A very special form of the energy balance equation will arise under certain limiting assumptions.

The **Bernoulli equation** is obtained from the energy
balance equation when we have a steady, incompressible, and
inviscid flow in which there is no shaft work, no heat transfer,
and no change in internal energy occurs.

An **inviscid flow** is one in which viscous losses are
negligible (i.e., the fluid viscosity is very small).

Let's look at each of these limitations in turn:

*Steady Flow*: the left-hand side becomes zero as the
d/dt term vanishes

*Incompressible Fluid*: the density remains constant
everywhere

*Inviscid Flow*: there are no viscous work losses
($W_\mu=0$) **and** the flow is everywhere a "plug flow" (i.e.,
velocities are constant perpendicular to the main flow -- this
*also* means that the shear stresses are zero!)

*No Heat Transfer (adiabatic)*: q=0

*No Change in Internal Energy*: $\hat U$ drops out of
the equation

Under these simplifications our macroscopic energy balance becomes:

$\displaystyle{0 = - \int\int_{CS} \rho \left (\frac{1}{2}v^2 + gz + \frac{P}{\rho} \right ) (\mathbf{v}\cdot \mathbf{n})dA}$

Because of our constant velocities and constant densities, we can simplify this further to be

$\displaystyle{\left (\frac{1}{2}v^2 + gz + \frac{P}{\rho} \right ) = constant}$

The Bernoulli Equation (sometimes called Bernoulli's Law) is typically used because it is true along a streamline (i.e., for a CV that encircles a single streamline), although it will clearly also work for more complex CVs that satisfy the above criteria.

Explain the connection between a macroscopic energy balance and Bernoulli's Equation