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 = ρvd μ where ρ 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 1.35 cm with a speed of 0.0680 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.) (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? In the human body, the artery branches into approximately 10 billion capillaries, not the single capillary in part (b). The area of the sum of all capillaries is the area of the artery, and so the blood flows through the capillaries through the artery. In each capillary, given this speed v and the small diameter d of 5–10 micrometers, the Reynolds number is small and indicates laminar flow.

Answer 1