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From hyperbolic regularization to exact hydrodynamics for linearized Grad's equations.

Matteo Colangeli1, Iliya V Karlin, Martin Kröger

  • 1ETH Zürich, Department of Materials, Polymer Physics, Zürich, Switzerland.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 7, 2007
PubMed
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This study introduces a novel hyperbolic regularization method for linear hydrodynamics, enabling accurate modeling across Knudsen numbers. The approach ensures stable and hyperbolic equations for fluid dynamics, applicable to kinetic models like the 13 moment Grad system.

Area of Science:

  • Fluid dynamics
  • Kinetic theory
  • Mathematical physics

Background:

  • Burnett's hydrodynamic equations face challenges with hyperbolic regularization.
  • Existing methods may lack accuracy or stability for linear hydrodynamics across Knudsen numbers.

Purpose of the Study:

  • To develop a method for deriving hyperbolic linear hydrodynamic equations with arbitrary Knudsen number accuracy.
  • To ensure the stability and hyperbolicity of derived hydrodynamic equations.
  • To demonstrate the method using a 13 moment Grad system.

Main Methods:

  • Inspired by hyperbolic regularization of Burnett's equations.
  • Utilizing a dynamic invariance principle to derive exact constitutive relations for stress tensor and heat flux.
  • Applying a transformation to render exact hydrodynamic equations hyperbolic and stable.

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Main Results:

  • A systematic method to derive hyperbolic linear hydrodynamic equations to any desired accuracy in Knudsen number.
  • Exact constitutive relations for stress tensor and heat flux are obtained.
  • The method is successfully applied to a 13 moment Grad system, demonstrating its efficacy.

Conclusions:

  • The proposed method offers a robust framework for hyperbolic linear hydrodynamics.
  • This approach enhances the stability and accuracy of fluid dynamics modeling in kinetic regimes.
  • The dynamic invariance principle and transformation are key to achieving hyperbolic and stable equations.