Bernoulli Equation

The Bernoulli equation is fundamentally:

1. An energy conservation equation
2. A consequence of $F=ma$ along streamlines

$$p+\frac{1}{2}\rho V^2+\gamma z=c$$

Between any two points, (1) and (2), on a streamline in steady, inviscid, incompressible flow the Bernoulli equation can be applied in the form:

$$p_1+\frac{1}{2}\rho V_1^2+\gamma z_1=p_2+\frac{1}{2}\rho V_2^2+\gamma z_2$$

Bernoulli’s Law specifically applies along streamlines. For across streamlines, refer to N2L Normal to Streamline.

Physical Interpretation

To interpret Bernoulli’s equation in terms of work and energy, we can multiply each term by a volume element $L^3$:

$$p\cdot L^3+\frac{1}{2}\rho V^2\cdot L^3+\gamma z\cdot L^3=c^{\prime }$$

Rearranging:

$$p\cdot L^2\cdot L+\frac{1}{2}\rho L^3{V}^2+\gamma z\cdot L^2\cdot L=c^{\prime }$$

where:

• $p\cdot L^2\cdot L$ represents work done by pressure forces
• $\frac{1}{2}\rho L^3{V}^2$ represents kinetic energy
• $\gamma z\cdot L^2\cdot L$ represents work done by gravity

Thus, Bernoulli’s equation is just an energy balance along a streamline. The work done on a fluid particle by all forces acting on the particle is equal to the change of the kinetic energy of the particle.

We can also write in terms of energy per unit weight (“head”)

$$\frac{p}{\gamma }+\frac{V^2}{2g}+z=c$$

where:

• $p/\gamma$ is the pressure head – work done by pressure forces
• $V^2/2g$ is the velocity head – kinetic energy per unit weight
• $z$ is the elevation head – potential energy per unit weight

This explains why, for a fluid along a streamline:

• If velocity increases, pressure must decrease
• If elevation increases, pressure decreases

Static, Stagnation, Dynamic, Total Pressure

In the Bernoulli equation $p+\frac{1}{2}\rho V^2+\gamma z=c$, the total pressure is constant along a streamline. We can identify different terms that are part of the equation:

• $p$ is the static pressure.
• $\frac{1}{2}\rho V^2$ is the dynamic pressure.
• $\gamma z$ is the hydrostatic pressure.

We can see the interpretation of the dynamic pressure by considering the pressure at the end of a small tube inserted into the flow and pointing upstream. After the initial transient motion has died out, the liquid will fill the tube to a height of $H$ as shown. The fluid, including at its tip (2), will be stationary. That is, $V_2=0$ or point 2 is the stagnation point.