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Equations for the Conservation of Energy
Author: John M. Cimbala, Penn State University
Latest revision: 25 September 2007
Mechanical Energy Equation
Nonconservative form:
()
1
2
ij
ii
i i
i
j
D
uu
ug
u
Dt
x
τ
ρρ
∂
=+
∂
Conservative form:
( )
1
2
1
2
ij
ji
i
i
i
i
j
j
τ
ρ
u
ρ
ρ
u
tx
x
∂
∂
∂
+=
+
∂∂
∂
or
ij
i
i
j
j
τ
E
uE
ρ
u
x
∂
+
∂
where
2
1
2
mV
is the kinetic energy (the conserved quantity),
1
2
E
ρ
≡
is the kinetic energy per unit volume,
and
1
2
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
i
i
j
i
jj
u
E
ρ
τ
up
x
x
j
φ
∂
∂
+
+−
∂
∂
where
i
ij
j
u
σ
x
∂
≡
∂
= rate of viscous dissipation of kinetic energy per unit volume (increases
internal
energy at
expense of
kinetic
energy).
is always
positive
since friction is an
irreversible
process.
For a Newtonian fluid,
with Stokes’ assumption that
2
3
0
λμ
,
becomes
2
3
2
j
i
ij ij
ij
u
u
μ
ee
μ
x
x
∂
∂
=−
∂
∂
.
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This note was uploaded on 07/23/2008 for the course ME 521 taught by Professor Cimbala during the Fall '07 term at Pennsylvania State University, University Park.
 Fall '07
 CIMBALA

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