2014-20.md

course	course_year	question_number	tags	title	year
Fluid Dynamics	IB	20	IB 2014 Fluid Dynamics	Paper 2, Section I, B	2014

Consider the steady two-dimensional fluid velocity field

$$\mathbf{u}=\left(\begin{array}{l} u \\ v \end{array}\right)=\left(\begin{array}{ll} \epsilon & -\gamma \\ \gamma & -\epsilon \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)$$

where $\epsilon \geqslant 0$ and $\gamma \geqslant 0$. Show that the fluid is incompressible. The streamfunction $\psi$ is defined by $\mathbf{u}=\boldsymbol{\nabla} \times \boldsymbol{\Psi}$, where $\boldsymbol{\Psi}=(0,0, \psi)$. Show that $\psi$ is given by

$$\psi=\epsilon x y-\frac{\gamma}{2}\left(x^{2}+y^{2}\right)$$

Hence show that the streamlines are defined by

$$(\epsilon-\gamma)(x+y)^{2}-(\epsilon+\gamma)(x-y)^{2}=C$$

for $C$ a constant. For each of the three cases below, sketch the streamlines and briefly describe the flow. (i) $\epsilon=1, \gamma=0$, (ii) $\epsilon=0, \gamma=1$, (iii) $\epsilon=1, \gamma=1$.