| course |
course_year |
question_number |
tags |
title |
year |
Fluid Dynamics |
IB |
10 |
|
Paper 1, Section II, C |
2020 |
Steady two-dimensional potential flow of an incompressible fluid is confined to the wedge $0<\theta<\alpha$, where $(r, \theta)$ are polar coordinates centred on the vertex of the wedge and $0<\alpha<\pi$.
(a) Show that a velocity potential $\phi$ of the form
$$\phi(r, \theta)=A r^{\gamma} \cos (\lambda \theta),$$
where $A, \gamma$ and $\lambda$ are positive constants, satisfies the condition of incompressible flow, provided that $\gamma$ and $\lambda$ satisfy a certain relation to be determined.
Assuming that $u_{\theta}$, the $\theta$-component of velocity, does not change sign within the wedge, determine the values of $\gamma$ and $\lambda$ by using the boundary conditions.
(b) Calculate the shape of the streamlines of this flow, labelling them by the distance $r_{\min }$ of closest approach to the vertex. Sketch the streamlines.
(c) Show that the speed $|\mathbf{u}|$ and pressure $p$ are independent of $\theta$. Assuming that at some radius $r=r_{0}$ the speed and pressure are $u_{0}$ and $p_{0}$, respectively, find the pressure difference in the flow between the vertex of the wedge and $r_{0}$.
[Hint: In polar coordinates $(r, \theta)$,
$$\nabla f=\left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}\right) \quad \text { and } \quad \nabla \cdot \mathbf{F}=\frac{1}{r} \frac{\partial}{\partial r}\left(r F_{r}\right)+\frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta}$$
for a scalar $f$ and a vector $\mathbf{F}=\left(F_{r}, F_{\theta}\right)$.]