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In fluid dynamics, potential flow, also know as irrotational flow (of incompressible fluids) is steady flow defined by the equations
Equivalently,
where:
The equations above imply , or Laplace's equation, holds.
Together with the Navier-Stokes equations and 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 potential flow in 2d))
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".
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.
More precisely, potential flow cannot account for the behaviour of flows that include a boundary layer.
Nevertheless, understanding potential flow is important in many branches of fluid mechanics. In particular, simple potential flows (called elemental flow s) such as the free vortex and the point source possess ready analytical solutions. These solutions can be superposedThe 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. 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 dynamicsFor " Contract For Difference," the financial derivative, see CFDs. For the Wikipedia categories for deletion administrative page, see an administrative page on the English Wikipedia. Fluid dynamics Computational fluid dynamics is the use of computers to, 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.
Since the flow is inviscid and free of shear force s, this means that any streamline can be replaced with 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 solidIn fluid mechanics a Riabouchinsky solid is a technique used for approximating boundary layer separation from a bluff body using potential flow. It is named for Dimitri Pavlovitch Riabouchinsky. Riabouchinsky solids are typically used for analysing the bes.
Potential flow in two dimensions is simple to analyse using complex numbers, viewed for convenience on the Argand diagram.
The basic idea is to define a holomorphic function . If we write
then the Cauchy-Riemann equationsPartial differential equations Complex analysis In complex analysis, the Cauchy-Riemann differential equations are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic. Let f u + iv be a fu show that
(it is conventional to regard all symbols as real numbers; and to write
and ).The velocity field , specified by
then satisfies the requirements for potential flow:
and
Lines of constant are known as streamlines and lines of constant are known as equipotential lines (see equipotential surface).
The two sets of curves intersect at right angles, for
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