Lecture 10

Question 1

What key feature characterizes the decay of velocity in Stokes flow past a sphere?

Answer 1

The velocity field decays as ∼𝑟^−1 for large 𝑟. This is due to the dominant term in the streamfunction solution.

Question 2

What is the streamfunction 𝜓 useful for in axisymmetric Stokes flow?

Answer 2

It automatically satisfies incompressibility ∇⋅𝑢=0 and reduces the governing equations to a biharmonic form, simplifying analysis.

Question 3

Why is the vorticity 𝜔 not a scalar in axisymmetric spherical flows?

Answer 3

Because the flow is three-dimensional and the vorticity has components in multiple directions, making it necessary to treat ∇×𝜔 as a vector quantity, not just a scalar curl.

Question 4

What boundary conditions are applied in the sphere frame?

Answer 4

u=0 on the sphere surface (no-slip at 𝑟=𝑎)
𝑢→−𝑈_0 as r→∞ (flow far away is uniform).
Pressure vanishes at infinity: 𝑃(∞)=0.

Question 5

After nondimensionalizing the Stokes equations for the sphere problem, what becomes the key dimensionless parameter?

Answer 5

The problem becomes entirely governed by the geometry—there’s no Reynolds number (Re ≪ 1), and hence the problem becomes linear and quasi-steady.

Question 6

What is the form of the velocity components for axisymmetric Stokes flow past a sphere?

Answer 6

Radial:
𝑈_𝑟(𝑅,𝜃)=𝑓(𝑅)cos𝜃
Angular:
𝑈_𝜃(𝑅,𝜃)=𝑔(𝑅)sin𝜃

Question 7

What are the 3 main ODEs solved for in the velocity-pressure formulation?

Answer 7

Continuity equation (relates 𝑓 and 𝑔)
𝑅-momentum (relates 𝑓, 𝑔, and ℎ)
𝜃-momentum (used to close the system with pressure)

Question 8

What is the final expression for the pressure field around the sphere?

Answer 8

P(R,θ)= 3/2R^2 cosθ
This shows high pressure at the front of the sphere and low at the rear.

Question 9

What is the hydrodynamic drag force on the sphere?

Answer 9

F_H=−6πμaU_0e^z
​This is Stokes’ drag law for low Reynolds number flow.

Question 10

How is the terminal velocity for a buoyant sphere derived?

Answer 10

y equating the net buoyancy force to the Stokes drag:

𝑉=2/9 (𝑎^2𝑔Δ𝜌/𝜇)