Variational solution for pulsatile flow in rigid tubes.

Abstract

The theoretical equation for determining the profile of velocity during sinusoidal flow through a rigid tube has been shown experimentally to give meaningful biomedical results. However, the solution is complex and does not lend itself to rapid numerical computation. In this study, a fourth order polynomial of even powers was assumed to fit the velocity profile. The unknown constants in the polynomial were obtained by fitting the boundary conditions and by applying the variational method to the Euler-Lagrange equation. The simplified velocity equation was shown to be directly proportional to the magnitude of the pressure gradient amplitude and inversely proportional to the viscosity. The variational solution was shown to reduce to the exact solution for steady flow through rigid tubes (parabolic velocity profile) in the limit as α approached zero. The velocities as predicted by this polynomial agree within 4 percent with the more complex equation for values of the unsteadiness parameter, that ranged between 0 and 4. The maximum error for values of α<6 can be approximated by [0.2 exp (0.71α)] percent. This polynomial solution takes only about one-half the computational time of the exact Bess el function solution to this problem. In conclusion, therefore, a modified and simplified solution of the pulsatile flow equation is possible and potentially useful.