(Solved): Biological fluid mechanics The dimensional analysis is a very useful tool for solving fluid mechan ...

Biological fluid mechanics

The dimensional analysis is a very useful tool for solving fluid mechanics problems. Consider the steady, fully developed, one-dimensional, laminar flow of an incompressible Newtonian fluid (of density \( \rho \) ) in a circular tube of diameter \( D \) and length \( L \), as a model system of the blood flow into a vein. For such a system the pressure drop \( (\Delta p) \) can be expressed as a function of the system variables: \( \Delta p=g(\rho, \eta, L, D,\langle v>) \); where \( \eta \) and \( \langle v> \) are the fluid viscosity and average velocity, respectively. (a) Write down the Buckingham Pi theorem and apply it to the above system to find the number of independent dimensionless group of variables related to the pressure drop. [5] (b) Write down the expressions of the independent dimensionless groups of variables related to the pressure drop and describe their physical meaning. [5] (c) Write down the expressions of the three independent dimensionless groups of variables related to the dimensionless expression of the Navier-Stokes equation and describe their physical meaning. [5] (d) Write down the expression of the Fanning friction factor \( (f) \) and sketch a double logarithmic graph showing the relationship between the Fanning friction factor and the Reynolds number \( (R e) \) for a fluid flowing into a smooth pipe, starting from the point \( (R e=100, f=0.2) \) and going to the point \( \left(R e=10^{6}\right. \), \( f=0.002 \) ). [5]