Editing Bernoulli's principle (section)

=== Unsteady potential flow ===

The Bernoulli equation for unsteady potential flow is used in the theory of [[Wind wave|ocean surface waves]] and [[acoustics]]. For an irrotational flow, the [[flow velocity]] can be described as the [[gradient]] {{math|∇''φ''}} of a [[velocity potential]] {{mvar|φ}}. In that case, and for a constant density {{mvar|ρ}}, the [[momentum]] equations of the [[Euler equations (fluid dynamics)|Euler equations]] can be integrated to:<ref name="Batchelor2000" />{{rp|page=383}}<math display="block">\frac{\partial \varphi}{\partial t} + \tfrac12 v^2 + \frac{p}{\rho} + gz = f(t),</math>

which is a Bernoulli equation valid also for unsteady—or time dependent—flows. Here {{math|{{sfrac|∂''φ''|∂''t''}}}} denotes the [[partial derivative]] of the velocity potential {{mvar|φ}} with respect to time {{mvar|t}}, and {{math|1=''v'' = {{abs|∇''φ''}}}} is the flow speed. The function {{math|''f''(''t'')}} depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment {{mvar|t}} applies in the whole fluid domain. This is also true for the special case of a steady irrotational flow, in which case {{mvar|f}} and {{math|{{sfrac|∂''φ''|∂''t''}}}} are constants so equation ({{EquationNote|A}}) can be applied in every point of the fluid domain.<ref name="Batchelor2000" />{{rp|page=383}} Further {{math|''f''(''t'')}} can be made equal to zero by incorporating it into the velocity potential using the transformation:<math display="block">\Phi = \varphi - \int_{t_0}^t f(\tau)\, \mathrm{d}\tau,</math>
resulting in:
<math display="block">\frac{\partial \Phi}{\partial t} + \tfrac12 v^2 + \frac{p}{\rho} + gz = 0.</math>

Note that the relation of the potential to the flow velocity is unaffected by this transformation: {{math|1=∇Φ = ∇''φ''}}.

The Bernoulli equation for unsteady potential flow also appears to play a central role in [[Luke's variational principle]], a variational description of free-surface flows using the [[Lagrangian mechanics]].