A common parameter that can be used to predict turbulence in...

A common parameter that can be used to predict turbulence in fluid flow is called the Reynolds number. The Reynolds number for fluid flow in a pipe is a dimensionless quantity defined as $\operatorname{Re}=\frac{\rho v d}{\eta}$ where $\rho$ is the density of the fluid, $v$ is its speed, $d$ is the inner diameter of the pipe, and $\eta$ is the viscosity of the fluid. The criteria for the type of flow are as follows: $\bullet$If $\operatorname{Re}<2300,$ the flow is laminar. $\bullet$If $2300<$ Re $<4000$, the flow is in a transition region between laminar and turbulent. $\bullet$If $\operatorname{Re}>4000,$ the flow is turbulent. (a) Let's model blood of density $1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and viscosity $3.00 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}$ as a pure liquid, that is, ignore the fact that it contains red blood cells. Suppose it is flowing in a large artery of radius $1.50 \mathrm{~cm}$ with a speed of $0.0670 \mathrm{~m} / \mathrm{s}$ Show that the flow is laminar. (b) Imagine that the artery ends in a single capillary so that the radius of the artery reduces to a much smaller value. What is the radius of the capillary that would cause the flow to become turbulent? (c) Actual capillaries have radii of about $5-10$ micrometers, much smaller than the value in part (b). Why doesn't the flow in actual capillaries become turbulent?

A common parameter that can be used to predict turbulence in fluid flow is called the Reynolds number. The Reynolds number for fluid flow in a pipe is a dimensionless quantity defined as $\operatorname{Re}=\frac{\rho v d}{\eta}$ where $\rho$ is the density of the fluid, $v$ is its speed, $d$ is the inner diameter of the pipe, and $\eta$ is the viscosity of the fluid. The criteria for the type of flow are as follows:
$\bullet$If $\operatorname{Re}<2300,$ the flow is laminar.
$\bullet$If $2300<$ Re $<4000$, the flow is in a transition region between laminar and turbulent.
$\bullet$If $\operatorname{Re}>4000,$ the flow is turbulent.
(a) Let's model blood of density $1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and viscosity $3.00 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}$ as a pure liquid, that is, ignore the fact that it contains red blood cells. Suppose it is flowing in a large artery of radius $1.50 \mathrm{~cm}$ with a speed of $0.0670 \mathrm{~m} / \mathrm{s}$ Show that the flow is laminar. (b) Imagine that the artery ends in a single capillary so that the radius of the artery reduces to a much smaller value. What is the radius of the capillary that would cause the flow to become turbulent?
(c) Actual capillaries have radii of about $5-10$ micrometers, much smaller than the value in part (b). Why doesn't the flow in actual capillaries become turbulent?

Key Concepts

Characteristic Dimension and Velocity

The characteristic dimension (such as the diameter of a pipe or vessel) and the fluid's velocity are essential parameters in assessing flow regimes. They help scale the problem and determine the relative impact of viscous versus inertial forces in a system.

Fluid Viscosity

Viscosity is a measure of a fluid's resistance to deformation and flow. It quantifies the internal friction within the fluid, affecting how easily it can flow and influencing the balance between viscous and inertial forces.

Transition Region

The transition region refers to the intermediate state between laminar and turbulent flow. In this region, the flow intermittently shifts between orderly and chaotic behavior, making it challenging to predict precisely.

Fluid Density

Fluid density, defined as mass per unit volume, plays a vital role in the dynamics of fluid flow by contributing to the inertial forces. It is one of the key parameters in calculating the Reynolds Number.

Laminar Flow

Laminar flow is characterized by smooth, orderly fluid motion where layers slide past one another with minimal mixing. This type of flow typically occurs at lower Reynolds numbers when viscous forces dominate the flow dynamics.

Reynolds Number

The Reynolds Number is a dimensionless parameter used to predict the behavior of flow in fluids. It compares the magnitude of inertial forces to viscous forces and is crucial in determining whether the flow will be laminar, turbulent, or in transition.

Turbulent Flow

Turbulent flow involves chaotic and irregular fluid motion with significant mixing and eddies. This regime occurs at high Reynolds numbers when inertial forces overcome the stabilizing influence of viscosity.

Transcript

00:01 Hi there, so for this problem, we are given that the reynolds number is equal to the density of the fluid, row, the speed b, d is the inner diameter of the pipe, and eta is the viscosity of the fluid.

00:15 So, for part a of this problem, the question is, less model blood of density that we are given in there, 1 .06, thanks then to the 3, and viscosity as a pure liquid, that is in order to fat that it contains red blood cells, so it's supposed it is flowing in a large artery of radius that we are also given in there, which is 1 .5 centimeters with a speed that is also given.

00:41 So we need to determine that the flow is laminar.

00:46 Okay, so we just need to substitute these values into the expression that we are given, into the reynolds expression, reynolds number expression.

00:56 So the density that we are given for blood is 1 .06 times 10, to the 3, this in units of kilograms per cubic meter, this times the speed that is 6 .7 times 10 to the minus 2.

01:12 This is in units of meters per second.

01:15 Now the diameter is two times the radius that we are given.

01:19 So that will be 3 centimeters.

01:21 That is 3 times 10 to the minus 2 in meters.

01:25 And this divided by the viscosity that we are also given that value, and that value is 3 times 10.

01:31 To the minus 3 in units of pascal times seconds.

01:37 So, simplifying this, using our calculator, we obtain a value of 710.

01:43 Now, since this value is less than 2 ,300, then with that, we can conclude that the flow is laminar.

01:56 So that's the solution for part a of this problem.

02:00 Now, for part b, we are asked about, imagine that the artery ends in a single capillary so that the radius of the artery reduces to a much smaller value.

02:14 What is the radius of the capillary that will cause the flow to become turbulent? so we are going to denote the situation in part 8 using subscripts 1 and in the expression for the reynolds number for the capillary, which we denotate as the situation too.

02:32 So we incorporate the continuity equation for fluids as the red blood flows into the smaller blood vessel.

02:40 So we will have the renal numbers 2 is equal to the density times this speed 2 times the diameter 2.

02:49 This divided by the meal, what we're going to call mu.

03:02 And then this, we can write this as the density times the speed 1.

03:12 This is the area 1 divided by the area 2.

03:15 In here we have used the continuity equation.

03:18 These 2 times the radius that we are given, the radius 2.

03:23 This divided by mu.

03:28 Then, and we know that the area 1 is pi times the radius 1 to the square, and the area 2 is pi times the radius 2 to the square.

03:40 So canceling and simplifying everything, we will find that this is 2, times the density times the speed 1 times the radius 1 and after the square this divided by mu times the radius 2...

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