Poiseuille's equation does not hold if the flow velocity is...

Poiseuille's equation does not hold if the flow velocity is high enough that turbulence sets in. The onset of turbulence occurs when the $\textbf{Reynolds number}$, $Re$, exceeds approximately 2000. Re is defined as $$Re = \frac{2\overline{v}r\rho}{\eta} ,$$ $\overline{v}$ where is the average speed of the fluid, $\rho$ is its density, $\eta$ is its viscosity, and $r$ is the radius of the tube in which the fluid is flowing. ($a$) Determine if blood flow through the aorta is laminar or turbulent when the average speed of blood in the aorta ($r$ = 0.80 cm) during the resting part of the heart's cycle is about ($b$) During exercise, the blood-flow speed approximately doubles. Calculate the Reynolds number in this case, and determine if the flow is laminar or turbulent.

Transcript

0:00 Aloha.

00:01 For this problem, we're going to be calculating the reynolds number, which is a parameter to help us determine whether a fluid is flowing in a laminar flow or a turbulent flow.

00:14 We're told in the problem that reynolds number, r .e, is greater than 2000 if the flow is turbulent.

00:23 And we're going to be using this formula for the reynolds number.

00:28 This is something we're given.

00:30 Two times the average velocity times the radius of the tube of the fluid times the density divided by aida which is the viscosity and we're going to be doing this for the case of blood throw through the aorta and in part a part a told this is sort of like for person resting and that means that the velocity we're given 35 centimeters per second this is our average velocity that's the average heart rate of rest.

01:03 So that's what we're given in the problem.

01:07 We're told that the radius of the aorta is 0 .8 centimeters.

01:12 And we know that we're talking about blood.

01:14 So we can find the density of blood and the viscosity of blood as known constants.

01:23 And we can find them in tables in chapter 10.

01:27 So the density is 1 .05 times 10 to 3 kilograms.

01:32 A meter cubed.

01:35 So you can just find these in tables in the text.

01:40 And the viscosity is four times 10 to minus 3 pascal seconds.

01:47 So we're just going to, we just asked if this blood flowing of the order is turbulent or a laminar flow.

01:55 And that's what it means if it's not turbulent, just a smooth flow.

01:59 So we just have to calculate the random number and just see whether or not it's more or less than 2 ,000.