What Distance Negates the Dipole's Influence in Stokes Flow Around a Sphere?

[funcosed]

Homework Statement

The flow due to translation of a sphere in a Newtonian fluid at rest is given by the following streamfunction,

ψ(r,Θ) = (1/4)Ua²(3r/a - a/r)sin²Θ

which consists of a stokeslet and a potential dipole. If the contribution of the dipole is less than 1% it can be considered negligible. At what distance is it negligible?

Homework Equations

The origin is at the centre of the sphere and the axis Θ=0 is parallel to velocity vector U
U = |U| is the magnitude of the velocity
a is the radius of the sphere

The Attempt at a Solution

No idea where to start with this one, any tips greatly appreciated.

 

[funcosed]

So have got started at least, decomposing ψ into

ψ_s = Crsin²Θ the stokeslet

ψ_d = (D/r)³sin²Θ the dipole

C = (3/4)*Ua D = (-3/4)*Ua

then using the stokes streamfunction

u_r = 2(C/r + D/r³)cosΘ
and
u_Θ = (-C/r - D/r³)sin²Θ
Not sure if these are right?

then, u = (2C/r)cosΘ - (C/r)sinΘ

and large r behavior is like 1/r.

but this still doesn't really answer the question?

 

[funcosed]

Can I simplt compare the C/r and D/r³ terms,
i.e. dipole is important for D/r³ > 0.01(C/r) ?

 

[funcosed]

Was way off here its simply compare (a³U/4r)sin²Θ and 0.01(3raU/4)sin²Θ to get

r > a²/0.03