The stability of a shear flow imposed along a diffusive interface that separates two miscible liquids (a heavier liquid lies underneath) is studied using direct numerical simulations. The phase-field approach is employed for description of a thermo-and hydrodynamic evolution of a heterogeneous binary mixture. The approach takes into account the dynamic interfacial stresses at a miscible interface and uses the extended Fick's law for setting the diffusion transport (the diffusion flux is proportional to the gradient of chemical potential). The shear flow is unstable to two kinds of instabilities: (i) the Kelvin-Helmholtz instability, with an immovable vortex formed in the middle of an interface (in the vertical direction), and (ii) the Holmboe instability, with travelling waves along the interfacial boundary. The development of the Holmboe instability results in a stronger enhancement of molecular mixing between the mixture components. Earlier, the boundaries of these instabilities were determined using the linear stability analysis and employing the concept of a 'frozen interface'. In the current work, through the solution of full equations, we obtain the stability boundaries for several sets of governing parameters, showing a greater variety of the possible shapes of the stability diagrams. The Kelvin-Helmholtz instability always occurs at lower gravity effects (lower density contrasts), while the Holmboe instability occurs when gravity is stronger. We show that for some parameters these two instabilities are separated by a zone where the shear flow is stable, and this zone disappears for the other sets of parameters.
We model the buoyancy-driven motion of a liquid droplet in an ambient liquid, assuming that the liquids are miscible. The classical representation of miscible liquids as a single-phase fluid with impurity (neglecting surface tension effects) cannot describe all experimental observations of moving droplets in a miscible environment, in particular, the tendency of droplets to pull to a spherical shape. In the framework of the classical approach, we show that the motion of a miscible droplet results in its instant dispersion (except for a very slow rise). We also model the motion of a miscible droplet in the framework of the phase-field approach, taking into account surface tension forces. We vary the value of the surface tension coefficient within a very wide range, modelling a droplet that rises preserving a spherical shape, or a droplet which dynamically becomes indistinguishable from the droplet with an interface endowed with no surface tension. We also show that by employing the concept of dynamic surface tension, one may reproduce the motion of a droplet that pulls into a sphere in the initial period of its evolution and that disintegrates similar to a droplet with zero surface tension at the later stages.
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 ...
Thermal convection in a flat horizontal layer of a porous medium with solid impermeable boundaries on which the heat flow is given is considered. The porous medium is saturated with a viscous incompressible fluid pumped along the layer. In the system under discussion, with a vertical heat flow inhomogeneous along the layer, localized convective structures may occur in the region where the heat flow exceeds the critical value corresponding to homogeneous heating from below and corresponding to the beginning of convection in the layer. With an increase in the rate of longitudinal pumping of the liquid through the layer, a transition from a state in which the localized convective structures are stable to a state in which the localized convective flow is completely washed out of the region of its excitation occurs. Calculations were performed in the framework of Darcy-Boussinesq. Results of the numerical calculation of the process removal of localized convective structures from the zone of its excitation with an increase in the rate of longitudinal pumping of liquid through the layer are presented. The map of the system state modes is obtained.
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