Lattice Boltzmann models for binary solutions: Models for diffusion between species with unequal masses and models for flow of immiscible species in a Hele-Shaw cell

In this thesis we investigate lattice BGK models for diffusion between species with unequal masses and models for viscous displacement of a more viscous fluid by a less viscous fluid in a Hele-Shaw cell. Lattice BGK, which is based on a discretization of the Boltzmann equation in the relaxation time approximation, is a promising method for performing computational fluid dynamical simulations and it is ideal for massively parallel computations and easily extendable to complex fluid phenomena. We formulate a one-dimensional model for simulating a binary diffusion couple containing species with different masses and find some unexpected oscillations in the movement of the center of mass. We show that these oscillations are not a discretization artifact but result from traveling waves in the number density and barycentric velocity that allow for momentum exchange as they reflect from the ends of the couple. Next, we consider immiscible displacement in a Hele-Shaw cell where the fluid with low viscosity is used to displace the other which has higher viscosity, a situation that is subject to the Saffman-Taylor instability of the interface that separates the fluids. We formulate a two-dimensional lattice BGK model for this problem which models two nearly immiscible fluid by using a regular binary solution and a gradient energy on the mole fraction. We test our model for static problems and successfully recover the miscibility gap as well as interfacial properties such as surface tension and interfacial width. By performing a series of simulations with domain widths that are very wide compared to the linear-stability prediction of the natural wavelength, we measure the natural wavelength of our model and find that it differs from the sharp-interface quasi-steady-state linear stability result for strictly incompressible and immiscible fluids by 17%. We numerically measure the dispersion relation (logarithmic growth rate as a function of wavelength) of our model by simulating a half-wavelength disturbance for a range of domain widths and find reasonable agreement with the sharp-interface quasi-steady-state linear stability analysis. We extend our simulations to the strongly non-linear regime and discuss the dynamics of finger competition, interfacial singularities such as finger pinch-off and reconnection and the emergence of a single ringer solution for long times, whose shape compares well with the Saffman and Taylor single finger solution. We find that the dynamics of finger competition are related to the dynamics of vortices and stagnation points in the flow field. From a linear stability analysis of this problem in a radial geometry, we make conjectures on the dynamics of pattern formation. By using au implementation of our model in a radial geometry, we find that many aspects of our non-linear results, such as generation of harmonics and tip-splitting, can be explained in terms of the conjectures we made based on linear stability.