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We consider a layer of fluid of depth have infinite horizontal dimensions. The
temperature at the bottom and the top are maintained at and
respectively. We use tensor notation in writing down the equations of motion. Thus
we use cartesian tensor with
and velocity
for j=1,2,3.
[Note: We frequently mix vector and tensor notation, e.g. the velocity vector can
be written as
]
 Equation of continuity
or in equaivalent form

(1) 
where is the density of the fluid.
 Equations of momentum

(2) 
where
is the stress tensor where is the shear viscosity. Strictly speaking, the
pressure term in the stress tensor is the mechanical tensor which is different
than the thermodynamic pressure term. But their difference is so small that we
assume they are same for all the practical case.
 Equation of energy

(3) 
internal energy per unit mass of fluid [ ( specific heat at
constant volume) for gases and ( specific heat) for liquid], thermal
diffusivity, temperature and rate of viscous dissipation is
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