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<p>This paper presents a diffuse interface model for the evolution of two incompressible fluids within a porous medium. The model is governed by the Cahn–Hilliard equations, which incorporate a Flory–Huggins logarithmic potential, and is coupled with the evolutionary Stokes equations at the pore scale through a surface tension term. A family of regular potentials <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mo stretchy="false">{</mo> <msub> <mi>F</mi> <mi>δ</mi> </msub> <mo stretchy="false">}</mo> </math></EquationSource> <EquationSource Format="TEX">$\{ F_{\delta }\} $</EquationSource> </InlineEquation> that approximate the singular logarithmic potential <i>F</i> is introduced. The existence of solutions for the pore-scale problem with the regular potential <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>δ</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$F_{\delta }$</EquationSource> </InlineEquation> is established using Galerkin’s method. Subsequently, the two-scale convergence approach and the unfolding operator technique are applied to derive the homogenized model from the microscopic description. In the homogenized model equations, associated with the regularized potentials <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mo stretchy="false">{</mo> <msub> <mi>F</mi> <mi>δ</mi> </msub> <mo stretchy="false">}</mo> </math></EquationSource> <EquationSource Format="TEX">$\{ F_{\delta }\} $</EquationSource> </InlineEquation>, the limit as <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>δ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\delta \to 0 $</EquationSource> </InlineEquation> is taken to recover the upscaled model associated with the original logarithmic functional.</p>

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Micro-scale diffuse interface modeling with regularized logarithmic potential in porous media

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月8日

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  • Nitu Lakhmara,
  • Hari Shankar Mahato

摘要

This paper presents a diffuse interface model for the evolution of two incompressible fluids within a porous medium. The model is governed by the Cahn–Hilliard equations, which incorporate a Flory–Huggins logarithmic potential, and is coupled with the evolutionary Stokes equations at the pore scale through a surface tension term. A family of regular potentials { F δ } $\{ F_{\delta }\} $ that approximate the singular logarithmic potential F is introduced. The existence of solutions for the pore-scale problem with the regular potential F δ $F_{\delta }$ is established using Galerkin’s method. Subsequently, the two-scale convergence approach and the unfolding operator technique are applied to derive the homogenized model from the microscopic description. In the homogenized model equations, associated with the regularized potentials { F δ } $\{ F_{\delta }\} $ , the limit as δ → 0 $\delta \to 0 $ is taken to recover the upscaled model associated with the original logarithmic functional.