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Numerical Solution of the Onset of Natural Convection in a Rotating Nanofluid Layer Induced by Purely Internal Heating

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Abstract

In this paper, the influence of rotation on the onset of natural convection made by purely internal heating in a horizontal layer of nanofluid is investigated. The boundaries are taken to be rigid-rigid and the normal flux of volumetric fraction of nanoparticles under the thermophoretic effects is supposed to be zero on the boundaries. The model used for nanofluid combines the effects of Brownian motion and thermophoresis. The purely internal heating problem determines that there is no applied temperature difference across the layer and so the external Rayleigh number is no longer applicable. Therefore, here the important parameter is an internal Rayleigh number, one based on the heat source strength. Linear stability theory based upon normal mode technique is used to find the critical internal Rayleigh number. The influence of rotation, the Lewis number, the modified diffusivity ratio and the nanoparticle Rayleigh number on the onset of convection are observed numerically using the higher order Galerkin Method. It is found that the effect of increasing the Lewis number, the modified diffusivity ratio and the nanoparticle Rayleigh number is to hasten the onset of convection, while rotation has a stabilizing effect on the stability of the system.

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Abbreviations

a :

Dimensionless wave number

\(a_c \) :

Critical wave number

c :

Specific heat

\(D_B \) :

Brownian diffusion coefficient

\(D_T \) :

Thermophoretic diffusion coefficient

\({\hat{\mathbf{e}}}_\mathbf{z}\) :

Unit vector in z-direction

\({\vec {\mathbf{g}}}\) :

Acceleration due to gravity

\({\vec {\mathbf{j}}}_p \) :

Diffusion mass flux for the nanoparticle

k :

Thermal conductivity

L :

Thickness of nanofluid layer

\(L_e \) :

Lewis number

\(N_A \) :

Modified diffusivity ratio

\(\hbox {N}_{\mathrm{B}} \) :

Modified specific heat increment

p :

Pressure

\(P_r \) :

Prandtl number

\(Q_0^{*}\) :

Strength of internal heat source

\(R_I \) :

Internal Rayleigh number

\(R_{I,c} \) :

Critical internal Rayleigh number

\(T_a \) :

Rotation parameter

t :

Time

T :

Temperature

\({\vec {\mathbf{v}}}\) :

Velocity of nanofluid

\(\left( {x,y,z} \right) \) :

Space co-ordinates

\(\alpha \) :

Thermal diffusivity

\(\beta \) :

Coefficient of thermal expansion

\({\vec {\varvec{\Omega }^{*}}}\) :

Angular velocity

\(\mu \) :

Viscosity

\(\rho \) :

Density of the nanofluid

\(\rho _0 \) :

Reference density of nanofluid

\(\rho _p \) :

Density of nanoparticles

\(\left( {\rho c} \right) \) :

Heat capacity

\(\phi \) :

Volume fraction of the nanoparticles

\(\phi _0 ^{*}\) :

Reference scale for the nanoparticle fraction

\(\nabla _{\mathrm{P}}^{2} \) :

Horizontal Laplacian operator

\(\nabla ^{2}\) :

Laplacian operator

*:

Dimensional variables

’:

Perturbed quantities

0:

Reference scale

f:

Base fluid

p:

Particle

b:

Basic state

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  1. Athabasca River Basin Research Institute, Athabasca University, Edmonton, Canada

    Dhananjay Yadav

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Yadav, D. Numerical Solution of the Onset of Natural Convection in a Rotating Nanofluid Layer Induced by Purely Internal Heating. Int. J. Appl. Comput. Math 3, 3663–3681 (2017). https://doi.org/10.1007/s40819-017-0319-3

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