Hydrodynamic interaction of particles

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Homework Statement

The flow of a single sphere translating at speed U in a fluid at rest far away from the sphere is given by the streamfunction,

ψ(r,Θ) = (1/4)Ua2(3r/a - a/r)sin2Θ a=radius of sphere

the origin is at the centre of the sphere and the axis Θ = 0 is parallel to U.

It has already been established that the streamfunction consists of a stokeslet and a dipole and at a distance D the dipoles influence is negligible.
Now consisder sedimentation of two spheres A and B separated by a distance d (d >> a and d > D). Both spheres move under the influence of gravity (g = ge_z)at the same speed.
The angle between a line joining the centre of the spheres and the vertical is α (alpha).

The velocity is U = U₀+(a/d)U₁+...

The neglected terms are of order O((a/d)²). What is value of U₀ and show that the next order each sphere induces a velocity (a/d)U₁ on the other one. Provide an expression for U₁ and a rough plot for each sphere.

Homework Equations

Velocity components given by Stokes stream function:

U_r = (1/r²sinΘ)∂ψ/∂Θ

U_Θ = (-1/rsinΘ)∂ψ/∂r

The Attempt at a Solution

Taking the origin to be at the centre of sphere A its stream function is the expression given above. The stream function for B is the same but with r translated by d, i.e. r = r+d. Is this correct?
Then I think the superposition of the two can then be used to calculate the velocity via stokes stream function given above. Is this the right approach?
The I take it U₀ will be the part of the expression that has no a/d term and U₁ will be the part with an a/d term ??

 

[funcosed]

and continuing .....

Ur = (1/r2sinΘ)∂ψ/∂Θ
UΘ = (-1/rsinΘ)∂ψ/∂r

for this I get
U = (1/2)UcosΘ[6a/r + 3ad/r² - a³/r³ - a³/(r³+dr³)]
- (1/4)UsinΘ[6a/r + a³/r³ - a³/r(r+dr)²]


Taking terms without a/d for U ...not sure if this is right?

U = (1/2)UcosΘ[6a/r + 3ad/r² - a³/r³] - (1/4)UsinΘ[6a/r + a³/r³]

4. Show that at the next order each sphere induces a velocity (a/d)U₁ on the other. Provide an expression for U₁ and a rough plot of (a/d)U₁ for each of the two spheres.

Can get expression from part 3, (1/2)UcosΘ[-a^3/(r^3+dr^2)]
but this doesn't seem right to me!
Have I gone wrong in the algebra or is it the wrong approach?

5. Show that the angle α is conserved,
require expression for α and then dα/dt = 0
I think I use cos∝ = U₀*t / d
then sub in expression for U0 ? Is this the right idea?

6. Show that the pair of spheres fall down a path making an angle γ with the vertical. Calculate γ as a function of a, d and α. Hence show that the path of spheres is vertical only if the line connecting there centres is vertical or horizontal.

Not sure about this one but if someone could nudge me in the right direction I'd appreciate it.