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For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
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Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier-Stokes/Allen-Cahn System.

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This study demonstrates the Navier-Stokes/Allen-Cahn system converges to a sharp interface model for two-phase fluid flow. Error estimates are provided using a relative entropy method under specific mobility conditions.

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Area of Science:

  • Multiphase flow dynamics
  • Partial differential equations
  • Fluid mechanics

Background:

  • Modeling two-phase flow in viscous, incompressible fluids is crucial for understanding complex fluid behaviors.
  • The Allen-Cahn equation is often used to model interfaces, but its connection to sharp interface models requires rigorous analysis.
  • Bridging diffuse interface models (like Allen-Cahn) with sharp interface models (like Navier-Stokes) is a key challenge in fluid dynamics.

Purpose of the Study:

  • To establish the mathematical convergence of the Navier-Stokes/Allen-Cahn system to a classical sharp interface model.
  • To analyze the two-phase flow of two viscous, incompressible fluids with identical viscosities in a bounded domain.
  • To derive error estimates quantifying the approximation accuracy.

Main Methods:

  • Utilizing a relative entropy method to analyze the mathematical convergence.
  • Demonstrating that the solution to the Navier-Stokes/Allen-Cahn system stays close to a perturbed two-phase flow problem.
  • Analyzing the behavior of the Allen-Cahn mobility parameter as it tends to zero in a subcritical manner.

Main Results:

  • Proven convergence of the Navier-Stokes/Allen-Cahn system to a sharp interface model for two-phase flow.
  • Obtained error estimates for the convergence using the relative entropy method.
  • Established that convergence holds under specific subcritical conditions for the Allen-Cahn mobility parameter.

Conclusions:

  • The Navier-Stokes/Allen-Cahn system accurately approximates sharp interface models for two-phase flow under specified conditions.
  • The relative entropy method is effective for deriving error estimates in such fluid dynamics models.
  • The findings provide a rigorous mathematical foundation for using diffuse interface models in simulating two-phase flows.