# Thermodynamically Consistent Hydrodynamic Phase Field Models and Numerical Approximation for Multi-Component Compressible Viscous Fluid Mixtures

#### Abstract

Material systems comprising of multi-component, some of which are compressible, are ubiquitous in nature and industrial applications. In the compressible fluid flow, the material compressibility comes from two sources. One is the material compressibility itself and another is the mass-generating source. For example, the compressibility in the binary fluid flows of non-hydrocarbon (e.g. Carbon dioxide) and hydrocarbons en- countered in the enhanced oil recovery (EOR) process, comes from the compressibility of the gas-liquid mixture itself. Another example of the mixture of compressible flu- ids is growing tissue, in which cell proliferation and cell migration make the material volume changes so that it cannot be described as incompressible.

We present a systematic derivation of thermodynamically consistent hydrody- namic phase field models for compressible viscous fluid mixtures using the general- ized Onsager principle along with the one fluid multi-component formulation. By maintaining momentum conservation while enforcing mass conservation at differ- ent levels, we obtain two compressible models. When the fluid components in the mixture are incompressible, we show that one compressible model reduces to the quasi-incompressible model via a Lagrange multiplier approach. Several different approaches to arriving at the quasi-incompressible model are discussed. Then, we conduct a linear stability analysis on all the binary models derived in the thesis and show the differences of the models in near equilibrium dynamics.

We present a linear, second order fully discrete numerical scheme on a staggered grid for a thermodynamically consistent hydrodynamic phase field model of binary compressible flows of fluid mixtures derived from the generalized Onsager Principle.

The hydrodynamic model not only possesses the variational structure, but also war- rants the mass, linear momentum conservation as well as energy dissipation. We first reformulate the model in an equivalent form using the energy quadratization method and then discretize the reformulated model to obtain a semi-discrete partial differential equation system using the Crank-Nicolson method in time. The numer- ical scheme so derived preserves the mass conservation and energy dissipation law at the semi-discrete level. Then, we discretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using the 2nd order finite difference method, which respects a discrete energy dissipation law. We prove the unique solvability of the linear system resulting from the fully discrete scheme. Mesh refinements are presented to show the convergence property of the new scheme.

In the compressible polymer mixtures, we first construct a Flory-Huggins type of free energy and explore the phase separation phenomena due to spinodal decom- position. We investigate the phase separation with and without hydrodynamics, respectively. It tells us that hydrodynamics indeed changes local densities, the path of phase evolution and even the final energy steady states of fluid mixtures. This is alarming, indicating that hydrodynamic effects are instrumental in determining the correct spatial phase diagram for the binary fluid mixture.

Finally, we study the interface dynamics and investigate the mass adsorption phe- nomena of one component at the interface, to show the performance of our model and the numerical scheme in simulating hydrodynamics of the hydrocarbon mix- tures.

*This paper has been withdrawn.*