A large open tank of water is emptied slowly through a slot in its floor

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

Fluids and Continuum Physics, Ch Kinematics of flow: fields, streamlines and the material derivative

Problem

A large open tank of water is emptied slowly through a slot in its floor. Take

z

upward from the undisturbed free surface (so

z

is negative in the water) and

x

horizontal. In the region of interest the steady velocity field is

u = -Sx, \qquad v = 0, \qquad w = Sz, \qquad S = 0.40\ \mathrm{s^{-1}},

with

x

and

z

in metres. The water is a continuum of uniform density

\rho = 1000\ \mathrm{kg\,m^{-3}}

g = 9.8\ \mathrm{m\,s^{-2}}

, the atmosphere above the free surface is at

1.013\times10^{5}\ \mathrm{Pa}

, and the pressure in the tank may be treated as hydrostatic,

p = p_{\mathrm{atm}} - \rho g z

(absolute). ((a)) Test the velocity field for incompressibility. ((b)) Find the absolute pressure at the point

P

x = 0.50\ \mathrm{m}

z = -1.20\ \mathrm{m}

. ((c)) Find

Dp/Dt

for the fluid parcel that is at

P

, and say whether the pressure it carries is rising or falling. ((d)) The probe at

P

hangs from a vertical steel wire of length

2.0\ \mathrm{m}

and cross-sectional area

0.20\ \mathrm{mm^{2}}

under a tension of

30\ \mathrm{N}

. The steel is elastic with Young modulus

200\ \mathrm{GPa}

and the weight of the wire itself is ignored. Find the extension of the wire.

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