Biomedical Engineering Reference
In-Depth Information
the pressure, velocity, and height in a steady motion of an ideal fluid, but its energy
form has extended its domain of application to viscous fluids. However, there is an
important restriction to the applicability of Bernoulli's equation: the flow has to be
one-dimensional. This is not very restrictive in microfluidics, because microflows in
capillary and microchannels are mostly one-dimensional.
There are three forms of Bernoulli's equations. The first form derives directly
from the Navier-Stokes equation under rather severe restrictions. First, suppose
that the fluid is an ideal fluid, so that the diffusion terms in the NS equation may be
neglected (Euler's form of the NS equations)
�
�
∂
+ Ñ =
u
� �
1
u
u
F
P
.
- Ñ
(2.64)
∂
t
ρ
Equation (2.64) is a vector equation having the dimension of the flow field. Suppose
also that the velocity field derives from a potential
u
= -Ñ
φ
If the external force field is conservative, that is, derives from a potential
�
F
= -Ñ
Ω
and if the fluid is supposed incompressible
ρ
=
const
.
then (2.64) may be cast under the form
∂
-Ñ + Ñ = -Ñ
Ω
- Ñ
∂
)
� �
P
(
u
.
u
φ
t
ρ
Then the following gradient is identically zero
2
é
ù
∂
φ
u
P
Ñ -
+
+
Ω
+
=
0
ê
ú
∂
t
2
ρ
ê
ú
ë
û
Thus, the function must be constant
2
∂
φ
u
P
C
-
+
+
Ω
+
=
∂
t
2
ρ
If we assume that the flow is stationary, we obtain the Bernoulli's equation
2
u
P
C
+
Ω
+
=
(2.65)
2
ρ
This first approach requires quite severe conditions; other forms of Bernoulli's
equations have been derived with less restricting hypotheses.
The second form of Bernoulli's equation arises from the fact that in a steady
flow, the particles of fluid move along streamlines, as on rails, and are accelerated
or decelerated by the forces acting tangent to the streamlines (Figure 2.28).
