Streamfunctions, Vorticity & Lubrication

Question 1

How do we define a streamfunction?

Answer 1

Question 2

Prove that, for incompressable fluids, the path doesn’t affect the streamfunction.

Answer 2

Question 3

What is the advantage of working with streamfunctions?

Answer 3

We don’t need to use the continuity equation. It’s built straight into the mathematics.

Question 4

By considering the change in streamfunctions, derive the Cauchy-Riemann relations. State them in polar coordinates too.

Answer 4

Question 5

How do we define vorticity?

Answer 5

Question 6

By taking the curl of the momentum equation, state the 3D vorticity equation.

Answer 6

Question 7

What equations do we work with using only 2D vorticity & streamfunctions?

Answer 7

Question 8

What is the 3D vorticity equation in dimensionless units?

Answer 8

Question 9

By setting Reynold’s number to zero, derive the Biharmonic equation.

Answer 9

Question 10

What is non-uniformity?

Answer 10

If you’re working dimensionlessly, but one of the dimension scales isn’t defined, then your analysis only works above or below that scale.

Question 11

Describe the situation in which Reynold’s lubrication equation is used.

Answer 11

Question 12

Given the lubrication equation, what is the volume flux?

Answer 12

Question 13

Given the lubrication equation, what is the volume flux?

Answer 13