ENERGETIC VARIATIONAL APPROACH IN COMPLEX FLUIDS: MAXIMUM DISSIPATION PRINCIPLE By Yunkyong

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 laws. We will discuss the important roles of MDP in designing numerical method for computations of hydrodynamic system 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 system. 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. 1. Introduction. The energetic variational approaches of hydrodynamic systems in complex fluids are the direct consequence of the second law of thermodynam-ics. The complex fluids in our interests are the fluids with micro-structures (molecular configurations), for instance, viscoelastic polymer models such as Hookean model, finite extensible nonlinear elastic (FENE) dumbbell models, rod like liquid crystal models , and multi-phase The interaction/coupling between different scales or phases, plays a crucial role in understanding complex fluids. can be described by the macro-scopic deformation to the microscopic structure through kinematic transport and the macroscopic elastic stresses induced by the molecular configurations in microscopic level. A competition in multi-phase fluids [1, 19, 31, 32] can be described by the macroscopic kinetic energy and the internal " elastic " energy through the kinematic transport. The complex fluids thus are basically described by multiscale-multiphysics model. We illustrate the energetic variational approach for one of complex fluid model using the least action principle (LAP) [15] and the maximum/minimum dissipation principle (MDP) [22, 23, 24] to understand complex fluids. In computational viewpoint the MDP shows a way to achieve an efficient method for numerical computations.