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Stream function

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Streamlines – lines with a constant value of the stream function – for the incompressible potential flow around a circular cylinder in a uniform onflow.

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.

Two-dimensional stream function[edit]


The two-dimensional stream function is based on the following assumptions:

  • The space domain is three-dimensional.
  • The flow field can be described as two-dimensional plane flow, with velocity vector

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 .


The test surface[edit]

Consider two points and in the plane, and a curve , also in the plane, that connects them. Then every point on the curve has coordinate . Let the total length of the curve be .

Suppose a ribbon-shaped surface is created by extending the curve upward to the horizontal plane , where is the thickness of the flow. Then the surface has length , width , and area . Call this the test surface.

Flux through the test surface[edit]

The volume flux through the test surface connecting the points and

The total volumetric flux through the test surface is

where is an arc-length parameter defined on the curve , with at the point and at the point . Here is the unit vector perpendicular to the test surface, i.e.,

where is the rotation matrix corresponding to a anticlockwise rotation about the positive axis:

The integrand in the expression for is independent of , so the outer integral can be evaluated to yield

Classical definition[edit]

Lamb and Batchelor define the stream function as follows.[3]

Using the expression derived above for the total volumetric flux, , this can be written as


In words, the stream function is the volumetric flux through the test surface per unit thickness, where thickness is measured perpendicular to the plane of flow.

The point is a reference point that defines where the stream function is identically zero. Its position is chosen more or less arbitrarily and, once chosen, typically remains fixed.

An infinitesimal shift in the position of point 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. Consequently if two incompressible flow fields are superimposed, then the stream function of the resultant flow field is the algebraic sum of the stream functions of the two original fields.

Effect of shift in position of reference point[edit]

Consider a shift in the position of the reference point, say from to . Let denote the stream function relative to the shifted reference point :

Then the stream function is shifted by

which implies the following:

  • A shift in the position of the reference point effectively adds a constant (for steady flow) or a function solely of time (for nonsteady flow) to the stream function at every point .
  • The shift in the stream function, , is equal to the total volumetric flux, per unit thickness, through the surface that extends from point to point . Consequently if and only if and lie on the same streamline.

In terms of vector rotation[edit]

The velocity can be expressed in terms of the stream function as

where is the rotation matrix corresponding to a anticlockwise rotation about the positive axis. Solving the above equation for produces the equivalent form

From these forms it is immediately evident that the vectors and are

  • perpendicular:
  • of the same length: .

Additionally, the compactness of the rotation form facilitates manipulations (e.g., see Condition of existence).

In terms of vector potential and stream surfaces[edit]

Using the stream function, one can express the velocity in terms of the vector potential

where , and is the unit vector pointing in the positive direction. This can also be written as the vector cross product

where we've used the vector calculus identity

Noting that , and defining , one can express the velocity field as

This form shows that the level surfaces of and the level surfaces of (i.e., horizontal planes) form a system of orthogonal stream surfaces.

Alternative (opposite sign) definition[edit]

An alternative definition, sometimes used in meteorology and oceanography, is

Relation to vorticity[edit]

In two-dimensional plane flow, the vorticity vector, defined as , reduces to , where


These are forms of Poisson's equation.

Relation to streamlines[edit]

Consider two-dimensional plane flow with two infinitesimally close points and lying in the same horizontal plane. From calculus, the corresponding infinitesimal difference between the values of the stream function at the two points is

Suppose takes the same value, say , at the two points and . Then this gives

implying that the vector is normal to the surface . Because 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 ( coordinate).

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.

Condition of existence[edit]

It's straightforward to show that for two-dimensional plane flow satisfies the curl-divergence equation

where is the rotation matrix corresponding to a anticlockwise rotation about the positive axis. This equation holds regardless of whether or not the flow is incompressible.

If the flow is incompressible (i.e., ), then the curl-divergence equation gives


Then by Stokes' theorem the line integral of over every closed loop vanishes

Hence, the line integral of is path-independent. Finally, by the converse of the gradient theorem, a scalar function exists such that


Here represents the stream function.

Conversely, if the stream function exists, then . Substituting this result into the curl-divergence equation yields (i.e., the flow is incompressible).

In summary, the stream function for two-dimensional plane flow exists if and only if the flow is incompressible.

Potential flow[edit]

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 part of a two-dimensional Helmholtz decomposition, while the velocity potential accounts for the irrotational part.

Summary of properties[edit]

The basic properties of two-dimensional stream functions can be summarized as follows:

  1. The x- and y-components of the flow velocity at a given point are given by the partial derivatives of the stream function at that point.
  2. The value of the stream function is constant along every streamline (streamlines represent the trajectories of particles in steady flow). That is, in two dimensions each streamline is a level curve of the stream function.
  3. The difference between the stream function values at any two points gives the volumetric flux through the vertical surface that connects the two points.

Two-dimensional stream function for flows with time-invariant density[edit]

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 is defined such that

and represents the mass flux (rather than volumetric flux) per unit thickness through the test surface.

See also[edit]



  1. ^ Lagrange, J.-L. (1868), "Mémoire sur la théorie du mouvement des fluides (in: Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin, année 1781)", Oevres de Lagrange, vol. Tome IV, pp. 695–748
  2. ^ Stokes, G.G. (1842), "On the steady motion of incompressible fluids", Transactions of the Cambridge Philosophical Society, 7: 439–453, Bibcode:1848TCaPS...7..439S
    Reprinted in: Stokes, G.G. (1880), Mathematical and Physical Papers, Volume I, Cambridge University Press, pp. 1–16
  3. ^ Lamb (1932, pp. 62–63) and Batchelor (1967, pp. 75–79)


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