| course |
course_year |
question_number |
tags |
title |
year |
Fluid Dynamics |
IB |
20 |
|
Paper 1, Section II, A |
2021 |
A two-dimensional flow is given by a velocity potential
$$\phi(x, y, t)=\epsilon y \sin (x-t)$$
where $\epsilon$ is a constant.
(a) Find the corresponding velocity field $\mathbf{u}(x, y, t)$. Determine $\boldsymbol{\nabla} \cdot \mathbf{u}$.
(b) The time-average $\langle\psi\rangle(x, y)$ of a quantity $\psi(x, y, t)$ is defined as
$$\langle\psi\rangle(x, y)=\frac{1}{2 \pi} \int_{0}^{2 \pi} \psi(x, y, t) d t .$$
Show that the time-average of this velocity field is zero everywhere. Write down an expression for the acceleration of fluid particles, and find the time-average of this expression at a fixed point $(x, y)$.
(c) Now assume that $|\epsilon| \ll 1$. The material particle at $(0,0)$ at $t=0$ is marked with dye. Write down equations for its subsequent motion. Verify that its position $(x, y)$ for $t>0$ is given (correct to terms of order $\epsilon^{2}$ ) by
$$\begin{aligned}
x &=\epsilon^{2}\left(\frac{1}{4} \sin 2 t+\frac{t}{2}-\sin t\right) \\
y &=\epsilon(\cos t-1)
\end{aligned}$$
Deduce the time-average velocity of the dyed particle correct to this order.