Editing Potential flow (section)

==Description and characteristics==
[[File:Construction of a potential flow.svg|thumb|A potential flow is constructed by adding simple [[elementary flow]]s and observing the result.]]
[[Image:Potential cylinder.svg|thumb|right|[[Streamlines, streaklines, and pathlines|Streamlines]] for the incompressible [[potential flow around a circular cylinder]] in a uniform onflow.]]
In potential or irrotational flow, the vorticity vector field is zero, i.e., 

<math display="block">\boldsymbol\omega \equiv \nabla\times\mathbf v=0,</math>

where <math>\mathbf v(\mathbf x,t)</math> is the velocity field and <math>\boldsymbol\omega(\mathbf x,t)</math> is the [[vorticity]] field. Like any vector field having zero curl, the velocity field can be expressed as the gradient of certain scalar, say <math>\varphi(\mathbf x,t)</math> which is called the '''velocity potential''', since the curl of the gradient is always zero. We therefore have<ref name=B_99_101>Batchelor (1973) pp. 99–101.</ref>

<math display="block"> \mathbf{v} = \nabla \varphi.</math>

The velocity potential is not uniquely defined since one can add to it an arbitrary function of time, say <math>f(t)</math>, without affecting the relevant physical quantity which is <math>\mathbf v</math>. The non-uniqueness is usually removed by suitably selecting appropriate initial or boundary conditions satisfied by <math>\varphi</math> and as such the procedure may vary from one problem to another.

In potential flow, the [[Circulation (physics)|circulation]] <math>\Gamma</math> around any [[Simply connected space|simply-connected contour]] <math>C</math> is zero. This can be shown using the [[Stokes theorem]], 

<math display="block">\Gamma \equiv \oint_C \mathbf v\cdot d\mathbf l = \int \boldsymbol\omega\cdot d\mathbf f=0</math>

where <math>d\mathbf l</math> is the line element on the contour and <math>d\mathbf f</math> is the area element of any surface bounded by the contour. In multiply-connected space (say, around a contour enclosing solid body in two dimensions or around a contour enclosing a torus in three-dimensions) or in the presence of concentrated vortices, (say, in the so-called [[irrotational vortices]] or point vortices, or in smoke rings), the circulation <math>\Gamma</math> need not be zero. In the former case, Stokes theorem cannot be applied and in the later case, <math>\boldsymbol\omega</math> is non-zero within the region bounded by the contour. Around a contour encircling an infinitely long solid cylinder with which the contour loops <math>N</math> times, we have 

<math display="block">\Gamma = N \kappa</math>

where <math>\kappa</math> is a cyclic constant. This example belongs to a doubly-connected space. In an <math>n</math>-tuply connected space, there are <math>n-1</math> such cyclic constants, namely, <math>\kappa_1,\kappa_2,\dots,\kappa_{n-1}.</math>