It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881.
The other important consequence will be realized by looking at the incompressible vorticity equation where
are parallel, the non-linear terms in the above equation are identically zero
Thus Beltrami flows satisfies the linear equation When
, the components of vorticity satisfies a simple heat equation.
Viktor Trkal considered the Beltrami flows without any external forces in 1919[3] for the scalar function
, i.e., Introduce the following separation of variables then the equation satisfied by
The generalized Beltrami flow satisfies the condition[4] which is less restrictive than the Beltrami condition
For steady generalized Beltrami flow, we have
Introduce the stream function Integration of
So, complete solution is possible if it satisfies all the following three equations A special case is considered when the flow field has uniform vorticity
Wang (1991)[5] gave the generalized solution as assuming a linear function for
Substituting this into the vorticity equation and introducing the separation of variables
results in The solution obtained for different choices of
represents a flow downstream a uniform grid,
represents a flow created by a stretching plate,
represents an Asymptotic suction profile etc.
Here, G. I. Taylor gave the solution for a special case where
satisfies the equation and also Taylor also considered an example, a decaying system of eddies rotating alternatively in opposite directions and arranged in a rectangular array which satisfies the above equation with
is the length of the square formed by an eddy.
Therefore, this system of eddies decays as O. Walsh generalized Taylor's eddy solution in 1992.
Marris and Aswani (1977)[8] showed that the only possible solution is
represents a flow due to two opposing rotational stream on a parabolic surface,
represents rotational flow on a plane wall,
represents a flow ellipsoidal vortex (special case – Hill's spherical vortex),
represents a type of toroidal vortex etc.
A special case of the above solution is Poiseuille flow for cylindrical geometry with transpiration velocities on the walls.
Chia-Shun Yih found a solution in 1958 for Poiseuille flow into a sink when
[10] Beltrami fields are a classical steady solution to the Euler equation.
Beltrami fields play an important role in (ideal) fluid mechanics in equilibrium, as complexity is only expected for these fields.