В данной статье представлен новый подход к анализу динамической устойчивости прямоугольных ортотропных пластин. В частности, в приближении теории плоских сечений исследуется проблема флаттера для ортотропной панели в сверхзвуковом потоке газа, которая сводится к краевой задаче для несимметричного дифференциального оператора. С целью улучшения стандартной процедуры вычислений методом Бубнова-Галеркина предлагается в качестве базисных функций этого метода использовать собственные формы колебаний прямоугольной ортотропной пластины в вакууме, для которых автором получены новые аналитические представления. Согласно данному подходу краевая задача сводится к однородной бесконечной системе линейных алгебраических уравнений. На основе асимптотического анализа и теории регулярных бесконечных систем линейных алгебраических уравнений разработан точный и эффективный алгоритм построения собственных форм пластины в вакууме. Таким образом, в статье обсуждаются как алгоритм построения базисных функций метода Бубнова-Галеркина, так и алгоритм определения критического значения параметра скорости, при котором имеет место динамическая неустойчивость. Численно изучается сходимость метода Бубнова-Галеркина в зависимости от параметров задачи. Результаты численного моделирования показывают, что при изменении значений сил в плоскости пластины и упругих свойств материала хорошая сходимость метода может быть достигнута при первых 16-ти базисных функциях. Аналогичная сходимость метода наблюдается и для удлиненной пластины. Вычислительная эффективность метода иллюстрируется примерами.Ключевые слова: прямоугольная пластина, флаттер, метод Бубнова-Галеркина, собственные формы колебаний FLUTTER OF CLAMPED ORTHOTROPIC RECTANGULAR PLATE S.O. Papkov Sevastopol State University, Sevastopol, Russian FederationA new approach for dynamic stability analysis of rectangular orthotropic plates is presented. In particular, in the approximation of the theory of planar sections the problem of the flutter of a panel in a supersonic gas flow is reduced to a boundary value problem for nonsymmetric differential operator. To improve standard technique of the Bubnov-Galerkin method, it is proposed to use new analytical representations of the eigenmodes of vibrations of a rectangular orthotropic plate in a vacuum as the basis functions of this method. According to this approach, the boundary value problem is essentially reduced to a homogeneous infinite system of linear algebraic equations. By using the asymptotic analysis and theory of regular infinite systems of linear algebraic equations, the effective and accurate algorithm for constructing the mode shapes in vacuum is developed. So, both the algorithm for constructing basis functions and the algorithm for determining the critical value of the velocity parameter are presented in this paper. The convergence of the Bubnov-Galerkin method is studied numerically for different problem parameters. The results of numerical modeling show that good convergence of the method can be achieved with first 16 basis functions ...

A class of closed-form exact solutions for the Navier–Stokes equation written in the Boussinesq approximation is discussed. Solutions describe the motion of a non-homogeneous reacting fluid subjected to harmonic vibrations of low or finite frequency. Inhomogeneity of the medium arises due to the transversal density gradient which appears as a result of the exothermicity and chemical transformations due to a reaction. Ultimately, the physical mechanism of fluid motion is the unequal effect of a variable inertial field on laminar sublayers of different densities. We derive the solutions for several problems for thermo- and chemovibrational convections including the viscous flow of heat-generating fluid either in a plain layer or in a closed pipe and the viscous flow of fluid reacting according to a first-order chemical scheme under harmonic vibrations. Closed-form analytical expressions for fluid velocity, pressure, temperature, and reagent concentration are derived for each case. A general procedure to derive the exact solution is discussed.

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The authors consider the problem of determining the stability boundary of a two-layer system of miscible liquids placed in a gravity field. Liquids are aqueous solutions of non-reacting substances with different diffusion coefficients, which are linear functions of concentrations. At the very beginning of the evolution, the solutions are separated from each other by an infinitely thin horizontal contact surface. Such a configuration can be easily realized experimentally, although it is more difficult for theoretical analysis since the base state of the system is non-stationary. Once brought into contact, the solutions begin to mix penetrating each other and creating conditions for the development of the double-diffusive instability since the initial configuration of the system is assumed to be statically stable. The problem of the convective instability of a mixture includes the equation of motion written in the Darcy and Boussinesq approximations, the continuity equation, and two transport equations for the concentrations. We apply the linearization method suggested by Wiedeburg (1890) to find a closed-form solution to the non-stationary base state problem including concentration-dependent diffusion laws for species. We derive analytical expressions for neutral stability curves and study corrections introduced by nonlinear diffusion to the stability analysis.

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