course |
course_year |
question_number |
tags |
title |
year |
Fluid Dynamics |
IB |
23 |
|
Paper 3, Section II, C |
2019 |
A cubic box of side $2 h$, enclosing the region $0<x<2 h, 0<y<2 h,-h<z<h$, contains equal volumes of two incompressible fluids that remain distinct. The system is initially at rest, with the fluid of density $\rho_{1}$ occupying the region $0<z<h$ and the fluid of density $\rho_{2}$ occupying the region $-h<z<0$, and with gravity $(0,0,-g)$. The interface between the fluids is then slightly perturbed. Derive the linearized equations and boundary conditions governing small disturbances to the initial state.
In the case $\rho_{2}>\rho_{1}$, show that the angular frequencies $\omega$ of the normal modes are given by
$$\omega^{2}=\left(\frac{\rho_{2}-\rho_{1}}{\rho_{1}+\rho_{2}}\right) g k \tanh (k h)$$
and express the allowable values of the wavenumber $k$ in terms of $h$. Identify the lowestfrequency non-trivial mode $(\mathrm{s})$. Comment on the limit $\rho_{1} \ll \rho_{2}$. What physical behaviour is expected in the case $\rho_{1}>\rho_{2}$ ?