An open cylindrical tank of cross-sectional area is draining through a small hole of…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

Fluid Dynamics, Ch Euler's equation in Lamb's form and the four Bernoulli theorems

Problem

An open cylindrical tank of cross-sectional area

A = 1.0\,\mathrm{m^2}

is draining through a small hole of diameter

2.0\,\mathrm{cm}

in its base. At the instant considered the depth of water above the hole is

h = 1.0\,\mathrm{m}

. The usual treatment of this problem replaces the true unsteady flow by Torricelli's steady, irrotational-flow result

U = \sqrt{2gh}

for the speed through the hole, i.e. it uses Bernoulli's theorem for steady irrotational flow (uniform

B

) rather than the unsteady form

\partial\phi/\partial t + B = F(t)

. ((a)) Using mass conservation between the falling free surface and the escaping jet, find an expression for

\mathrm{d}U/\mathrm{d}t

in terms of

g

, the hole area

a

and the tank area

A

. ((b)) Taking the length scale of the flow between the surface and the hole to be

h

, estimate the size of the unsteady term

\partial\phi/\partial t

relative to the steady term

\tfrac12U^2

, and hence decide whether the steadiness condition of the theorem used is satisfied.

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