In an irrotational fluid region,

$$\nabla\times\mathbf v=\mathbf0.$$

On a suitable simply connected domain, a curl-free velocity field can be represented as the gradient of a scalar velocity potential $\phi$:

$$\boxed{\mathbf v=\nabla\phi}.$$

Some texts use the opposite sign convention. Here the velocity is defined as the positive gradient of $\phi$.

Recovering velocity from one scalar field

In Cartesian coordinates,

$$\boxed{u=\frac{\partial\phi}{\partial x},\qquad v=\frac{\partial\phi}{\partial y},\qquad w=\frac{\partial\phi}{\partial z}}.$$

The scalar potential therefore replaces three velocity components by one function, while irrotationality is satisfied automatically because

$$\nabla\times\nabla\phi=\mathbf0.$$

Adding a constant does not change the velocity:

$$\phi\rightarrow\phi+C.$$

The zero level of velocity potential is therefore arbitrary.

Incompressibility gives Laplace's equation

For incompressible flow,

$$\nabla\cdot\mathbf v=0.$$

Substituting

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

gives

$$\nabla\cdot\nabla\phi=0,$$

so

$$\boxed{\nabla^2\phi=0}.$$

Thus the velocity potential of an incompressible irrotational flow is a harmonic function.

Potential-flow problems can therefore be posed as Laplace boundary-value problems.

Solid-wall boundary condition

For an impermeable wall moving with local velocity $\mathbf V_w$, no penetration requires

$$\boxed{(\mathbf v-\mathbf V_w)\cdot\mathbf n=0}.$$

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

$$\boxed{\frac{\partial\phi}{\partial n} =\mathbf V_w\cdot\mathbf n}.$$

For a stationary wall,

$$\boxed{\frac{\partial\phi}{\partial n}=0}.$$

This is a Neumann boundary condition: potential flow enforces the normal velocity of an impermeable boundary.

It does not impose no slip. An inviscid potential-flow model can have nonzero tangential velocity at a stationary wall.

Equipotential surfaces

Because

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

the velocity is normal to surfaces of constant $\phi$. In a steady flow, streamlines are tangent to the velocity, so they cross regular equipotential surfaces orthogonally.

Worked example

Consider the two-dimensional velocity field

$$u=2Ax,$$

$$v=-2Ay,$$

where $A$ is constant.

Its vorticity is

$$\omega_z =\frac{\partial v}{\partial x} -\frac{\partial u}{\partial y} =0-0=0,$$

so the field is irrotational.

To find a velocity potential, require

$$\frac{\partial\phi}{\partial x}=2Ax.$$

Integrating with respect to $x$,

$$\phi=Ax^2+f(y).$$

Now require

$$\frac{\partial\phi}{\partial y}=-2Ay,$$

so

$$f'(y)=-2Ay,$$

and therefore

$$f(y)=-Ay^2+C.$$

Thus

$$\boxed{\phi=A(x^2-y^2)+C}.$$

The flow is also incompressible because

$$\frac{\partial u}{\partial x} +\frac{\partial v}{\partial y} =2A-2A=0.$$

Correspondingly,

$$\nabla^2\phi =2A-2A=0.$$

The potential is harmonic exactly as incompressible irrotational flow requires.

Scope of potential flow

A velocity potential exists locally wherever a smooth velocity field is curl-free. A single globally defined potential can fail on domains containing holes or singularities with nonzero circulation.

Potential flow therefore describes irrotational regions, not every fluid flow. It is especially useful outside boundary layers, wakes, and vortex cores, where the vorticity may be negligible even though nearby viscous regions remain essential to the complete physical problem.