A potential flow is characterized by an irrotational velocity field. This property allows the description of the velocity field as the gradient of a scalar function (because taking the curl of the gradient is equivalent to make the cross product of two parallel vectors which is always zero). The flow can be compressible or incompressible, stationary or unstationary. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ...

The free-vortex flow is solution of the vorticity transport equation (obtained by taking the curl of the Euler momentum equation) only if the sources terms are zero, that is that the flow is barotropic (the pressure varies only with the density) and that the external forces are negligible or derives from a potential (the force fields are described by taking the gradient of scalar functions). Vorticity is a mathematical concept used in fluid dynamics. ...

In incompressible, steady or unsteady, fluid dynamics, potential flow obeys the following equations Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ...

(zero rotation) (zero divergence = volume conservation) Equivalently,		where: is the fluid velocity vector Φ is the fluid flow potential, scalar "" is curl "" is divergence. "" is gradient

Note that "" is not formally equivilent to "". In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... For other uses, see Gradient (disambiguation). ...

The equations above imply , or Laplace's equation, holds. In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ...

Together with the Navier-Stokes equations or the Euler equations, these equations can be used to calculate solutions to many practical flow situations. In two dimensions, potential flow reduces to a very simple system that is analysed using complex numbers (see below). The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations which describe the motion of fluid substances such as liquids and gases. ... In fluid dynamics, the Euler equations govern the motion of a compressible, inviscid fluid. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

Potential flow does not include all the characteristics of flows that are encountered in the real world. For example, potential flow excludes turbulence, which is commonly encountered in nature. Richard Feynman considered potential flow to be so unphysical that the only fluid to obey the assumptions was "dry water". In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic, stochastic property changes. ... Richard Phillips Feynman (May 11, 1918 – February 15, 1988; surname pronounced ) was an American physicist known for expanding the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and particle theory. ...

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. DAlemberts paradox states that an inviscid (non-viscous), incompressible flow produces no drag on an object surrounded by such fluid, yet it does produce lift. ...

More precisely, potential flow cannot account for the behaviour of flows that include a boundary layer. In physics and fluid mechanics, the boundary layer is that layer of fluid in the immediate vicinity of a bounding surface. ...

Nevertheless, understanding potential flow is important in many branches of fluid mechanics. In particular, simple potential flows (called elemental flows) such as the free vortex and the point source possess ready analytical solutions. These solutions can be 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. Vortex created by the passage of an aircraft wing, revealed by coloured smoke A vortex is a spinning turbulent flow (or any spiral whirling motion) with closed streamlines. ... The term superposition can have several meanings: Quantum superposition Law of superposition in geology and archaeology Superposition principle for vector fields Superposition Calculus is used for equational first-order reasoning This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

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 equations inside the boundary layer. A computer simulation of high velocity air flow around the Space Shuttle during re-entry. ... In physics and fluid mechanics, the boundary layer is that layer of fluid in the immediate vicinity of a bounding surface. ...

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 solids. In fluid mechanics a Riabouchinsky solid is a technique used for approximating boundary layer separation from a bluff body using potential flow. ...

Analysis

Potential flow in two dimensions is simple to analyse using complex numbers, viewed for convenience on the Argand diagram. In fluid dynamics, potential flow in two dimensions is simple to analyse using complex numbers. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

The basic idea is to define a holomorphic function f. If we write Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

f(x + iy) = φ + iψde

then the Cauchy-Riemann equations show that In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not sufficient condition for a function to be holomorphic. ...

(it is conventional to regard all symbols as real numbers; and to write z = x + iy and w = φ + iψ).

Also notice that (Some steps are missing, try calculating yourself)

The velocity field , specified by

then satisfies the requirements for potential flow:

and

ψ is defined as the stream function. Lines of constant ψ are known as streamlines and lines of constant φ are known as equipotential lines (see equipotential surface). In fluid dynamics, the stream function is defined for two-dimensional flows. ... In fluid dynamics, a streamline is a line which is everywhere tangent to the velocity of the flow. ... In fluid mechanics, equipotentials are lines or surfaces of equal head that are in direct relation to pressure. ...

The two sets of curves intersect at right angles, for

Examples: general considerations

Any differentiable function may be used for f. The examples that follow use a variety of elementary functions; special functions may also be used. In mathematics, several functions are important enough to deserve their own name. ... In mathematics, several functions are important enough to deserve their own name. ...

Note that multi-valued functions such as the natural logarithm may be used, but attention must be confined to a single Riemann surface. This diagram does not represent a true function; because the element 3, in X, is associated with two elements b and c, in Y. In mathematics, a multivalued function is a total relation; i. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ...

Examples: Power laws

If

w = Azⁿ

then, writing x + iy = re^iθ, we have

φ = Arⁿcosnθ

and

ψ = Arⁿsinnθ

Power law with n = 1

If w = Az¹, that is, a power law with n = 1, the streamlines (ie lines of constant ψ) are a system of straight lines parallel to the x-axis. This is easiest to see by writing in terms of real and imaginary components:

thus giving φ = Ax and ψ = Ay.

Power law with n = 2

If n = 2, then w = Az² and the streamline corresponding to a particular value of ψ are those points satisfying

ψ = Ar²sin2θ

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

ψ = 2Axy.

The velocity field is given by , or

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 f(z) = z² at z = 0). A point in a flow where the velocity is zero, where any streamline touches a solid surface at an angle. ...

The ψ = 0 streamline is particularly interesting: it has two (or four) branches, following the coordinate axes, ie x = 0 and y = 0.

As no fluid flows across the x-axis, it (the x-axis) may be treated as a solid boundary. It is thus possible to ignore the flow in the lower half-plane where y < 0 and to focus on the flow in the upper half-plane.

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 < 0 and y < 0 are ignored.

Power law with n = 3

If n = 3 the resulting flow is a sort of hexagonal version of the n = 2 case considered above. Streamlines are given by 3x²y − y³ = ψ.

Power law with n = − 1

if , the streamlines are given by

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

Thus the streamlines are circles that are tangent to the x-axis at the origin. The velocity field is given by Circle illustration This article is about the shape and mathematical concept of circle. ...

The circles in the upper half-plane thus flow clockwise, those in the lower half-plane flow anticlockwise. Note that speeds go as r ^{− 2}; and the speed at the origin is infinite.

Power law with n = − 2

{this section is to be completed}

See also

• Stream function
• Laplacian field
• Conformal mapping
• Flownet
• Velocity potential

In fluid dynamics, the stream function is defined for two-dimensional flows. ... In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. ... In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ... The construction of a Flownet is a graphical method used to solve two-dimensional steady-state groundwater flow problems through aquifers. ... A velocity potential is used in fluid dynamics, when a fluid is irrotational. ...

External links

• http://www.idra.unige.it/~irro/lecture_e.html