This study focuses on performing a linear instability analysis to explore the onset of convective instability arising from the combined effects of vertical throughflow and thermal dispersion in a fluid-saturated porous layer governed by the Darcy–Brinkman model. The considered system is subjected to a temperature gradient, with both walls maintained at distinct constant temperatures. This temperature difference generates buoyancy forces, which act as a driving mechanism for the onset of convection within the system. The stability characteristics of the flow are influenced by several dimensionless parameters, including the Darcy number (Da), the thermal dispersion coefficient (Di), and the Péclet number (Pe). We employed the Chebyshev-tau method, combined with the QZ-algorithm, to numerically solve the generalized eigenvalue problem. The investigation is exemplified through results that highlight scenarios where advection dominates diffusion as well as cases where diffusion prevails over advection. We found that the behavior of thermal convective instability is highly dependent on the Péclet number (Pe). For values of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Pe \le 2.5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mi>e</mi> <mo>≤</mo> <mn>2.5</mn> </mrow> </math></EquationSource> </InlineEquation>, the system exhibits destabilizing behavior, indicating a greater likelihood of instability. However, for Péclet numbers exceeding 2.5, the system transitions to a stabilizing regime, where the onset of instability is suppressed. This demonstrates a clear threshold at <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$Pe = 2.5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mi>e</mi> <mo>=</mo> <mn>2.5</mn> </mrow> </math></EquationSource> </InlineEquation>, distinguishing two distinct dynamic behaviors within the system. Additionally, our results indicate that both the Darcy number (Da) and the thermal dispersion coefficient (Di) contribute to stabilizing thermal convective instability. These factors act to reduce the growth rate of disturbances, thereby enhancing the stability of the system under thermal gradients.

Thermal Dispersion and Vertical Throughflow Effects on the Onset of Convection in Darcy–Brinkman Porous Media

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发表日期：

2026年1月7日

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Pappu Kumar Mourya,
Ankush Ankush,
Gautam Kumar

摘要

This study focuses on performing a linear instability analysis to explore the onset of convective instability arising from the combined effects of vertical throughflow and thermal dispersion in a fluid-saturated porous layer governed by the Darcy–Brinkman model. The considered system is subjected to a temperature gradient, with both walls maintained at distinct constant temperatures. This temperature difference generates buoyancy forces, which act as a driving mechanism for the onset of convection within the system. The stability characteristics of the flow are influenced by several dimensionless parameters, including the Darcy number (Da), the thermal dispersion coefficient (Di), and the Péclet number (Pe). We employed the Chebyshev-tau method, combined with the QZ-algorithm, to numerically solve the generalized eigenvalue problem. The investigation is exemplified through results that highlight scenarios where advection dominates diffusion as well as cases where diffusion prevails over advection. We found that the behavior of thermal convective instability is highly dependent on the Péclet number (Pe). For values of $Pe \le 2.5$ $P e \leq 2.5$ , the system exhibits destabilizing behavior, indicating a greater likelihood of instability. However, for Péclet numbers exceeding 2.5, the system transitions to a stabilizing regime, where the onset of instability is suppressed. This demonstrates a clear threshold at $Pe = 2.5$ $P e = 2.5$ , distinguishing two distinct dynamic behaviors within the system. Additionally, our results indicate that both the Darcy number (Da) and the thermal dispersion coefficient (Di) contribute to stabilizing thermal convective instability. These factors act to reduce the growth rate of disturbances, thereby enhancing the stability of the system under thermal gradients.