Fluid dynamics question (Poiseuille)

[Andrew Jacobson]

Hi, I'm attempting to do a question involving blood flowing through a blood vessel and I'm incredibly stuck and would appreciate some help. The question is as follows:
'A simplified model of blood flow through the human body makes the approximation that the flow I is proportional to the pressure differential ΔP between any two points of the system ΔP∝I.
Take a length of vessel L with diameter d and viscosity η and let v(r) be the velocity as measured by r from the central axis. Assuming that the blood flow is laminar we can then model:
v(r)=ΔP((d²/4)-r²)/4ηL
The viscous force F_v acts on any cylindrical element due to the slow moving blood outside the element. The magnitude is given by:
F_v=-ηAdv/dr where A=2πrL.'
'(a) Sketch v(r) and then calculate the average velocity through the vessel'
I sketched it as a negative x^2 graph except it doesn't go below the x axis. For the average velocity I integrated to find the area under the curve and then divided by the range. This gave me (ΔPd²)/(24ηL)
'(b) calculate the flow through the vessel'
I said that the flow = ∫v(r)2πrdr between d/2 and 0 and get the answer (ΔPπd⁴)/128ηL
'(c) Calculate the force on the walls of the vessel'
Here I was a little less sure. I worked out dv/dr to be -(ΔPr)/(2ηL) and then substituted it into the given equation to get F=ΔPπr² and then subbed in r=d/2 (because it's the force at the wall) to get F=(ΔPπd²/4
'(d) What is the net force on the vessel? Show this is consistent with your answer to (c)'
Here is where I'm completely stuck. I've tried making ΔP=dF/dA and solving differentially but I'm extremely confused. If somebody could help with this part (and correct me if I've gone wrong beforehand) then it'd be greatly appreciated.
Thanks in advance.

 

[TSny]

Hi, Andrew. Welcome to PF!

Your work looks excellent. In part (d) I guess they want an alternate way to get the force on the vessel that's different than the method of part (c).

Suppose you consider the entire volume of fluid in the vessel of length L. What external forces act on this volume of fluid?

 

[Andrew Jacobson]

Hi,external w. Welcome to PF!

Your work looks excellent. In part (d) I guess they want an alternate way to get the force on the vessel that's different than the method of part (c).

Suppose you consider the entire volume of fluid in the vessel of length L. What external forces act on this volume of fluid?

Thank you for the quick reply! Uhm... would the external forces be the ones from the walls acting on the blood thus making the net force 0?

 

[TSny]

The total force from the cylindrical wall of the vessel would be one of the forces on the on the cylindrical section of blood of length L. But there is also a force on each circular end of the cylindrical section of blood.

 

[Chestermiller]

Hi, I'm attempting to do a question involving blood flowing through a blood vessel and I'm incredibly stuck and would appreciate some help. The question is as follows:
'A simplified model of blood flow through the human body makes the approximation that the flow I is proportional to the pressure differential ΔP between any two points of the system ΔP∝I.
Take a length of vessel L with diameter d and viscosity η and let v(r) be the velocity as measured by r from the central axis. Assuming that the blood flow is laminar we can then model:
v(r)=ΔP((d²/4)-r²)/4ηL
The viscous force F_v acts on any cylindrical element due to the slow moving blood outside the element. The magnitude is given by:
F_v=-ηAdv/dr where A=2πrL.'
'(a) Sketch v(r) and then calculate the average velocity through the vessel'
I sketched it as a negative x^2 graph except it doesn't go below the x axis. For the average velocity I integrated to find the area under the curve and then divided by the range. This gave me (ΔPd²)/(24ηL)

The average velocity is usually calculated as the volumetric throughput rate (the flow through the vessel) divided by the cross sectional area of the tube.

'(b) calculate the flow through the vessel'
I said that the flow = ∫v(r)2πrdr between d/2 and 0 and get the answer (ΔPπd⁴)/128ηL
'(c) Calculate the force on the walls of the vessel'
Here I was a little less sure. I worked out dv/dr to be -(ΔPr)/(2ηL) and then substituted it into the given equation to get F=ΔPπr² and then subbed in r=d/2 (because it's the force at the wall) to get F=(ΔPπd²/4

This just shows that the shear force at the wall is equal to the pressure difference times the cross sectional area; thus the net force on the fluid is zero.

'(d) What is the net force on the vessel? Show this is consistent with your answer to (c)'

This is a little confusing. The net force on the vessel is equal to the shear force on the wall. The pressure forces are not acting on the vessel axially.

 

[TSny]

This is a little confusing. The net force on the vessel is equal to the shear force on the wall. The pressure forces are not acting on the vessel axially.

My interpretation of part (d) is to find the shear force on the vessel "indirectly" by using the fact that the net force on the fluid is zero and Newton's third law. But I could be mistaken.