Editing Potential flow (section)

==Incompressible flow==
In case of an [[incompressible flow]] — for instance of a [[liquid]], or a [[gas]] at low [[Mach number]]s; but not for [[sound]] waves — the velocity {{math|'''v'''}} has zero [[divergence]]:<ref name=B_99_101/>

<math display="block">\nabla \cdot \mathbf{v} =0 \,,</math>

Substituting here <math>\mathbf v = \nabla\varphi</math> shows that <math>\varphi</math> satisfies the [[Laplace equation]]<ref name=B_99_101/>

<math display="block">\nabla^2 \varphi = 0 \,,</math>

where {{math|∇<sup>2</sup> {{=}} ∇ ⋅ ∇}} is the [[Laplace operator]] (sometimes also written {{math|Δ}}). Since solutions of the Laplace equation are [[harmonic function]]s, every harmonic function represents a potential flow solution. As evident, in the incompressible case, the velocity field is determined completely from its [[kinematics]]: the assumptions of irrotationality and zero divergence of flow. [[Dynamics (physics)|Dynamics]] in connection with the momentum equations, only have to be applied afterwards, if one is interested in computing pressure field: for instance for flow around airfoils through the use of [[Bernoulli's principle]].

In incompressible flows, contrary to common misconception, the potential flow indeed satisfies the full [[Navier–Stokes equations]], not just the [[Euler equations (fluid dynamics)|Euler equations]], because the viscous term 

<math display="block">\mu\nabla^2\mathbf v = \mu\nabla(\nabla\cdot\mathbf v)-\mu\nabla\times\boldsymbol\omega=0</math>

is identically zero. It is the inability of the potential flow to satisfy the required boundary conditions, especially near solid boundaries, makes it invalid in representing the required flow field. If the potential flow satisfies the necessary conditions, then it is the required solution of the incompressible Navier–Stokes equations.

In two dimensions, with the help of the harmonic function <math>\varphi</math> and its conjugate harmonic function <math>\psi</math> (stream function), incompressible potential flow reduces to a very simple system that is analyzed using [[complex analysis]] (see below).