Generalized Helical Flows

This paper studies the compatibility conditions for a system of equations describing nonuniform helical flows of an inviscid incompressible fluid. The class of flows considered traces back to the works of I. S. Gromeka and E. Beltrami, who independently discovered stationary solutions of the Euler equations satisfying the collinearity condition between the velocity and vorticity vectors. Their results laid the foundation for the theory of helical flows, identifying special solution classes of hydrodynamic equations. The system under consideration comprises the Euler equations supplemented by differential constraints that relate the velocity and vorticity vectors. Gromeka showed that if the function α is constant, the system becomes involutive. However, when α(x, y, z) is variable, the analysis becomes significantly more complex, and in general, the system is not involutive. A group analysis is performed for the resulting closed nonlinear system relating the velocity components and the function α\alpha. An optimal system of subgroups of the six-dimensional Lie algebra admitted by the system is constructed. Invariant solutions with respect to one-parameter subgroups are derived and are described by quasilinear equations with two independent variables.

ideal incompressible fluid, generalized helical flows, invariant solutions