# Relating portions of Poiseille's Law to concepts of pressure/radius

• oddjobmj
In summary, the equation states that for the flux to remain constant, the pressure and radius of an artery must be related by a power of four. If the radius of an artery is reduced to three-fourths of its former value, then the pressure is more than tripled.
oddjobmj

## Homework Statement

High blood pressure results from constriction of arteries. To maintain a normal flow rate (flux), the heart has to pump harder, thus increasing the blood pressure. Use Poiseille's Law to show that if R0 and P0 are normal value sof the radius and pressure in an artery and the constricted values are R and P, then for the flux to remain constant, P and R are related by the equation

P/P0 = (R0/R)^4

Deduce that if the radius of an artery is reduced to three-fourths of its former value, then the pressure is more than tripled.

## Homework Equations

P/P0 = (R0/R)^4

and what I know of Poiseille's Law

F = ((pi)(P)(R^4))/((8)(n)(l))

Where R=volume, P=pressure, l=length of blood vessel, F=flux, and n=viscosity of blood

## The Attempt at a Solution

Firstly, I've worded and formatted the question exactly as listed. I did this because I'm a little confused at what it's even asking me to do. Does it read like there are two parts; 1) Proving P and R are related by the given equation and 2) deducing the given specific relationship between a change in radius and a change in volume?

I really don't know where to start. I don't want the answer given to me but any hints on how to get started and what they're really asking me for would be greatly appreciated.

so the initial radius R0 and pressure P0 give a flux F
$$F = \frac{\pi P_0 R_0^4}{8nl}$$

we assume at a new reduced radius R, we can find a pressure P which gives the same flux F
$$F = \frac{\pi P R^4}{8nl}$$

now equate the RHS of each and rearrange to get P/P0

Thank you for your reply Lanedance. The way you worded it makes much more sense. However, I'm not sure what you mean by RHS.

Two instances of the equivilant of F will result in all known variables to cancel out on both sides leaving the variables we're working with. Now I see how they derive the first equation I listed. Now if I plug in some reasonable values I can solve for the unknown.

I guess I'm mostly confused because this now seems like a rather simple algebra problem and this is a calculus II class.

EDIT: Also, would you recommend I place reasonable values for R and P in and solve or should I leave it as is with all the variables?

Last edited:
yeah it looks like algebra mainly to me

RHS - right hand side

personally i always prefer to leave the variables in until the last step or as late as possible. I find it more flexible & more audiatble

the question asks you to subsititute in $R = \frac{3}{4}R_0$

Awesome, after inserting the 3/4 and adjusting all to the RHS I essentially came down to 256/81=P/4000 (4000=P0, the value I picked as a common P value; used in a previous example).

Then I simply said 256/81 > 3 and did a little explaining.

Thank you for your help!

## 1. What is Poiseille's Law and how is it related to pressure and radius?

Poiseille's Law is a mathematical equation that describes the relationship between the flow rate and pressure gradient in a fluid flowing through a cylindrical pipe. It states that the flow rate is directly proportional to the pressure gradient and the fourth power of the radius of the pipe.

## 2. How does an increase in pressure affect the flow rate according to Poiseille's Law?

An increase in pressure gradient will result in an increase in the flow rate, assuming all other factors remain constant. This is because the flow rate is directly proportional to the pressure gradient in Poiseille's Law.

## 3. What effect does a change in radius have on the flow rate according to Poiseille's Law?

A change in the radius of the pipe will have a significant impact on the flow rate according to Poiseille's Law. Increasing the radius by a small amount can result in a significant increase in flow rate, as the flow rate is directly proportional to the fourth power of the radius.

## 4. How does Poiseille's Law relate to the concept of fluid viscosity?

Poiseille's Law takes into account the viscosity of the fluid in its equation. Viscosity is a measure of a fluid's resistance to flow, and it affects the flow rate by creating a frictional force that opposes the flow. A higher viscosity fluid will result in a lower flow rate, as seen in the equation for Poiseille's Law.

## 5. Can Poiseille's Law be applied to non-Newtonian fluids?

Poiseille's Law is derived specifically for Newtonian fluids, which have a constant viscosity regardless of the shear stress applied. Therefore, it cannot be directly applied to non-Newtonian fluids, which have variable viscosities depending on the applied force. However, it can be used as an approximation for laminar flow in some non-Newtonian fluids under certain conditions.

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