Talk:Stream function

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Hello everyone,

Could someone explain if there is something like a 3d-stream function and if not, why not? (unsigned)

3D stream functions[edit]

In fluid mechanics most "real" problems do not have analytic solutions due to the nonlinearity of the Navier-Stokes equations (which are also hyperbolic) when expressed in (useful) Eulerian form. In general very few PDE's have analytic solutions. According to pg. 140 of I.G.Currie's Fundamental Mechanics of Fluids, in general it is not possible to satisfy the three dimensional continuity equation by a single scalar function. Hence for fluid mechanics, generally speaking there is no 3-D stream function. However, in the case of axis-symetric flow there is a Stoke's stream function. --Emptybeeker (talk) 04:39, 18 January 2008 (UTC)

Article should be more general[edit]

Stream functions do not just describe fluid. These functions apply to equipotential lines for a variety of phenomena. This article should be much more general, since these functions are used in electrical engineering, physics, etc.... Anyone?Tparameter 04:56, 8 November 2006 (UTC)
going... going... Tparameter 17:49, 9 November 2006 (UTC)

I think that the article is very general but there is too much maths and not enough explaining. It is easy to define something but harder to write an encyclopaedic article about it. I will have a go when I have more time. I must say that I have not come across stream functions outside of fluids, they appear in aeronautics and hydrodynamics, where else have you used them? Rex the first talk | contribs 21:05, 10 November 2006 (UTC)

Suggested expansion for fluid dynamics[edit]

I was not sure where to include this. It is mostly specific to fluid dynamics. Refer to comments section for more details.

Stream functions are defined so that they satisfy the continuity equation at all times. This is useful because it decreases the number of equations and variables, one needs to handle. Also once we find the stream function for a particular flow we are assured that the continuity equation is satisfied. This is most easily accomplished in 2D, steady, incompressible flow, where the continuity equation has only two terms. [1] It is also possible to define stream function for 2D, steady, compressible flow [2] as follows:


\rho u =  \frac{\partial \psi}{\partial y},\qquad
\rho v = -\frac{\partial \psi}{\partial x}

The trade-off for the decreased number of terms and equations is in the increased order of the velocity terms.

In steady, 2D flows, stream function can be assigned a physical meaning by noting that [3],

  1. Lines with constant value of stream function form the streamlines of the flow. Across these lines there is no mass flow.
  2. The difference in the value of the stream function on any two streamlines is numerically equal to the mass flow between those two streamlines.

Few comments[edit]

  • Stream function are defined to satisfy the continuity equation. The fact they also represent the streamlines in certain cases is secondary.
  • Stream functions are generally used in cases where the continuity equation can be reduced to two terms. In such cases the use of stream function decreases the number of variables and equations by one.
  • I have not encountered stream functions in 3D flows, but I think it should be possible to define a stream function in such cases too.

Notes[edit]

  1. ^ The fluid dynamics continuity equation for a 2D, steady, incompressible flow:
    
\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0
  2. ^ The fluid dynamics continuity equation for a 2D, steady, compressible flow:
    
\frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} = 0
  3. ^ Consider a 2D, steady, compressible flow.
    
\begin{align}
d \psi & = \frac{\partial \psi}{\partial x} dx + \frac{\partial \psi}{\partial y} dy \\
       & = -\rho v dx + \rho u dy \\
       & = \rho \vec V \cdot d \vec A \\
       & = d \dot m \\
\end{align}

myth 19:25, 20 December 2006 (UTC)

n.b. vorsity formula is not correct. check it out. here should be +\omega, not -\omega —Preceding unsigned comment added by 81.5.107.247 (talk) 20:15, 6 May 2008 (UTC)

Sign is incorrect[edit]

This article defines the streamfunction with the opposite sign to every fluid dynamics text and article I have read (I am a fluid dynamicist by trade, so I know my stuff).

\psi is normally defined via \mathbf{u}=\mathbf{z}\times\nabla\psi=(-\psi_y,\psi_x), which is the opposite sign to the definition given here (\mathbf{z} is a unit vector in the +z direction).

If you fix the sign of \psi you will also get the usual vorticity formula \nabla^2\psi=\omega (see comment above).

I have edited the article to reflect this, using a dashed \psi' for this alternative definition to avoid any ambiguity with the rest of the text.

Reference: [1] —Preceding unsigned comment added by 131.236.1.5 (talk) 07:34, 29 May 2009 (UTC)

Strange, since (u,v)=(+\partial_y \psi,-\partial_x\psi) is in agreement with most authorative textbooks I know, e.g.
  • Batchelor, "An introduction to fluid dynamics", p. 76
  • Landau & Lifshitz, "Fluid mechanics", p. 19
  • Courant & Friedrichs, "Supersonic flow and shock waves", p.248
  • Lord Rayleigh "The theory of sound", §238
  • Lin, "The theory of hydrodynamic stability", p. 28
  • Csanady, "Circulation in the coastal ocean", p. 193
  • Kevorkian, "Partial differential equations", p. 64
Only Lamb, "Hydrodynamics", p. 63, uses the opposite sign. And of course both signs can be used.
But there is a rationale for this sign use. The above is until now all about two-dimensional (2D) flow. In 3D flow, there is a generalization of the stream function, called the 'vector stream function' or 'vector potential' (in analogy with the vector potential of magnetic induction in electromagnetic field theory), \boldsymbol{\Psi}, see Batchelor pp. 77 & 79:
\boldsymbol{u}=\boldsymbol{\nabla}\times\boldsymbol{\Psi}
with 3D velocity vector \boldsymbol{u}. In Cartesian coordinates, \boldsymbol{u} has components (u,v,w) and \boldsymbol{\Psi} has components (\Psi_x,\Psi_y,\Psi_z). Then for 2D flow, with w=0 and \Psi_x=\Psi_y=0, the 2D streamfunction is \psi=\Psi_z with the signs as in the first sentence of my comment. The above vectorial description of the 3D velocity \boldsymbol{u} in terms of vector stream function \boldsymbol{\Psi} has the advantage that it is independent of the coordinate system used. In a straightforward way, also Stokes stream functions for e.g. cylindrical and spherical coordinates derive from it (see Batchelor, pp. 77-79).
While on the other side the above definition \boldsymbol{u}=\boldsymbol{z}\times\boldsymbol{\nabla}\psi, with \psi=F(x,y), is restricted to 2D flow in Cartesian coordinates. In terms of the vector stream function, for 2D flow in the (x,y) plane the definition would be:
\boldsymbol{u}=\boldsymbol{\nabla}\times(\boldsymbol{z}\psi),
explaining the opposite sign.
So the article was not "incorrect": it is only an equally valid definition different from what you expect. And the most common one as far as I can see. But there is nothing wrong with mentioning both sign options in the article. -- Crowsnest (talk) 15:47, 29 May 2009 (UTC)