Question

The aorta has a diameter, d. Blood has a viscosity, ?, 4.00 x 10^-3 Pa.s and a density of 1025 kg.m^-3.

1. Using Poiseuille's equation, calculate the pressure drop per unit length along the aorta. Assume that the aorta has a diameter of 2.63 cm and that blood flows along the aorta at an average speed of 0.255 m.s^-1.

2. Poiseuille's equation does not hold if the flow velocity is high enough that turbulence sets in. The onset of turbulence occurs when the Reynolds number, Re, exceeds a certain limit. Using the data given in the information above and in the previous question, calculate the Reynolds number for blood flowing through the aorta.

Answer 1

1. volume flow rate, V is given as V=Pr⁴/(8L)

where P is change of pressure, r=radius, =viscosity, and L=length of aorta

therefore change of pressure per uni length can be written as

P/L=8V/(r⁴)

now volume flow is V= cross sectional areaxvelocity=r²xv=(1.36x10^-2)²x0.255=1.48x10^-4m³/s

substituting all the given values in the above equation we get

P/L=44.06 Pa/m

2. reynold's number, Re=vD/

where = coefficient of viscosity,=4 x 10^-3Pa.s

=density=1025kg/m³

D=diameter of orifice=2.63cm=2.63x10^-2m

and v=velocity=0.255m/s

substituting we get Re=1718.54 which is well within the laminar flow.