Elementary flow is a collection of basic flows from which it is possible to construct more complex flows by
superposition. Some of the flows reflect specific cases and constraints such as
incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An e ...
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 any path between two points does not c ...
flows, or both, as in the case of
potential flow
In fluid dynamics, potential flow (or ideal 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 appr ...
.
Two-dimensional uniform flow
Given a uniform velocity of a fluid at any position in space:
:
This flow is incompressible because the velocity is constant, the first derivatives of the velocity components are zero, and the total divergence is zero:
Given the
circulation is always zero the flow is also irrotational, we can derive this from the
Kelvin's circulation theorem
In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin who published it in 1869) states:In a barotropic, ideal fluid with conservative body forces, the circulation around a closed curve (which encloses the ...
and from the explicit computation of the
vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wit ...
:
:
Being incompressible and two-dimensional, this flow is constructed from a
stream function
The stream function is defined for incompressible flow, incompressible (divergence-free) fluid flow, flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of t ...
:
:
:
from which
:
and in cylindrical coordinates:
:
:
from which
:
As usual the stream function is defined up to a constant value which here we take as zero. We can also confirm that the flow is irrotational from:
:
Being irrotational, the potential function is instead:
:
:
and therefore
:
and in
cylindrical coordinates
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
:
:
:
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 dimension on the orthogonal plane.
Line sources and line sinks (below) are important elementary flows because they play the role of monopole(s) for incompressible fluids (which can also be considered examples of
solenoidal fields 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. Similarly ...
s, in the same manner as for
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and
magnetic
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particle ...
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:
:
Where the total outgoing flux is constant
:
Therefore,
:
This is derived from a stream function
:
or from a potential function
:
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.
:
This is derived from a stream function
:
or from a potential function
:
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
:
The last approximation is to the first order in d.
Given
:
It remains
:
The velocity is then
:
:
And the potential instead
:
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 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 any path between two points does not c ...
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 any path between two points does not c ...
and
incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An e ...
and therefore a case of
potential flow
In fluid dynamics, potential flow (or ideal 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 appr ...
.
This is characterized by a cylindrical symmetry:
:
Where the total circulation is constant for every closed line around the central vortex
:
and is zero for any line not including the vortex.
Therefore,
:
This is derived from a stream function
:
or from a potential function
:
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:
:
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 function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
:
We look for a solution with separated variables:
:
which gives
:
Given the left part depends only on ''r'' and the right parts depends only on
, the two parts must be equal to a constant independent from ''r'' and
. The constant shall be positive.
Therefore,
:
:
The solution to the second equation is a linear combination of
and
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
:
The potential is instead given by
:
References
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Further reading
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External links
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{{DEFAULTSORT:Elementary Flow
Fluid dynamics