IMPROVEMENT OF THE MATHEMATICAL MODEL OF LAMINAR FLOW WITH OPPOSITELY-ROTATING COAXIAL LAYERS

Subject: this paper relates to the field of hydrodynamics and scientific foundations of hydraulic engineering construction, and is devoted to investigation of the laminar flow of an incompressible fluid in which two coaxial layers moving together along a cylindrical tube rotate in opposite directions. In literature, this flow is called counter-vortex. Research objectives: improvement of the theoretical model of the counter-vortex laminar flow. In the turbulent range, the flow is characterized by intensive mixing of the moving medium and dissipation of its mechanical energy. Both these properties find practical application: the first one - in technologies that involve mixing of heterogeneous and multiphase media; the second one - for dissipation of mechanical energy of flows of liquids or gases in high-head hydraulic spillways and for suppressing the noise of aircraft engines and propellers. Theoretical study of laminar counter-vortex flows with oppositely rotating layers allows us to reveal the general physical laws of their hydrodynamics. Materials and methods: the theoretical model of the laminar counter-vortex flow is based on the method of expansion of the Navier-Stokes differential equations into Fourier-Bessel series. The improvement of the theoretical model of the laminar counter-vortex flow consists in elimination of the previously used assumption that the second partial derivative with respect to the axial coordinate on the right-hand side of equation (9) of this article is negligible. Results: a refined theoretical model of the laminar counter-vortex flow is obtained, based on the removal of this assumption. More accurate formulas for calculating the radial-longitudinal distributions of azimuthal and axial velocities in the studied flow are obtained in the form of Fourier-Bessel series or products of these series. The distributions of azimuthal and axial velocities are represented graphically in the form of their profiles. Conclusions: it is shown that taking into account the second partial derivative with respect to the axial coordinate on the right-hand side of equation (9) leads to a more accurate theoretical model of the laminar counter-vortex flow and makes it possible to substantially improve the accuracy of the analytical solution for velocity fields in such flows, namely by more than 2.3 % for small Reynolds numbers (less than Re = 100).