Derivation,analysis And Numerical Approximations To Thermodynamic And Nonisothermal Hydrodynamic Models Of Binary Fluid Flows

The research of multi-component material system has been one of the most important issues in the field of science and engineering.The phase field method,especially the phase field method of the multiphase material system,has been proved to be one of the effective methods to describe the dynamic evolution process of the multiphase material system from non-equilibrium state to equilibrium state.Allen-Cahn model is a commonly used thermody-namic consistency model in the modeling of multi-component material systems,but it cannot guarantee the conservation property of the physical quantity described by the phase variable after integration over the whole region.In the first part of this paper,nonlocal constraints with a penalizing potential and nonlocal constraints with a Lagrange multiplier are added to the classical Allen-Cahn model to make the modified Allen-Cahn model satisfy the proper-ty of multi-component volume conservation of the system,which is called the Allen-Cahn model with nonlocal constraints.For the above two kinds of Allen-Cahn models with non-local constraints,we propose four linear and unconditional energy stable numerical schemes with second-order in time and space by using energy quadratic method(EQ)and scalar aux-iliary variable method(SAV)respectively,and mainly analyze the error estimates of the four numerical schemes.In addition to rigorous error estimates for each numerical scheme,we also show that the linear systems produced by these energy-stable numerical schemes are uniquely solvable.Finally,the numerical simulation verifies the accuracy of the proposed numerical schemes and its characteristics of energy dissipation and volume conservation in droplet fusion simulation.Multi-component fluid convection under non-isothermal conditions plays an important role in engineering applications and is one of the central topics in fluid dynamics and heat transfer research.The characteristics of the thermocapillary convection in two-layer,liquid-liquid systems are generally more complex due to the hydrodynamics and thermal interaction between fluid motions in the two adjacent liquid layers.In the second part of this paper,for nonisothermal multiphase flow problems,we propose the nonisothermal hydrodynamic phase field model consisting of the phase transport Cahn-Hilliard equation based on general-ized Onsager principle,Navier-Stokes equation with the extra stress and energy conservation equation derived from a non-equilibrium thermodynamic framework.This hydrodynamic model is new in that it not only allows one to employ more accurate approximations for the in-ternal energy as a function of the absolute temperature and phase variable as well as the latent heat but also takes into account the hydrodynamic effects and the thermal-hydrodynamic cou-pling.Then,according to the entropy quadratic method(EQ)and a newly proposed method,the supplementary variable method(SVM),two kinds of second order structure-preserving numerical schemes are proposed for the above thermodynamic consistency model of non-isothermal incompressible binary viscous fluid flows.Both of these two numerical schemes maintain the property of volume conservation for each phase liquid component,energy con-servation and positive entropy production rate.EQ scheme is weakly nonlinear,and we use the fast Fourier algorithm to solve it;The SVM scheme is linear and the main idea is to in-troduce a supplementary variable to the thermodynamically consistent model to enforce the entropy-production-rate after discretization.Then we use the finite difference method to carry out spatial discretization of the above two semi-discrete numerical schemes on the staggered grid and obtain two fully discrete numerical schemes.Then,the convergence order of the two numerical schemes is verified by the mesh refinement experiment.Finally,we simulate the thermocapillarity effect in a two-layer incompressible viscous fluid system in an adiabatic(insulated)container by using the thermodynamic consistency model of incompressible two-layer incompressible viscous fluid flow and the proposed numerical scheme and algorithm.The numerical studies demonstrated some weak,thermocapillary effects at the fluid interface of the immiscible binary fluid.