Consider the flow of 37 degree C blood in a 1.50 mm radius artery. Take the density of blood to be 1025 kg/m^3, and the viscosity to be (2.084 x 10^-3) Pa(s). What is the greatest average speed v of the blood if the flow is to remain laminar? What is the corresponding flow rate Q?
To determine the greatest average speed v of the blood, we can use the formula for the flow rate Q in a laminar flow through a pipe:
Q = (pi * r^4 * (P1 - P2))/(8 * n * L)
Where:
Q = Flow rate
r = Radius of the artery = 1.50 mm = 0.0015 m
P1 = Pressure at the beginning of the artery
P2 = Pressure at the end of the artery
n = Viscosity of blood = 2.084 x 10^-3 Pa(s)
L = Length of the artery (assumed to be 1 m for simplicity)
Rearranging the formula for flow rate:
Q = (pi * r^4 * (P1 - P2))/(8 * n * L)
We can use the formula for average speed v in laminar flow:
v = Q/A
Where:
v = Average speed
A = Cross-sectional area of the artery
Given that the radius of the artery is 1.5 mm, the cross-sectional area A can be calculated as:
A = pi * r^2
A = pi * (0.0015)^2
A = pi * 2.25 x 10^-6
A = 7.07 x 10^-6 m^2
Now, substituting the area A and flow rate Q back into the average speed formula:
v = Q/A
v = [(pi * r^4 * (P1 - P2))/(8 * n * L)] / [pi * r^2]
v = (r^2 * (P1 - P2))/(8 * n * L)
v = (0.0015^2 * (P1 - P2))/(8 * 2.084 x 10^-3 * 1)
Now, to determine the greatest average speed v, we need to find the maximum pressure difference (P1 - P2) that still allows for laminar flow. This involves the critical Reynolds number, which is approximately 2000 for the onset of turbulent flow.
Re = (2 * r * v * density) / viscosity
2000 = (2 * 0.0015 * v * 1025) / 2.084 x 10^-3
Solving for v:
v = (2000 * 2.084 x 10^-3) / (2 * 0.0015 * 1025)
v ≈ 2.47 m/s
Therefore, the greatest average speed v of the blood to remain in laminar flow is approximately 2.47 m/s.
To determine the corresponding flow rate Q, we can substitute the calculated value of v back into the flow rate formula:
Q = (pi * r^4 * (P1 - P2))/(8 * n * L)
Q = (pi * 0.0015^4 * (P1 - P2))/(8 * 2.084 x 10^-3 * 1)
Q = (9.415 x 10^-12 * (P1 - P2))/(1.667 x 10^-2)
Q = 5.646 x 10^-10 (P1 - P2)
Since P1 and P2 are not given in the problem statement, the exact flow rate Q cannot be calculated without this information.