In this dissertation we present an energetic variational approach to modeling complex fluids. The principle of least action or the principle of virtual work is applied to derive the conservative (Hamiltonian) part of the system, including the non-Newtonian stress tensor. This follows by an application of Onsager’s principle of minimum dissipation, which introduces the dissipative terms in the resulting time-dependent system of coupled partial differential equations. The final coupled system is the result of a total force balance and the dynamics for the microscopic configurations/patterns.

An existence result of one particular hydrodynamical system is presented in the third chapter. The system is derived from the energetic variational approach and is essentially a simplified Ericksen-Leslie model for nematic liquid crystals. The key point in the mathematical proof is to take advantage of a high order energy law of the system, which provides the needed regularity for the local solutions. Other techniques include a modified Galerkin approximation, some interpolating inequalities of Ladyzhenskaya’s type, and an Aubin-Lions compactness argument.

We also applied our energetic variational approach to model mixtures involving multiple (two or more) different phases. To demonstrate the applicability of the proposed model, a few two-dimensional simulations have been carried out, including (1) the force balance at the triple line in equilibrium, (2) a rising bubble penetrating a fluid-fluid interface, and (3) a solid particle falling in a binary fluid. The effects of slip at solid surface have been examined in connection with contact line motion and a pinch-off phenomenon.