In fluid dynamics, potential flow or irrotational flow refers to the idealised, frictionless flow of a fluid. Flows of two kinds are visualised in this way:
The flow of an inviscid fluid The flow of a fluid of low viscosity, in regions that do not contain a boundary layer. See Prandtl hypothesis. 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 irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of the gradient of a 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 flows and Hele-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 for aerofoils, water waves, electroosmotic flow, and groundwater flow. A region containing friction forces (described as shear forces and viscous forces) can be described as a region containing vorticity. For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable. In flow regions where vorticity is known to be important, such as wakes and boundary layers, potential flow theory is not able to provide reasonable predictions of the flow. 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 around aircraft, groundwater flow, acoustics, water waves, and electroosmotic flow.
Description and characteristics
In potential or irrotational flow, the vorticity vector field is zero, i.e.,
ω ≡ ∇ × v = 0 , {\displaystyle {\boldsymbol {\omega }}\equiv \nabla \times \mathbf {v} =0,}
where v ( x , t ) {\displaystyle \mathbf {v} (\mathbf {x} ,t)} is the velocity field and ω ( x , t ) {\displaystyle {\boldsymbol {\omega }}(\mathbf {x} ,t)} is the vorticity field. Like any vector field having zero curl, the velocity field can be expressed as the gradient of certain scalar, say φ ( x , t ) {\displaystyle \varphi (\mathbf {x} ,t)} which is called the velocity potential, since the curl of the gradient is always zero. We therefore have
v = ∇ φ . {\displaystyle \mathbf {v} =\nabla \varphi .}
The velocity potential is not uniquely defined since one can add to it an arbitrary function of time, say f ( t ) {\displaystyle f(t)} , without affecting the relevant physical quantity which is v {\displaystyle \mathbf {v} } . The non-uniqueness is usually removed by suitably selecting appropriate initial or boundary conditions satisfied by φ {\displaystyle \varphi } and as such the procedure may vary from one problem to another. In potential flow, the circulation Γ {\displaystyle \Gamma } around any simply-connected contour C {\displaystyle C} is zero. This can be shown using the Stokes theorem,
Γ ≡ ∮ C v ⋅ d l = ∫ ω ⋅ d f = 0 {\displaystyle \Gamma \equiv \oint _{C}\mathbf {v} \cdot d\mathbf {l} =\int {\boldsymbol {\omega }}\cdot d\mathbf {f} =0}
where d l {\displaystyle d\mathbf {l} } is the line element on the contour and d f {\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-called irrotational vortices or point vortices, or in smoke rings), the circulation Γ {\displaystyle \Gamma } need not be zero. In the former case, Stokes theorem cannot be applied and in the later case, ω {\displaystyle {\boldsymbol {\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 N {\displaystyle N} times, we have
Γ = N κ {\displaystyle \Gamma =N\kappa }
where κ {\displaystyle \kappa } is a cyclic constant. This example belongs to a doubly-connected space. In an n {\displaystyle n} -tuply connected space, there are n − 1 {\displaystyle n-1} such cyclic constants, namely, κ 1 , κ 2 , … , κ n − 1 . {\displaystyle \kappa _{1},\kappa _{2},\dots ,\kappa _{n-1}.}
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