Stream function
In fluid dynamics, two types of stream function are defined: The properties of stream functions make them useful for analyzing and graphically illustrating flows.
The remainder of this article describes the two-dimensional stream function.
The two-dimensional stream function is based on the following assumptions: Although in principle the stream function doesn't require the use of a particular coordinate system, for convenience the description presented here uses a right-handed Cartesian coordinate system with coordinates
Suppose a ribbon-shaped surface is created by extending the curve
is the unit vector perpendicular to the test surface, i.e., where
, so the outer integral can be evaluated to yield Lamb and Batchelor define the stream function
[3] Using the expression derived above for the total volumetric flux,
is a reference point that defines where the stream function is identically zero.
results in the following change of the stream function: From the exact differential so the flow velocity components in relation to the stream function
must be Notice that the stream function is linear in the velocity.
denote the stream function relative to the shifted reference point
: Then the stream function is shifted by which implies the following: The velocity
are Additionally, the compactness of the rotation form facilitates manipulations (e.g., see Condition of existence).
Using the stream function, one can express the velocity in terms of the vector potential
, one can express the velocity field as This form shows that the level surfaces of
(i.e., horizontal planes) form a system of orthogonal stream surfaces.
An alternative definition, sometimes used in meteorology and oceanography, is In two-dimensional plane flow, the vorticity vector, defined as
Consider two-dimensional plane flow with two infinitesimally close points
From calculus, the corresponding infinitesimal difference between the values of the stream function at the two points is Suppose
everywhere (e.g., see In terms of vector rotation), each streamline corresponds to the intersection of a particular stream surface and a particular horizontal plane.
Consequently, in three dimensions, unambiguous identification of any particular streamline requires that one specify corresponding values of both the stream function and the elevation (
The development here assumes the space domain is three-dimensional.
The concept of stream function can also be developed in the context of a two-dimensional space domain.
In that case level sets of the stream function are curves rather than surfaces, and streamlines are level curves of the stream function.
Consequently, in two dimensions, unambiguous identification of any particular streamline requires that one specify the corresponding value of the stream function only.
), then the curl-divergence equation gives Then by Stokes' theorem the line integral of
Finally, by the converse of the gradient theorem, a scalar function
For two-dimensional potential flow, streamlines are perpendicular to equipotential lines.
In other words, the stream function accounts for the solenoidal part of a two-dimensional Helmholtz decomposition, while the velocity potential accounts for the irrotational part.
The basic properties of two-dimensional stream functions can be summarized as follows: If the fluid density is time-invariant at all points within the flow, i.e., then the continuity equation (e.g., see Continuity equation#Fluid dynamics) for two-dimensional plane flow becomes In this case the stream function
