UC Riverside

Simulation of fluids can be performed using a background grid (an Eulerian method), particles (a Lagrangian method), or a combination of both (a hybrid method). Simulating multiphase flows requires different approaches depending on how the phases interact. This thesis presents two distinct approaches: a phase-field model in an Eulerian setting, obtained by coupling the Cahn-Hilliard and Navier-Stokes equations, and a buoyancy-driven phase separation scheme in a hybrid setting using the Material Point Method (MPM).In the phase-field model, a variable describes the concentration of each phase, and its evolution is governed by the Cahn-Hilliard equation, a fourth-order partial differential equation. We present a novel simplified discretization of the Cahn-Hilliard equation coupled with the Navier-Stokes equations describing the flow of the mixture, that solves the fourth-order equation as two coupled second-order Helmholtz equations.In the second part of the thesis, we introduce a buoyancy-driven phase separation scheme for MPM. MPM is a hybrid method that uses particles to represent the state of the system and a background grid to solve the equations of motion, and is capable of simulating a wide range of materials. However, it struggles to simulate multiphase flows: a single shared background grid causes what we call the sticking issue, where particles of different phases positioned close together move with similar velocities. Our scheme resolves this by introducing phase-specific background grids, allowing particles of different phases to move independently and separate by density in the presence of gravity. Unlike the Cahn-Hilliard equation, where separation arises from minimizing a free-energy functional via the chemical potential, separation here is driven directly by buoyancy forces.Finally, we present an implicit time integration scheme for our MPM phase separation method. Explicit schemes usually require smaller time step size for stability. By instead solving a nonlinear system for the state at each time step, the implicit scheme allows for larger time steps, reducing the number of steps needed and resulting in faster simulations overall.