UC Riverside

Fluid dynamics is the field of study that examines the motion of fluids such as liquids and gases. It can be used to investigate large-scale phenomena, such as ocean currents, as well as small-scale systems, like blood circulation. Fluid flows can be classified into two broad categories: laminar and turbulent flows. Laminar flows are smooth and streamlined, while turbulent flows are irregular and unpredictable. One of the fundamental tasks in analyzing fluid flows is to determine the flow rates and pressure values in a flow network, given its topology, channel dimensions, fluid properties, and boundary conditions.

In the first project, we study fluid mixing in microfluidic chips (MFCs), which are micro-scale fluid systems. In MFCs, flows are laminar, and for laminar flows, computing flow rates and pressure values are straightforward, but simulating the mixing process is computationally challenging. We present an approach for modeling concentration profiles in grid-based MFCs. Our algorithm outperforms COMSOL Multiphysics® software --- commercial software that uses finite element analysis method to model physics processes --- in terms of runtime while producing results that approximate those of COMSOL.

In the second project, we study turbulent flows in large-scale pipe systems such as water distribution systems and sewage networks. Unlike laminar flow systems, solving flows in turbulent models involves a system of nonlinear equations, and iterative algorithms have been widely applied in practice. We focus on the Hardy Cross loop-based algorithm (HC-loop) and the Newton-Raphson loop-based algorithm (NR-loop). We provide a mathematical analysis of the local convergence of these two algorithms, showing that, under certain conditions, NR-loop algorithm achieves quadratic convergence while HC-loop algorithm only converges linearly. This confirms earlier experimental observations reported in the literature.

In the third project, we investigate the minimum spanning tree congestion problem (STC), motivated by its application to improve the efficiency of the NR-loop algorithm for pipe flows analysis. We study the complexity of K-STC (STC for a fixed integer K) and prove that K-STC is NP-complete for K >= 5, improving the earlier hardness result and leaving only the case = 4 open. We also investigate K-STC restricted to graphs of radius 2, establishing that this variant is NP-complete for K >= 6. Additionally, we explore a variant of STC, denoted K-STC-D, where the objective is to determine if a graph has a depth-D spanning three of congestion K. We provide a tight bound for bipartite graphs by proving that 6-STC-2 is NP-complete, while 5-STC-2 is solvable in polynomial time. Finally, we present polynomial-time algorithms for two special cases involving bipartite graphs with restrictions on vertex degrees.