Thermal gradients in many natural environments, such as undersea hot springs and hydrothermal vents, are often accompanied by variations in the concentration of chemical compounds carried into the surrounding seawater. This interaction gives rise to a phenomenon known as thermosolutal mixed convection. To investigate the underlying physical mechanisms in such systems, a vertical pipe filled with saline water through a porous medium is considered. The non-Darcy Brinkman–Forchheimer extended model is employed, assuming that the saturated porous medium is in a thermal equilibrium state. The instability boundary curve reveals three distinct regimes: (i) the Rayleigh–Taylor (R–T) phenomenon, (ii) a nonlinear variation of mixed convection on a log–log scale and (iii) a linear variation on a log–log scale in the pipe. These regimes occur at specific Reynolds numbers. The instability mechanism of the base flow is examined using the spectral collocation technique. The present study focuses on the instability of a dense porous medium, where the Darcy–Forchheimer drag is incorporated in the momentum equation. A linear stability analysis is performed for a wide range of Darcy numbers (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$10^{-8}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>8</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$10^{-5}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>). Similar to cross-diffusive free convection in a vertical slot, a relationship is observed between the solutal Rayleigh number (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Ra_S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <msub> <mi>a</mi> <mi>S</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>) and the thermal Rayleigh number (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Ra_T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <msub> <mi>a</mi> <mi>T</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>). A hyperbolic relationship between the threshold values of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Ra_S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <msub> <mi>a</mi> <mi>S</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and Da is found at <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Re = 1000$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>e</mi> <mo>=</mo> <mn>1000</mn> </mrow> </math></EquationSource> </InlineEquation> for the upper limit of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$Ra_S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <msub> <mi>a</mi> <mi>S</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> in the first regime, expressed as <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$Ra_S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <msub> <mi>a</mi> <mi>S</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> Da =4.9 x <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$10^{-4}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. Simulations of secondary flow profiles are also presented at the critical state for all three regimes.

Stability of Double-diffusive Mixed Convection in a Vertical Pipe Filled by Porous Medium

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发表日期：

2026年2月1日

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Saurabh Kapoor,
Durgaprasad Nayak

摘要

Thermal gradients in many natural environments, such as undersea hot springs and hydrothermal vents, are often accompanied by variations in the concentration of chemical compounds carried into the surrounding seawater. This interaction gives rise to a phenomenon known as thermosolutal mixed convection. To investigate the underlying physical mechanisms in such systems, a vertical pipe filled with saline water through a porous medium is considered. The non-Darcy Brinkman–Forchheimer extended model is employed, assuming that the saturated porous medium is in a thermal equilibrium state. The instability boundary curve reveals three distinct regimes: (i) the Rayleigh–Taylor (R–T) phenomenon, (ii) a nonlinear variation of mixed convection on a log–log scale and (iii) a linear variation on a log–log scale in the pipe. These regimes occur at specific Reynolds numbers. The instability mechanism of the base flow is examined using the spectral collocation technique. The present study focuses on the instability of a dense porous medium, where the Darcy–Forchheimer drag is incorporated in the momentum equation. A linear stability analysis is performed for a wide range of Darcy numbers ( $10^{-8}$ $10^{- 8}$ to $10^{-5}$ $10^{- 5}$ ). Similar to cross-diffusive free convection in a vertical slot, a relationship is observed between the solutal Rayleigh number ( $Ra_S$ $R a_{S}$ ) and the thermal Rayleigh number ( $Ra_T$ $R a_{T}$ ). A hyperbolic relationship between the threshold values of $Ra_S$ $R a_{S}$ and Da is found at $Re = 1000$ $R e = 1000$ for the upper limit of $Ra_S$ $R a_{S}$ in the first regime, expressed as $Ra_S$ $R a_{S}$ Da =4.9 x $10^{-4}$ $10^{- 4}$ . Simulations of secondary flow profiles are also presented at the critical state for all three regimes.