Stokes flow past 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

 

[paranormal]

I would approach this problem by first understanding the physical significance of the streamfunction given in the homework statement. The streamfunction represents the velocity potential of the fluid flow around a translating sphere in a Newtonian fluid. The first term in the streamfunction, (1/4)Ua2(3r/a - a/r), represents the stokeslet, which is the flow due to a point force acting on the fluid. The second term, sin2Θ, represents the potential dipole, which is the flow due to two equal and opposite point forces acting on the fluid.

To determine the distance at which the contribution of the potential dipole becomes negligible, we can compare the magnitudes of the two terms in the streamfunction. Since the question states that the contribution of the dipole is less than 1%, we can set up the following inequality:

|sin2Θ| < 0.01|(1/4)Ua2(3r/a - a/r)|

Using the trigonometric identity sin2Θ = 2sinΘcosΘ, we can simplify the inequality to:

|sinΘcosΘ| < 0.005|(1/4)Ua2(3r/a - a/r)|

Since sinΘ and cosΘ are bounded between -1 and 1, we can further simplify the inequality to:

|1/4(Ua2)(3r/a - a/r)| < 0.005|(1/4)Ua2(3r/a - a/r)|

This means that the contribution of the potential dipole becomes negligible when:

|3r/a - a/r| < 0.005

Solving for r, we get:

2.99a < r < 3.01a

Therefore, the distance at which the contribution of the potential dipole becomes negligible is approximately within 2.99a and 3.01a from the centre of the sphere. Beyond this distance, the stokeslet dominates the flow and the contribution of the potential dipole can be considered negligible.