Variational Modeling and Complex Fluids

In this chapter, a general energetic variational framework for modeling the dynamics of complex fluids is introduced. The approach reveals and focuses on the couplings and competitions between different mechanisms involved for specific materials, including energetic contributions vs. kinematic transport relations, conservative parts vs. dissipative parts and kinetic parts vs. free energy parts of the systems, macroscopic deformation or flows vs. microscopic deformations, bulk effects vs. boundary conditions, etc. One has to notice that these variational approaches are motivated by the seminal works of Rayleigh (Proc Lond Math Soc 1(1):357–368, 1871) and Onsager (Phys Rev 37(4):405, 1931; Phys Rev 38(12):2265, 1931). In this chapter, the underlying physical principles and background, as well as the limitations of these approaches, are demonstrated. Besides the classical models for ideal fluids and elastic solids, these approaches are employed for models of viscoelastic fluids, diffusion, and mixtures.