Bernoulli’s equation is an energy equation that is derived from a force balance along streamlines. The force balance starts with Newton’s second law, and the two main forces are the weight of the differential element and the force due to pressure. The force balance then becomes the following equation.

\[ \partial ma_s = \partial W_s + \partial F_{Ps} \]

where \( \partial m \) is the differential mass
\(a_s \) is the acceleration along the streamline
\(\partial W_s\) is the differential weight along the streamline
\( \partial F_{Ps}\) is the differential pressure force along the streamline 

The force balance then simplifies to the following equation:

\[ \rho \nu \frac{d\nu}{ds} = -\gamma \; \mathrm{sin}\theta – \frac{dP}{ds}\]

where \( \rho \) is the fluid density
\( \nu \) is the fluid velocity along the streamline
\(ds\) is the differential distance along the streamline
\( \gamma \) is the specific weight
\( \theta \) is the angle of the element in relation to gravity
\( P \) is pressure

After applying assumptions and integrating the force balance, we obtain the typical form of Bernoulli’s equation.

\[ P + \frac{1}{2}\rho\nu^2 + \rho g z = constant \]

where \(g\) is the acceleration due to gravity
\(z\) is the height at a certain point 

In some situations, it is beneficial to do a mass balance at steady state over two points to apply Bernoulli’s equation. 

\[\nu_1A_1 = \nu_2A_2 \]

where \(\nu_1\) is the fluid velocity at point 1
\(A_1\) is the cross sectional area at point 1
\( \nu_2\) is the fluid velocity at point 2
\(A_2\) is the cross sectional area at point 2