SIB: Bernoulli Equation

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

DEFINITIONS:

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.

DEFINITIONS:

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

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

NOTE:

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.

OUTCOME:

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