MATH 292: Sp. Topics – Fluid Dynamics - HW 2 Spring 2010
Due before class on 03 February 2010. List any resources used in your solution and the names of
students you work with.
Problem 1:
(a) Consider a volume V(t)with surface S(t)with outward normal n(x, t)at time t, and Smoves
with the velocity field u(x, t). In class, we showed that for any vector or scalar field φ(x, t)
(Reynolds Transport Theorem),
D
Dt ZV(t)
φdV =ZV(t)
∂φ
∂t dV +ZS(t)
φu·ndS. (1)
Assuming that φ(x, t)is a scalar, show that this can be written
D
Dt ZV(t)
φdV =ZV(t)Dφ
Dt +φ∇ · udV.
(b) Consider the Eulerian form of the momentum equation:
ZV(t)
∂(ρu)
∂t dV +ZS(t)
(ρu)u·ndS =F,(2)
for external forcing F. Show that Eq. 2 can be written as simply
ZV(t)
ρDu
Dt dV =F.(3)
Starting with Eq. 2, you will want to apply the Divergence theorem for dyads.
Problem 2: If V(t)is a material volume and xis the position vector relative to a fixed origin, show
that D
Dt ZV(t)
ρ(x×u)dV =ZV(t)
ρ(x×Du
Dt )dV.
This result shows that the rate of change of angular momentum of the fluid is equal to its moment of
fluid acceleration.
Problem 3: As an example of the difference between streamlines, streaklines, and pathlines, con-
sider the two-dimensional flow field u(x, t) = (u(x, t), v(x, t),0), where
u=x(1 + 2 t), v =y.
Compare the streamline through the point (1,1) at time t= 0 with the pathline for a particle
released at (1,1) at time t= 0, and also with the streakline at t= 0 produced by tracers released
from (1,1). Show your results in a comparative graphical plot.
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