Editing Stream function (section)

==== Classical definition ====
[[Horace Lamb|Lamb]] and [[George Batchelor|Batchelor]] define the stream function <math>\psi</math> as follows.<ref>{{harvtxt|Lamb|1932|pages=62–63}} and {{harvtxt|Batchelor|1967|pages=75–79}}</ref>   

:<math>\psi(x,y,t) = \int_A^P \left( u\, \mathrm{d} y - v\, \mathrm{d} x \right)</math>

Using the expression derived above for the total volumetric flux, <math>Q</math>, this can be written as

:<math>\psi(x,y,t) = \frac{Q(x, y, t)}{b}</math>.

In words, the stream function <math>\psi</math> is the volumetric flux through the test surface per unit thickness, where thickness is measured perpendicular to the plane of flow.

The point <math>A</math> 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 <math>\mathrm{d} P=(\mathrm{d} x,\mathrm{d} y)</math> in the position of point <math>P</math> results in the following change of the stream function:

:<math>\mathrm{d} \psi = u\, \mathrm{d} y - v\, \mathrm{d} x</math>.

From the [[exact differential]] 

:<math>\mathrm{d} \psi = \frac{\partial \psi}{\partial x}\, \mathrm{d} x + \frac{\partial \psi}{\partial y}\, \mathrm{d} y,</math>

so the flow velocity components in relation to the stream function <math>\psi</math> must be

:<math>
u= \frac{\partial \psi}{\partial y},
\qquad
v = -\frac{\partial \psi}{\partial x}.
</math>

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.