Consider steady laminar flow of a Newtonian fluid in a long, cylindrical, elastic tube of length L. The radius of the tube at any cross section is a=a(x). Poiseuille's formula for the flow rate is a good approximation in this case.

1. Develop an expression for the outlet pressure p(L) in terms of the higher inlet pressure, the flow rate , fluid viscosity ?, and a(x)

2. For a pulmonary blood vessel, we may assume that the pressure-radius relationship is linear: , where a ₀ is the tube radius when the transmural pressure is zero and ? is the compliance of the tube. For a tube of length L, show that

   
   

   where a( 0) and a(L) are the values of a(x) at x=0 and x=L, respectively.

For pulsatile flow in a rigid cylindrical tube of length ?, the pressure drop ? p may be expressed as: ? p=f(L, a, ?, ?, ?, U), where a is tube radius, ? is density, ? is viscosity, ? is frequency, and U is the average velocity of flow. Using dimensional analysis, show that

where C ₁, C ₂, C ₃, and C ₄ are constants, Re is Reynolds number, and St is Strouhal number defined as a ?/U

Localized narrowing of an artery may be caused by the formation of arthero sclerotic plaque in...