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Darcy–Bénard Convection with Internal Heating and a Thermal Nonequilibrium—A Numerical Study

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Advances in Fluid Dynamics

Part of the book series: Lecture Notes in Mechanical Engineering ((LNME))

Abstract

The impact of local thermal nonequilibrium (LTNE) in the presence of a uniform internal heating in both the fluid and solid phases of the porous medium on the onset of Darcy–Bénard convection is investigated. Emphasis is laid on LTNE effect on the steady-state heat conduction in analyzing the onset criterion. The Galerkin method is used to carry out the parametric study on the instability characteristics of the system by numerically computing the critical stability parameters. The presence of LTNE effect on the steady-state heat conduction is found to advance the onset in comparison with its absence and also to increase the dimension of convection cells. The exiting results are obtained as a particular case from the present study.

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Abbreviations

\(a\) :

Wavenumber in the \(x\)-direction

\(c\) :

Specific heat

\(d\) :

Width of the layer

\(D = {\text{d}}/{\text{d}}z\) :

Differential operator

\(\vec{g}\) :

Acceleration due to gravity

\(h\) :

Interphase heat transfer coefficient

\(H\) :

Dimensionless interphase heat transfer coefficient

\(k\) :

Thermal conductivity

K :

Permeability

\(l\,\,\& \,\,m\) :

Horizontal wave number

p :

Pressure

\(\vec{q}\) :

Velocity vector \((u,v,w)\)

\(q^{{{\prime \prime \prime }}}\) :

Uniform heat source per unit volume

\(Q\) :

Dimensionless heat source strength

\(R_{D}\) :

Darcy-Rayleigh number

\(\Pr_{D}\) :

Darcy-Prandtl number

t :

Time

T :

Temperature

\(W\) :

Amplitude of perturbed vertical velocity

\((x,y,z)\) :

Cartesian coordinates

\(\alpha\) :

Ratio of thermal diffusivity

\(\gamma\) :

Porosity modified conductivity ratio

\(\beta\) :

Coefficient of thermal expansion of the fluid

\(\varepsilon\) :

Porosity

\(\mu_{f}\) :

Dynamic viscosity of the fluid

\(\nabla^{2}\) :

Laplacian operator

\(\rho_{f}\) :

Fluid density

\(\rho_{0\,}\) :

Fluid density at \(T = T_{0}\)

\(\phi\) :

Solid temperature

\(\varPhi\) :

Amplitude of solid temperature

\(\theta\) :

Fluid temperature

\(\varTheta\) :

Amplitude of fluid temperature

\(\omega\) :

Growth factor \(( = \omega_{r} + i\omega_{i} )\)

\(b\) :

Basic state

f :

fluid

s :

solid

\({\prime }\) :

Perturbed variable

\(*\) :

Dimensionless variable

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Acknowledgements

One of the authors CH is grateful to the SC/ST cell of Bangalore University, Bengaluru, for granting him a scholarship to carry out his research work.

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Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Jnana Bharathi Campus, Bangalore, 560056, India

    C. Hemanthkumar & I. S. Shivakumara

  2. Department of Mathematics, VIT University, Vellore, 632014, India

    B. Rushikumar

Authors
  1. C. Hemanthkumar
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  2. I. S. Shivakumara
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  3. B. Rushikumar
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Corresponding author

Correspondence to I. S. Shivakumara .

Editor information

Editors and Affiliations

  1. Vellore Institute of Technology, Vellore, Tamil Nadu, India

    B. Rushi Kumar

  2. Vellore Institute of Technology, Vellore, Tamil Nadu, India

    R. Sivaraj

  3. University of Botswana, Gaborone, Botswana

    J. Prakash

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Hemanthkumar, C., Shivakumara, I.S., Rushikumar, B. (2021). Darcy–Bénard Convection with Internal Heating and a Thermal Nonequilibrium—A Numerical Study. In: Rushi Kumar, B., Sivaraj, R., Prakash, J. (eds) Advances in Fluid Dynamics. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-15-4308-1_49

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  • DOI: https://doi.org/10.1007/978-981-15-4308-1_49

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  • Online ISBN: 978-981-15-4308-1

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