Influid dynamics,potential flow orirrotational flow refers to a description of a fluid flow with novorticity in it. Such a description typically arises in the limit of vanishingviscosity, i.e., for aninviscid fluid and with no vorticity present in the flow.

Potential flow describes thevelocity field as thegradient of a scalar function: thevelocity potential. As a result, a potential flow is characterized by anirrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to thecurl of the gradient of ascalar always being equal to zero.

In the case of anincompressible flow the velocity potential satisfiesLaplace's equation, andpotential theory is applicable. However, potential flows also have been used to describecompressible flows andHele-Shaw flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows.

Applications of potential flow include: the outer flow field foraerofoils,water waves,electroosmotic flow, andgroundwater flow. For flows (or parts thereof) with strongvorticity effects, the potential flow approximation is not applicable. In flow regions where vorticity is known to be important, such aswakes andboundary layers, potential flow theory is not able to provide reasonable predictions of the flow.^[1] However, there are often large regions of a flow in which the assumption of irrotationality is valid, allowing the use of potential flow for various applications; these include flow aroundaircraft,groundwater flow,acoustics,water waves, andelectroosmotic flow.^[2]

In potential or irrotational flow, the vorticity vector field is zero, i.e.,

$${\displaystyle \mathit{\omega }\equiv \mathrm{\nabla }\times \mathbf{v}=0,}$$

where${\displaystyle \mathbf{v}(\mathbf{x},t)}$ is the velocity field and${\displaystyle \mathit{\omega }(\mathbf{x},t)}$ is thevorticity field. Like any vector field having zero curl, the velocity field can be expressed as the gradient of certain scalar, say${\displaystyle \phi (\mathbf{x},t)}$ which is called thevelocity potential, since the curl of the gradient is always zero. We therefore have^[3]

$${\displaystyle \mathbf{v}=\mathrm{\nabla }\phi .}$$

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

In potential flow, thecirculation${\displaystyle \mathrm{\Gamma }}$ around anysimply-connected contour${\displaystyle C}$ is zero. This can be shown using theStokes theorem,

$${\displaystyle \mathrm{\Gamma }\equiv {\oint }_C\mathbf{v}\cdot d\mathbf{l}=\int \mathit{\omega }\cdot d\mathbf{f}=0}$$

where${\displaystyle d\mathbf{l}}$ is the line element on the contour and${\displaystyle d\mathbf{f}}$ 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-calledirrotational vortices or point vortices, or in smoke rings), the circulation${\displaystyle \mathrm{\Gamma }}$ need not be zero. In the former case, Stokes theorem cannot be applied and in the later case,${\displaystyle \mathit{\omega }}$ is non-zero within the region bounded by the contour. Around a contour encircling an infinitely long solid cylinder with which the contour loops${\displaystyle N}$ times, we have

$${\displaystyle \mathrm{\Gamma }=N\kappa }$$

where${\displaystyle \kappa }$ is a cyclic constant. This example belongs to a doubly-connected space. In an${\displaystyle n}$-tuply connected space, there are${\displaystyle n-1}$ such cyclic constants, namely,${\displaystyle {\kappa }_1,{\kappa }_2,\dots ,{\kappa }_{n-1}.}$

In case of anincompressible flow — for instance of aliquid, or agas at lowMach numbers; but not forsound waves — the velocityv has zerodivergence:^[3]

$${\displaystyle \mathrm{\nabla }\cdot \mathbf{v}=0,}$$

Substituting here${\displaystyle \mathbf{v}=\mathrm{\nabla }\phi }$ shows that${\displaystyle \phi }$ satisfies theLaplace equation^[3]

$${\displaystyle {\mathrm{\nabla }}^2\phi =0,}$$

where∇² = ∇ ⋅ ∇ is theLaplace operator (sometimes also writtenΔ). Since solutions of the Laplace equation areharmonic functions, every harmonic function represents a potential flow solution. As evident, in the incompressible case, the velocity field is determined completely from itskinematics: the assumptions of irrotationality and zero divergence of flow.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 ofBernoulli's principle.

In incompressible flows, contrary to common misconception, the potential flow indeed satisfies the fullNavier–Stokes equations, not just theEuler equations, because the viscous term

$${\displaystyle \mu {\mathrm{\nabla }}^2\mathbf{v}=\mu \mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{v})-\mu \mathrm{\nabla }\times \mathit{\omega }=0}$$

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${\displaystyle \phi }$ and its conjugate harmonic function${\displaystyle \psi }$ (stream function), incompressible potential flow reduces to a very simple system that is analyzed usingcomplex analysis (see below).

Potential flow theory can also be used to model irrotational compressible flow. The derivation of the governing equation for${\displaystyle \phi }$ fromEulers equation is quite straightforward. The continuity and the (potential flow) momentum equations for steady flows are given by

$${\displaystyle \rho \mathrm{\nabla }\cdot \mathbf{v}+\mathbf{v}\cdot \mathrm{\nabla }\rho =0,(\mathbf{v}\cdot \mathrm{\nabla })\mathbf{v}=-\frac{1}{\rho }\mathrm{\nabla }p=-\frac{c^2}{\rho }\mathrm{\nabla }\rho }$$

where the last equation follows from the fact thatentropy is constant for a fluid particle and that square of thesound speed is${\displaystyle c^2=(\mathrm{\partial }p/\mathrm{\partial }\rho {)}_s}$. Eliminating${\displaystyle \mathrm{\nabla }\rho }$ from the two governing equations results in

$${\displaystyle c^2\mathrm{\nabla }\cdot \mathbf{v}-\mathbf{v}\cdot (\mathbf{v}\cdot \mathrm{\nabla })\mathbf{v}=0.}$$

The incompressible version emerges in the limit${\displaystyle c\to \mathrm{\infty }}$. Substituting here${\displaystyle \mathbf{v}=\mathrm{\nabla }\phi }$ results in^[4][5]

$${\displaystyle (c^2-{\phi }_x^2){\phi }_{xx}+(c^2-{\phi }_y^2){\phi }_{yy}+(c^2-{\phi }_z^2){\phi }_{zz}-2({\phi }_x{\phi }_y{\phi }_{xy}+{\phi }_y{\phi }_z{\phi }_{yz}+{\phi }_z{\phi }_x{\varphi }_{zx})=0}$$

where${\displaystyle c=c(v)}$ is expressed as a function of the velocity magnitude${\displaystyle v^2=(\mathrm{\nabla }\varphi {)}^2}$. For apolytropic gas,${\displaystyle c^2=(\gamma -1)(h_0-v^2/2)}$, where${\displaystyle \gamma }$ is thespecific heat ratio and${\displaystyle h_0}$ is thestagnation enthalpy. In two dimensions, the equation simplifies to

$${\displaystyle (c^2-{\phi }_x^2){\phi }_{xx}+(c^2-{\phi }_y^2){\phi }_{yy}-2{\phi }_x{\phi }_y{\phi }_{xy}=0.}$$

Validity: As it stands, the equation is valid for any inviscid potential flows, irrespective of whether the flow is subsonic or supersonic (e.g.Prandtl–Meyer flow). However in supersonic and also in transonic flows, shock waves can occur which can introduce entropy and vorticity into the flow making the flow rotational. Nevertheless, there are two cases for which potential flow prevails even in the presence of shock waves, which are explained from the (not necessarily potential) momentum equation written in the following form

$${\displaystyle \mathrm{\nabla }(h+v^2/2)-\mathbf{v}\times \mathit{\omega }=T\mathrm{\nabla }s}$$

where${\displaystyle h}$ is thespecific enthalpy,${\displaystyle \mathit{\omega }}$ is thevorticity field,${\displaystyle T}$ is the temperature and${\displaystyle s}$ is the specific entropy. Since in front of the leading shock wave, we have a potential flow, Bernoulli's equation shows that${\displaystyle h+v^2/2}$ is constant, which is also constant across the shock wave (Rankine–Hugoniot conditions) and therefore we can write^[4]

$${\displaystyle \mathbf{v}\times \mathit{\omega }=-T\mathrm{\nabla }s}$$

1) When the shock wave is of constant intensity, the entropy discontinuity across the shock wave is also constant i.e.,${\displaystyle \mathrm{\nabla }s=0}$ and therefore vorticity production is zero. Shock waves at the pointed leading edge of two-dimensional wedge or three-dimensional cone (Taylor–Maccoll flow) has constant intensity. 2) For weak shock waves, the entropy jump across the shock wave is a third-order quantity in terms of shock wave strength and therefore${\displaystyle \mathrm{\nabla }s}$ can be neglected. Shock waves in slender bodies lies nearly parallel to the body and they are weak.

Nearly parallel flows: When the flow is predominantly unidirectional with small deviations such as in flow past slender bodies, the full equation can be further simplified. Let${\displaystyle U{\mathbf{e}}_x}$ be the mainstream and consider small deviations from this velocity field. The corresponding velocity potential can be written as${\displaystyle \phi =xU+\varphi }$ where${\displaystyle \varphi }$ characterizes the small departure from the uniform flow and satisfies the linearized version of the full equation. This is given by

$${\displaystyle (1-M^2)\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{z}^2}=0}$$

where${\displaystyle M=U/c_{\mathrm{\infty }}}$ is the constantMach number corresponding to the uniform flow. This equation is valid provided${\displaystyle M}$ is not close to unity. When${\displaystyle |M-1|}$ is small (transonic flow), we have the following nonlinear equation^[4]

$${\displaystyle 2{\alpha }_{\ast }\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }x}\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{x}^2}=\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{z}^2}}$$

where${\displaystyle {\alpha }_{\ast }}$ is the critical value ofLandau derivative${\displaystyle \alpha =(c^4/2{\upsilon }^3)({\mathrm{\partial }}^2\upsilon /\mathrm{\partial }{p}^2{)}_s}$^[6][7] and${\displaystyle \upsilon =1/\rho }$ is thespecific volume. The transonic flow is completely characterized by the single parameter${\displaystyle {\alpha }_{\ast }}$, which for polytropic gas takes the value${\displaystyle {\alpha }_{\ast }=\alpha =(\gamma +1)/2}$. Underhodograph transformation, the transonic equation in two-dimensions becomes theEuler–Tricomi equation.

The continuity and the (potential flow) momentum equations for unsteady flows are given by

$${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\rho \mathrm{\nabla }\cdot \mathbf{v}+\mathbf{v}\cdot \mathrm{\nabla }\rho =0,\frac{\mathrm{\partial }\mathbf{v}}{\mathrm{\partial }t}+(\mathbf{v}\cdot \mathrm{\nabla })\mathbf{v}=-\frac{1}{\rho }\mathrm{\nabla }p=-\frac{c^2}{\rho }\mathrm{\nabla }\rho =-\mathrm{\nabla }h.}$$

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

$${\displaystyle \frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}+\frac{v^2}{2}+h=f(t),\Rightarrow \frac{\mathrm{\partial }h}{\mathrm{\partial }t}=-\frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{t}^2}-\frac{1}{2}\frac{\mathrm{\partial }{v}^2}{\mathrm{\partial }t}+\frac{df}{dt}}$$

where${\displaystyle f(t)}$ is an arbitrary function. Without loss of generality, we can set${\displaystyle f(t)=0}$ since${\displaystyle \phi }$ is not uniquely defined. Combining these equations, we obtain

$${\displaystyle \frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{t}^2}+\frac{\mathrm{\partial }{v}^2}{\mathrm{\partial }t}=c^2\mathrm{\nabla }\cdot \mathbf{v}-\mathbf{v}\cdot (\mathbf{v}\cdot \mathrm{\nabla })\mathbf{v}.}$$

Substituting here${\displaystyle \mathbf{v}=\mathrm{\nabla }\phi }$ results in

$${\displaystyle {\phi }_{tt}+({\phi }_x^2+{\phi }_y^2+{\phi }_z^2{)}_t=(c^2-{\phi }_x^2){\phi }_{xx}+(c^2-{\phi }_y^2){\phi }_{yy}+(c^2-{\phi }_z^2){\phi }_{zz}-2({\phi }_x{\phi }_y{\phi }_{xy}+{\phi }_y{\phi }_z{\phi }_{yz}+{\phi }_z{\phi }_x{\varphi }_{zx}).}$$

Nearly parallel flows: As in before, for nearly parallel flows, we can write (after introducing a recaled time${\displaystyle \tau =c_{\mathrm{\infty }}t}$)

$${\displaystyle \frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{\tau }^2}+2M\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }x\mathrm{\partial }\tau }=(1-M^2)\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{z}^2}}$$

provided the constant Mach number${\displaystyle M}$ is not close to unity. When${\displaystyle |M-1|}$ is small (transonic flow), we have the following nonlinear equation^[4]

$${\displaystyle \frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{\tau }^2}+2\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }x\mathrm{\partial }\tau }=-2{\alpha }_{\ast }\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }x}\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{z}^2}.}$$

Sound waves: In sound waves, the velocity magnitude${\displaystyle v}$ (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,${\displaystyle c}$ is a constant (for example, in polytropic gas${\displaystyle c^2=(\gamma -1)h_0}$), we have^[8][4]

$${\displaystyle \frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{t}^2}=c^2{\mathrm{\nabla }}^2\phi ,}$$

which is a linearwave equation for the velocity potentialφ. Again the oscillatory part of the velocity vectorv is related to the velocity potential byv = ∇φ, while as beforeΔ is theLaplace operator, andc is the average speed of sound in thehomogeneous medium. Note that also the oscillatory parts of thepressurep anddensityρ each individually satisfy the wave equation, in this approximation.

Potential flow does not include all the characteristics of flows that are encountered in the real world. Potential flow theory cannot be applied for viscousinternal flows,^[1] except forflows between closely spaced plates.Richard Feynman considered potential flow to be so unphysical that the only fluid to obey the assumptions was "dry water" (quotingJohn von Neumann).^[9] Incompressible potential flow also makes a number of invalid predictions, such asd'Alembert's paradox, which states that the drag on any object moving through an infinite fluid otherwise at rest is zero.^[10] More precisely, potential flow cannot account for the behaviour of flows that include aboundary layer.^[1] Nevertheless, understanding potential flow is important in many branches of fluid mechanics. In particular, simple potential flows (calledelementary flows) such as thefree vortex and thepoint source possess ready analytical solutions. These solutions can besuperposed 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, incomputational fluid dynamics, one technique is to couple a potential flow solution outside theboundary layer to a solution of theboundary 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 ofRiabouchinsky solids.^{[dubious –discuss]}

Potential flow in two dimensions is simple to analyze usingconformal mapping, by the use oftransformations of thecomplex 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 usingcomplex numbers in three dimensions.^[11]

The basic idea is to use aholomorphic (also calledanalytic) ormeromorphic functionf, which maps the physical domain(x,y) to the transformed domain(φ,ψ). Whilex,y,φ andψ are allreal valued, it is convenient to define the complex quantities

Now, if we write the mappingf as^[11]

Then, becausef is a holomorphic or meromorphic function, it has to satisfy theCauchy–Riemann equations^[11]

The velocity components(u,v), in the(x,y) directions respectively, can be obtained directly fromf by differentiating with respect toz. That is^[11]

$${\displaystyle \frac{df}{dz}=u-iv}$$

So the velocity fieldv = (u,v) is specified by^[11]

Bothφ andψ then satisfyLaplace's equation:^[11]

Soφ can be identified as the velocity potential andψ is called thestream function.^[11] Lines of constantψ are known asstreamlines and lines of constantφ are known as equipotential lines (seeequipotential surface).

Streamlines and equipotential lines are orthogonal to each other, since^[11]

$${\displaystyle \mathrm{\nabla }\phi \cdot \mathrm{\nabla }\psi =\frac{\mathrm{\partial }\phi }{\mathrm{\partial }x}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }x}+\frac{\mathrm{\partial }\phi }{\mathrm{\partial }y}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}=\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }x}-\frac{\mathrm{\partial }\psi }{\mathrm{\partial }x}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}=0.}$$

Thus the flow occurs along the lines of constantψ and at right angles to the lines of constantφ.^[11]

Δψ = 0 is also satisfied, this relation being equivalent to∇ ×v =0. So the flow is irrotational. The automatic condition⁠∂²Ψ/∂x ∂y⁠ =⁠∂²Ψ/∂y ∂x⁠ then gives the incompressibility constraint∇ ·v = 0.

Any differentiable function may be used forf. The examples that follow use a variety ofelementary functions;special functions may also be used. Note thatmulti-valued functions such as thenatural logarithm may be used, but attention must be confined to a singleRiemann surface.

Examples of conformal maps for the power laww =Azⁿ

Examples of conformal maps for the power laww =Azⁿ, for different values of the powern. Shown is thez-plane, showing lines of constant potentialφ and streamfunctionψ, whilew =φ +iψ.

In case the followingpower-law conformal map is applied, fromz =x +iy tow =φ +iψ:^[12]

$${\displaystyle w=A{z}^n,}$$

then, writingz in polar coordinates asz =x +iy =re^iθ, we have^[12]

$${\displaystyle \phi =A{r}^n\cos n\theta \text{and}\psi =A{r}^n\sin n\theta .}$$

In the figures to the right examples are given for several values ofn. 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 powersn are:^[12]

• n =⁠1/2⁠: this corresponds with flow around a semi-infinite plate,
• n =⁠2/3⁠: flow around a right corner,
• n = 1: a trivial case of uniform flow,
• n = 2: flow through a corner, or near a stagnation point, and
• n = −1: flow due to a source doublet

The constantA is a scaling parameter: itsabsolute value|A| determines the scale, while itsargumentarg(A) introduces a rotation (if non-zero).

Ifw =Az¹, that is, a power law withn = 1, the streamlines (i.e. lines of constantψ) are a system of straight lines parallel to thex-axis. This is easiest to see by writing in terms of real and imaginary components:

$${\displaystyle f(x+iy)=A(x+iy)=Ax+iAy}$$

thus givingφ =Ax andψ =Ay. This flow may be interpreted asuniform flow parallel to thex-axis.

Ifn = 2, thenw =Az² and the streamline corresponding to a particular value ofψ are those points satisfying

$${\displaystyle \psi =A{r}^2\sin 2\theta ,}$$

which is a system ofrectangular hyperbolae. This may be seen by again rewriting in terms of real and imaginary components. Noting thatsin 2θ = 2 sinθ cosθ and rewritingsinθ =⁠y/r⁠ andcosθ =⁠x/r⁠ it is seen (on simplifying) that the streamlines are given by

$${\displaystyle \psi =2Axy.}$$

The velocity field is given by∇φ, or

$${\displaystyle \left(\begin{array}{c}u\\ v\end{array}\right)=\left(\begin{array}{c}\frac{\mathrm{\partial }\phi }{\mathrm{\partial }x}\\ \frac{\mathrm{\partial }\phi }{\mathrm{\partial }y}\end{array}\right)=\left(\begin{array}{c}+\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}\\ -\frac{\mathrm{\partial }\psi }{\mathrm{\partial }x}\end{array}\right)=\left(\begin{array}{c}+2Ax\\ -2Ay\end{array}\right).}$$

In fluid dynamics, the flowfield near the origin corresponds to astagnation point. Note that the fluid at the origin is at rest (this follows on differentiation off(z) =z² atz = 0). Theψ = 0 streamline is particularly interesting: it has two (or four) branches, following the coordinate axes, i.e.x = 0 andy = 0. As no fluid flows across thex-axis, it (thex-axis) may be treated as a solid boundary. It is thus possible to ignore the flow in the lower half-plane wherey < 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)x,y < 0 are ignored.

Ifn = 3, the resulting flow is a sort of hexagonal version of then = 2 case considered above. Streamlines are given by,ψ = 3x²y −y³ and the flow in this case may be interpreted as flow into a 60° corner.

Ifn = −1, the streamlines are given by

$${\displaystyle \psi =-\frac{A}{r}\sin \theta .}$$

This is more easily interpreted in terms of real and imaginary components:

Thus the streamlines arecircles 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 tor⁻²; and their values at the origin is infinite. This flow pattern is usually referred to as adoublet, ordipole, 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

$${\displaystyle (u,v)=\left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y},-\frac{\mathrm{\partial }\psi }{\mathrm{\partial }x}\right)=\left(A\frac{y^2-x^2}{{\left(x^2+y^2\right)}^2},-A\frac{2xy}{{\left(x^2+y^2\right)}^2}\right).}$$

or in polar coordinates:

$${\displaystyle (u_r,u_{\theta })=\left(\frac{1}{r}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\theta },-\frac{\mathrm{\partial }\psi }{\mathrm{\partial }r}\right)=\left(-\frac{A}{r^2}\cos \theta ,-\frac{A}{r^2}\sin \theta \right).}$$

Ifn = −2, the streamlines are given by

$${\displaystyle \psi =-\frac{A}{r^2}\sin 2\theta .}$$

This is the flow field associated with aquadrupole.^[13]

A line source or sink of strength${\displaystyle Q}$ (${\displaystyle Q>0}$ for source and for sink) is given by the potential

$${\displaystyle w=\frac{Q}{2\pi }\ln z}$$

where${\displaystyle Q}$ in fact is the volume flux per unit length across a surface enclosing the source or sink. The velocity field in polar coordinates are

$${\displaystyle u_r=\frac{Q}{2\pi r},u_{\theta }=0}$$

i.e., a purely radial flow.

A line vortex of strength${\displaystyle \mathrm{\Gamma }}$ is given by

$${\displaystyle w=\frac{\mathrm{\Gamma }}{2\pi i}\ln z}$$

where${\displaystyle \mathrm{\Gamma }}$ is thecirculation around any simple closed contour enclosing the vortex. The velocity field in polar coordinates are

$${\displaystyle u_r=0,u_{\theta }=\frac{\mathrm{\Gamma }}{2\pi r}}$$

i.e., a purely azimuthal flow.

For three-dimensional flows, complex potential cannot be obtained.

The velocity potential of a point source or sink of strength${\displaystyle Q}$ (${\displaystyle Q>0}$ for source and for sink) in spherical polar coordinates is given by

$${\displaystyle \varphi =-\frac{Q}{4\pi r}}$$

where${\displaystyle Q}$ in fact is the volume flux across a closed surface enclosing the source or sink. The velocity field in spherical polar coordinates are

$${\displaystyle u_r=\frac{Q}{4\pi r^2},u_{\theta }=0,u_{\varphi }=0.}$$

• Potential flow around a circular cylinder
• Aerodynamic potential-flow code
• Conformal mapping
• Darwin drift
• Flownet
• Laplacian field
• Laplace equation for irrotational flow
• Potential theory
• Stream function
• Velocity potential
• Helmholtz decomposition

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• Chanson, H. (2009),Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows, CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages,ISBN 978-0-415-49271-3
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• Chanson, H. (2007),"Le potentiel de vitesse pour les écoulements de fluides réels: la contribution de Joseph-Louis Lagrange [Velocity potential in real fluid flows: Joseph-Louis Lagrange's contribution]",La Houille Blanche (in French),93 (5):127–131,Bibcode:2007LHBl...93..127C,doi:10.1051/lhb:2007072
• Wehausen, J.V.; Laitone, E.V. (1960), "Surface waves", inFlügge, S.;Truesdell, C. (eds.),Encyclopedia of Physics, vol. IX, Springer Verlag, pp. 446–778, archived fromthe original on 2009-01-05, retrieved2009-03-29

Wikimedia Commons has media related toPotential flow.

• "Irrotational flow of an inviscid fluid".University of Genoa, Faculty of Engineering. Retrieved2009-03-29.
• "Conformal Maps Gallery". 3D-XplorMath. Retrieved2009-03-29. — Java applets for exploring conformal maps
• Potential Flow Visualizations - Interactive WebApps

• Fluid dynamics

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