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Power required to rotate a disc in a fluid

  1. Aug 31, 2015 #1
    1. The problem statement, all variables and given/known data

    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

    2. Relevant 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
     
  2. jcsd
  3. Aug 31, 2015 #2
    This is a dimensional analysis problem. Try the pi theorem.

    Chet
     
  4. Aug 31, 2015 #3
    Thanks for the hint Chet. I will give Pi theorem a try
     
    Last edited: Aug 31, 2015
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