Fluid Dynamics | Part IB, 2004

State Bernoulli's equation for unsteady motion of an irrotational, incompressible, inviscid fluid subject to a conservative body force χ-\nabla \chi.

A long vertical U-tube of uniform cross section contains an inviscid, incompressible fluid whose surface, in equilibrium, is at height hh above the base. Derive the equation

hd2ζdt2+gζ=0h \frac{d^{2} \zeta}{d t^{2}}+g \zeta=0

governing the displacement ζ\zeta of the surface on one side of the U-tube, where tt is time and gg is the acceleration due to gravity.

ηt+Uηx=ϕy,ϕt+Uϕx+gη=0 on y=0ϕy=0 on y=h,\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}

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