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Long wavelength asymptotics of self-oscillations of viscous incompressible fluid. (English) Zbl 1479.35631

Kusraev, Anatoly G. (ed.) et al., Operator theory and differential equations. Selected papers based on the presentations at the 15th conference on order analysis and related problems of mathematical modeling, Vladikavkaz, Russia, July 15–20, 2019. Cham: Birkhäuser. Trends Math., 185-203 (2021).
Summary: We obtain the long wavelength asymptotics of a secondary regime formed at stability loss of a stationary spatially periodic shear flow with non-zero average as one of the spatial periods tends to infinity (the wave number vanishes). It is known that if certain non-degeneracy conditions are satisfied, then from the basic solution a self-oscillatory regime branches. Recurrence formulas for kth term of the asymptotics of this secondary solution are obtained. To study the bifurcations of basic flow we obtain the scheme of Lyapunov-Schmidt method proposed by V. I. Yudovich [J. Appl. Math. Mech. 35, 587–603 (1971; Zbl 0247.76044); translation from Prikl. Mat. Mekh. 35, 638–655 (1971); ibid. 36, 424–432 (1972; Zbl 0257.34040); translation from Prikl. Mat. Mekh. 36, 450–459 (1972)]. At each step of the Lyapunov-Schmidt method series expansion in the small parameter \(\alpha\) is applied.
For the entire collection see [Zbl 1470.47003].

MSC:

35Q30 Navier-Stokes equations
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B10 Periodic solutions to PDEs
35B32 Bifurcations in context of PDEs
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[1] Troshkin, O.V.: Algebraic structure of the two-dimensional stationary Navier-Stokes equations and nonlocal uniqueness theorems. Dokl. Akad. Nauk SSSR 298(6), 1372-1376 (1988) · Zbl 0659.76036
[2] Meshalkin, L.D., Sinai, Ya.G.: Stability analysis of the stationary solution to a system of equations governing viscous incompressible planar flows. Prikl. Mat. Mekh. 25(6), 1140-1143 (1961) · Zbl 0108.39501
[3] Yudovich, V.I.: Investigation of continuum self-oscillations arising from loss of stability of a steady-state regime. Prikl. Mat. Mekh. 36(3), 450-459 (1972)
[4] Yudovich, V.I.: Onset of self-oscillations in a liquid. Prikl. Mat. Mekh. 35(4), 638-655 (1971)
[5] Kirichenko, O.V., Revina, S.V.: On the stability of two-dimensional flows close to the shear. Vestnik YuUrGU. Ser. Mat. Model. Progr. 12(3), 28-41 (2019) · Zbl 1430.35188
[6] Revina, S.V.: Stability of the Kolmogorov flow and its modifications. Comput. Math. Math. Phys. 57(6), 995-1012 (2017) · Zbl 1465.76036
[7] Revina, S.V.: On the problem of shear flow stability with respect to long-wave perturbations. Vladikavkaz Math. J. 18(4), 50-60 (2016) (in Russian) · Zbl 1469.76046
[8] Revina, S.V.: Recurrence formula for long wavelength asymptotics in the problem of shear flow stability. Comput. Math. Math. Phys. 53(8), 1207-1220 (2013) · Zbl 1299.76078
[9] Melekhov, A.P., Revina, S.V.: Onset of self-oscillations upon the loss of stability of spatially periodic two-dimensional viscous fluid flows relative to long-wave perturbations. Fluid Dyn. 45, 203-216 (2008) · Zbl 1210.76083
[10] Revina, S.V., Yudovich, V.I.: Initiation of self-oscillations at loss of stability of spatially-periodic three-dimensional viscous flows with respect to long-wave perturbations. Fluid Dyn. 36, 192-203 (2001) · Zbl 1101.76328
[11] Yudovich, V.I.: Instability of long-wave viscous flows. Fluid Dyn. 25, 516-521 (1990) · Zbl 0729.76034
[12] Yudovich, V.I.: Natural oscillations arising from loss of stability in parallel flows of a viscous liquid under long-wavelength periodic disturbances. Fluid Dyn. 8, 26-29 (1973)
[13] Yudovich, V.I.: Instability of viscous incompressible parallel flows with respect to spatially periodic perturbations. In: Numerical Methods for Problems in Mathematical Physics, pp. 242-249. Nauka, Moscow (1966, in Russian)
[14] Fylladitakis, E.D.: Kolmogorov flow: seven decades of history. J. Appl. Math. Phys. 6, 2227-2263 (2018). https://doi.org/10.4236/jamp.2018.611187
[15] Oparina, E.I., Troshkin, O.V.: Stability of Kolmogorov flow in a channel with rigid walls. Dokl. Phys. 49, 583-587 (2004)
[16] Yudovich, V.I.: Example of generation of a secondary steady or periodic flow when there is a loss of a stability of the laminar flow of a viscous incompressible fluid. Prikl. Mat. Mekh. 29(3), 455-467 (1965) · Zbl 0148.22307
[17] Arnold, V.I., Meshalkin, L.D.: A.N. Kolmogorov’s seminar on selected problems in analysis (1958-1959). Usp. Mat. Nauk. 15(1), 247-250 (1960)
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