Applied Math Seminar
                        2-3pm, March 13, Thursday
                       Mathematics Conference Room
                

   A complete set of eigenfunctions for the stability of pipe pulsatile flow

     
                               Francesco Fedele,
                 Dept. of Civil & Environmental Engineering, UVM

                        Darren Hitt & Rachakonda Prabhu,
                      Dept. of Mechanical Engineering, UVM


The study of pulsatile tube flow appears to have been first considered in the 
context of arterial hemodynamics in the mid-1950s. Womersley and co-workers 
obtained an exact solution of the Navier-Stokes equations for the fully-developed 
velocity profile of a oscillatory, incompressible flow in a circular tube. 
Numerical investigations of several authors have shown that the flow is stable 
for infinitesimal perturbances.  The main goal of our analysis is studying the 
hydrodynamic stability of the flow by solving for the stream function of 
axisymmetric perturbances, which  satisfies the fourth order Orr-Sommerfeld 
equation. We present a semi-analytical approach based on the Galerkin projection 
of the Orr-Sommerfeld equation onto an approximation functional space solution, 
spanned by a finite set of the eigenfunctions of the streamwise longwave limit 
of the Orr-Sommerfeld operator. Convergence to the exact solution is obtained as 
the number of eigenfunctions approaches infinity.  In this limit an infinite 
Floquet system is derived. In the streamwise longwave limit a multiscale 
perturbation method is applied to obtain an uniform solution of the infinite 
system. The analytical predictions are finally compared to the numerical solution 
of the truncated Floquet system using Runge-Kutta method.