Question

A flow inside a centrifuge can be approximated by a combination of a central cylinder and a radial line source flow, giving the following potential function:

Ø= a2/r -cosØ + aßlnr = r
Where a is the radius of the central base of the centrifuge and ß is a constant.
a) Provide expressions for the velocities Vr and vo .
b) Find the expression for the stream function.

Answer 1

a) Vr = - a^2/r cosθ + aß / r

Vθ = 1/r [ -a^2/r * sinθ ]

b) attached below

Step-by-step explanation:

potential function

Ø= a^2 /r cosØ + aßlnr ----- ( 1 )

a = radius , ß = constant

a) Expressions for Vr and Vθ

Vr = dØ / dr ----- ( 2 )

hence expression : Vr = - a^2/r cosθ + aß / r

Vθ = 1/r dØ / dθ ------ ( 3 )

back to equation 1

dØ / dr = - a^2/r sinθ + 0 --- ( 4 )

Resolving equations 3 and 4

Vθ = 1/r [ -a^2/r * sinθ ]

b) expression for stream function

attached below