Solutions of Burgers' Equation for Modeling Pulsatile Blood Flow in Arteries using Homotopy Analysis Method

Abstract: The Burgers' equation, a fundamental partial differential equation that combines both non-linear advection and diffusion effects, has been widely utilized to model fluid dynamics in various contexts. In this study, we focus on the solutions of the Burgers' equation to model pulsatile blood flow in arteries, accounting for the complex interplay between the non-linear convective transport and viscous diffusion of blood under pulsatile pressure gradients. We derive both analytical and numerical solutions to the modified Burgers' equation that incorporates a sinusoidal source term to represent the oscillatory nature of blood flow driven by the cardiac cycle. The analytical solutions are obtained using perturbation methods, providing insights into the zeroth-order steady-state flow and higher-order corrections due to pulsatility and non-linearity. The results illustrate the formation of wave-like structures in the velocity profiles, highlighting the impact of varying parameters such as viscosity, pressure gradient frequency, and amplitude on the blood flow patterns. The proposed model offers a simplified yet effective approach to understanding arterial blood flow dynamics, with potential applications in predicting hemodynamic conditions in normal and pathological states. This work underscores the versatility of the Burgers' equation in modeling complex biological flows and contributes to the development of more accurate and efficient models for cardiovascular fluid dynamics.

Keywords: Burgers' Equation, Pulsatile Blood Flow, Arterial Flow Modeling, Sinusoidal Pressure Gradient, Wave-like Structures, Homotopy Analysis Method