2014-19.md

course	course_year	question_number	tags	title	year
Fluid Dynamics	IB	19	IB 2014 Fluid Dynamics	Paper 1, Section I, B	2014

Constant density viscous fluid with dynamic viscosity $\mu$ flows in a two-dimensional horizontal channel of depth $h$. There is a constant pressure gradient $G>0$ in the horizontal $x$-direction. The upper horizontal boundary at $y=h$ is driven at constant horizontal speed $U>0$, with the lower boundary being held at rest. Show that the steady fluid velocity $u$ in the $x$-direction is

$$u=\frac{-G}{2 \mu} y(h-y)+\frac{U y}{h}$$

Show that it is possible to have $d u / d y<0$ at some point in the flow for sufficiently large pressure gradient. Derive a relationship between $G$ and $U$ so that there is no net volume flux along the channel. For the flow with no net volume flux, sketch the velocity profile.