N.V. Burmasheva, E.Yu. Prosviryakov. Exact solution of Navier–Stokes equations describing spatially inhomogeneous flows of a rotating fluid ... P. 79-87

We study an overdetermined system consisting of the Navier-Stokes equations and the incompressibility equation. The system of equations describes steady spatially inhomogeneous shear flows of a viscous incompressible fluid. The nontrivial exact solution of the system under consideration is determined in the Lin-Sidorov-Aristov class. A condition for the solvability of the system for the velocity field of the form
$$V_x=U\left(z\right)+u_1\left(z\right)x+u_2\left(z\right)y, \quad V_y=V\left(z\right)+v_1\left(z\right)x+v_2\left(z\right)y, \quad V_z=0$$
is obtained. In the study of the exact solution, it is stated that the solvability of the system of equations is possible under an algebraic connection between the horizontal gradients (spatial accelerations) of the velocities $u_1, u_2, v_1, v_2$ and the pressure components $P_{11}, P_{12}, P_{22}$. Pressure is a quadratic form with respect to the coordinates $x$ and $y$. It is established that the pressure components and spatial accelerations are constant. In this case, depending on the values of the parameters, an exact solution is obtained for the velocities $U$ and $V$. The exact solutions obtained can describe the inhomogeneous Poiseuille-Couette-Ekman flow.

Keywords: layered flows, shear flows, exact solutions, Coriolis parameter, overdetermined system, compatibility conditions

Received February 20, 2020

Revised March 26, 2020

Accepted April 27, 2020

Natalya Vladimirovna Burmasheva, Cand. Eng. Sci., Institute of Engineering Sciences of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, 620049 Russia; Ural federal university, Ekaterinburg, 620002 Russia, e-mail: nat_burm@mail.ru

Evgeniy Yur’evich Prosviryakov, Dr. Phys.-Math. Sci., Institute of Engineering Sciences of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, 620049 Russia; Ural federal university, Ekaterinburg, 620002 Russia, e-mail: evgen_pros@mail.ru

REFERENCES

1.   Aristov S.N., Shvarts K.G. Vikhrevye techeniya advektivnoi prirody vo vrashchayushchemsya sloe zhidkosti [Vortical flows of the advective nature in a rotating fluid layer]. Perm: Perm State Univ. Publ., 2006, 155 p.

2.   Zyryanov V.N. Teoriya ustanovivshikhsya okeanicheskikh techenii [Theory of steady-state oceanic currents]. Leningrad: Gidrometeoizdat Publ., 1985, 248 p.

3.   Korotayev G.K., Mikhaylova E.N., Shapiro N.B. Teoriya ekvatorial’nykh protivotechenii v Mirovom okeane [Theory of equatorial countercurrents in the World Ocean]. Kiev: Naukova Dumka Publ., 1986, 208 p.

4.   Aristov S.N., Prosviryakov E.Y. Inhomogeneous Couette flow. Rus. J. Nonlin. Dyn., 2014, vol. 10, no. 2, pp. 177–182 (in Russian) . doi: 10.20537/nd1402004 

5.   Aristov S.N., Prosviryakov E.Y. Large-scale flows of viscous incompressible vortical fluid. Rus. Aeronautics, 2015, vol. 58, no. 4, pp. 413–418. doi: 10.3103/S1068799815040091 

6.   Zubarev N.M., Prosviryakov E.Yu. Exact solutions for layered three-dimensional unsteady isobaric flows of a viscous incompressible fluid. J. Appl. Mechanics and Technical Physics, 2019, vol. 60, no. 6, pp. 1031–1037. doi: 10.1134/S0021894419060075 

7.   Berker R. Sur quelques cas d’integration des equations du mouvement d’un fluide visquex incompressible. Paris-Lille: Taffin-Lefort, 1936, 161 p.

8.   Shmyglevskii Yu.D. On isobaric planar flows of a viscous incompressible liquid. USSR Comput. Mathematics and Math. Physics, 1985, vol. 25, no. 6, pp. 191–193.
doi: 10.1016/0041-5553(85)90030-8 

9.   Lin C.C. Note on a class of exact solutions in magneto-hydrodynamics. Arch. Rational Mech. Anal., 1957, vol. 1, no. 1, pp. 391–395. doi: 10.1007/BF00298016 

10.   Sidorov A.F. Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory. J. Appl. Mech. Tech. Phy., 1989, vol. 30, no. 2, pp. 197–203. doi: 10.1007/BF00852164 

11.   Aristov S.N. Vikhrevie techeniya v tonkikh sloyakh zhidkosti [Vortical flows in thin liquid layers]. Dissertation, Dr. Sci. (Phys. & Math.), Vladivostok, 1990, 303 p.

12.   Privalova V.V., Prosviryakov E.Yu., Simonov M.A. Nonlinear gradient flow of a vertical vortex fluid in a thin layer. Russian J. Nonlinear Dynamics, 2019, vol. 15, no. 3, pp. 271–283. doi: 10.20537/nd190306 

13.   Burmasheva N.V., Prosviryakov E.Yu. Thermocapillary Convection of a Vertical Swirling Liquid. Theoret. Foundations of Chemical Engineering, 2020, vol. 54, no. 1, pp. 230–239.
doi: 10.1134/S0040579519060034 

14.   Wheeler M.H. On stratified water waves with critical layers and Coriolis forces. Discrete and Continuous Dynamical Systems – A, 2019, vol. 39, no. 8, pp. 4747–4770. doi: 10.3934/dcds.2019193 

15.   Sarja A., Singh P., Ekkad S. Parallel rotation for negating Coriolis force effect on heat transfer. Aeronautical J., 2020, vol. 124, no. 1274, pp. 581–596. doi:10.1017/aer.2020.1 

16.   Fein Y.Y., Kialka F., Geyer P., Gerlich S., Arndt M. Coriolis compensation via gravity in a matter-wave interferometer. New J. Physics, 2020, vol. 22. doi: 10.1088/1367-2630/ab73c5 

17.   Mills C. Calibrating and operating Coriolis flow meters with respect to process effects. Flow Measurement and Instrumentation, 2020, vol. 71. doi: 10.1016/j.flowmeasinst.2019.101649 

18.   Landau L.D., Lifshitz E.M. Fluid mechanics. Oxford: Pergamon Press, 1987, 539 p. ISBN: 9781483161044 . Original Russian text published in Landau L.D., Lifshits E.M. Gidrodinamika. 6-e. izd. Moscow: Fizmatlit Publ., 2006, 736 p.

19.   Polyanin A.D., Zaitsev V.F. Handbook of exact solutions for ordinary differential equations, 2nd ed. Boca Raton: Chapman&Hall/CRC, 2003, 803 p. ISBN: 1584882972 .

Cite this article as: N.V. Burmasheva, E.Yu. Prosviryakov. Exact solution of Navier–Stokes equations describing spatially inhomogeneous flows of a rotating fluid. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 2, pp. 79–87.