Equations of Change for Isothermal Systems
The equation of continuity
Mass balance over a fixed elementary volume
()
xx
x
yy
y
zz
z
xyz
t
yz
u
u
xz
u
u
xy
u
u
ρ
ρρ
+Δ
∂
ΔΔΔ
∂
⎡⎤
=Δ Δ
−
⎣⎦
+Δ Δ
−
⎢⎥
−
( )
.
y
u
uu
tx
y
z
⎛⎞
∂
∂∂
∂
⎜⎟
=−
+
+
=− ∇
∂
∂
⎝⎠
u
( )
is called divergence of
, and for incompressible fluids
.0
∇∇
=
Mass influx
(inward arrows)
Mass outflux
(outward
arrows)
x
y
z
∆
x
∆
y
∆
z
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View Full DocumentEquations of Change for Isothermal Systems
The equation of continuity
Mass balance over a moving elementary volume
()
(
)
(
)
(
)
D
..
.
tD
t
t
ρ
ρρ
∂∂
=− ∇
=−
∇ +
∇
⇔
≡
+
∇ =− ∇
⎡⎤
⎣⎦
uu
u
u
u
is called material (also, substantial/particle) derivative
is called local rate of change of
at a given point
D/
/
Dt
t
∇
u
is called convective (or advective ) derivative, which is the
change of
as a result of convection of the particle from one
location to another where
is different
(
)
Df
.
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 Fall '08
 Peters
 Equations, Force, Mass, Isothermal Systems

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