({\bar\mu}\) :

Effective viscosity of the porous medium

ρ f :

Fluid density

ρ p :

Nanoparticle mass density

(ρ c)f :

Heat capacity of the fluid

(ρ c)m :

Effective heat capacity of the porous medium

(ρ c) p :

Effective heat capacity of the nanoparticle material

γ :

Parameter defined as \({\frac{(\rho c)_{\rm m}}{(\rho c)_{\rm f}}}\)

\({\phi}\) :

Nanoparticle volume fraction

ν :

Kinematic viscosity μ/ρ f

ψ :

Stream function

α :

Wave number

ω :

Frequency of oscillation

b :

Basic solution

*:

Dimensional variable

′:

Perturbation variable

\({\nabla^2}\) :

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

\({\nabla_1^2}\) :

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

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Authors and Affiliations

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

    B. S. Bhadauria & Shilpi Agarwal

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  1. B. S. Bhadauria
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  2. Shilpi Agarwal
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Correspondence to B. S. Bhadauria.

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Bhadauria, B.S., Agarwal, S. Natural Convection in a Nanofluid Saturated Rotating Porous Layer: A Nonlinear Study. Transp Porous Med 87, 585–602 (2011). https://doi.org/10.1007/s11242-010-9702-9

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