The study of internal heat and variable gravity field on the onset of convection in a sparsely packed porous medium
Ravi Ragoju, Suman Shekhar, Gundlapally Shiva Kumar Reddy, Gali Janardhana Reddy
Abstract
The qualitative influence of internal heat and variable gravity field on the onset of convection in a sparsely packed porous layer with horizontal fluid-saturated are investigated. Linear stability analysis is performed using the normal mode technique. The dimensionless governing equations with four cases of gravity field variation: (1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mi>z</mml:mi> </mml:math> , (2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , (3) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> , and (4) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>z</mml:mi> </mml:msup> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> are solved using bvp4c in MATLAB R2020a and the corresponding eigenvalue problem is calculated for three types of boundaries, namely free-free, rigid-free, and rigid-rigid, respectively. The effect of an internal heat parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , Darcy number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mi>a</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , and gravity variation parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>δ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> for the stability of the system is investigated graphically. The critical value of the Rayleigh number and corresponding wavenumber are calculated for other fixed parameters. It is observed that internal heat parameters enhance the onset of convection, whereas the gravity variation parameter delays the same. It is also observed that the system becomes more stable for the exponential variable gravity function and less stable for the cubic variable gravity function.