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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