In fluid dynamics Jeffery–Hamel flow is a flow created by a converging or diverging channel with a source or sink of fluid volume at the point of intersection of the two plane walls.
It is named after George Barker Jeffery(1915)[1] and Georg Hamel(1917),[2] but it has subsequently been studied by many major scientists such as von Kármán and Levi-Civita,[3] Walter Tollmien,[4] F. Noether,[5] W.R. Dean,[6] Rosenhead,[7] Landau,[8] G.K. Batchelor[9] etc.
A complete set of solutions was described by Edward Fraenkel in 1962.
[10] Consider two stationary plane walls with a constant volume flow rate
is injected/sucked at the point of intersection of plane walls and let the angle subtended by two walls be
representing point of intersection and
The resulting flow is two-dimensional if the plates are infinitely long in the axial
direction, or the plates are longer but finite, if one were neglect edge effects and for the same reason the flow can be assumed to be entirely radial i.e.,
Then the continuity equation and the incompressible Navier–Stokes equations reduce to The boundary conditions are no-slip condition at both walls and the third condition is derived from the fact that the volume flux injected/sucked at the point of intersection is constant across a surface at any radius.
, the function is defined as Different authors defines the function differently, for example, Landau[8] defines the function with a factor
momentum equations reduce to and substituting this into the previous equation(to eliminate pressure) results in Multiplying by
are constants to be determined from the boundary conditions.
The above equation can be re-written conveniently with three other constants
as roots of a cubic polynomial, with only two constants being arbitrary, the third constant is always obtained from other two because sum of the roots is
The solution can be expressed in terms of elliptic functions.
Source:[13] The equation takes the same form as an undamped nonlinear oscillator(with cubic potential) one can pretend that
is velocity of a particle with unit mass, then the equation represents the energy equation(
The rich structure of this dynamical interpretation can be found in Rosenhead(1940).
, integration of governing equation gives and the boundary conditions becomes The equations can be simplified by standard transformations given for example in Jeffreys.
The limiting condition is obtained by noting that pure outflow is impossible when
Thus beyond this critical conditions, no solution exists.
is the complete elliptic integral of the first kind.
The corresponding critical Reynolds number or volume flux is given by where
is the complete elliptic integral of the second kind.
, the critical Reynolds number or volume flux becomes
For pure inflow, the implicit solution is given by and the boundary conditions becomes Pure inflow is possible only when all constants are real
is the complete elliptic integral of the first kind.
becomes larger), the flow tends to become uniform(thus approaching potential flow solution), except for boundary layers near the walls.
everywhere except in the boundary layer of thickness
and the boundary layers have classical thickness