A model of pulsatile flow in a uniform deformable vessel

Simulations of blood flow in natural and artificial conduits usually require large computers for numerical solution of the Navier-Stokes equations. Often, physical insight into the fluid dynamics is lost when the solution is purely numerical. An alternative to solving the most general form of the Navier-Stokes equations is described here, wherein a functional form of the solution is assumed in order to simplify the required computations. The assumed forms for the axial pressure gradient and velocity profile arc chosen such that conservation of mass is satisfied for fully established pulsatile flow in a straight, deformable vessel. The resulting equations are cast in finite-difference form and solved explicitly. Results for the limiting cases of rigid wall and zero applied pressure are found to be in good agreement with analytical solutions. Comparison with the experimental results of Klanchar et al. [ Circ. Res., 66, 1624–1635 (1990)] also shows good agreement. Application of the model to realistic physiological parameter values provides insight as to the influence of the pulsatile nature of the flow field on wall shear development in the presence of a moving wall boundary. Specifically, the model illustrates the dependence of flow rate and shear rate on the amplitude of the vessel wall motion and the phase difference between the applied pressure difference and the oscillations of the vessel radius. The present model can serve as a useful tool for experimentalists interested in quantifying the magnitude and character of velocity profiles and shearing forces in natural and artificial biologic conduits.