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Convection between two plates

  1. Nov 12, 2014 #1
    1. The problem statement, all variables and given/known data

    From my lecture notes, here are the equations for convection between two plates. I have derived equations 9.6, 9.7 and 9.8. But for 9.4 there's a problem when gravity becomes involved.

    rcltuh.png

    2. Relevant equations

    Navier stokes: ## \rho \frac{D \vec u}{D t} = -\nabla p + \mu \nabla^2 \vec u + \vec F ##

    3. The attempt at a solution

    However, I was reading through Tritton's book on flows where he detailed the derivation:

    Starting from the navier-stokes equation:

    [tex]\rho \frac{D \vec u}{D t} = -\nabla p + \mu \nabla^2 \vec u + \vec F [/tex]

    where ##\vec F## represents contribution of other forces (such as gravity).

    They then begin to define ##\vec F##:

    By letting density vary, we have ##\rho = \rho_0 + \Delta \rho##. Gravitational acceleration can be defined through a potential: ##\vec g = -\nabla \phi = -\nabla gz##. Thus,

    [tex]\vec F = -(\rho_0 + \Delta \rho)\nabla \phi = -\nabla(\rho_0 \phi) + \Delta \rho \vec g[/tex]

    Introducing ##P = p + \rho_0 \phi##, navier stokes becomes:

    [tex] \rho_0 \frac{D\vec u}{D t} = -\nabla P + \mu \nabla^2 \vec u + \Delta \rho \vec g [/tex]
     
  2. jcsd
  3. Nov 12, 2014 #2
    You haven't told us what your problem is.

    Chet
     
  4. Nov 13, 2014 #3
    Equations 9.5 from the lecture notes and eqn from the book doesn't match
     
  5. Nov 13, 2014 #4
    What it is about them that you feel doesn't match?

    Chet
     
  6. Nov 13, 2014 #5
    Substituting ##P## inside and changing ##\nabla## to ##\frac{\partial}{\partial z}##, it gives:

    [tex]\rho_0 \frac{D\vec u}{D t} = -\nabla P + \mu \nabla^2 \vec u + \Delta \rho \vec g[/tex]
    [tex]\rho_0 \frac{D\vec u}{D t} = - \frac{\partial}{\partial z}(p + rho_0 \phi) + \mu \nabla^2 \vec u + \Delta \rho \vec g [/tex]
    [tex]\rho_0 \frac{D\vec u}{D t} = - \frac{\partial}{\partial z}(p - rho_0 z \vec g) + \mu \nabla^2 \vec u + \Delta \rho \vec g [/tex]
    [tex]\rho_0 \frac{D\vec u}{D t} = - \frac{\partial p}{\partial z} + \rho_0 \vec g + \mu \nabla^2 \vec u + (\rho - \rho_0) \vec g [/tex]
    [tex]\rho_0 \frac{D\vec u}{D t} = - \frac{\partial p}{\partial z} + \mu \nabla^2 \vec u + \rho \vec g [/tex]
    [tex]\frac{D\vec u}{D t} = - \frac{1}{\rho_0} \frac{\partial p}{\partial z} + \frac{1}{\rho_0}\mu \nabla^2 \vec u + \frac{\rho}{\rho_0} \vec g [/tex]
     
  7. Nov 13, 2014 #6
    It appears that the p's in Eqns. 9 are what you are calling P. The ρ0g has apparently been absorbed into the pressure term.

    Chet
     
  8. Nov 13, 2014 #7
    I don't think that's right, as applying the same equation in the horizontal direction (w) gives eqn 9.4. The small ##p## in eqn 9.4 should not include ##\rho_0\phi##.
     
  9. Nov 13, 2014 #8
    The derivative of ##\rho_0g## is zero in the horizontal direction.
     
  10. Nov 13, 2014 #9
    Ah that's true. Quite annoying when the lecture notes don't specify the derivation, but this makes sense! Thanks alot.
     
  11. Nov 15, 2014 #10
    Got the answer, thanks alot!
     
    Last edited: Nov 15, 2014
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