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The onset of MHD nanofluid convection between a porous layer in the presence of purely internal heat source and chemical reaction

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Abstract

In many processes such as microwave heating, chemical reaction, pasteurization, sterilization, heat transfer convection is occurred by means of purely internal heating sources, instead of heating solid surfaces. This paper deals with the analysis of the onset of MHD nanofluid convection inside a porous layer in the presence of purely internal heat source and chemical reaction. The nanofluid is enclosed between two solid surfaces and also incorporates the effect of Brownian motion along with thermophoresis. The simulation is performed for three cases of thermal boundary conditions, namely (I) isothermal condition for both surfaces, (II) isothermal condition for upper surface and insulated condition for lower surface, and (III) isothermal condition for lower surface and insulated condition for upper surface. Also, for all case studies, the zero nanoparticle flux condition under the thermophoretic effects is considered at the boundaries. In this study, the effect of chemical reaction and porosity parameters on the critical heat source Rayleigh number and critical wave number is investigated. It is found that the critical heat source Rayleigh number increases with an increase in the magnetic Chandrasekhar number, chemical reaction parameter, and porosity parameter.

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Abbreviations

a i, b i , c i :

Coefficients

c :

Specific heat capacity \(( {\text{J}}\,{\text{kg}}^{ - 1} \,{\text{K}}^{ - 1} )\)

\(c_{1} ,c_{2} ,c_{3} ,c_{4}\) :

Constant parameters

\({\text{D}}_{\text{B}}\) :

Brownian diffusion coefficient

\({\text{D}}_{\text{T}}\) :

Thermophoretic diffusion coefficient

\(e_{\text{x}} ,e_{\text{y}} ,e_{\text{z}}\) :

Unit vector in \(x,y,z\) direction

g :

Acceleration due to gravity \(( {\text{m}}\,{\text{s}}^{ - 2} )\)

\(H,H_{0}\) :

Magnetic field \(( {\text{Wb }}\,{\text{m}}^{ - 2} )\)

k :

Thermal conductivity (\({\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1}\))

\(k_{\text{x}} ,k_{\text{y}}\) :

Wave number along the \(x\) and \(y\) directions

\(K_{\text{p}}\) :

Permeability of porous medium

\({K}_{\text{c}}\) :

Constant chemical reaction parameter

\({\rm K}_{\text{c}}\) :

Chemical reaction parameter

\(L\) :

Nanofluid layer thickness (m)

\({\text{Le}}\) :

Lewis number

\({\text{N}}_{\text{B}}\) :

Modified specific heat increment

\({\text{N}}_{\text{d}}\) :

Diffusivity ratio

p :

Pressure (\({\text{pa}}\))

P:

Porosity parameter

\({ \Pr }\) :

Prandtl number

\({ \Pr }_{\text{M}}\) :

Magnetic Prandtl number

Q :

Chandrasekhar number

Ra:

Rayleigh number

\({\text{Ra}}_{\text{d}}\) :

Density Rayleigh number

\({\text{Ra}}_{\text{s}}\) :

Heat source Rayleigh number

\({\text{Ra}}_{\text{np}}\) :

Nanoparticle Rayleigh number

\(S_{0}\) :

Strength of internal heat source \(( {\text{W}}\,{\text{m}}^{ - 2} )\)

t :

Time (s)

T :

Temperature (K)

\(u,v,w\) :

Darcy velocity components \(( {\text{m s}}^{ - 1} )\)

\(V_{\text{D}}\) :

Darcy velocity \(( {\text{m s}}^{ - 1} )\)

\(x,y,z\) :

Cartesian coordinate (m)

\(\alpha\) :

Parameter defined as \(k\rho^{ - 1} c^{ - 1}\)

\(\beta_{\text{T}}\) :

Volumetric coefficient of expansion

\(\rho\) :

Density \(( {\text{Kg}}\,{\text{m}}^{ - 3} )\)

\(\varphi\) :

Nanoparticle concentration

\(\mu\) :

Dynamic viscosity (\({\text{Pa s}}\))

\(\mu_{\text{e}}\) :

Magnetic permeability of the nanofluid

\(\varepsilon\) :

Porosity of porous medium

\(\eta\) :

Parameter defined as \(\left( {4\pi \mu_{\text{e}} \sigma } \right)^{ - 1}\)

\(\sigma\) :

Electrical conductivity of the nanofluid (\(\Omega ^{ - 1} \,{\text{m}}^{ - 1}\))

\(\psi\) :

Dimensionless nanoparticle concentration

\(\theta\) :

Dimensionless temperature

\(\omega\) :

Dimensionless frequency

\(\nabla^{2}\) :

Laplacian operator

\(\nabla_{\text{H}}^{2}\) :

Horizontal Laplacian operator

crit:

Critical

f:

Base fluid

nf:

Nanofluid

L:

Lower

p:

Nanoparticle

U:

Upper

W :

Wall

\(\infty\) :

Reference condition

\(\text{b}\) :

Basic state

pr:

Perturbation part

¯ :

Dimensional variables

\(\overrightarrow {{}}\) :

Vectorial variables

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Acknowledgements

The author would like to acknowledge the Shahrood University of Technology, which supported this project.

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Authors and Affiliations

  1. School of Mechanical Engineering, Shahrood University of Technology, P. O. Box: 3619995161, Shahrood, Iran

    Pooria Akbarzadeh

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Akbarzadeh, P. The onset of MHD nanofluid convection between a porous layer in the presence of purely internal heat source and chemical reaction. J Therm Anal Calorim 131, 2657–2672 (2018). https://doi.org/10.1007/s10973-017-6710-9

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