We are solving blood flow in an artery with stenosis. Blood flow is considered to be unidirectional and axi-symmeric in a rigid tube of circular cross section, with porous walls. We have modeled the pressure gradient, taken from medical literataure, by Fourier Series Approximation. Slip boundary condition is assumed. Obviously, wall distensibility is neglected. We have employed Laplace (wrt the variable t) and Finite Hankel Transform (wrt to the variable r). As Laplace Transform is used, we need the initial condition u(r, t) at t =0. Any suggestion? Can I assume it to be zero? Justification? Some authors have taken Poiseuille Flow. Is it justifiable?
I would not think Poiseuille Flow would be necessarily applicable through an artery with stenosis since Poiseuille Flow is based on laminar flow--any arterial flow through any area of stenosis usually causes turbulent flow negating any laminar flow properties.
To Wayne Schmedel; Thank you for your professional answer from the realistic medical perspective. Right now, we are modeling the blood flow only as laminar and that is the reason we are employing the transform techniques. This is a primitive model to get some physical insight.
Is there any relevant aperiodic component of the solution? Indeed, it would seem more natural to look for t-periodic solutions (thus getting rid of the initial condition, too).
Even if you split the solution as steady and oscillatory, you may not need an initial condition since you can make the unsteady solution periodic.
Dear Prof.Girija Jayaraman, You may kindly recollect that I met you in PSGR Krishnammal College, Coimbatore and you suggested me to read Fung. My Ph.D. scholar is working on pulsatile blood flow following the lines of McDonald's model of developing Fourier Series for the pressure gradient - as was followed by Prof.Chaturani of IIT Bombay. We have got some good and noteworthy results. We have solved the above problem by splitting the solution into steady and oscillatory part and solving the resulting equations by a straight forward variable separable technique for each harmonic component. Earlier we were trying to solve by simultaneously applying Laplace and Hankel transform, in which case I wanted the initial condition. But, still I would like to think about solving the same problem by employing these transforms. Similarly, while working on blood flow through elliptic cross section (Pulmonary artery), we are stuck with the numerical intricacies of Matthieu Functions.
Calculation of series solution using Matthieu functions etc. will involve as much computer time and approximation as a numerical solution. Elliptic cross section takes away symmetry in all directions except the major/minor axis. So, the problems will be tougher with or without transforms.
I wonder why you are struck with the transform techniques. Select a meaningful problem and be open minded about the choice of the techniques.
I am likely to be in Coimbatore by the end of June. Shall let you know if confirmed.
Dear Prof.Girija Jayaraman, We are aware of the computational complexity involved in Matthieu Functions. Right now the Ph.D. candidate is winding up her work and we shall take up elliptic cross section only after she submits her thesis. We are planning to move on to flow of Cerebrospinal Spinal Fluid for which this work will form the basis. We will be greatly honored to meet you when you visit Coimbatore in June.
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