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Homework Statement
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.
Homework Equations
Navier stokes: ## \rho \frac{D \vec u}{D t} = -\nabla p + \mu \nabla^2 \vec u + \vec F ##
The Attempt at a Solution
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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]