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This is the first course of a two-semester fluid mechanics sequence for graduate students in the thermal sciences. This course deals with solutions of these equations, both exact and approximate. Key points of this lecture are: Equations Conservation Energy, Conservation of Energy, Equations for the Conservation of Energy, Mechanical Energy Equation, Non-Conservative Form, Alternate Conservative Form, Thermal Energy Equation, Heat Equation, Real Energy Equation, Law of Thermodynamics
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Non-conservative form:
1
2
ij
i i i i i
j
u u u g u
Dt x
Conservative form:
1
2 1
2
i i ij
j i i i i i
j j
ρ u u τ
u ρ u u ρ u g u
t x x
or
ij
j i i i
j j
τ E
u E ρ u g u
t x x
where
2 1
2
mV is the kinetic energy (the conserved quantity),
1
2 i i
and
1
2 i i
u u is the kinetic energy per unit mass. The terms on the right are sources (or sinks) of kinetic energy per
unit volume.
Alternate conservative form:
j
j i i ij i
j j
u E
u E ρ u g τ u p
t x x x j
where
i
ij
j
u
σ
x
= rate of viscous dissipation of kinetic energy per unit volume (increases internal energy at
with Stokes’ assumption that
2
3
2
3
j i
ij ij
i j
u u
μ e e μ
x x
First Law of Thermodynamics for a Material Volume:
1
2 i i i i ij i j i i
A A
ρ e u u d ρ g u d τ u dA q dA
Dt
V V
where e = internal energy per unit mass (what most Thermodynamics books call u). Or, using the RTT,
First Law of Thermodynamics for a Control Volume:
1 1
2 2 i i i i j j i i ij i j i i
CV CS CV CS CS
e u u d ρ e u u u dA ρ g u d τ u dA q dA
t
ρ
First Law of Thermodynamics in Differential form:
1
2
i
i i i i ij i
j i
D q
ρ e u u ρ u g τ u
Dt x x
Or, combining this with the mechanical energy equation, after some algebra we get an alternate form:
Differential Thermal Energy Equation (Heat Equation):
i i
i i
De q u
ρ p
Dt x x
φ
where the terms are explained below:
I Rate of increase of internal energy of a fluid element per unit volume (following the fluid element). This
term can be positive or negative.
II Rate of heat flux into the fluid element per unit volume (negative because q
is defined as positive outward).
This term can be positive or negative.
III Rate of increase of internal energy per unit volume due to volumetric compression (negative because u i,i
volumetric expansion, which is defined as positive, but compression increases the internal energy). This
term can be positive or negative.
IV Rate of increase of internal energy per unit volume due to viscous dissipation. This term is always positive.