CHE 527 Advanced Applied Mathematical Analysis in Chemical Engineering Fall 2006
Homework 9
Due in class on Tuesday, November 21.
1. Potential flow past a sphere is governed by
∇2φ=1
r2
∂
∂r r2∂φ
∂r +1
r2sin θ
∂
∂θ sin θ∂φ
∂θ = 0
where
v=∇φ=∂φ
∂r er+1
r
∂φ
∂θ eθ
is the velocity. Use separation of variables to solve for φ(r, θ) in the domain a≤r≤ ∞ subject to the
B.C.’s ∂φ
∂r = 0 on r=a, and ∂φ
∂r =U∞cos θas r→ ∞.
2. Consider the diffusive mass transfer (∇2c= 0) of a solute inside a liquid drop subject to a mass transfer
boundary condition (−D ∂c
∂r =k(c−c0) at r=a). For an axisymmetric case c0=c1+c2cos θ, where
c1and c2are known constants, find c(r,θ ) inside the drop. Do not leave any integrals unevaluated.
3. Use a Laplace transform to solve the wave equation:
∂2φ
∂x2=1
c2
∂2φ
∂t2
subject to the boundary condition
φ(0, t) = 0
and to the initial conditions
φ(x, 0) = 0,
∂φ(x, 0)
∂t = 1.
4. The motion of a mass on a spring with damping is governed by
d2x
dt2+ 2αdx
dt + (α2+β2)x=1
mf(t)
where α=c/2m,β=pω2
0−α2, and ω2
0=k/m. For unforced motion (f(t) = 0), starting from rest
at x=x0, use LaPlace transforms to determine x(t) for β= 0 and β > 0 (real). For each case, make
a qualitative sketch of xvs. t.
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