Direct Energy, 2018a

242 10.6 Electrouidics
However, energy is conserved, so
(P 1 − P 2 )ΔV + 1 2 (ρ densΔV) ( | −→ v 1 | 2 −| −→ v 2 | 2) =0. (10.5)
This expression can be simplied algebraically.
P 1 − P 2 + 1 2 ρ dens| −→ v 1 | 2 − 1 2 ρ dens| −→ v 2 | 2 =0 (10.6)
Both pressure P and velocity −→ v are functions of location. The only way
this expression can be true for all locations is if it is true for each location
and a constant.
P 1 + 1 2 ρ dens| −→ v 1 | 2 = P 2 + 1 2 ρ dens| −→ v 2 | 2 = constant (10.7)
Bernoulli's equation with the rather severe assumptions above becomes
P + 1 2 ρ dens| −→ v | 2 = constant. (10.8)
Bernoulli's equation is also used to describe the lift of an air foil or the path
of a curve ball in baseball [162, p. 350]. In some EHDs electricity induces
changes in the pressure or volume of a microuidic channel. The uid in
these devices may be conductive or insulating. As seen by Eq. 10.8, this
change in pressure induces a change in uid velocity.
In other EHDs, applied voltages exert forces on conductive uids. A
charged object in an external electric eld −→ E feels a force in the direction
of the electric eld. A current in an external magnetic eld −→ B feels a force.
The direction of this force is perpendicular to both the direction of the
current and the direction of the external magnetic eld. These eects are
summarized by the Lorentz force equation
−→ F = Q
( −→E +
−→ v ×
−→ B
)
(10.9)
which was discussed in Chapter 5. In that chapter, Hall eect devices
and magnetohydrodynamic devices were discussed, both of which can be
understood by the Lorentz force equation with an external magnetic eld
but no electrical eld. This type of EHD can be understood by the Lorentz
force equation with an external electric eld but no magnetic eld. The
liquid in these devices must be conductive. When a voltage is applied
across this type of EHD, an electric eld is induced which causes the liquid
to ow, and this eect is said to be due to a streaming potential [159]. A
related eect called electrophoresis occurs in liquids which contain charged