In fluid dynamics, two types of stream function are defined:

• The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange in 1781,^[1] is defined for incompressible (divergence-free), two-dimensional flows.
• The Stokes stream function, named after George Gabriel Stokes,^[2] is defined for incompressible, three-dimensional flows with axisymmetry.

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

Assumptions

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

${\displaystyle \quad \mathbf {u} ={\begin{bmatrix}u(x,y,t)\\v(x,y,t)\\0\end{bmatrix}}.}$

• The velocity satisfies the continuity equation for incompressible flow:

${\displaystyle \quad \nabla \cdot \mathbf {u} =0.}$

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 ${\displaystyle (x,y,z)}$.

Derivation

The test surface

Consider two points ${\displaystyle A}$ and ${\displaystyle P}$ in the ${\displaystyle xy}$ plane, and a curve ${\displaystyle AP}$, also in the ${\displaystyle xy}$ plane, that connects them. Then every point on the curve ${\displaystyle AP}$ has ${\displaystyle z}$ coordinate ${\displaystyle z=0}$. Let the total length of the curve ${\displaystyle AP}$ be ${\displaystyle L}$.

Suppose a ribbon-shaped surface is created by extending the curve ${\displaystyle AP}$ upward to the horizontal plane ${\displaystyle z=b}$ ${\displaystyle (b>0)}$, where ${\displaystyle b}$ is the thickness of the flow. Then the surface has length ${\displaystyle L}$, width ${\displaystyle b}$, and area ${\displaystyle b\,L}$. Call this the test surface.

Flux through the test surface

The total volumetric flux through the test surface is

${\displaystyle Q(x,y,t)=\int _{0}^{b}\int _{0}^{L}\mathbf {u} \cdot \mathbf {n} \,\mathrm {d} s\,\mathrm {d} z}$

where ${\displaystyle s}$ is an arc-length parameter defined on the curve ${\displaystyle AP}$, with ${\displaystyle s=0}$ at the point ${\displaystyle A}$ and ${\displaystyle s=L}$ at the point ${\displaystyle P}$. Here ${\displaystyle \mathbf {n} }$ is the unit vector perpendicular to the test surface, i.e.,

${\displaystyle \mathbf {n} \,\mathrm {d} s=-R\,\mathrm {d} \mathbf {r} ={\begin{bmatrix}\mathrm {d} y\\-\mathrm {d} x\\0\end{bmatrix}}}$

where ${\displaystyle R}$ is the ${\displaystyle 3\times 3}$ rotation matrix corresponding to a ${\displaystyle 90^{\circ }}$ anticlockwise rotation about the positive ${\displaystyle z}$ axis:

${\displaystyle R=R_{z}(90^{\circ })={\begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix}}.}$

The integrand in the expression for ${\displaystyle Q}$ is independent of ${\displaystyle z}$, so the outer integral can be evaluated to yield

${\displaystyle Q(x,y,t)=b{\int <!--\; \int \; -->}_{}^{}}$Zdroj:https://en.wikipedia.org?pojem=Stream_function
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