Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Energetic variational approach in complex fluids: Maximum dissipation principle

Pages: 1291 - 1304, Volume 26, Issue 4, April 2010      doi:10.3934/dcds.2010.26.1291

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Yunkyong Hyon - Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN 55455, United States (email)
Do Young Kwak - Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 305-701, South Korea (email)
Chun Liu - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (email)

Abstract: We discuss the general energetic variational approaches for hydrodynamic systems of complex fluids. In these energetic variational approaches, the least action principle (LAP) with action functional gives the Hamiltonian parts (conservative force) of the hydrodynamic systems, and the maximum/minimum dissipation principle (MDP), i.e., Onsager's principle, gives the dissipative parts (dissipative force) of the systems. When we combine the two systems derived from the two different principles, we obtain a whole coupled nonlinear system of equations satisfying the dissipative energy law. We will discuss the important roles of MDP in designing numerical method for computations of hydrodynamic systems in complex fluids. We will reformulate the dissipation in energy equation in terms of a rate in time by using an appropriate evolution equations, then the MDP is employed in the reformulated dissipation to obtain the dissipative force for the hydrodynamic systems. The systems are consistent with the Hamiltonian parts which are derived from LAP. This procedure allows the usage of lower order element (a continuous $C^0$ finite element) in numerical method to solve the system rather than high order elements, and at the same time preserves the dissipative energy law. We also verify this method through some numerical experiments in simulating the free interface motion in the mixture of two different fluids.

Keywords:  Energetic variational approach, dissipation energy law, least action principle, maximum dissipation principle, Navier-Stokes equation, phase field equations.
Mathematics Subject Classification:  Primary: 76A05, 76M99; Secondary: 65C30.

Received: November 2008;      Revised: April 2009;      Available Online: December 2009.