In a deep water tank a slow upwelling is driven above a diffuser and spreads out…

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

In a deep water tank a slow upwelling is driven above a diffuser and spreads out beneath the free surface. Take

z

upward from the free surface, so

z

is negative in the water, and

x

y

horizontal. The steady field measured near the axis is

u = Sx, \qquad v = Sy, \qquad w = -2Sz, \qquad S = 0.15\ \mathrm{s^{-1}},

with coordinates in metres. The water has uniform density

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

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

, the atmosphere above the surface is at

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

, and the pressure may be treated as hydrostatic,

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

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

P

x = 0.30\ \mathrm{m}

y = -0.20\ \mathrm{m}

z = -0.80\ \mathrm{m}

. ((c)) Find

Dp/Dt

for the parcel at

P

and say whether the pressure it carries rises or falls. ((d)) The sensor at

P

hangs from a nylon line of length

5.0\ \mathrm{m}

and diameter

1.2\ \mathrm{mm}

under a tension of

12\ \mathrm{N}

. The nylon is elastic with Young modulus

3.0\ \mathrm{GPa}

and the weight of the line itself is ignored. Find its extension.

Browse the problem library