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 $$\operatorname{Re}=\frac{\rho v d}{\eta}$$ 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: $\cdot $ If Re $<2300$, the flow is laminar. $\cdot $ If $2300<\mathrm{Re}<4000$, the flow is in a transition region between laminar and turbulent. $\cdot $ 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:
$\cdot $ If Re $<2300$, the flow is laminar.
$\cdot $ If $2300<\mathrm{Re}<4000$, the flow is in a transition region between laminar and turbulent.
$\cdot $ 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?

Added by Stephanie C.

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