course |
course_year |
question_number |
tags |
title |
year |
Fluid Dynamics |
IB |
16 |
|
3.I.10C |
2004 |
State Bernoulli's equation for unsteady motion of an irrotational, incompressible, inviscid fluid subject to a conservative body force $-\nabla \chi$.
A long vertical U-tube of uniform cross section contains an inviscid, incompressible fluid whose surface, in equilibrium, is at height $h$ above the base. Derive the equation
$$h \frac{d^{2} \zeta}{d t^{2}}+g \zeta=0$$
governing the displacement $\zeta$ of the surface on one side of the U-tube, where $t$ is time and $g$ is the acceleration due to gravity.
$$\begin{aligned}
& \frac{\partial \eta}{\partial t}+U \frac{\partial \eta}{\partial x}=\frac{\partial \phi}{\partial y}, \quad \frac{\partial \phi}{\partial t}+U \frac{\partial \phi}{\partial x}+g \eta=0 \quad \text { on } \quad y=0 \\
& \frac{\partial \phi}{\partial y}=0 \quad \text { on } \quad y=-h,
\end{aligned}$$