Serway & Jewett — Fluid Mechanics: Problem 38

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 \[Re = \frac{\rho v d}{\eta}\] 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: If \(Re If \(2300 If \(Re > 4000\), the flow is turbulent. (a) Let’s model blood of density \(1.06 \times 10^{3} \text{ kg/m}^{3}\) and viscosity \(3.00 \times 10^{-3} \text{ Pa} \cdot \text{s}\) 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 \(1.50 \text{ cm}\) with a speed of \(0.0670 \text{ m/s}\). 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?