The velocity, \(c\), at which pressure pulses travel through arteries (pulse- wave velocity) is a function of the artery diameter, \(D\) and wall thickness, \(h,\) the density of blood, \(\rho,\) and the modulus of elasticity, \(E\), of the arterial wall. Determine a set of nondimensional parameters that can be used to study experimentally the relationship between the pulse-wave velocity and the variables listed. Form the nondimensional parameters by inspection.

First identify the units of each variable. velocity \(c\) is in meter per second (m/s), diameter \(D\) and wall thickness \(h\) are in meters (m), blood density \(\rho\) is in kilograms per cubic meter (kg/m^3), and modulus of elasticity \(E\) is in pascal (Pa or kg/(m*s^2)).

The nondimensional parameters can be determined by considering the ratios of the variables that will lead to a dimensionless quantity:1. Reynolds number, Re = \( \frac{\rho c D}{\mu} \) which represents the ratio of the inertial forces to the viscous forces. 2. Elastic modulus number, Em = \( \frac{E h}{\rho c^{2}} \) represents the ratio of the elastic forces in the arterial wall to the kinetic energy of the blood flow per unit volume. 3. Diameter thickness ratio, Dt = \( \frac{D}{h} \) represents the ratio of the diameter to the thickness of the arteries. The above parameters are dimensionless and can be used to characterize or study experimentally the relationship between the pulse-wave velocity and the variables listed.

To validate, confirm that the unit of each term in the nondimensional parameters cancels out correctly. Check the units for each of:1. Re = \( \frac{\rho c D}{\mu} \) = \( \frac{(kg/m^3)*(m/s)*m}{kg/(ms)} \) = Dimensionless 2. Em = \( \frac{E h}{\rho c^{2}} \) = \( \frac{(kg/m*s^2)*m}{(kg/m^3)*(m/s)^2} \) = Dimensionless 3. Dt = \( \frac{D}{h} \) = \( \frac{m}{m} \) = Dimensionless As each of the parameters is indeed dimensionless, these can be considered as valid nondimensional parameters.