Pulsatile viscous flow in flexible-elastic tubes
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The flow of an incompressible viscous fluid through a circular tube has received considerable attention during the past fifteen years, both for steady flow and for pulsatile flow. The practical application of blood flow through vascular channels has served as impetus for these studies. For the case of steady flow through an elastic tube, there is a pressure drop along the longitudinal axis due to viscous energy dissipation. The diameter of this tube is theoretically determined by the equality of the pressure difference across the wall and the elastic stress. Due to this equality there would be an expected variation in the diameter of the tube along the flow axis. However, it has been shown that this variation can be neglected for the case of laminar flow when the physical properties of the system are in the ranges of the physiological counterparts. The non-steady flow field is a much more complicated one, both theoretically and experimentally. The mathematical model describing the system consists of the usual Navier- Stokes equations, and equations of continuity, and, in addition, the equations of motion of the boundary. This is a simultaneous set of non-linear, second order, partial differential equations representing the flow field and boundary displacement field. The boundary conditions complicate natters as the no-slip condition for velocity must be applied to a moving wall whose position is unknown. Thus a complete unsimplified theoretical solution would be very difficult and at the present time has not been obtained. The first major progress was made in 1955 by two mathematicians working independently, Womersley and Uchida. They obtained exact solutions of the Navier-Stokes equations for the case of pulsatile flow in rigid tubes. This greatly simplifies the mathematical model as the system reduces to one linear ordinary differential equation which can be integrated. In 1957 Womersley" obtained an approximate solution for the case of a non-rigid tube. To date, this remains as the most comprehensive solution to the problem, although there have been many variations on Womersley's original model. The main disadvantage to these solutions is that they are very laborious to apply. The solutions are given in terms of complex Bessel functions and in general require considerable calculation. To overcome this hardship another model was proposed by Fry. He proposed a simplified model to which a solution can be conveniently obtained to the differential equation instantaneously on an analog computer or can be found in closed form with one integration. One of the main objectives of this study will be to investigate the aforementioned models by comparing them to experimentally measured flow fields and determine the merits and/or, disadvantages of each.
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