The structure and stabilization by boundary conditions of an annular flow of Kolmogorov type. (English) Zbl 06760709
Summary: A system of Navier-Stokes equations with a right-hand side is considered in the case when the system approximately describes the motion of a thin layer of a viscous incompressible fluid in an annular domain under the action of external electromagnetic force. The problem possesses an unstable two-stream nonstationary main flow and a set of quasistationary solutions of vortex type for the tested range of parameters. A method of study of the general dynamic pattern is proposed in the paper. The method is based on the construction of control boundary conditions specified on the internal boundary of the annulus and providing the stabilization of considered unstable modes. The problem of boundary stabilization of the main and secondary flows is also solved numerically and we obtained that it is sufficient to take into account only a part of unstable modes in the construction of stabilizing conditions for the main flow. The method based on the partial stabilization of the main flow is first proposed for stabilization of secondary flows, which essentially simplifies the implementation of the algorithm. Formulations of the problems and numerical algorithms are presented.
MSC:
| 65P40 | Numerical nonlinear stabilities in dynamical systems |
| 37N35 | Dynamical systems in control |
| 76D55 | Flow control and optimization for incompressible viscous fluids |
References:
| [1] | N. F. Bondarenko, M. Z. Gak, and F. V. Dolzhanskii, Laboratory and theoretical models of a flat periodic flow. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana15 (1979), No. 10, 1017-1026 (in Russian).; Bondarenko, N. F.; Gak, M. Z.; Dolzhanskii, F. V., Laboratory and theoretical models of a flat periodic flow, Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana, 15, 10, 1017-1026 (1979) |
| [2] | V. A. Dovzhenko, Ur. V. Novikov, and A. M. Obukhov, Modelling of the process of generation of vortices in an axial symmetric azimuthal field magnetohydrodynamic method. Fiz. Atmos. Okeana15 (1979), No. 11, 1199-1202 (in Russian).; Dovzhenko, V. A.; Ur Novikov, V.; Obukhov, A. M., Modelling of the process of generation of vortices in an axial symmetric azimuthal field magnetohydrodynamic method, Fiz. Atmos. Okeana 15, 11, 1199-1202 (1979) |
| [3] | A. V. Fursikov and A. A. Kornev, Feedback stabilization for Navier-Stokes equations: Theory and calculations. In: Mathematical Aspects of Fluid Mechanics. Lecture notes series, Cambridge Univ. Press, 2012, pp. 130-172.; Fursikov, A. V.; Kornev, A. A., Feedback stabilization for Navier-Stokes equations: Theory and calculations, In: Mathematical Aspects of Fluid Mechanics., 130-172 (2012) · Zbl 1296.76049 |
| [4] | A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control. J. Math. Fluid Mech. 3 (2001), 259-301.; Fursikov, A. V., Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control, J. Math. Fluid Mech., 3, 259-301 (2001) · Zbl 0983.93021 |
| [5] | E. Z. Gak, The Magnetic Field and Water-Electrolytes in Nature, Scientific Research, Technology. Elmore, St. Petersburg, 2013 (in Russian).; Gak, E. Z., The Magnetic Field and Water-Electrolytes in Nature, Scientific Research, Technology. (2013) |
| [6] | A. A. Ivanchikov, A. A. Kornev, and A. V. Ozeritskii, On a new approach to asymptotic stabilization problems. Comp. Math. Math. Phys49 (2009), No. 12, 2070-2084.; Ivanchikov, A. A.; Kornev, A. A.; Ozeritskii, A. V., On a new approach to asymptotic stabilization problems, Comp. Math. Math. Phys, 49, 12, 2070-2084 (2009) · Zbl 07818356 |
| [7] | A. A. Kornev, Classification of methods of approximate projection onto a stable manifold. Doklady Akad. Nauk. 400 (2005), 736-738.; Kornev, A. A., Classification of methods of approximate projection onto a stable manifold, Doklady Akad. Nauk., 400, 736-738 (2005) |
| [8] | V. M. Ponomarev, Stability of one class of axisymmetric flows of incompressible liquid. Izvestiya Akad. Nauk SSSR, Mekhan. Zhidk. Gaza (1980), No. 1, 3-9 (in Russian).; Ponomarev, V. M., Stability of one class of axisymmetric flows of incompressible liquid, Izvestiya Akad. Nauk SSSR, Mekhan. Zhidk. Gaza, 1, 3-9 (1980) · Zbl 0449.76032 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
