A method for replicating exact solutions of the Euler equations for incompressible Beltrami flows

Abstract

In the paper, Beltrami flows or helical flows are flows in which the vorticity and velocity vectors are collinear, and the proportionality coefficient between these vectors is nonzero and is the same at all points of the flow. A method is proposed that allows using known helical solutions to obtain new helical solutions of the Euler equations for an incompressible fluid. Some of these new solutions cannot be obtained by the known methods of replicating solutions by shifting and rotating the coordinate system, symmetry, scaling, cyclic permutation of the velocity and coordinate components, vector summation. The new replication method is applied to such parametric families of exact solutions in which the proportionality coefficient between velocity and vorticity remains unchanged for different values of the parameter. The essence of the method is that for such families the derivative of the velocity with respect to the parameter is also the helical velocity. The sequential differentiation of the speed of a new solution with respect to a parameter gives an endless chain of new exact solutions.

Keywords

• helical solutions of the navier–stokes equations
• exact solutions of the euler equations
• beltrami flows