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{{about||potential flow around a cylinder|Potential flow around a circular cylinder}}
[[Image:Streamlines around a NACA 0012.svg|thumb|300px|right|Potential-flow [[Streamlines, streaklines, and pathlines|streamlines]] around a [[NACA airfoil|NACA 0012 airfoil]] at 11° [[angle of attack]], with upper and lower [[streamtube]]s identified.]]
In [[fluid dynamics]], '''potential flow''' describes the [[velocity field]] as the [[gradient]] of a scalar function: the [[velocity potential]]. As a result, a potential flow is characterized by an [[Conservative vector field#Irrotational vector fields|irrotational velocity field]], which is a valid approximation for several applications. The irrotationality of a potential flow is due to the [[Curl (mathematics)|curl]] of the gradient of a [[Scalar (physics)|scalar]] always being equal to zero.

In the case of an [[incompressible flow]] the velocity potential satisfies [[Laplace's equation]], and [[potential theory]] is applicable. However, potential flows also have been used to describe [[compressible flow]]s. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows.
Applications of potential flow are for instance: the outer flow field for [[airfoil|aerofoil]]s, [[ocean surface wave|water waves]], [[electroosmotic flow]], and [[groundwater flow equation|groundwater flow]]. For flows (or parts thereof) with strong [[vorticity]] effects, the potential flow approximation is not applicable.

==Characteristics and applications==
[[File:Construction of a potential flow.svg|thumb|A potential flow is constructed by adding simple elementary flows 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.]]

===Description and characteristics===
In fluid dynamics, a potential flow is described by means of a velocity potential {{mvar|φ}}, being a [[Function (mathematics)|function]] of space and time. The [[flow velocity]] {{math|'''v'''}} is a [[vector field]] equal to the gradient, {{math|∇}}, of the velocity potential {{mvar|φ}}:<ref name=B_99_101>Batchelor (1973) pp. 99–101.</ref>

:<math> \mathbf{v} = \nabla \varphi.</math>

Sometimes, also the definition {{math|'''v''' {{=}} −∇''φ''}}, with a minus sign, is used. But here we will use the definition above, without the minus sign. From [[vector calculus]] it is known that the [[Vector calculus identities#Curl of the gradient|curl of a gradient]] is equal to zero:<ref name=B_99_101/>

:<math>\nabla \times \nabla \varphi = \mathbf{0} \,,</math>

and consequently the [[vorticity]], the [[curl (mathematics)|curl]] of the velocity field {{math|'''v'''}}, is zero:<ref name=B_99_101/>

:<math>\nabla \times \mathbf{v} = \mathbf{0} \,.</math>

This implies that a potential flow is an [[irrotational flow]]. This has direct consequences for the applicability of potential flow. In flow regions where vorticity is known to be important, such as [[wake]]s and [[boundary layer]]s, potential flow theory is not able to provide reasonable predictions of the flow.<ref name=B_378_380>Batchelor (1973) pp. 378–380.</ref> Fortunately, there are often large regions of a flow where the assumption of irrotationality is valid which is why potential flow is used for various applications. For instance in: flow around [[aircraft]], [[groundwater flow]], [[acoustics]], [[water wave]]s, and [[electroosmotic flow]].<ref name=Kirby>{{Citation| author=Kirby, B.J. | title=Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices.| url=http://www.kirbyresearch.com/textbook| year=2010| publisher=Cambridge University Press | isbn=978-0-521-11903-0}}</ref>

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

with the dot denoting the [[inner product]]. As a result, the velocity potential {{mvar|φ}} has to satisfy [[Laplace's equation]]<ref name=B_99_101/>

:<math>\nabla^2 \varphi = 0 \,,</math>

where {{math|∇<sup>2</sup> {{=}} ∇ ⋅ ∇}} is the [[Laplace operator]] (sometimes also written {{math|Δ}}). In this case the flow can be determined completely from its [[kinematics]]: the assumptions of irrotationality and zero divergence of flow. [[Dynamics (physics)|Dynamics]] only have to be applied afterwards, if one is interested in computing pressures: for instance for flow around airfoils through the use of [[Bernoulli's principle]].

In two dimensions, potential flow reduces to a very simple system that is analyzed using [[complex analysis]] (see below).

===Compressible flow===
====Steady flow====<!-- [[Full potential equation]] redirects here -->
Potential flow theory can also be used to model irrotational compressible flow. The '''full potential equation''', describing a [[steady flow]], is given by:<ref name=Anderson>{{cite book | first=J. D. | last=Anderson | author-link=John D. Anderson | title=Modern compressible flow | year=2002 | publisher=McGraw-Hill | isbn=0-07-242443-5 |page= 358–359}}</ref>

:<math>\left( 1 - M_x^2 \right) \frac{\partial^2 \Phi}{\partial x^2} + \left( 1 - M_y^2 \right) \frac{\partial^2 \Phi}{\partial y^2} + \left( 1 - M_z^2 \right) \frac{\partial^2 \Phi}{\partial z^2} - 2 M_x M_y \frac{\partial^2 \Phi}{\partial x\, \partial y} - 2 M_y M_z \frac{\partial^2 \Phi}{\partial y\, \partial z} - 2 M_z M_x \frac{\partial^2 \Phi}{\partial z\, \partial x} = 0 \,,</math>

with [[Mach number]] components

:<math>M_x = \frac{1}{a} \frac{\partial \Phi}{\partial x} \,, \qquad M_y = \frac{1}{a} \frac{\partial \Phi}{\partial y} \,, \qquad \text{and} \qquad M_z = \frac{1}{a} \frac{\partial \Phi}{\partial z} \,,</math>

where {{mvar|a}} is the local [[speed of sound]]. The flow velocity {{math|'''v'''}} is again equal to {{math|∇Φ}}, with {{math|Φ}} the velocity potential. The full potential equation is valid for [[Speed of sound|sub-]], [[transonic|trans-]] and [[Supersonic speed|supersonic flow]] at arbitrary [[angle of attack]], as long as the assumption of irrotationality is applicable.<ref name=Anderson/>

In case of either subsonic or supersonic (but not transonic or [[hypersonic]]) flow, at small angles of attack and thin bodies, an additional assumption can be made: the velocity potential is split into an undisturbed onflow velocity {{math|''V''<sub>∞</sub>}} in the {{mvar|x}}-direction, and a small [[perturbation theory|perturbation]] velocity {{math|∇''φ''}} thereof. So:<ref name=Anderson/>

:<math>\nabla \Phi = V_\infty x + \nabla \varphi \,.</math>

In that case, the ''linearized small-perturbation potential equation'' — an approximation to the full potential equation — can be used:<ref name=Anderson/>

:<math>\left(1-M_\infty^2\right) \frac{\partial^2 \varphi}{\partial x^2} + \frac{\partial^2 \varphi}{\partial y^2} + \frac{\partial^2 \varphi}{\partial z^2} = 0,</math>

with {{math|''M''<sub>∞</sub> {{=}} {{sfrac|''V''<sub>∞</sub>|''a''<sub>∞</sub>}}}} the Mach number of the incoming free stream. This linear equation is much easier to solve than the full potential equation: it may be recast into Laplace's equation by a simple coordinate stretching in the {{mvar|x}}-direction.

:{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Derivation of the full potential equation
|-
|For a steady inviscid flow, the [[Euler equations (fluid dynamics)|Euler equations]] — for the mass and momentum density — are, in subscript notation and in non-[[conservation form]]:<ref name=Lamb_3_6>Lamb (1994) §6–§7, pp. 3–6.</ref>
:<math>\begin{align}
\frac{\partial}{\partial x_i} \left( \rho\, v_i \right) &= 0 \,, \\
\rho\, v_j\, \frac{\partial v_i}{\partial x_j} &= - \frac{\partial p}{\partial x_i} \,,
\end{align}</math>
while using the [[summation convention]]: since {{mvar|j}} occurs more than once in the term on the left hand side of the momentum equation, {{mvar|j}} is summed over all its components (which is from 1 to 2 in two-dimensional flow, and from 1 to 3 in three dimensions). Further:
* {{mvar|ρ}} is the fluid [[density]],
* {{mvar|p}} is the [[pressure]],
* {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) {{=}} (''x'', ''y'', ''z'')}} are the coordinates and
* {{math|(''v''<sub>1</sub>, ''v''<sub>2</sub>, ''v''<sub>3</sub>)}} are the corresponding components of the velocity vector {{math|'''v'''}}.
The speed of sound squared {{math|''a''<sup>2</sup>}} is equal to the derivative of the pressure {{mvar|p}} with respect to the density {{mvar|ρ}}, at constant [[entropy]] {{mvar|S}}:<ref>Batchelor (1973) p. 161.</ref>

:<math>a^2 = \left[ \frac{\partial p}{\partial \rho} \right]_S.</math>

As a result, the flow equations can be written as:

:<math> v_i\, \frac{\partial \rho}{\partial x_i} + \rho\, \frac{\partial v_i}{\partial x_i} = 0  \qquad \text{and} \qquad \rho\, v_j\, \frac{\partial v_i}{\partial x_j} = -a^2\, \frac{\partial \rho}{\partial x_i} \,.</math>

Multiplying (and summing) the momentum equation with {{math|''v<sub>i</sub>''}}, and using the mass equation to eliminate the density gradient gives:

:<math>\rho\, v_i\, v_j\, \frac{\partial v_i}{\partial x_j} = \rho\, a^2\, \frac{\partial v_i}{\partial x_i} \,.</math>

When divided by {{mvar|ρ}}, and with all terms on one side of the equation, the compressible flow equation is:

:<math>\frac{\partial v_i}{\partial x_i} - \frac{v_i\, v_j}{a^2} \frac{\partial v_i}{\partial x_j} = 0 \,.</math>

Note that until this stage, no assumptions have been made regarding the flow (besides that it is a [[steady flow]]).

Now, for irrotational flow the velocity {{math|'''v'''}} is the gradient of the velocity potential {{math|Φ}}, and the local Mach number components {{math|''M<sub>i</sub>''}} are defined as:

:<math>v_i = \frac{\partial \Phi}{\partial x_i} \qquad \text{and} \qquad M_i = \frac{v_i}{a} = \frac{1}{a} \frac{\partial \Phi}{\partial x_i} \,.</math>

When used in the flow equation, the full potential equation results:

:<math>\frac{\partial^2 \Phi}{\partial x_i\, \partial x_i} - M_i\, M_j\, \frac{\partial^2 \Phi}{\partial x_i\, \partial x_j} = 0 \,.</math>

Written out in components, the form given at the beginning of this section is obtained. When a specific [[equation of state]] is provided, relating pressure {{mvar|p}} and density {{mvar|ρ}}, the speed of sound can be determined. Subsequently, together with adequate boundary conditions, the full potential equation can be solved (most often through the use of a [[computational fluid dynamics]] code).
|}

====Sound waves====
{{main|Sound|Acoustics|Wave equation}}

Small-amplitude sound waves can be approximated with the following potential-flow model:<ref>Lamb (1994) §287, pp. 492–495.</ref>

:<math>\frac{\partial^2 \varphi}{\partial t^2} = \overline{a}^2 \Delta \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|ā}} 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.

===Applicability and limitations===
Potential flow does not include all the characteristics of flows that are encountered in the real world.  Potential flow theory cannot be applied for viscous [[internal flow]]s.<ref name=B_378_380/> [[Richard Feynman]] considered potential flow to be so unphysical that the only fluid to obey the assumptions was "dry water" (quoting John von Neumann).<ref>{{citation| author1-link=Richard Feynman | first1=R. P. | last1=Feynman | first2=R. B. | last2=Leighton | author2-link=Robert B. Leighton | first3=M. | last3=Sands | author3-link=Matthew Sands | year=1964 | title=[[The Feynman Lectures on Physics]] | publisher=Addison-Wesley | volume=2 }}, p. 40-3. Chapter 40 has the title: ''The flow of dry water''.</ref> Incompressible potential flow also makes a number of invalid predictions, such as [[d'Alembert's paradox]], which states that the drag on any object moving through an infinite fluid otherwise at rest is zero.<ref name=B_404_405>Batchelor (1973) pp. 404–405.</ref> More precisely, potential flow cannot account for the behaviour of flows that include a [[boundary layer]].<ref name=B_378_380/> Nevertheless, understanding potential flow is important in many branches of fluid mechanics. In particular, simple potential flows (called [[elementary flow]]s) such as the [[free vortex]] and the [[wikt:point source|point source]] possess ready analytical solutions.  These solutions can be [[Superposition principle|superposed]] to create more complex flows satisfying a variety of boundary conditions.  These flows correspond closely to real-life flows over the whole of fluid mechanics; in addition, many valuable insights arise when considering the deviation (often slight) between an observed flow and the corresponding potential flow. Potential flow finds many applications in fields such as aircraft design. For instance, in [[computational fluid dynamics]], one technique is to couple a potential flow solution outside the [[boundary layer]] to a solution of the [[Boundary layer#Boundary layer equations|boundary layer equations]] inside the boundary layer. The absence of boundary layer effects means that any streamline can be replaced by a solid boundary with no change in the flow field, a technique used in many aerodynamic design approaches. Another technique would be the use of [[Riabouchinsky solid]]s.{{dubious|date=March 2009}}

==Analysis for two-dimensional flow==<!-- [[Potential flow in two dimensions]] redirects here -->
{{main|Conformal map}}
'''Potential flow in two dimensions''' is simple to analyze using [[conformal mapping]], by the use of [[transformation (geometry)|transformation]]s of the [[complex plane]]. However, use of complex numbers is not required, as for example in the classical analysis of fluid flow past a cylinder.  It is not possible to solve a potential flow using [[complex number]]s in three dimensions.<ref name=B_106_108>Batchelor (1973) pp. 106–108.</ref>

The basic idea is to use a [[Holomorphic function|holomorphic]] (also called [[analytic function|analytic]]) or [[meromorphic function]] {{mvar|f}}, which maps the physical domain {{math|(''x'', ''y'')}} to the transformed domain {{math|(''φ'', ''ψ'')}}. While {{mvar|x}}, {{mvar|y}}, {{mvar|φ}} and {{mvar|ψ}} are all [[real number|real valued]], it is convenient to define the complex quantities 

:<math>z=x+iy \qquad \text{and} \qquad w=\varphi+i\psi \,.</math>

Now, if we write the mapping {{mvar|f}} as<ref name=B_106_108/>

:<math> f(x+iy)=\varphi+i\psi \qquad \text{or} \qquad f(z)=w \,.</math>

Then, because {{mvar|f}} is a holomorphic or meromorphic function, it has to satisfy the [[Cauchy–Riemann equations]]<ref name=B_106_108/>

:<math>\frac{\partial\varphi}{\partial x}=\frac{\partial\psi}{\partial y} \,, \qquad \frac{\partial\varphi}{\partial y}=-\frac{\partial\psi}{\partial x} \,.</math>

The velocity components {{math|(''u'', ''v'')}}, in the {{math|(''x'', ''y'')}} directions respectively, can be obtained directly from {{mvar|f}} by differentiating with respect to {{mvar|z}}. That is<ref name=B_106_108/>

:<math>\frac{df}{dz}=u-iv</math>

So the velocity field {{math|'''v''' {{=}} (''u'', ''v'')}} is specified by<ref name=B_106_108/>

:<math>u=\frac{\partial\varphi}{\partial x}=\frac{\partial\psi}{\partial y},\qquad v=\frac{\partial\varphi}{\partial y}=-\frac{\partial\psi}{\partial x} \,. </math>

Both {{mvar|φ}} and {{mvar|ψ}} then satisfy [[Laplace's equation]]:<ref name=B_106_108/>

:<math>\Delta\varphi = \frac{\partial^2\varphi}{\partial x^2} + \frac{\partial^2\varphi}{\partial y^2} = 0 \qquad \text{and} \qquad \Delta\psi = \frac{\partial^2\psi}{\partial x^2} + \frac{\partial^2\psi}{\partial y^2} = 0 \,.</math>

So {{mvar|φ}} can be identified as the velocity potential and {{mvar|ψ}} is called the [[stream function]].<ref name=B_106_108/> Lines of constant {{mvar|ψ}} are known as [[Streamlines, streaklines, and pathlines#Streamlines|streamlines]] and lines of constant {{mvar|φ}} are known as equipotential lines (see [[equipotential surface]]). 

Streamlines and equipotential lines are orthogonal to each other, since<ref name=B_106_108/>

:<math>\nabla \varphi \cdot \nabla \psi = \frac{\partial\varphi}{\partial x} \frac{\partial\psi}{\partial x} + \frac{\partial\varphi}{\partial y} \frac{\partial\psi}{\partial y} = \frac{\partial \psi}{\partial y} \frac{\partial \psi}{\partial x} - \frac{\partial \psi}{\partial x} \frac{\partial \psi}{\partial y} = 0 \,.</math>

Thus the flow occurs along the lines of constant {{mvar|ψ}} and at right angles to the lines of constant {{mvar|φ}}.<ref name=B_106_108/>

{{math|Δ''ψ'' {{=}} 0}} is also satisfied, this relation being equivalent to {{math|∇ × '''v''' {{=}} '''0'''}}. So the flow is irrotational. The automatic condition {{math|{{sfrac|∂<sup>2</sup>Ψ|∂''x'' ∂''y''}} {{=}} {{sfrac|∂<sup>2</sup>Ψ|∂''y'' ∂''x''}}}} then gives the incompressibility constraint {{math|∇ · '''v''' {{=}} 0}}.

==Examples of two-dimensional flows==
{{main|Potential flow around a circular cylinder|Rankine half body}}
Any differentiable function may be used for {{mvar|f}}. The examples that follow use a variety of [[elementary function]]s; [[special function]]s may also be used. Note that [[multi-valued function]]s such as the [[natural logarithm]] may be used, but attention must be confined to a single [[Riemann surface]].

===Power laws===
{| class="infobox bordered" style="width: 250px;"
|-
| [[Image:Conformal power half.svg|right|250px]]
|-
| [[File:Conformal power two third.svg|right|250px]]
|-
| [[File:Conformal power one.svg|right|250px]]
|-
| [[File:Conformal power one and a half.svg|right|250px]]
|-
| [[File:Conformal power two.svg|right|250px]]
|-
| [[File:Conformal power three.svg|right|250px]]
|-
| [[File:Conformal power minus one.svg|right|250px]]
|-
| Examples of conformal maps for the power law {{math|''w'' {{=}} ''Az<sup>n</sup>''}}, for different values of the power {{mvar|n}}. Shown is the {{mvar|z}}-plane, showing lines of constant potential {{mvar|φ}} and streamfunction {{mvar|ψ}}, while {{math|''w'' {{=}} ''φ'' + ''iψ''}}.
|}

In case the following [[power (mathematics)|power]]-law conformal map is applied, from {{math|''z'' {{=}} ''x'' + ''iy''}} to {{math|''w'' {{=}} ''φ'' + ''iψ''}}:<ref name=B_409_413>Batchelor (1973) pp. 409–413.</ref>

:<math>w=Az^n \,,</math>

then, writing {{mvar|z}} in polar coordinates as {{math|''z'' {{=}} ''x'' + ''iy'' {{=}} ''re<sup>iθ</sup>''}}, we have<ref name=B_409_413/>

:<math>\varphi=Ar^n\cos n\theta \qquad \text{and} \qquad \psi=Ar^n\sin n\theta \,.</math>

In the figures to the right examples are given for several values of {{mvar|n}}. The black line is the boundary of the flow, while the darker blue lines are streamlines, and the lighter blue lines are equi-potential lines. Some interesting powers {{mvar|n}} are:<ref name=B_409_413/>
*{{math|''n'' {{=}} {{sfrac|1|2}}}}: this corresponds with flow around a semi-infinite plate,
*{{math|''n'' {{=}} {{sfrac|2|3}}}}: flow around a right corner,
*{{math|''n'' {{=}} 1}}: a trivial case of uniform flow,
*{{math|''n'' {{=}} 2}}: flow through a corner, or near a stagnation point, and
*{{math|''n'' {{=}} −1}}: flow due to a source doublet

The constant {{mvar|A}} is a scaling parameter: its [[absolute value]] {{math|{{abs|''A''}}}} determines the scale, while its [[arg (mathematics)|argument]] {{math|arg(''A'')}} introduces a rotation (if non-zero).

==== Power laws with {{math|''n'' {{=}} 1}}: uniform flow ==== <!-- [[Uniform flow]] redirects here -->
If {{math|''w'' {{=}} ''Az''<sup>1</sup>}}, that is, a power law with {{math|''n'' {{=}} 1}}, the streamlines (i.e. lines of constant {{mvar|ψ}}) are a system of straight lines parallel to the {{mvar|x}}-axis. This is easiest to see by writing in terms of real and imaginary components:

:<math>f(x+iy) = A\, (x+iy) = Ax + i Ay </math>

thus giving {{math|''φ'' {{=}} ''Ax''}} and {{math|''ψ'' {{=}} ''Ay''}}. This flow may be interpreted as '''uniform flow''' parallel to the {{mvar|x}}-axis.

==== Power laws with {{math|''n'' {{=}} 2}} ====
If {{math|''n'' {{=}} 2}}, then {{math|''w'' {{=}} ''Az''<sup>2</sup>}} and the streamline corresponding to a particular value of {{mvar|ψ}} are those points satisfying

:<math>\psi=Ar^2\sin 2\theta \,,</math>

which is a system of [[hyperbola|rectangular hyperbolae]]. This may be seen by again rewriting in terms of real and imaginary components. Noting that [[List of trigonometric identities#Multiple-angle formulae|{{math|sin 2''θ'' {{=}} 2 sin ''θ'' cos ''θ''}}]] and rewriting {{math|sin ''θ'' {{=}} {{sfrac|''y''|''r''}}}} and {{math|cos ''θ'' {{=}} {{sfrac|''x''|''r''}}}} it is seen (on simplifying) that the streamlines are given by

:<math>\psi=2Axy \,.</math>

The velocity field is given by {{math|∇''φ''}}, or

:<math>\begin{pmatrix} u \\ v \end{pmatrix} =  \begin{pmatrix} \frac{\partial \varphi}{\partial x} \\[2px] \frac{\partial \varphi}{\partial y} 
\end{pmatrix} =  \begin{pmatrix} + {\partial \psi \over \partial y} \\[2px] - {\partial \psi \over \partial x} \end{pmatrix} = \begin{pmatrix} +2Ax \\[2px] -2Ay \end{pmatrix} \,.</math>

In fluid dynamics, the flowfield near the origin corresponds to a [[stagnation point]].  Note that the fluid at the origin is at rest (this follows on differentiation of {{math|''f''(z) {{=}} ''z''<sup>2</sup>}} at {{math|''z'' {{=}} 0}}). The {{math|''ψ'' {{=}} 0}} streamline is particularly interesting: it has two (or four) branches, following the coordinate axes, i.e. {{math|''x'' {{=}} 0}} and {{math|''y'' {{=}} 0}}. As no fluid flows across the {{mvar|x}}-axis, it (the {{mvar|x}}-axis) may be treated as a solid boundary. It is thus possible to ignore the flow in the lower half-plane where {{math|''y'' < 0}} and to focus on the flow in the upper halfplane. With this interpretation, the flow is that of a vertically directed jet impinging on a horizontal flat plate. The flow may also be interpreted as flow into a 90 degree corner if the regions specified by (say) {{math|''x'', ''y'' < 0}} are ignored.

==== Power laws with {{math|''n'' {{=}} 3}} ====
If {{math|''n'' {{=}} 3}}, the resulting flow is a sort of hexagonal version of the {{math|''n'' {{=}} 2}} case considered above.  Streamlines are given by, {{math|''ψ'' {{=}} 3''x''<sup>2</sup>''y'' − ''y''<sup>3</sup>}} and the flow in this case may be interpreted as flow into a 60° corner.

==== Power laws with {{math|''n'' {{=}} −1}}: doublet ==== <!-- [[Doublet (potential flow)]] redirects here]] -->
If {{math|''n'' {{=}} −1}}, the streamlines are given by

:<math>\psi=-\frac{A}{r}\sin\theta.</math>

This is more easily interpreted in terms of real and imaginary components:
:<math> \psi = \frac{-A y}{r^2} = \frac{-A y}{x^2 + y^2} \,, </math>
:<math> x^2 + y^2 + \frac{A y}{\psi} = 0 \,, </math>
:<math> x^2+\left(y+\frac{A}{2\psi}\right)^2=\left(\frac{A}{2\psi}\right)^2 \,.</math>

Thus the streamlines are [[circle]]s that are tangent to the x-axis at the origin. The circles in the upper half-plane thus flow clockwise, those in the lower half-plane flow anticlockwise.  Note that the velocity components are proportional to {{math|''r''<sup>−2</sup>}}; and their values at the origin is infinite. This flow pattern is usually referred to as a '''doublet''', or '''dipole''', and can be interpreted as the combination of a source-sink pair of infinite strength kept an infinitesimally small distance apart. The velocity field is given by 

:<math>(u,v)=\left( \frac{\partial \psi}{\partial y}, - \frac{\partial \psi}{\partial x} \right) =  \left(A\frac{y^2-x^2}{\left(x^2+y^2\right)^2},-A\frac{2xy}{\left(x^2+y^2\right)^2}\right) \,.</math>

or in polar coordinates:

:<math>(u_r, u_\theta)=\left( \frac{1}{r} \frac{\partial \psi}{\partial \theta}, - \frac{\partial \psi}{\partial r} \right) = \left(-\frac{A}{r^2}\cos\theta, -\frac{A}{r^2}\sin\theta\right) \,.</math>

==== Power laws with {{math|''n'' {{=}} −2}}: quadrupole ====
If {{math|''n'' {{=}} −2}}, the streamlines are given by

:<math>\psi=-\frac{A}{r^2}\sin 2 \theta \,.</math>

This is the flow field associated with a [[quadrupole]].<ref>{{Cite book| publisher = Wiley-Interscience| isbn = 9780471511298| last = Kyrala| first = A.| title = Applied Functions of a Complex Variable| year = 1972| pages = 116–117}}</ref>

===Line source and sink===
A line source or sink of strength <math>Q</math> (<math>Q>0</math> for source and <math>Q<0</math> for sink) is given by the potential

:<math>w = \frac{Q}{2\pi} \ln z</math>

where <math>Q</math> in fact is the volume flux per unit length across an surface enclosing the source or sink. The velocity field in polar coordinates are

:<math>u_r = \frac{Q}{2\pi r},\quad u_\theta=0</math>

i.e., a purely radial flow.

===Line vortex===
A line vortex of strength <math>\Gamma</math> is given by

:<math>w=\frac{\Gamma}{2\pi i}\ln z</math>

where <math>\Gamma</math> is the [[circulation (fluid dynamics)|circulation]] around any simple closed contour enclosing the vortex. The velocity field in polar coordinates are

:<math>u_r = 0,\quad u_\theta=\frac{\Gamma}{2\pi r}</math>

i.e., a purely azimuthal flow.

==Analysis for three-dimensional flow==
For three-dimensional flows, complex potential cannot be obtained.

===Point source and sink===
The velocity potential of a point source or sink of strength <math>Q</math> (<math>Q>0</math> for source and <math>Q<0</math> for sink) in spherical polar coordinates is given by

:<math>\phi = -\frac{Q}{4\pi r}</math>

where <math>Q</math> in fact is the volume flux across a closed surface enclosing the source or sink.


== See also ==
*[[Potential flow around a circular cylinder]]
*[[Aerodynamic potential-flow code]]
*[[Conformal mapping]]
*[[Darwin drift]]
*[[Flownet]]
*[[Laplacian field]]
*[[Potential theory]]
*[[Stream function]]
*[[Velocity potential]]

==Notes==
{{reflist}}

==References==
*{{citation| first=G.K. | last=Batchelor | authorlink=George Batchelor | title=An introduction to fluid dynamics | publisher=Cambridge University Press | year=1973 | isbn=0-521-09817-3 }}
*{{Citation|last=Chanson|first=H.|authorlink=Hubert Chanson|title=Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows| url=http://espace.library.uq.edu.au/view/UQ:191112|publisher=CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages| year=2009| isbn=978-0-415-49271-3}}
*{{citation
 | first=H. | last=Lamb | authorlink=Horace Lamb
 | title=Hydrodynamics
 | edition=6th
 | publisher=Cambridge University Press
 | year=1994
 | origyear=1932
 | isbn=978-0-521-45868-9
}}
*{{citation
 | first=L.M. | last=Milne-Thomson
 | authorlink=L. M. Milne-Thomson
 | title=Theoretical hydrodynamics
 | edition=5th
 | publisher=Dover
 | year=1996
 | origyear=1968
 | isbn=0-486-68970-0 
}}

==Further reading==
*{{citation
 | first=H. | last=Chanson | authorlink=Hubert Chanson
 | title=Le potentiel de vitesse pour les écoulements de fluides réels: la contribution de Joseph-Louis Lagrange <nowiki>[Velocity potential in real fluid flows: Joseph-Louis Lagrange's contribution]</nowiki>
 | url=http://espace.library.uq.edu.au/view/UQ:119883
 | journal=La Houille Blanche
 | volume= | issue=5 | pages=127–131
 | year=2007
 | doi=10.1051/lhb:2007072 
|language=fr}}  
*{{citation
 | contribution=Surface waves
 | first1=J.V. | last1=Wehausen | author1-link=John V. Wehausen
 | first2=E.V. | last2=Laitone
 | editor1-first=S. | editor1-last=Flügge | editor1-link=Siegfried Flügge
 | editor2-first=C. | editor2-last=Truesdell | editor2-link=Clifford Truesdell
 | title=Encyclopedia of Physics
 | url=http://www.coe.berkeley.edu/SurfaceWaves
 | volume=IX | pages=446–778
 | publisher=Springer Verlag
 | year=1960
}}

==External links==
{{Commons category}}
* {{cite web
 | title=Irrotational flow of an inviscid fluid
 | url=http://www.diam.unige.it/~irro/lecture_e.html
 | publisher=[[University of Genoa]], Faculty of Engineering
 | year = 
 | accessdate=2009-03-29
}}
* {{cite web
 | url=http://3d-xplormath.org/j/applets/en/index.html
 | title=Conformal Maps Gallery
 | publisher=3D-XplorMath
 | year=
 | accessdate=2009-03-29
}} — Java applets for exploring conformal maps
* [http://airfoil.dimanov.com/ Potential Flow Visualizations - Interactive WebApps]

{{DEFAULTSORT:Potential Flow}}
[[Category:Fluid dynamics]]