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<p>In this study, we present a rigorous model for analyzing the behavior of three immiscible components within incompressible viscous flows. By combining the Cahn–Hilliard free energy approach with the Stokes equation, we address both thermodynamic and hydrodynamic aspects of the fluid mixture. For a particular consistent selection of the bulk-free energy function, we enhance the Stokes equation to account for capillary forces and surface tension effects, and the Cahn–Hilliard equation with the convection effects. At the pore scale, we show the existence of weak solution in a bounded and sufficiently smooth domain <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10665_2024_10418_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega \subset {\mathbb {R}}^d, d= 2, 3 \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. Using two-scale convergence techniques, we derive an upscaled model that effectively characterizes the averaged behavior of the entire system.</p>

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Analysis of a Cahn–Hilliard model for a three-phase flow problem

  • 期刊论文

  • 发表日期：

    2025年1月12日

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  • Nitu Lakhmara,
  • Hari Shankar Mahato

摘要

In this study, we present a rigorous model for analyzing the behavior of three immiscible components within incompressible viscous flows. By combining the Cahn–Hilliard free energy approach with the Stokes equation, we address both thermodynamic and hydrodynamic aspects of the fluid mixture. For a particular consistent selection of the bulk-free energy function, we enhance the Stokes equation to account for capillary forces and surface tension effects, and the Cahn–Hilliard equation with the convection effects. At the pore scale, we show the existence of weak solution in a bounded and sufficiently smooth domain \(\Omega \subset {\mathbb {R}}^d, d= 2, 3 \) Ω ⊂ R d , d = 2 , 3 . Using two-scale convergence techniques, we derive an upscaled model that effectively characterizes the averaged behavior of the entire system.