Power required to rotate a disc in a fluid

[Achievementhunt]

Homework Statement

This is an optional question given to Fluid mechanic students to work on for leisure.

P = power
ρ = fluid density, rho
ω= angular velocity omega
μ= dynamic viscosity, mu
D= diameter

Homework Equations

Show the that the power required to rotate the disc is given by:

P/(ρ * ω^3 * D^5) =F[(ρ* D^2 * ω)/μ)]

3. My attempt at a solution

The mass flow rate ( upsilon/m-dot) of the fluid flowing over the disc:

υ = ρAv

A= area = (Π * D^2)/4

V = Velocity = (Dω)/2

ω = 2Πf ?

The shearing force from the viscous fluid pressure onto the disc:

F= υv (mass flow rate x velocity)

F= (Π * D^3 * ω)/8

Power = rate of fluid doing work onto disc= Force x Fluid velocity

P = Fv = (Π * D^4 * ω^2)/16

This is where I am stuck, I don't know how to use dynamic viscosity if a thickness, z, of the disc is not given, therefore a velocity gradient cannot be found. If given an alternative method is:

Velocity gradient = dv/dz Therefore the shearing stress is (Tau) τ= μ * dv/dz

Where the inital velocity is zero and z is a constant, replace dv for V in terms of D/2 and dD, differentiate with respect to D to find τ, shear stress.

F= τA

Therefore P = Fv.

Any suggestions? Thanks

 

[Chestermiller]

This is a dimensional analysis problem. Try the pi theorem.

Chet

 

[Achievementhunt]

Thanks for the hint Chet. I will give Pi theorem a try