SIB: Bernoulli Equation

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