Weekly Problem Set 4 - University of Cape Town

University of Cape Town
Department of Physics
PHY209S
2005
Weekly Problem Set 4VA
Instructions: Attempt all these problems. You may work with your friends, but the work you hand in
must be your own effort. Full solutions should be handed on Friday 19 August 2005.
1. The velocity distribution for the flow of an incompressible fluid is given by
v = (3 − x) î + (4 + 2y) ĵ + (2 − z) ˆk m s -1
Show that this satisfies the requirements of the continuity equation.
2. For the velocity field given by v = 6x î − 2yz ĵ + 3 ˆk m s -1 , determine where the flow field is
incompressible.
3. The stream function for a two dimensional, incompressible flow field is given by Ψ = 2x − 2y.
Sketch the streamlines for this fluid flow.
4. The velocity field in a flow is given by v = x 2 y î + x2 t ĵ m s-1
(a)
(b)
Is this flow steady or unsteady? Why?
Plot the streamline through the origin at times t = 0, 1 and 2 seconds.
8. A velocity field is given by v = Ax î − Ay ĵ where A = 0.3 s-1 and x, y are in metres.
(a) Obtain an equation for the streamlines in the xy-plane.
(b) Determine the velocity of a particle at (2,8,0).
(c) If the particle passing through the point (2,8,0) is marked at time t = 0, determine the location of
the particle at time t = 6 s.
(d) What is the velocity of the particle at t = 6 s?
(e) Obtain an equation for the pathlines.
5. Give the velocity field
2
v = 2 t ˆi + xz ˆj − t y kˆ
m s -1 ,
(a)
(b)
(c)
(d)
Find the acceleration D v following a fluid particle.
Dt
Is the flow field steady? Explain your answer.
Is this flow field incompressible? Explain your answer.
Is this flow field irrotational? Explain your answer.

6. A Venturi meter is used to measure the flow speed
of a fluid in a pipe. The meter is connected
between two sections of the pipe where the crosssectional
area A of the entrance and exit of the
meter matches the pipe’s cross-sectional area.
Between the entrance and exit, the fluid flows
from the pipe with speed v and then through a
narrow “throat” of cross-sectional area a with
speed V. A manometer connects the wider portion
of the meter to the narrower portion.
V
v
The change in the fluid’s speed is accompanied by a change ∆p in the fluid’s pressure, which
causes a height difference h of the liquid in the two arms of the manometer.
h
manometer
(a)
By applying Bernoulli’s equation and the equation of continuity to points 1 and 2 in the figure, show
2
2a
∆p
that
v = where ρ is the density of the fluid.
2 2
ρ A − a
( )
(b) Suppose now that the fluid is water, that the cross-sectional areas are 93 cm 2 in the pipe and 51 cm 2
in the throat, and that the pressure is 75 mmHg in the pipe and 29 mmHg in the throat. What is the
rate of water flow in the pipe?
7. Two dimensional steady state flow is described by
v = (y − z + 2) î + (4xy + 4z) ĵ m s -1 .
Calculate the circulation of this field within the rectangle delimited by x = 1 m to 3 m and
y = 0 m to 2 m.
ab/08/05