1
Fluid has Mass
As a fluid moves, it has interia from its mass,
which cannot be neglected in more formal
descriptions of stress. Total mass in
volume V is: m =
ρ
dV where we must
conserve mass, i.e.:
Dm/Dt = D/Dt (
ρ
dV)
∂ρ
/
∂
tdV+
ρ
v
i
v
j
dS = 0
ds
v
i
v
j
Density(t) of
material
Movement of dS
(surface) along a
path
The final material derivative:
D
ρ
/Dt =
∂ρ
/
∂
t + v
i
∂ρ
/
∂
x
j
Amount of “stuff” moving along a path
the path follows the fluid
current described by the
fluid's velocity field
v
time
position
Continuity Equation for Mass
the rate at which mass enters a system is
equal to the rate at which mass leaves the
system:
If d
ρ
/dt is constant, the mass continuity
equation simplifies to a volume continuity
equation:
•v = 0; i.e. the change in rate of
local volume changes is zero.
***Depends on the biofluid as to whether this
assumption holds or not
D
ρ
/Dt +
ρ ∂
v
i
/
∂
x
i
= 0
ds
v
i
v
j
Empirical Data on Water to
Validate Assumptions
Tait (1888):
V(p) = V(p=1) – 0.31 V
o
log
10
(B+p/B+1)
•
V(p) is volume of water at pressure p
•
B = 2668+19t -0.3t
2
+0.0017t
3
; t = temperature
**For t = 40
o
C: V = 1.0079 @ 1atm
0.988 @ 483 atm
0.97 @ 967 atm

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