If we introduce a function \(\Phi'\) by \( \Phi' = \Phi - \int^t f(t')\,dt'\), we obtain

\[ \nabla \Phi' = \nabla \Phi = \mathbf{u}, \quad \partial_t \Phi' = \partial_t \Phi - f(t) \]

Then, \(\partial_t \Phi + \frac{1}{2} u^2 + h + \chi = f(t)\) reduces to \[ \partial_t \Phi' + \frac{1}{2}u^2 + h + \chi = \text{const.} \] This is called the integral of the motion, where \[ u^2 = (\partial_x \Phi)^2 + (\partial_y \Phi)^2 + (\partial_z \Phi)^2 \]

In the above integral, the function \(\Phi\) is used instead of \(\Phi'\) \[ \partial_t \Phi + \frac{1}{2}u^2 + h + \chi = \text{const.} \]

When the flow is steady, the first term \(\partial_t \Phi\) vanishes. Then we obtain the same expression as Bernoulli’s equation \(H := \frac{u^2}{2} + h + \chi = \text{const.}\) However, in the present potential flow, the constant on the right holds at all points (not restricted to a single stream-line)

If the gravitational potential is written as \(\chi = gz\), the integral of motion for a fluid of uniform density \(\rho_0\) is given as \[ \partial_t \Phi + \frac{1}{2}u^2 + \left(\frac{p}{\rho_0}\right) + gz = \text{const.} \]

When the velocity potential \(\Phi\) is known, this equation gives the pressure \(p\) since \(u^2\) is also known from \(\boxed{u^2 = (\partial_x \Phi)^2 + (\partial_y \Phi)^2 + (\partial_z \Phi)^2}\).

¹ Tsutomu Kambe. (2007). Elementary Fluid Mechanics. World Scientific, Singapore.