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New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows.
1Center for Research in Scientific Computation & Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA.
East Asian Journal on Applied Mathematics
|June 25, 2013
Summary
New finite difference methods simulate inextensible interfaces in fluid dynamics, crucial for modeling red blood cell deformation. These methods accurately conserve interface properties by treating surface tension as an augmented variable.
Area of Science:
- Computational fluid dynamics
- Mathematical biology
- Numerical analysis
Background:
- Simulating inextensible interfaces in incompressible flows is vital for understanding phenomena like red blood cell deformation.
- Existing immersed boundary and interface methods struggle with the nonlinear, coupled nature of these problems, leading to complex systems.
- Accurately determining unknown surface tension is critical for conserving interface properties.
Purpose of the Study:
- To develop novel finite difference methods for simulating inextensible moving interfaces in 2D incompressible flows.
- To address the challenges posed by unknown surface tension and nonlinear coupling in governing equations.
- To provide accurate and efficient simulation tools for problems in mathematical biology.
Main Methods:
- Augmented immersed interface method (IIM) to treat unknown surface tension as an augmented variable.
- Regularization strategy involving a controlled tangential force to solve the inverse problem for surface tension.
- Modified projection method for Navier-Stokes equations to enforce pressure jump conditions.
- Solving multiple Poisson equations for the Stokes flow case.
Main Results:
- The proposed methods successfully simulate inextensible moving interfaces.
- Numerical experiments demonstrate good agreement with existing literature results.
- The methods reveal interesting phenomena related to interface dynamics and surface tension effects.
Conclusions:
- The augmented IIM provides an effective framework for simulating inextensible interfaces with unknown surface tension.
- The developed methods offer accurate and robust solutions for incompressible flow problems with moving interfaces.
- This work contributes significantly to computational fluid dynamics and mathematical biology applications.
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