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 Current Archive Rules for authors Send an article 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]. ← Contents of issue


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