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> Fluids Problem, Flow due to spheres falling in fluid
Posted: Feb 28 2011, 07:54 PM


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This is a long one but interesting.

In this problem the influence of the presence of a particle on the path of a pair of particles falling vertically under gravity in a viscous newtonian fluid is studied. Viscosity, μ, and density, ρ, of the fluid are constant and pressure denoted p. The flow is 3d. The three particles are solid homogeneous spheres, with the same density ρs > ρ. and the same radius a.

1. The flow due to a single sphere in translation is at speed U in a fluid at rest far away from the sphere is given by the following streamfunction in spherical coordinates.

ψ(r,Θ) = (1/4)Ua^2(3r/a - a/r)sin^2Θ

U = |U|

The origin is the centre of the sphere and the axis Θ=0 is parallel to U. The stream function consists of two terms, the stokeslet and a dipole. The contribution of the potential dipole is negligible when it is less than 1% of the stokeslet. From what distance D from the centre of the sphere does the contribution of the dipole become negligible.

My ans
stokeslet = (3/4)Uar(sin^2Θ)
dipole = -(1/4)Ua^3(sin^2Θ)

Dipole negligible when (1/4)Ua^3(sin^2Θ) <= 0.01(3/4)UaD(sin^2Θ)

so D=(a^2)/0.03

2. If the sphere moves simply under the effect of gravity g=-gz where z is the unit vector in the z direction., what is its falling speed.

My ans Get speed from terminal velocity ie when gravity force (fg) = drag force (fd)
fd = fg
6∏μUa = volume[4∏a^3] x density difference[ρs-ρ]x gravity[g
U = (2/9)(a^2(ρs-ρ])/μ)gz

3. Consider two spheres, A and B, falling and seperated by a distance d ( d>>a and d>D). We define ∝ as the angle between a straight line joining the centre of the two spheres and the vertical. Each sphere falls with velocity U

U =U0 + (a/d)U1 +...

where the neglected terms are of order (a/d)^2 or higher.
Find U0.

My ans Taking the origin at centre sphere A its streafunction is simply the function given in part 1. The stream function for sphere B is the same with r = r+d .........Is this right??

The stream function is the sum of the two stream functions and can be used to calculate the velocity components via stokes stream function,

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

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

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

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

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 doesnt seem right to me!!!!
Have I gone wrong in the algebra or is it the wrong approach???

5. Show that the angle ∝ is conserved,
require expression for ∝ and then d∝/dt = 0
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 ∝. 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.

7 - 10 Still to come................. ohmy.gif

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