Microflows - Microfluidics for Biotechnology - page 47

Figure 2.28 Bernoulli'sequationalongastreamline.

This formulation does not require the irrotational hypothesis for the flow, and
we are left with the equation

du d Ω 1 d P

u d s = - - (2.66)

d s ρ d s

where s is the distance along the streamline and u is the velocity directed along the
streamline. The integration of (2.66) yields

2

u P

+ Ω + = C

2 ρ

The third form of Bernoulli's equation is derived from the conservation of en-
ergy. Energy balance is often a powerful and elegant method in physics and that
was the initial Bernoulli's approach. In the case described in Figure 2.29, an element
of fluid is transferred from one point to another in a duct with impermeable, rigid
boundaries. We can look at it as if there were imaginary pistons moving with the
speed of the fluid.
The energies per unit volume, made up of kinetic, potential, and pressure, are
equated to obtain

2 2

ρ u ρ u

1 2

U = + ρ g z + P = + P (2.67)

1 1 2

2 2

so that, by taking arbitrary points

2

ρ u

U = + ρ g z P + = const .

(2.68)

2

The advantage of the energy approach is that it is general. First, it contains the
streamline equation, just by taking imaginary pistons corresponding to the stream-
line as shown in Figure 2.30, second, the fluid may be assumed compressible; and

Figure 2.29 Sketchofthedisplacementofaluidelementinaduct.