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<p>We study thermal convection in a porous layer governed by the Darcy-Brinkman model, incorporating internal heating and a higher-order thermal diffusion (bi-Laplacian) term that models hyper-diffusive heat transport. Linear normal-mode analysis and an energy method are used to determine, respectively, the instability threshold and a nonlinear stability bound across Brinkman coefficient, internal heating, and boundary-type parameters. For stability analysis, we employ complementary solvers for linear-onset convection and nonlinear energy stability, integrating the Moore-Penrose pseudoinverse with golden section search and eigenvalue tracking to determine critical Rayleigh numbers. To solve the eigenvalue systems resulting from stability analyses, two numerical methods are formulated: Method&#xa0;I uses conventional shifted Chebyshev polynomials, whereas the new method, Method&#xa0;II, employs boundary-satisfying trial functions that remove boundary rows and columns, reduce algebraic dimension, and improve conditioning. A residual-based audit measures the residuals of the equations and the boundary conditions across systematic variations in the number of trial functions, yielding a reliable accuracy-cost metric. The two numerical methods produce nearly identical eigenvalues and eigenfunctions and agree within numerical tolerance across all evaluations. The new method attains the required accuracy with fewer polynomials, typically about half as many as in Method&#xa0;I, thereby offering a clear advantage over the traditional approach. We expect that the suggested numerical method will have a big impact on future studies. Under stress-free and rigid-rigid velocity boundaries, increases in the Brinkman coefficient and the presence of rigid walls raise both stability margins, whereas stronger internal heating lowers the linear-onset and energy-stability thresholds.</p>

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Linear instability and nonlinear stability of thermal convection in porous medium with internal heat source and higher temperature gradient

  • 期刊论文

  • 发表日期：

    2026年2月9日

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  • Sanaa L. Khalaf,
  • Akil J. Harfash

摘要

We study thermal convection in a porous layer governed by the Darcy-Brinkman model, incorporating internal heating and a higher-order thermal diffusion (bi-Laplacian) term that models hyper-diffusive heat transport. Linear normal-mode analysis and an energy method are used to determine, respectively, the instability threshold and a nonlinear stability bound across Brinkman coefficient, internal heating, and boundary-type parameters. For stability analysis, we employ complementary solvers for linear-onset convection and nonlinear energy stability, integrating the Moore-Penrose pseudoinverse with golden section search and eigenvalue tracking to determine critical Rayleigh numbers. To solve the eigenvalue systems resulting from stability analyses, two numerical methods are formulated: Method I uses conventional shifted Chebyshev polynomials, whereas the new method, Method II, employs boundary-satisfying trial functions that remove boundary rows and columns, reduce algebraic dimension, and improve conditioning. A residual-based audit measures the residuals of the equations and the boundary conditions across systematic variations in the number of trial functions, yielding a reliable accuracy-cost metric. The two numerical methods produce nearly identical eigenvalues and eigenfunctions and agree within numerical tolerance across all evaluations. The new method attains the required accuracy with fewer polynomials, typically about half as many as in Method I, thereby offering a clear advantage over the traditional approach. We expect that the suggested numerical method will have a big impact on future studies. Under stress-free and rigid-rigid velocity boundaries, increases in the Brinkman coefficient and the presence of rigid walls raise both stability margins, whereas stronger internal heating lowers the linear-onset and energy-stability thresholds.