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I am interested in the applied analysis of partial differential equations arising from fluid dynamics as well as related numerical analysis and scientific computation. In particular, I am working on modeling, analysis and numerical simulations of multiphase flow by phase field approach and hybridizable discontinuous Galerkin methods; mathematical analysis of Prandtl boundary layer theory; flow instability and dynamical transitions. A unique feature of my research is the combination of rigorous mathematics with genuine physical applications.
Description of Research Projects
Modeling and numerical simulations of multiphase flow
Multiphase flow phenomena are ubiquitous. When two or more macroscopically immiscible fluids (e.g. oil and water) are in contact, they are in fact separated by a thin transitional layer. The thickness of this layer is typically small compared to other characteristic length scales of the flow, and the thin layer is often approximated by an interface of zero thickness. This approximation leads to the classical sharp interface model in which fluid equations are posed on each side of the interface and jump conditions are prescribed across it. However, the sharp interface model breaks down (for instance, curvature becomes singular) when a fluid interface undergoes topological transitions such as pinchoff or reconnection. Moreover, it has been shown numerically that the transitional layer plays an important role when the radius of curvature or distance between interfaces becomes comparable to the interface thickness. One alternative modeling approach is the diffuse interface method, also referred to as phase field approach. In a phase field model, the interface is diffusive thanks to chemical diffusion between different fluid components, across which the field variables (e.g., order parameter) vary continuously. The phase field model is capable of capturing smooth topological transitions of fluid interface. The model typically consists of a Cahn-Hillard type equation coupled with the fluid equations with built-in capillary forcing. Numerically, the simulation of a phase field model can be carried out on a fixed grid without explicit interface tracking. The major challenge in solving a phase field fluid model is that one has to deal with a coupled high order nonlinear system which is stiff due to the sharp transition of field variables over the diffused interface. In this direction, we focus on the design and analysis of unconditionally stable, high order, decoupled numerical schemes for solving phase field fluid models. We also develop phase field models for complex multiphase flow in various contexts.
Multiphase flow phenomena are ubiquitous. When two or more macroscopically immiscible fluids (e.g. oil and water) are in contact, they are in fact separated by a thin transitional layer. The thickness of this layer is typically small compared to other characteristic length scales of the flow, and the thin layer is often approximated by an interface of zero thickness. This approximation leads to the classical sharp interface model in which fluid equations are posed on each side of the interface and jump conditions are prescribed across it. However, the sharp interface model breaks down (for instance, curvature becomes singular) when a fluid interface undergoes topological transitions such as pinchoff or reconnection. Moreover, it has been shown numerically that the transitional layer plays an important role when the radius of curvature or distance between interfaces becomes comparable to the interface thickness. One alternative modeling approach is the diffuse interface method, also referred to as phase field approach. In a phase field model, the interface is diffusive thanks to chemical diffusion between different fluid components, across which the field variables (e.g., order parameter) vary continuously. The phase field model is capable of capturing smooth topological transitions of fluid interface. The model typically consists of a Cahn-Hillard type equation coupled with the fluid equations with built-in capillary forcing. Numerically, the simulation of a phase field model can be carried out on a fixed grid without explicit interface tracking. The major challenge in solving a phase field fluid model is that one has to deal with a coupled high order nonlinear system which is stiff due to the sharp transition of field variables over the diffused interface. In this direction, we focus on the design and analysis of unconditionally stable, high order, decoupled numerical schemes for solving phase field fluid models. We also develop phase field models for complex multiphase flow in various contexts.
Simulation: Spinodal decomposition and coarsening [Video]
Simulation: Rayleigh-Taylor instability [Video]
Simulation: Saffman-Taylor instability [Video]
Fig. 1: Rayleigh-Taylor instability: pinchoff of a light fluid layer sandwiched by heavy fluids
Fig. 2: Mutiphase flow in karstic geometry: a light bubble passing the domain interface at y=0 separating a porous medium (top) and a conduit (bottom).
Fig. 3: Saffman-Taylor instability by Cahn-Hilliard-Darcy model
2. Prandtl boundary layer theory and applications
In many applications of fluid mechanics, the Reynolds number is very large, resulting in a small diffusion coefficient. Thus the flow is often regarded as inviscid and governed by the Euler equations, except in a thin layer near the solid boundary where viscous effects are dominant (fluid particles sticking to the solid surface). This is the Prandtl boundary layer theory in which the boundary layer functions satisfy the so-called Prandtl type equations. The mathematical justification of Prandtl boundary layer theory, i.e., if the flow can be well approximated by inviscid flow plus boundary layers, remains a conundrum in applied mathematics. Indeed, no general convergence result on the asymptotic limit is expected, since, the Prandtl equations can be ill-posed, and the asymptotic expansion is invalid if the Prandtl equations start with an unstable initial shear layer with nondegenerate critical points (in violation of Oleinik monotone condition). We work on constructing examples where convergence or invalidity can be shown rigorously, in attempt to shed light on the resolution of Prandtl boundary layer theory. We also apply the ideas and tools developed in boundary layer theory in deriving and justifying simplified physical models.
3. Flow instability and transitions
Flow instability and transitions is a central theme in fluid mechanics. The problems can often be put into the perspective of infinite-dimensional dynamic systems that exhibit attractors. We study flow instability and transitions using the dynamic transition theory developed by Ma and Wang. The main philosophy of this theory is to search for the full set of transition states often represented by a local attractor. Based on the Principle Exchange of Stability and Center Manifold Reduction, the theory identifies the transition states and classify them into three types: continuous, catastrophic, and random. Roughly speaking, a continuous transition means that the basic state bifurcates to a local attractor; a catastrophic transition means that a system will jump to another state, and a random transition indicates that both continuous and catastrophic transitions are possible depending on the initial perturbation. The type of transitions is in general determined by the sign of a so-called transition number that can be conveniently evaluated by numerical means.
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