MA330: Assignment 9
To be turned in May 17th at the start of class.
1. Suppose that the temperature of a heated object obeys the heat equation ut = K∆u everywhere in the domain Ω.
In addition, suppose that the surface of the object, ∂Ω, is insulated so that the heat flux through the surface is 0.
(a) Show that
d
dt
ZZZ
ZZZ
2
k∇uk dV = 2K
(∇u) · (∇∆u)dV
Ω
Ω
(b) Use the integration by parts technique derived in class to prove
ZZZ
ZZ
ZZZ
(∇u) · (∇∆u)dV =
(∆u)(n̂ · ∇u)dS −
(∆u)2 dV
Ω
∂Ω
Ω
(c) Show that
d
dt
ZZZ
ZZZ
2
k∇uk dV = −
K
Ω
2
u2t dV ≤ 0
Ω
RRR
Notice that f (t) =
k∇uk2 dV is never negative, and is 0 only if the temperature profile is perfectly flat.
Ω
This means f (t) is a measure of how flat the temperature profile is, that is, how uniformly the thermal energy
has spread throughout the object. What does the inequality above imply about temperature profiles as time
passes? Explain why this is intuitive.
2. Suppose fluid is placed between two concentric cylinders, with the outer cylinder spun at a constant rate of ω
revolutions per unit time and the inner cylinder held motionless. Let the outer cylinder have radius Ro and the
inner cylinder have radius Ri . Suppose the fluid is incompressible, the flow of fluid is steady, and the pressure only
depends on r.
(a) Let (r, θ, z) be a cylindrical coordinate system, with r = 0 corresponding to the center of the inner cyclinder.
Explain why the conditions described above lead to the following problem:
ρ(v · ∇)v = µ∆v − ∇P,
∇·v =0
v(inner boundary) = 0,
v(outer boundary) = Ro ωh− sin θ, cos θ, 0i.
where v = f (r)h− sin θ, cos θ, 0i.
(b) Recall from class the del operator in cylindrical coordinates.
∇ = ∇r
∂
∂
∂
1
∂
∂
∂
+ ∇θ
+ ∇z
= hcos θ, sin θ, 0i
+ h− sin θ, cos θ, 0i
+ h0, 0, 1i
∂r
∂θ
∂z
∂r r
∂θ
∂z
Use this to prove
∇ · v = 0,
(v · ∇)v = −
f (r)2
hcos θ, sin θ, 0i,
r
∇P = P 0 (r)hcos θ, sin θ, 0i.
(c) Recall from class the Laplacian operator in cylindrical coordinates.
∆=
1 ∂
∂2
1 ∂2
∂2
+ 2+ 2 2+ 2
r ∂r ∂r
r ∂θ
∂z
Use this to prove
∆v =
1 ∂
r ∂r
r
∂f
∂r
1
−
f (r)
r2
h− sin θ, cos θ, 0i.
(d) By applying parts a), b), and c), show that
f (r)2
1 ∂
∂f
f (r)
−ρ
h− sin θ, cos θ, 0i − P 0 (r)hcos θ, sin θ, 0i
hcos θ, sin θ, 0i = µ
r
− 2
r
r ∂r
∂r
r
and then use orthogonality to conclude
r2 f 00 (r) + rf 0 (r) − f (r) = 0,
P 0 (r) =
ρ
f (r)2 .
r
(e) Show that the general solution to the ODE for f is
f (r) =
c1
+ c2 r
r
Fit the boundary data, then integrate to find the pressure difference between the inner and outer cylinder.
2