Influence of Internal Heat Source on Thermal Instability in a Horizontal Porous Layer with Mass Flow and Inclined Temperature Gradient Abstract—An investigation has been presented to analyze the effect of internal heat source on the onset of Hadley-Prats flow in a horizontal fluid saturated porous medium. We examine a better understanding of the combined influence of the heat source and mass flow effect by using linear stability analysis. The resultant eigenvalue problem is solved by using shooting and Runga-Kutta methods for evaluate critical thermal Rayleigh number with respect to various flow governing parameters. It is identified that the flow is switch from stabilizing to destabilizing as the horizontal thermal Rayleigh number is enhanced. The heat source and mass flow increases resulting a stronger destabilizing effect. Keywords—Linear stability analysis, heat source, porous medium, mass flow. I. I NTRODUCTION I N the last few decades, the study on thermal convection driven by an internal heat source has been attracted by many researchers due to its importance in real life applications. The present problem is also in connection with the above study induced by horizontal mass flow. It has many practical applications such as underground energy transport, cooling of nuclear reactors, geophysical and environmental problems etc. Specific important areas are like the food processing, oil recovery, underground storage of waste products and thermal convection in clouds [1]. The mechanism of thermal convection has a great importance in environmental problem processes [2]. Some of the authors reported on convection by internal heat sources. Few papers are concerning to the experimental investigation by Schwiderski et al. [3] and Tritton et al. [4]. Roberts [5] and Thirlby [6] done the theoretical analysis on the above experimental results. Parthiban and Patil [7] investigated the thermal convection due to non-uniform heating boundaries with inclined thermal gradients in the presence of internal heat source, followed by the extension of anisotropic porous layer studied by Parthiban and Patil [8]. The effect of internal heat source with inclined porous layer for various flow parameters are analyzed by Barletta et al. [9], where both boundaries are isothermal and keep them at same temperature. Rionero and Straughan [10] investigated the linear and nonlinear instability in presence of heat generation and variable gravity. Anjanna Matta is with the Department of Mathematics, Indian Institute of Technology Hyderabad, ODF Estate, Telangana, 502205, India (e-mail: anjireddyiitm@gmail.com). P. A. L. Narayana is with the Department of Mathematics, Indian Institute of Technology Hyderabad, ODF Estate, Telangana, 502205, India (Corresponding author, phone: 91 (40) 2301 7050; e-mail: ananth@iith.ac.in). Extensive reviews of the theory and applications can be seen in the articles of Alex and Patil [11]. Hill [12] reported on a fluid-saturated porous layer with concentration based internal heat generation, in that, he studied the linear and energy stability analysis of thermosolutal convection. Chamka [13] analyzed the influence of an internal heat source or sink for hydromagnetic simultaneous heat and mass transfer by using similarity solutions. Thermosolutal convection in a saturated anisotropic porous medium with internal heat source is reported by Bhadauria [14]. Borujerdi et al. [15] examine the study state heat conduction with uniform heat source where solid and fluid phases are at different temperature. Then after, Borujerdi et al. [16] studied the influence of Darcy number on the critical thermal Rayleigh number in onset of convection with uniform internal heating. A collection of comprehensive theories and experiments of thermal convection in porous media on real life problems were surveyed in the recent book of Nield and Bejan [17]. The purpose of this theoretical study is to analyze the situation in which both the effects of heat source and mass flow are present simultaneously. The governing equations have been transformed into eigenvalue problem, and it is solved numerically by using Shooting and Runga-Kutta method for various modes of instability. We organize the paper in the following steps. Section II deals with the governing equations of the model considered and section III followed by basic state solution, linear analysis and numerical scheme described in section IV and section V. Results are analyzed in section VI. II. MATHEMATICAL ANALYSIS An infinite shallow horizontal fluid saturated porous medium with thickness d is considered. z ∗ -axis is taken vertically upward and there is a net flow along the direction of x ∗ -axis with magnitude M ∗ . The vertical thermal differences along the boundaries is ∇θ. Further, we imposed the horizontal thermal gradient vector as (β θx , β θy ) and the internal heat source is Q  . The linear Boussinesq approximation is applicable. The flow in porous layer is formed by the Darcy law and the governing equations in dimensional form are ∇  · q  =0 , (1) Anjanna Matta, P. A. L. Narayana World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:9, No:7, 2015 396 International Scholarly and Scientific Research & Innovation 9(7) 2015 scholar.waset.org/1307-6892/10002268 International Science Index, Mathematical and Computational Sciences Vol:9, No:7, 2015 waset.org/Publication/10002268