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Nonisothermal Parallel Plate Flow Problem

  1. Nov 6, 2011 #1
    1. The problem statement, all variables and given/known data

    The problem wants us to develop an equation for velocity distribution and the mass flow rate of a fluid, flowing between 2 parallel plates:

    The distance between the two plates is a.
    The top plate is maintained at constant T1; the bottom plate is maintained at constant T2.
    The coordinate normal to the plates is X; the coordinate in the direction of flow is Z.
    (Thus, the bottom plate is defined as x = 0 and the top plate x = a).

    To attempt the problem, we are told to solve for solve for [itex]\tau[/itex][itex]_{xz}[/itex] from the equation of motion, and then identify the velocity distribution and mass flow rate using the first equation below:

    2. Relevant equations

    [itex]\tau[/itex][itex]_{xz}[/itex] = - [itex]\mu[/itex][itex]\frac{dv_{z}}{dx}[/itex]

    [itex]\mu[/itex] = [itex]\mu[/itex][itex]_{o}[/itex] (1 + [itex]\beta[/itex] ( T[itex]_{o} - T_{1}[/itex]) x } / {T_{o}^2 a}[/itex] )

    3. The attempt at a solution

    Solving for [itex]\tau[/itex][itex]_{xz}[/itex] from the equation of motion:

    [itex]\tau[/itex][itex]_{xz}[/itex] = C1 + x [itex]\frac{dP}{dz}[/itex] = [itex]\mu[/itex] [itex]\frac{dv_{z}}{dx}[/itex]

    From here, I use no slip boundary conditions at x = 0 and x = a and solve the above equation plugging in for [itex]\mu[/itex].

    From this I get a very convoluted expression:

    v[itex]_{z}[/itex] = [itex]\mu[/itex][itex]_{o}[/itex] ( [itex]\frac{T_{o}^2 a}{\beta ( T_{o} - T_{1} )}[/itex] + ( [itex]\frac{-a^6 T_{o}^3}{ln(1 + \beta ( T_{o} - T_{1} ) / T_{o}^2) \beta^5 (T_{o} - T_{1})^5}[/itex]) ln(1 + \beta ( T_{o} - T_{1} x / (a T_{o}^2) [itex]\frac{dP}{dz}[/itex]

    However, plugging in T1 = To, should yield the simple equation for isothermal flow. Instead, it = 0.

    Any advice?

    PS. I accidentally solved the problem the first time through with [itex]\tau[/itex][itex]_{xz}[/itex] = [itex]\frac{1}{\mu}[/itex] [itex]\frac{dvz}{dx}[/itex], and it worked nicely. Is there a good way to justify this? Or am I simply insane.
     
    Last edited: Nov 6, 2011
  2. jcsd
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