Biomedical Engineering Reference
In-Depth Information
the fluid there are enough molecules to define statistical properties like velocity and
pressure. The continuum hypothesis works well at a microscopic scale for liquids;
for gases, the hypothesis breaks down at the nanoscopic scale where the character-
istic Knudsen number (Kn) becomes of the order of 1
λ
Kn
=
(2.1)
L
where
λ
is the mean free path of the molecules and
L
is the characteristic dimen-
sion of the channel. For gases, the limit for
L
is about 1
m
m. In liquids, the mean
free path is much smaller and the continuum hypothesis is applicable to any
microsystem.
In the most general point of view, fluid flows are determined by the knowledge
of velocities
U
= {
u
i
,
i
= 1, 3}, pressure
P
, density
ρ
, viscosity
m
, specific heat
Cp
, and
temperature
T
. For each fluid, density, viscosity, and specific heat are related to pres-
sure and temperature (or enthalpy) via characteristic equations of state (EOS)
ρ
= (
f f(P), T
,
)
µ
=
g P T
( ,
)
(2.2)
Cp
=
h P T
( ,
)
Pressure and temperature characterize the number and the state of the molecules
that are present in a given volume. Equations of state are generally complicated, but
they can be approximated by analytical functions if the domain of variation of the
parameters (
P
and
T
) is not too large. Thus we are left with five unknowns:
u
x
,
u
y
,
u
z
,
P
, and
T
. These unknowns are related by a system of three equations: (1) a scalar
equation for the mass conservation, (2) a vector equation for the conservation of
momentum, and (3) a scalar equation for the conservation of energy. In biotechnol-
ogy, fluid flows are often isothermal or variation of temperature is negligible. Note
that this is not the case for microchemistry where chemical reactions are seldom
isothermal. If temperature is constant or nearly constant, we have to deal with four
unknowns
u
x
,
u
y
,
u
z
, and
P
, with the help of the mass conservation equation and
the conservation of momentum equation, plus the EOS
ρ
= f(
P
),
m
= g(
P
). Some
authors give the name
Navier-Stokes equations
to the whole system; others confine
this name to the second equation (momentum).
2.2.1.1
GeneralCase:GoverningEquations
The first equation is the mass conservation (or continuity) equation. For simplicity
we demonstrate here only the two dimensional form of this equation. Assume a
velocity field (
u
,
v
) and an element of volume (D
x
, D
y
) as sketched in Figure 2.5.
The mass conservation equation requires
∂ ∆ ∆
ρ
x
y
é
∂
ρ
u
ù
é
∂
ρ
v
)
ù
(
)
(
)
(
=
ρ
u y
D +
ρ
v x
D -
ρ
u
+
D -
y
ρ
v
+
D
x
ê
ú
ê
ú
∂
t
∂
x
∂
y
ë
û
ë
û
ρ
∂
(
ρ
u
)
∂
(
ρ
v
)
∂
+
(2.3)
0
dividing by D
x
D
y
+
=
∂
t
∂
x
∂
y
