On the Problem of Shear Flow Stability with Respect to Long-Wave Perturbations

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2016.4.5994 On the Problem of Shear Flow Stability with Respect to Long-Wave Perturbations Revina S. V. Vladikavkaz Mathematical Journal 2016. Vol. 18. Issue 4. Abstract: To find secondary flow branching to the steady flow it is necessary to consider linear spectral problem and linear adjoint problem. Long-wave asymptotics of linear adjoint problem in two-dimensional case is under consideration. We assume the periodicity with spatial variables when one of the periods tends to infinity. Recurrence formulas are obtained for the $k$th term of the velocity and pressure asymptotics. If the deviation of the velocity from its period-average value is an odd function of spatial variable, the velocity coefficients are odd for odd $k$ and even for even \(k\). The relations between coefficients of linear adjoint problem and linear spectral problem are obtained. Keywords: stability of two-dimensional viscous flows, long-wave asymptotics, linear adjoint problem Language: Russian Download the full text + References 1. Yudovich V.I. Initiation of Self-Oscillations in Fluids. Prikladnaya Matematika i Mekhanika [J.Appl. Math. Mech.], 1971, vol. 35, no. 4, pp. 638-655 [in Russian]. 2. Yudovich V.I. Instability of Parallel Flows of a Viscous Incompressible Fluid Relative to Spatially Periodic Perturbations. Chislennye metody resheniya zadach matematicheskoy fiziki [Numerical Methods for Solving Problems in Mathematical Physics] Nauka, Moscow, 1966, pp. 242-249 [in Russian]. 3. 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 Dynamics, 2008, vol. 43, no. 2, pp. 203-216. 4. Revina S.V. Recurrence Formulas for Long Wavelength Asymptotics in the Problem of Shear Flow Stability. Computational Mathematics and Mathematical Physics, 2013, vol. 53, no. 8, pp. 1207-1220. 5. Revina S.V. Linear Adjoint Problem for the Problem of Stability of Two-Dimensional Shear Flows with Non-Zero Average. Paper no. 226-B2014. Deposed in VINITI 11.08.2014 [in Russian]. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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