Editing Potential flow (section)

=== Unsteady flow ===<!-- [[Full potential equation]] redirects here -->
The continuity and the (potential flow) momentum equations for unsteady flows are given by

<math display="block">\frac{\partial\rho}{\partial t} + \rho \nabla\cdot\mathbf v + \mathbf v\cdot\nabla \rho = 0, \quad \frac{\partial\mathbf v}{\partial t}+ (\mathbf v \cdot\nabla)\mathbf v= -\frac{1}{\rho}\nabla p =-\frac{c^2}{\rho}\nabla \rho=-\nabla h.</math>

The first integral of the (potential flow) momentum equation is given by 

<math display="block">\frac{\partial\varphi}{\partial t} + \frac{v^2}{2} + h = f(t), \quad \Rightarrow \quad \frac{\partial h}{\partial t} = -\frac{\partial^2\varphi}{\partial t^2} - \frac{1}{2}\frac{\partial v^2}{\partial t} + \frac{df}{dt}</math>

where <math>f(t)</math> is an arbitrary function. Without loss of generality, we can set <math>f(t)=0</math> since <math>\varphi</math> is not uniquely defined. Combining these equations, we obtain

<math display="block">\frac{\partial^2\varphi}{\partial t^2} + \frac{\partial v^2}{\partial t}=c^2\nabla\cdot\mathbf v - \mathbf v\cdot (\mathbf v \cdot \nabla)\mathbf v.</math>

Substituting here <math>\mathbf v=\nabla\varphi</math> results in

<math display="block">\varphi_{tt} + (\varphi_x^2+ \varphi_y^2+ \varphi_z^2)_t= (c^2-\varphi_x^2)\varphi_{xx}+(c^2-\varphi_y^2)\varphi_{yy}+(c^2-\varphi_z^2)\varphi_{zz}-2(\varphi_x\varphi_y\varphi_{xy}+\varphi_y\varphi_z\varphi_{yz}+\varphi_z\varphi_x\phi_{zx}).</math>

'''Nearly parallel flows:''' As in before, for nearly parallel flows, we can write (after introudcing a recaled time <math>\tau=c_\infty t</math>)

<math display="block">\frac{\partial^2\phi}{\partial \tau^2}  +  2M \frac{\partial^2\phi}{\partial x\partial\tau}= (1-M^2) \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2}</math>

provided the constant Mach number <math>M</math> is not close to unity. When <math>|M-1|</math> is small (transonic flow), we have the following nonlinear equation{{r|landau}}

<math display="block">\frac{\partial^2\phi}{\partial \tau^2}  +  2\frac{\partial^2\phi}{\partial x\partial\tau} = -2\alpha_*\frac{\partial\phi}{\partial x} \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2}.</math>

'''Sound waves:''' In sound waves, the velocity magntiude <math>v</math> (or the Mach number) is very small, although the unsteady term is now comparable to the other leading terms in the equation. Thus neglecting all quadratic and higher-order terms and noting that in the same approximation, <math>c</math> is a constant (for example, in polytropic gas <math>c^2=(\gamma-1)h_0</math>), we have<ref>Lamb (1994) §287, pp. 492–495.</ref>{{r|landau}}

<math display="block">\frac{\partial^2 \varphi}{\partial t^2} = c^2 \nabla^2 \varphi,</math>

which is a linear [[wave equation]] for the velocity potential {{mvar|φ}}. Again the oscillatory part of the velocity vector {{math|'''v'''}} is related to the velocity potential by {{math|'''v''' {{=}} ∇''φ''}}, while as before {{math|Δ}} is the [[Laplace operator]], and {{mvar|c}} is the average speed of sound in the [[transmission medium|homogeneous medium]]. Note that also the oscillatory parts of the [[pressure]] {{mvar|p}} and [[density]] {{mvar|ρ}} each individually satisfy the wave equation, in this approximation.