Bernoulli's equation is more than an energy formula for a pipe. Its scope and the location where its constant applies can be derived directly from the inviscid momentum equation.

For a constant-density inviscid fluid under a conservative body force

$$\mathbf f_b=-\nabla\Phi,$$

Euler's equation is

$$\frac{\partial\mathbf v}{\partial t} +(\mathbf v\cdot\nabla)\mathbf v =-\nabla\left(\frac p\rho+\Phi\right).$$

Use the vector identity

$$({\mathbf v}\cdot\nabla)\mathbf v =\nabla\left(\frac12|\mathbf v|^2\right) -\mathbf v\times\boldsymbol\omega,$$

where

$$\boldsymbol\omega=\nabla\times\mathbf v.$$

Then

$$\boxed{ \frac{\partial\mathbf v}{\partial t} -\mathbf v\times\boldsymbol\omega +\nabla B=\mathbf0 },$$

with the Bernoulli function

$$\boxed{B=\frac p\rho+\frac12|\mathbf v|^2+\Phi}.$$

This form makes clear when Bernoulli's quantity is constant along one streamline and when it is constant throughout an entire flow region.

Steady rotational flow: constant along streamlines

For steady flow,

$$\frac{\partial\mathbf v}{\partial t}=\mathbf0,$$

so

$$\nabla B=\mathbf v\times\boldsymbol\omega.$$

Take the dot product with $\mathbf v$:

$$\mathbf v\cdot\nabla B =\mathbf v\cdot(\mathbf v\times\boldsymbol\omega)=0.$$

The directional derivative of $B$ along the velocity is therefore zero. Hence

$$\boxed{B=\text{constant along each streamline}}.$$

Different streamlines can generally have different Bernoulli constants when the steady flow has nonzero vorticity.

For uniform gravity,

$$\Phi=gz,$$

and multiplying by $\rho$ gives the familiar form

$$\boxed{p+\frac12\rho v^2+\rho gz=\text{constant along a streamline}}.$$

Steady irrotational flow: one constant throughout the region

If the steady flow is also irrotational,

$$\boldsymbol\omega=\mathbf0,$$

then

$$\nabla B=\mathbf0.$$

Therefore

$$\boxed{B=\text{one constant throughout each connected irrotational region}}.$$

This stronger result is why the pressure at any point in a steady potential-flow field can be compared directly with one common far-field state.

It is not necessary to trace whether the two points lie on the same particular streamline.

Unsteady irrotational flow

Now suppose the flow is irrotational but may vary with time. Write

$$\mathbf v=\nabla\phi.$$

Then

$$\frac{\partial\mathbf v}{\partial t} =\nabla\left(\frac{\partial\phi}{\partial t}\right).$$

Since

$$\boldsymbol\omega=0,$$

Euler's equation becomes

$$\nabla\left( \frac{\partial\phi}{\partial t} +B \right)=\mathbf0.$$

Thus

$$\boxed{ \frac{\partial\phi}{\partial t} +\frac p\rho +\frac12|\mathbf v|^2 +\Phi =C(t) }.$$

The right side may depend on time but is spatially uniform throughout a connected potential-flow region.

Because adding a function of time to $\phi$ does not change

$$\mathbf v=\nabla\phi,$$

the function $C(t)$ can often be absorbed into the arbitrary time-dependent reference level of the velocity potential.

Worked comparison

Consider steady irrotational flow at constant elevation. Far from a body,

$$v=U,\qquad p=p_\infty.$$

Since the Bernoulli constant is global,

$$\frac{p_\infty}{\rho}+\frac12U^2 =\frac p\rho+\frac12v^2$$

at any regular point in the same connected potential-flow region.

Therefore

$$\boxed{p-p_\infty=\frac12\rho(U^2-v^2)}.$$

If the local speed is

$$v=2U,$$

then

$$p-p_\infty =\frac12\rho(U^2-4U^2) =-\frac32\rho U^2.$$

Equivalently, the pressure coefficient is

$$\boxed{C_p=-3}.$$

The hierarchy to remember

For constant-density inviscid flow with conservative body forces:

• steady flow: Bernoulli function is constant along each streamline;
• steady irrotational flow: one Bernoulli constant applies throughout a connected region;
• unsteady irrotational flow: the velocity potential contributes the additional term $\partial\phi/\partial t$.

These are not separate empirical formulas. They are increasingly specialized consequences of Euler's momentum equation.