The stream function is defined for
incompressible (
divergence-free)
flows in two dimensions – as well as in three dimensions with
axisymmetry. The
flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
components can be expressed as the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s of the
scalar stream function. The stream function can be used to plot
streamlines, which represent the trajectories of particles in a steady flow. The two-dimensional Lagrange stream function was introduced by
Joseph Louis Lagrange in 1781. The
Stokes stream function is for axisymmetrical three-dimensional flow, and is named after
George Gabriel Stokes
Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish English physicist and mathematician. Born in County Sligo, Ireland, Stokes spent all of his career at the University of Cambridge, where he was the Luc ...
.
Considering the particular case of
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, the difference between the stream function values at any two points gives the volumetric flow rate (or
volumetric flux) through a line connecting the two points.
Since streamlines are
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to the flow velocity vector of the flow, the value of the stream function must be constant along a streamline. The usefulness of the stream function lies in the fact that the flow velocity components in the ''x''- and ''y''- directions at a given point are given by the
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s of the stream function at that point.
For two-dimensional
potential flow, streamlines are perpendicular to
equipotential lines. Taken together with the
velocity potential, the stream function may be used to derive a
complex potential. In other words, the stream function accounts for the
solenoidal
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 ...
part of a two-dimensional
Helmholtz decomposition, while the velocity potential accounts for the
irrotational part.
Two-dimensional stream function
Definitions
Lamb and
Batchelor define the stream function
for an
incompressible flow
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 ...
velocity field
as follows.
[ and ] Given a point
and a point
,
:
is the integral of the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
of the
flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
vector
and the
normal to the curve element
In other words, the stream function
is the
volume flux through the curve
. The point
is simply a reference point that defines where the stream function is identically zero. A shift in
results in adding a constant to the stream function
at
.
An
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
shift
of the position
results in a change of the stream function:
:
.
From the
exact differential
:
the flow velocity components in relation to the stream function
have to be
:
in which case they indeed satisfy the condition of zero
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
resulting from flow incompressibility, i.e.
:
Definition by use of a vector potential
The sign of the stream function depends on the definition used.
One way is to define the stream function
for a two-dimensional flow such that the
flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
can be expressed through the
vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field.
Formally, given a vector field v, a ''vecto ...
:
Where
if the flow velocity vector
.
In
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
this is equivalent to
:
Where
and
are the flow velocity components in the cartesian
and
coordinate directions, respectively.
Alternative definition (opposite sign)
Another definition (used more widely in
meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...
and
oceanography
Oceanography (), also known as oceanology and ocean science, is the scientific study of the oceans. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynami ...
than the above) is
:
,
where
is a unit vector in the
direction and the subscripts indicate partial derivatives.
Note that this definition has the opposite sign to that given above (
), so we have
:
in Cartesian coordinates.
All formulations of the stream function constrain the velocity to satisfy the two-dimensional
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...
exactly:
:
The last two definitions of stream function are related through the
vector calculus identity
:
Note that
in this two-dimensional flow.
Derivation of the two-dimensional stream function
Consider two points A and B in two-dimensional plane flow. If the distance between these two points is very small: δn, and a stream of flow passes between these points with an average velocity, q perpendicular to the line AB, the volume flow rate per unit thickness, δΨ is given by:
:
As δn → 0, rearranging this expression, we get:
:
Now consider two-dimensional plane flow with reference to a coordinate system. Suppose an observer looks along an arbitrary axis in the direction of increase and sees flow crossing the axis from ''left to right''. A sign convention is adopted such that the flow velocity is ''positive''.
Flow in Cartesian coordinates
By observing the flow into an elemental square in an x-y
Cartesian coordinate
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
system, we have:
:
where u is the flow velocity parallel to and in the direction of the x-axis, and v is the flow velocity parallel to and in the direction of the y-axis. Thus, as δn → 0 and by rearranging, we have:
:
Continuity: the derivation
Consider two-dimensional plane flow within a Cartesian coordinate system.
Continuity states that if we consider incompressible flow into an elemental square, the flow into that small element must equal the flow out of that element.
The total flow into the element is given by:
:
The total flow out of the element is given by:
:
Thus we have:
:
and simplifying to:
:
Substituting the expressions of the stream function into this equation, we have:
:
Vorticity
The stream function can be found from
vorticity using the following
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with ...
:
:
or
:
where the vorticity vector
– defined as the
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was ...
of the flow velocity vector
– for this two-dimensional flow has
i.e. only the
-component
can be non-zero.
Proof that a constant value for the stream function corresponds to a streamline
Consider two-dimensional plane flow within a Cartesian coordinate system. Consider two infinitesimally close points
and
. From calculus we have that
:
Say
takes the same value, say
, at the two points
and
, then
is tangent to the curve
at
and
:
implying that the vector
is normal to the curve
. If we can show that everywhere
, using the formula for
in terms of
, then we will have proved the result. This easily follows,
:
Properties of the stream function
# The stream function
is constant along any streamline.
# For a continuous flow (no sources or sinks), the volume flow rate across any closed path is equal to zero.
# For two incompressible flow patterns, the algebraic sum of the stream functions is equal to another stream function obtained if the two flow patterns are super-imposed.
# The rate of change of stream function with distance is directly proportional to the velocity component perpendicular to the direction of change.
See also
*
Elementary flow
References
Citations
Sources
*
*
*
*
*
*
{{refend
Continuum mechanics
Fluid dynamics
External links
Joukowsky Transform Interactive WebApp