In fluid dynamics, Prandtl–Batchelor theorem states that if in a two-dimensional laminar flow at high Reynolds number closed streamlines occur, then the vorticity in the closed streamline region must be a constant.
A similar statement holds true for axisymmetric flows.
The theorem is named after Ludwig Prandtl and George Batchelor.
Prandtl in his celebrated 1904 paper stated this theorem in arguments,[1] George Batchelor unaware of this work proved the theorem in 1956.
[2][3] The problem was also studied in the same year by Richard Feynman and Paco Lagerstrom[4] and by W.W. Wood in 1957.
[5] At high Reynolds numbers, the two-dimensional problem governed by two-dimensional Euler equations reduce to solving a problem for the stream function
As it stands, the problem is ill-posed since the vorticity distribution
can have infinite number of possibilities, all of which satisfies the equation and the boundary condition.
The difficulty arises only when there are some closed streamlines inside the domain that does not connect to the boundary and one may suppose that at high Reynolds numbers,
is not uniquely defined in regions where closed streamlines occur.
is uniquely defined in such cases, through an examination of the limiting process
The steady, non-dimensional vorticity equation in our case reduces to Integrate the equation over a surface
The integrand in the left-hand side term can be written as
is the outward unit vector normal to the contour line element
The left-hand side integrand can be made zero if the contour
is taken to be one of the closed streamlines since then the velocity vector projected normal to the contour will be zero, that is to say
Thus one obtains This expression is true for finite but large Reynolds number since we did not neglect the viscous term before.
Unlike the two-dimensional inviscid flows, where
, in the first approximation (neglecting the small corrections), we have Since
is constant for a given streamline, we can take that term outside the integral, One may notice that the integral is negative of the circulation since where we used the Stokes theorem for circulation and
The only way the above equation can be satisfied is only if i.e., vorticity is not changing across these closed streamlines, thus proving the theorem.
Of course, the theorem is not valid inside the boundary layer regime.
This theorem cannot be derived from the Euler equations.