# RESEARCH

## MATHEMATICAL ANALYSIS OF BLOOD FLOW AND OXYGEN TRANSPORT IN MICROCIRCULATION

A computationally efficient model for three-dimensional transient convection-diffusion-consumption moving boundary problem. (Thesis)

**Abstract: **Capillaries are tiny vessels connecting arterioles with venules and forming networks through out the body. The diffusion of substrate, such as oxygen, from the microcirculation through capillaries and the its consumption by tissue cells are basic in human physiology. Blockage or shortage of blood in one or more capillaries due to thrombosis or systemic hypo-perfusion could lead to pathological conditions such as stroke. However, transient process of capillary-tissue oxygen transport is poorly understood. The object of this thesis is to develop a mathematical and computationally efficient model that estimates transient oxygen concentration in three dimensional striated muscles (e.g. cardiac muscle), which is a four dimensional unsteady convection-diffusion-consumption moving boundary problem. In particular, we aim to simulate the consequential stages of pathological conditions such as hyperoxia or hypoxia, by determining oxygen concentration levels and its transport in order to better understand and predict adverse health effects. Our computational method is simple, efficient and accurate. It can be applied to boundaries of arbitrary shape.

## Closed Form Solutions of Unsteady Two-Fluid Flow in a Tube

Published in Application and Applied Mathematics (AAM) paper link

**Abstract: **Exact closed-form solutions for the mathematical model of unsteady two-fluid flow in a circular tube are presented. The pressure gradient is assumed to be oscillatory or exponentially increasing or decreasing in time. The instantaneous velocity profiles and flow rates depend on the size of the core fluid, the density ratio, the viscosity ratio, and a parameter (e.g. the Womersley number) quantifying time changes. Applications include blood flow in small vessels.

## Exact closed-form solutions for the two-fluid oscillatory flow and Navierâ€™s partial slip boundary condition

To appear in Fluid Mechanics Research Journal (FMRJ)

**Abstract: **Two-fluid flows occur when immiscible moving fluids of different properties are in contact. Exact closed-form solutions are presented for oscillatory two-fluid flows. Exact solutions are important not only due for its applicability to specific problems but also serve as accuracy standards for approximate solutions. These oscillatory solutions are governed by geometry, the viscosity ratios, and the normalized frequency s. The velocity profiles for large s are different from those of small s. As s -> 0 the solutions may approach a steady-state. For non-zero s there is a phase difference between the two fluids. An important result is that Navier's partial slip condition fails for oscillatory flows.