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October  2010, 26(4): 1291-1304. doi: 10.3934/dcds.2010.26.1291

Energetic variational approach in complex fluids: Maximum dissipation principle

Yunkyong Hyon 1, , Do Young Kwak 2, and  Chun Liu 3,

1. 

Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN 55455, United States
2. 

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 305-701, South Korea
3. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

Received  November 2008 Revised  April 2009 Published  December 2009

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: dissipation energy law, Energetic variational approach, maximum dissipation principle, Navier-Stokes equation, phase field equations., least action principle.
Mathematics Subject Classification: Primary: 76A05, 76M99; Secondary: 65C3.
Citation: Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291
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