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Steady-state solutions for the Muskat problem.

Omar Sánchez1

  • 1Instituto de Ciencias Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain.

Collectanea Mathematica (Barcelona, Spain)
|April 28, 2023
PubMed
Summary

This study investigates stationary solutions for the Muskat problem, extending previous work to cases with large surface tension coefficients. Numerical simulations illustrate the behavior of these solutions beyond previously established limits.

Keywords:
Fingering patternsMuskat problemPeriodic solutionSteady state solutions

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Area of Science:

  • Fluid dynamics
  • Partial differential equations
  • Mathematical modeling

Background:

  • The Muskat problem describes fluid interfaces driven by pressure differences.
  • Previous research established solution existence for surface tensions below a finite threshold.
  • The behavior of solutions for large surface tension remains less understood.

Purpose of the Study:

  • To investigate the existence of stationary solutions for the Muskat problem with large surface tension coefficients.
  • To extend the understanding of Muskat problem solutions beyond previously known limits.
  • To explore the impact of high surface tension on fluid interface dynamics.

Main Methods:

  • Analytical study of stationary solutions for the Muskat problem.
  • Extension of existing theoretical frameworks to accommodate large surface tension.
  • Numerical simulations to visualize and analyze solution behavior.

Main Results:

  • Existence of stationary solutions demonstrated for large surface tension coefficients.
  • The study surpasses the finite value limitation previously identified.
  • Numerical examples provide insights into solution dynamics under high surface tension.

Conclusions:

  • Stationary solutions for the Muskat problem exist even with large surface tension.
  • This work expands the parameter range for which solutions are known.
  • Numerical simulations complement theoretical findings, offering a comprehensive view of solution behavior.