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In the larger context of the Navier-Stokes equations (and especially in the context of
potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely g ...
), elementary flows are basic flows that can be combined, using various techniques, to construct more complex flows. In this article the term "flow" is used interchangeably with the term "solution" due to historical reasons. The techniques involved to create more complex solutions can be for example by
superposition In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ...
, by techniques such as topology or considering them as local solutions on a certain neighborhood, subdomain or
boundary layer In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a Boundary (thermodynamic), bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces ...
and to be patched together. Elementary flows can be considered the basic building blocks (
fundamental solution In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not ...
s, local solutions and
solitons In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such locali ...
) of the different types of equations derived from the Navier-Stokes equations. Some of the flows reflect specific constraints such as
incompressible Incompressible may refer to: * Incompressible flow, in fluid mechanics * incompressible vector field, in mathematics * Incompressible surface, in mathematics * Incompressible string, in computing {{Disambig ...
or
irrotational In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not chan ...
flows, or both, as in the case of
potential flow In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity pre ...
, and some of the flows may be limited to the case of two dimensions. Due to the relationship between
fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
and field theory, elementary flows are relevant not only to
aerodynamics Aerodynamics () is the study of the motion of atmosphere of Earth, air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dynamics and its subfield of gas dynamics, and is an ...
but to all field theory in general. To put it in perspective boundary layers can be interpreted as
topological defect In mathematics and physics, solitons, topological solitons and topological defects are three closely related ideas, all of which signify structures in a physical system that are stable against perturbations. Solitons do not decay, dissipate, dispe ...
s on generic
manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, and considering fluid dynamics analogies and limit cases in
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
,
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
and
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
one can see how all these solutions are at the core of recent developments in theoretical physics such as the ads/cft duality, the SYK model, the physics of nematic liquids, strongly correlated systems and even to quark gluon plasmas.


Two-dimensional uniform flow

For steady-state, spatially uniform flow of a fluid in the plane, the velocity vector is :\mathbf = v_0 \cos(\theta_0)\, \mathbf_x +v_0 \sin(\theta_0)\, \mathbf_y where : v_0 is the absolute magnitude of the velocity (i.e., v_0 = , \mathbf, ); : \theta_0 is the angle the velocity vector makes with the positive axis (\theta_0 is positive for angles measured in a counterclockwise sense from the positive axis); and : \mathbf_x and \mathbf_y are the unit basis vectors of the coordinate system. Because this flow is incompressible (i.e., \nabla \cdot \mathbf = 0) and two-dimensional, its velocity can be expressed in terms of a
stream function In fluid dynamics, two types of stream function (or streamfunction) are defined: * The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange in 1781, is defined for incompressible flow, incompressible (divergence-free ...
, \psi: :v_x = \frac :v_y = - \frac where :\psi = \psi_0 - v_0 \sin (\theta_0)\, x + v_0 \cos (\theta_0)\, y and \psi_0 is a constant. In cylindrical coordinates: :v_r = - \frac 1 r \frac :v_\theta = \frac and :\psi = \psi_0 + v_0\, r \sin (\theta - \theta_0) This flow is irrotational (i.e., \nabla \times \mathbf = \mathbf) so its velocity can be expressed in terms of a potential function, \phi: :v_x = - \frac :v_y = - \frac where :\phi = \phi_0 - v_0 \cos (\theta_0)\, x - v_0 \sin (\theta_0)\, y and \phi_0 is a constant. In
cylindrical coordinates A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...
:v_r = \frac :v_\theta = \frac \frac :\phi = \phi_0 - v_0\, r \cos(\theta - \theta_0)


Two-dimensional line source

The case of a vertical line emitting at a fixed rate a constant quantity of fluid Q per unit length is a line source. The problem has a cylindrical symmetry and can be treated in two dimensions on the orthogonal plane. Line sources and line sinks (below) are important elementary flows because they play the role of monopole for incompressible fluids (which can also be considered examples of
solenoidal field In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: \nabla \cdot \mathbf ...
s i.e. divergence free fields). Generic flow patterns can be also de-composed in terms of
multipole expansion A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Multipo ...
s, in the same manner as for
electric Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
and
magnetic Magnetism is the class of physical attributes that occur through a magnetic field, which allows objects to attract or repel each other. Because both electric currents and magnetic moments of elementary particles give rise to a magnetic field, m ...
fields where the monopole is essentially the first non-trivial (e.g. constant) term of the expansion. This flow pattern is also both irrotational and incompressible. This is characterized by a cylindrical symmetry: :\mathbf = v_r(r) \mathbf_r Where the total outgoing flux is constant : \int_S \mathbf \cdot d \mathbf = \int_^ ( v_r(r) \, \mathbf_r ) \cdot ( \mathbf_r \, r \, d \theta ) = \! 2 \pi \, r \, v_r(r) = Q Therefore, :v_r = \frac This is derived from a stream function :\psi(r,\theta) = -\frac \theta or from a potential function :\phi(r,\theta) = -\frac \ln r


Two-dimensional line sink

The case of a vertical line absorbing at a fixed rate a constant quantity of fluid ''Q'' per unit length is a line sink. Everything is the same as the case of a line source a part from the negative sign. :v_r = - \frac This is derived from a stream function :\psi(r,\theta) = \frac \theta or from a potential function :\phi(r,\theta) = \frac \ln r Given that the two results are the same a part from a minus sign we can treat transparently both line sources and line sinks with the same stream and potential functions permitting ''Q'' to assume both positive and negative values and absorbing the minus sign into the definition of ''Q''.


Two-dimensional doublet or dipole line source

If we consider a line source and a line sink at a distance d we can reuse the results above and the stream function will be :\psi(\mathbf) = \psi_Q(\mathbf - \mathbf/2) - \psi_Q(\mathbf + \mathbf/2) \ \simeq \mathbf \cdot \nabla \psi_Q(\mathbf) The last approximation is to the first order in d. Given :\mathbf = d \cos (\theta_0) \mathbf_x + \sin (\theta_0) \mathbf_y= d \cos (\theta-\theta_0) \mathbf_r + \sin (\theta-\theta_0) \mathbf_\theta It remains : \psi(r,\theta) = - \frac \frac The velocity is then : v_r(r,\theta) = \frac \frac : v_\theta(r,\theta) = \frac \frac And the potential instead : \phi(r,\theta) = \frac \frac


Two-dimensional vortex line

This is the case of a vortex filament rotating at constant speed, there is a cylindrical symmetry and the problem can be solved in the orthogonal plane. Dual to the case above of line sources, vortex lines play the role of monopoles for
irrotational flow In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity pr ...
s. Also in this case the flow is also both
irrotational In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not chan ...
and
incompressible Incompressible may refer to: * Incompressible flow, in fluid mechanics * incompressible vector field, in mathematics * Incompressible surface, in mathematics * Incompressible string, in computing {{Disambig ...
and therefore a case of
potential flow In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity pre ...
. This is characterized by a cylindrical symmetry: :\mathbf = v_\theta(r) \, \mathbf_\theta Where the total circulation is constant for every closed line around the central vortex :\oint \mathbf \cdot d \mathbf = \int_^ (v_\theta(r) \, \mathbf_\theta) \cdot (\mathbf_\theta \, r \, d\theta) = \! 2 \pi \, r\, v_\theta(r) = \Gamma and is zero for any line not including the vortex. Therefore, :v_\theta = \frac This is derived from a stream function :\psi(r,\theta) = \frac \ln r or from a potential function :\phi(r,\theta) = - \frac \theta Which is dual to the previous case of a line source


Generic two-dimensional potential flow

Given an incompressible two-dimensional flow which is also irrotational we have: :\nabla^2 \psi = 0 Which is in cylindrical coordinates
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...
:\frac \frac \left(r \frac\right) + \frac \frac= 0 We look for a solution with separated variables: :\psi(r,\theta) = R(r) \Theta(\theta) which gives :\frac \frac \left(r \frac\right) = -\frac \frac Given the left part depends only on ''r'' and the right parts depends only on \theta, the two parts must be equal to a constant independent from ''r'' and \theta. The constant shall be positive. Therefore, :r \frac \left(r \frac R(r)\right) = m^2 R(r) :\frac = - m^2 \Theta(\theta) The solution to the second equation is a linear combination of e^ and e^ In order to have a single-valued velocity (and also a single-valued stream function) ''m'' shall be a positive integer. therefore the most generic solution is given by :\psi = \alpha_0 + \beta_0 \ln r + \sum_ The potential is instead given by :\phi = \alpha_0 - \beta_0 \theta + \sum_


References

* * ;Specific


Further reading

* * * *


External links

* * {{DEFAULTSORT:Elementary Flow Fluid dynamics