MAE 150A, Winter 2009
J. D. Eldredge
Homework 2 (5 problems + 1 OPTIONAL), due Friday, January 23
1. Problem 4.23, Wilcox (4.24 in 2nd edition).
2. A vortex has a velocity distribution given by
u
=
u
θ
e
θ
, where
u
θ
=
Γ
2
πr
±
1

e

r
2
/σ
2
²
and Γ and
σ
are constant.
(a) Find the vorticity ﬁeld corresponding to this velocity (see Wilcox
§
D.2 for vorticity in cylin
drical coordinates).
(b)
Accurately
plot both
u
θ
and
ω
versus
r
(taking Γ = 1 and
σ
= 1 for simplicity). Using Matlab
or some other mathematical software is recommended.
(c) Estimate the value of
r
(to two decimal places) at which
u
θ
is maximum (again,
σ
= 1).
3. A spherical air bubble in an incompressible ﬂuid has a radius
R
(
t
) that changes with time. Using
the integral form of conservation of mass, ﬁnd the radial velocity
u
r
(
r
) in the ﬂuid outside the
bubble, in terms of
R
and
dR/dt
. (Hint: consider a spherical control volume with an inner wall
just outside the bubble, and an outer wall at distance
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 Spring '09
 ELDRIDGE
 Fluid Dynamics, spherical air bubble

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