From Continuity Equation to Navier–Stokes: A Mathematical Description of Cardiovascular and Cardiopulmonary Bypass Blood Flow

Blood flow in the cardiovascular system and cardiopulmonary bypass (C.P.B.) circuits can be effectively described using the fundamental principles of fluid mechanics. This paper presents a simplified mathematical progression from the continuity equation to the Navier–Stokes equations in order to analyze blood flow under physiological and extracorporeal conditions. Starting from the conservation of mass, the continuity equation is derived and applied to incompressible fluid flow relevant to blood. The conservation of momentum is then introduced, leading to the formulation of the Navier–Stokes equations for viscous flow. To maintain relevance for undergraduate study, the analysis is restricted to steady, laminar, incompressible, and Newtonian flow assumptions. Simplified solutions of the Navier–Stokes equations are discussed for flow through cylindrical vessels, providing insight into velocity distribution and pressure gradients commonly encountered in arteries and C.P.B. tubing. The mathematical treatment highlights the importance of viscosity, flow rate, and vessel geometry in determining hemodynamic behavior. This study emphasizes the application of fundamental physics and mathematics in understanding cardiovascular blood flow and perfusion systems, thereby bridging basic fluid mechanics concepts with their clinical relevance in cardiac surgery and cardiopulmonary bypass technology.