Incompressible Flow 3 - Stokes Flow over a Sphere

Question 1

Examples of Stokes Flows

Answer 1

1cm diameter particle moving in oil at 1cm/s (Re ~ 0.1)
Aerosols/dust/droplets (e.g. 1um in diameter moving in the air can have Re«1)
Small droplets from an inhaler
PM2.5 - combustion related

Question 2

What is Stokes flow over a sphere?

Answer 2

Re «1
So either fluid velocity or sphere diameter is very small or viscosity is high

Question 3

What does a low Reynolds number flow indicate?

Answer 3

Inertial force of flow is small compared with viscous force, so convection (inertia) term is small and can be ignored

Question 4

What is balanced in the Stokes’ equation?

Answer 4

Pressure force is balanced by shear stress only

Question 5

What is notable about the magnitudes of the gauge pressure?

Answer 5

Symmetrical before and after, but opposite signs
(to find sphere surface pressure, set r = a)

Question 6

How does shear stress vary around a sphere?

Answer 6

Larger close to surface, diminishes away from the sphere

Question 7

Where does pressure peak around a sphere?

Answer 7

Stagnation points (i.e. if flow is from left to right, pressure peaks at left and right side of sphere)

Question 8

Where does shear stress/friction force peak around a sphere?

Answer 8

Where velocity gradient is highest (i.e. if flow is from left to right, shear peaks at top and bottom of sphere)

Question 9

Points about streamlines around a sphere

Answer 9

Streamlines (and therefore velocity distribution) are entirely independent of fluid viscosity because this will be balanced by pressure gradients
Streamlines are symmetrical before and after sphere due to weak convective acceleration (being ignored)

Question 10

Flow observations around the sphere

Answer 10

Linear pressure distribution on surface
Flow is retarded all the way to the far region
No flow acceleration near top of sphere