Question

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 Re = rhovd μ where rho is the density of the fluid, v is its speed, d is the inner diameter of the pipe, and μ is the viscosity of the fluid. Viscosity is a measure of the internal resistance of a liquid to flow and has units of Pa · s. The criteria for the type of flow are as follows. • If Re < 2,300, the flow is laminar. • If 2,300 < Re < 4,000, the flow is in a transition region between laminar and turbulent. • If Re > 4,000, the flow is turbulent. (a) Let's model blood of density 1.06 103 kg/m3 and viscosity 3.00 10-3 Pa · 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 1.25 cm with a speed of 0.0650 m/s. Show that the flow is laminar. (State the Reynolds number of the flow, which will be less than 2,300, indicating laminar flow.) (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? (Use the minimum Reynolds number for which flow is fully turbulent.)

Answer 1

a) Re= 574.17

b) r= 8.71 cm

Step-by-step explanation:

In order to solve this problem, we will need to use the formula for the Reynolds number:

Re=ρ*v*d/μ

All of the required data is already given in the problem, but before we use the above-mentioned formula, we need to convert the data to SI units, as follows:

• The density already has SI units ( 1.06 *10³ kg/m³)
• The radius is 1.25 cm, which converts to 0.0125 m. Thus, the inner diameter is 0.025 m
• The viscosity already has SI units (3.00 * 10⁻³ Pa · s)
• The speed already has SI units (0.0650 m/s)

a) Now we proceed to calculate Reynolds number:

Re=1.06 *10³ kg/m³ * 0.0650 m/s * 0.025 m / (3.00 * 10⁻³ Pa · s)

Re=574.17

Re<2,300 ; thus the flow is laminar.

b) To answer this question we use the same equation, and give the Reynolds number a value of 4,000 in order to find out d₂:

4,000= 1.06 *10³ kg/m³ * 0.0650 m/s * d₂ / (3.00 * 10⁻³ Pa · s)

We solve for d₂:

d₂=0.174 m

Thus the radius of the capillary that would cause the flow to become turbulent is 0.174 m / 2= 0.0871 m or 8.71 cm, given that neither the speed nor the viscosity change.

However, in your question you wrote that the artery ends in a capillary so that the radius reduces its value. But the lower the radius, the lower the Reynolds number. And as such, it would not be possible for the flow to turn from laminar to turbulent, if the other factors (such as speed, or density) do not change.