(a) To determine the net force on the blood in this section of the artery, we can use the Hagen-Poiseuille equation to calculate the flow rate through the artery and subsequently find the net force acting on the blood.
The Hagen-Poiseuille equation for pressure drop ΔP in a pipe of length L, radius r, and dynamic viscosity η, with laminar flow is given by:
ΔP = (8 * η * L * Q) / (π * r^4)
Given:
Pressure drop, ΔP = 100 Pa
Radius, r = 10 mm = 0.01 m
Average speed, v = 15 mm/s = 0.015 m/s
Dynamic viscosity of blood, η is needed but not provided in the given data
Length of the artery, L is not provided in the given data
First, we need to calculate the flow rate Q using the formula Q = A * v, where A is the cross-sectional area of the artery.
The cross-sectional area of the artery is:
A = π * r^2
A = π * (0.01 m)^2
A = π * 0.0001 m^2
The flow rate Q is:
Q = A * v = π * 0.0001 m^2 * 0.015 m/s = 1.5 x 10^-6 m^3/s
Now, we can rearrange the Hagen-Poiseuille equation to solve for the net force:
Net force = ΔP * (π * r^4) / (8 * η * L)
Without the dynamic viscosity of blood (η) and the length of the artery (L), we cannot calculate the exact net force. We would need these additional parameters to find the net force on the blood correctly.
(b) To determine the power expended maintaining the flow, we can use the formula for power:
Power = Flow rate * Pressure drop
Power = Q * ΔP
Power = 1.5 x 10^-6 m^3/s * 100 Pa
Power = 0.15 x 10^-4 W
Power ≈ 1.5 x 10^-3 W
Therefore, the power expended maintaining the flow is approximately 1.5 x 10^-3 Watts.