Modelling Liquid Flows in Diffusers by Reduced Equations

To know the dynamic characteristics of liquid in hydrocyclones and diffusers is important for optimizing the technical parameters of the liquid ends of turbine pumps on long-distance oil pipelines. It is possible to describe these characteristics by using the available analytic expressions for the solutions to the model equations of hydrodynamics or their simplified versions used in these problems.
It is known that the simplified systems of hydrodynamic type derived from the Navier–Stokes equation allow us to model quite precisely liquid flows in regions of arbitrary geometric shape. In this article we reduce the Helmholtz equation in the case of a flat diffuser flow to a boundary value problem for the Jeffrey–Hamel ODE by means of the Hamel substitution. At finite values of the Reynolds number we establish the possibility of constructing approximate solutions to the reduced equation via nonlinear Ritz–Galerkin approximation using a variational version of the Lyapunov–Schmidt method. With this approximation, we can determine the liquid velocity field to arbitrary precision. The article includes examples of approximately computed velocity diagrams for the flows close to $n$-modal with $n \le 5$.