Fluid Dynamics 3
2011/12
Sheet 4
Questions 1,2,3 to be handed in on 11th November
1. Consider a vortex with centre along the ˆ
z
axis, so the flow field is of the form
u
=
f
(
r
)
ˆ
θ
in
cylindrical polars.
(i) Show that
∇ ·
u
= 0.
(ii) Using Euler’s equation in cylindrical coordinates, show that the pressure satisfies
p
′
(
r
) =
ρ
f
2
(
r
)
r
.
(iii) Compute the vorticity
ω
=
∇ ×
u
and show that it vanishes if
f
=
A/r
. Compute the
pressure for this case.
2. Consider the impingement and subsequent spreading of a horizontal, twodimensional jet of
water of speed
U
and width
d
onto a plane surface, inclined at an angle
α
to the horizontal.
Sufficiently far from the point of impingement the jet flow becomes smooth, uniform and
parallel to the inclined plane and the pressure returns to atmospheric pressure.
U
U
2
U
1
b
2
b
1
d
α
(i) Use an expression of the conservation of mass to show that
U
1
b
1
+
U
2
b
2
=
Ud,
where
U
1
and
b
1
are the velocity and breath of the flow along the plane on one side of
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 Fall '11
 Eggers
 Fluid Dynamics, Euler’s Equation, suitably constructed control, momentum integral theorem

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