2012-19.md

course	course_year	question_number	tags	title	year
Fluid Dynamics	IB	19	IB 2012 Fluid Dynamics	Paper 1, Section I, A	2012

Viscous fluid, with viscosity $\mu$ and density $\rho$ flows along a straight circular pipe of radius $R$. The average velocity of the flow is $U$. Define a Reynolds number for the flow.

The flow is driven by a constant pressure gradient $-G>0$ along the pipe and the velocity is parallel to the axis of the pipe with magnitude $u(r)$ that satisfies

$$\frac{\mu}{r} \frac{\mathrm{d}}{\mathrm{d} r}\left(r \frac{\mathrm{d} u}{\mathrm{~d} r}\right)=-G,$$

where $r$ is the radial distance from the axis.

State the boundary conditions on $u$ and find the velocity as a function of $r$ assuming that it is finite on the axis $r=0$. Hence, show that the shear stress $\tau$ at the pipe wall is independent of the viscosity. Why is this the case?