In fluid dynamics, two types of stream function (or streamfunction) are defined:
The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange in 1781, is defined for incompressible (divergence-free), two-dimensional flows. The Stokes stream function, named after George Gabriel Stokes, 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 flow field can be described as two-dimensional plane flow, with velocity vector
u = [ u ( x , y , t ) v ( x , y , t ) 0 ] . {\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:
∇ ⋅ u = 0. {\displaystyle \quad \nabla \cdot \mathbf {u} =0.}
The domain has no holes, or only has holes that have no net flux inwards or outwards. 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 ( x , y , z ) {\displaystyle (x,y,z)} .
Derivation
The test surface Consider two points A {\displaystyle A} and P {\displaystyle P} in the x y {\displaystyle xy} plane, and a continuous curve A P {\displaystyle AP} , also in the x y {\displaystyle xy} plane, that connects them where every point on the curve A P {\displaystyle AP} has z {\displaystyle z} coordinate z = 0 {\displaystyle z=0} . Let the total length of the curve A P {\displaystyle AP} be L {\displaystyle L} . Suppose a ribbon-shaped surface is created by extending the curve A P {\displaystyle AP} upward to the horizontal plane z = b {\displaystyle z=b} ( b > 0 ) {\displaystyle (b>0)} , where b {\displaystyle b} is the thickness of the flow. Then the surface has length L {\displaystyle L} , width b {\displaystyle b} , and area b L {\displaystyle b\,L} . Call this the test surface.
Flux through the test surface
The total volumetric flux through the test surface is
Q ( x , y , t ) = ∫ 0 b ∫ 0 L u ⋅ n ^ d s d z {\displaystyle Q(x,y,t)=\int _{0}^{b}\int _{0}^{L}\mathbf {u} \cdot {\hat {\mathbf {n} }}\,\mathrm {d} s\,\mathrm {d} z}
where s {\displaystyle s} is an arc-length parameter defined on the curve A P {\displaystyle AP} , with s = 0 {\displaystyle s=0} at the point A {\displaystyle A} and s = L {\displaystyle s=L} at the point P {\displaystyle P} . Here n ^ {\displaystyle {\hat {\mathbf {n} }}} is the unit vector perpendicular to the test surface, i.e.,
n ^ d s = − R d r = [ d y − d x 0 ] {\displaystyle {\hat {\mathbf {n} }}\,\mathrm {d} s=-R\,\mathrm {d} \mathbf {r} ={\begin{bmatrix}\mathrm {d} y\\-\mathrm {d} x\\0\end{bmatrix}}}
… excerpt ends here. Continue reading the full article.
