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Variational principle for the Navier-Stokes equations.

R R Kerswell1

  • 1Department of Mathematics, University of Bristol, Bristol BS8 1TW, England, United Kingdom.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
Summary

This study introduces a new variational principle for Navier-Stokes equations, simplifying complex fluid dynamics problems. By decomposing velocity, it enables estimation of fluid dissipation through solvable linear problems.

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Area of Science:

  • Fluid Dynamics
  • Mathematical Physics
  • Nonlinear Dynamics

Background:

  • The Navier-Stokes equations describe fluid motion but are notoriously difficult to solve.
  • Existing variational principles for these equations are often overconstrained and intractable.
  • Estimating fluid dissipation is crucial for understanding complex flow behaviors.

Purpose of the Study:

  • To develop a tractable variational principle for the Navier-Stokes equations.
  • To introduce a novel velocity decomposition method to simplify the variational problem.
  • To establish complementary variational problems for bounding fluid dissipation.

Main Methods:

  • A nonunique velocity decomposition (u = φ + ν) is introduced.
  • The variational problem is transformed into finding a saddle point over an extended domain.

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  • Complementary minimization and maximization problems are constructed to bound dissipation.
  • The method relies on solving a series of linear problems.

    • Main Results:

    • The complex variational problem is reformulated into a more tractable saddle-point problem.
    • Dual variational principles are derived to provide upper and lower bounds on viscous dissipation.
    • These bounds are obtained by solving sequences of linear problems, avoiding direct nonlinear equation solving.
    • The unique intersection of these bounds is conjectured to yield the exact solution.

    Conclusions:

    • The proposed variational approach offers a computationally feasible method for analyzing Navier-Stokes equations.
    • This framework generalizes existing bounding techniques for fluid dissipation.
    • The reliance on linear problems makes this approach practical for estimating bounds in complex fluid systems.