A small blood vessel near the skin surface has a radius of 10 um, a length of 1m
ID: 1459389 • Letter: A
Question
A small blood vessel near the skin surface has a radius of 10 um, a length of 1mm. and the pressure drop along the blood vessel is 2.5 pa= 19mmHg. The viscosity of the blood is 2.7*10^-3 N s/m^2.
a . What is the volume flow rae of blood through this blood vessel? what is the velocity of blood flow?
b. Vasodilation causes the radius of this blood vessel to in crease to 12 um, while leaving the pressure drop along the vessel unchanged. what is the volume flow rate through this blood vessel now? What is the velocity of the blood flow?
Explanation / Answer
Pressure drop for a fluid flowing through a pipe is given by -
?P = 8?µ?L?Q / (??R?)
where,
µ - dynamic viscosity = 2.7*10^-3 N s/m^2
L - pipe length =1*10^-3 m
Q - volumetric flow rate ?
R - pipe radius = 10*10^-6 m
?P - Pressure Drop
2.5 = 8*2.7*10^-3*1*10^-3*Q / (?*(10*10^-6)^4)
Q = 3.64 *10^-15 m^3/s
Volume flow rate of blood through this blood vessel, Q = 3.64 *10^-15 m^3/s
Velocity * Area = Volume flow rate of Blood
Velocity = Volume flow rate of Blood / Area
Velocity = 3.64 *10^-15 m^3/s / 3.14 * (10*10^-6)^2
Velocity = 1.16 * 10^-5 m/s
(b)
Volume flow rate through this blood vessel will remain same = 1.16 * 10^-5 m/s
We can use Equation of Continuity ,
A1 * v1 = A2 * v2
3.14 * r1^2 * 1.16 * 10^-5 = 3.14 * r2^2 * v2
3.14 * (10*10^-6)^2 * 1.16 * 10^-5 = 3.14 * (12*10^-6))^2 * v2
v2 = 8.1 * 10^-6 m/s
Velocity of the blood flow, v2 = 8.1 * 10^-6 m/s