# Modelling Liquid Flows in Diffusers by Reduced Equations

Yu.I. SapronovTo 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.Full text

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.

- Keywords
- Navier - Stokes equations; Helmholtz equations; diffuser current; Hamel substitution; Lyapunov - Schmidt variation method; velocity diagram.
- References
- 1. Jeffery G.B. The Two-Dimensional Steady Notion of a Viskous fluid. Phil. Mag. Ser. 6, 1915, vol. 29, no. 172, pp. 455-465. DOI: 10.1080/14786440408635327

2. Hamel G. Spiralformige Bewegungen zaher Flussigkeiten. Jahresber. Detsch. Math. Ver., 1917, Bd 25, pp. 34-60.

3. Akulenko L.D., Kumakschev S.A. Bifurcation of a Main Steady-State Viscous Fluid Flow in a Plane Divergent Channel. Fluid Dynamics, 2005, no. 3, pp. 359-368. DOI: 10.1007/s10697-005-0076-6

4. Akulenko L.D., Kumakschev S.A. Bifurcation of Multimode Flows of a Viscous Fluid in a Plane Diverging channel. Journal of Applied Mathematics and Mechanics, 2008, vol. 72, issue 3, pp. 296-302. DOI: 10.1016/j.jappmathmech.2008.07.007

5. Kochin N.E., Kibel I.A., Rose N.V. Teoreticheskaya gidrodinamika. Ch. 2 [Theoretical Hydrodynamics. Part 2]. Moscow, Fismatgiz, 1963. 736 p.

6. Landau L.D., Lifschic E.I. Teoreticheskaya fizika. T. VI. Gidrodinamika [Theoretical Physics. V. VI. Hydrodynamics]. Moscow, Nauka, 1986. 736 p.

7. Darinskii B.M., Sapronov Yu.I., Tsarev S.L. Bifurcations of Extremals of Fredholm Functionals. Journal of Mathematical Sciences, 2007, vol. 145, no. 6, pp. 5311-5453. DOI: 10.1007/s10958-007-0356-2

8. Sviridyuk G.A. Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type. Russian Academy of Sciences. Izvestiya Mathematics, 1994, vol. 42, no. 3, pp. 601-614. DOI: 10.1070/IM1994v042n03ABEH001547

9. Sviridyuk G.A., Zagrebina S.A. On the Verigin Problem for the Generalized Boussinesq Filtration Equation. Russian Mathematics (Izvestiya VUZ. Matematika), 2003, vol. 47, no. 7, pp. 55-59.

10. Zagrebina S.A. On the Showalter-Sidorov Problem. Russian Mathematics (Izvestiya VUZ. Matematika), 2007, vol. 51, no. 3, pp. 19-24. DOI: 10.3103/S1066369X07030036

11. Krasnoselskii M.A., Zabreiko P.P. Geometricheskie metody nelineinogo analiza [Geometric Methods of Nonlinear Analysis]. Moscow, Nauka, 1975. 512 p.

12. Borzakov A.Yu., Lemeschko A.A., Sapronov Yu.I. [Nonlinear Approximation and Visualization Rittsevskie Extremals Bifurcation]. Vestnik VGU. Seriya: Fizika. Matematika [Proceedings of the Voronezh State University. Series: Physics. Mathematics], 2003, issue 2, pp. 100-112. (in Russian)

13. Borzakov A.Yu. [Application of Methods of Finite-Dimensional Reduction to Global Analysis of Boundary Value Problems for the Equation Duffing]. Sbornik trudov matematicheskogo fakul'teta VGU [Proceedings of the Math. Faculty of VSU], 2005, issue 9, pp. 9-22. (in Russian)

14. Kostin D.V. Analysis Scheme for Bimodal Deflections of a Weakly Inhomogeneous Elastic Beam. Doklady Mathematics, 2008, vol. 77, issue 1, pp. 46-50. DOI: 10.1134/S1064562408010122

15. Kostina T.I. [Nonlocal Calculation of the Key Functions in the Problem of Periodic Solutions of Variational Equations]. Vestnik VGU. Seriya: Fizika. Matematika [Proceedings of the Voronezh State University. Series: Physics. Mathematics], 2011, no. 1, pp. 181-186. (in Russian)