Computing Three-dimensional Free-surface Flows of Polymer Solutions with Macroscopic Models Based on the Conformation Tensor

摘要

Flows with free surfaces and free boundaries arise in many industrial and biological applications. In most of these flows there are two distinguishing features: (1) the fluid is a complex one, i.e., it has microstructural features; thus, the Cauchy stress is not merely composed of viscous and pressure forces, but includes a visco-elastic term due to the microstructure which is important and sometimes controlling, and (2) the surface forces (which are related to the shape of the free surfaces and free boundaries) are comparable to or larger than the viscous and elastic forces due to the flow of the liquid and the subsequent deformation of the microstructure. Examples are coating flows of polymer solutions, where the flow-induced deformation of the polymer molecules can generate steep layers of elastic stress, and the flow-induced deformation of leukocytes, where the elasticity of the cell membrane is coupled to the viscoelasticity of the cytoskeleton.Because surface and viscoelastic forces are comparable or more important than viscous ones, there are large non-diagonal contributions in the momentum equations that come from the deformation of the free surfaces or elastic boundaries, and from the microstructural elastic stress. Conversely, the equations that describe the shape of the boundaries are strongly affected by the coupling to the velocity field at the free surfaces (non-diagonal contribution), and the equations that describe the evolution of the microstructure are strongly dependent of the velocity gradient (non-diagonal terms), which accounts for the deformation of the microstructure induced by the flow. Thus, fully-coupled algorithms for solving the steady as well as time-dependent flow equations are desirable. I will discuss recent developments in applying mesoscopic models of microstructured liquids to three-dimensional free surface flows. In such models, the dynamic microstructure of the liquid is accounted for by one or more tensors which obey convection-diffusion-generation equations. Such tensors can represent the gyration tensor of local ensembles of polymer molecules (in the case of polymeric liquids), or the shape of liquid droplets (in the case of emulsions), or the shape of red blood cells (in the case of blood). Recent findings based on mesoscopic non-equilibrium thermodynamics showed that velocity-gradient-dependent terms in the evolution equations of microstructure are related uniquely to the elastic stress in the momentum equation. This has permitted the development of general theories which can account for disparate microstructural models while at the same time guaranteeing compatibility with macroscopic transport phenomena and thermodynamics. I will show how such theories can be incorporated into full three-dimensional finite element codes based on fully coupled formulations and full Newton's method with analytical Jacobian. I will show results on some model flows of dilute polymer solutions, and discuss recent developments and connections to fine-grain, microscopic models of microstructured liquids where the liquid microstructure is accounted for by using stochastic differential equations.