This article provides the results of a theoretical and experimental study of buoyancy-driven instabilities triggered by a neutralization reaction in an immiscible two-layer system placed in a vertical Hele–Shaw cell. Flow patterns are predicted by a reaction-induced buoyancy number K ρ , which we define as the ratio of densities of the reaction zone and the lower layer. In experiments, we observed the development of cellular convection ( K ρ ≤ 1 ), the fingering process with an aligned line of fingertips at a slightly denser reaction zone ( K ρ ≥ 1 ) and the typical Rayleigh–Taylor convection for K ρ > 1 . A mathematical model includes a set of reaction–diffusion–convection equations written in the Hele–Shaw approximation. The model’s novelty is that it accounts for the water produced during the reaction, a commonly neglected effect. The persisting regularity of the fingering during the collapse of the reaction zone is explained by the dynamic release of water, which compensates for the heavy fluid falling and stabilizes the pattern. Finally, we present a stability map on the plane of the initial concentrations of solutions. Good agreement between the experimental data and theoretical results is observed. This article is part of the theme issue ‘New trends in pattern formation and nonlinear dynamics of extended systems’.
We study the centrifugal buoyancy-driven chemoconvection in a Hele-Shaw cell that uniformly rotates around e perpendicular axis. The slot is considered thin enough to neglect the influence of the Coriolis force. The initial configuration of the system consists of two aqueous reacting solutions separated by a concentric boundary. The acid solution fills the center of the cavity, while the base solution is in the periphery. Bringing liquids into contact initiates a neutralization reaction to form a salt. We show that reaction-diffusion processes produce a potential well near the reaction front, which determines the pattern formation of the system. For some ratios of initial concentrations, there appears a periodic sequence of chemoconvective vortices in the well, while for others, when the well collapses, a shock-like density wave occurs. When the density of the acid solution is higher, the Rayleigh-Taylor instability develops in the system. We found that an increase in the rotation speed leads to a gradual disruption of the structure periodicity. It can even result in the ejection of some vortices from the potential well. We show that the density wave is extremely sensitive to the magnitude of the centrifugal force, occurring only at some critical value. Finally, we obtained a stability map of the system by performing direct numerical simulations for increasing the centrifugal Rayleigh number and the dimensionless distance of the initial contact surface between the solutions from the axis of rotation.
Исследуются хемоконвективные структуры в системе двух смешивающихся реагирующих жидкостей, помещенных в ячейку Хеле-Шоу цилиндрической формы, совершающей равномерные вращения вокруг оси симметрии. Ранее формирование структур в подобных условиях изучалось авторами экспериментально и теоретически при наличии статического поля силы тяжести. Радиально направленное инерционное поле, создаваемое центробежной силой, меняется по пространству (вдоль радиуса) и может регулироваться частотой вращения, что дает системе новые степени свободы. Начальная конфигурация представляет собой два концентрических слоя водных растворов, разделенных по пространству бесконечно тонкой диффузионной границей. Раствор кислоты расположен ближе к оси вращения, а раствор основания -на периферии ячейки. Концентрации веществ подобраны таким образом, что гарантируется начальная устойчивость конфигурации по отношению к возмущениям Релея-Тейлора. При приведении жидкостей в контакт начинается реакция нейтрализации, которая сопровождается выделением соли. Важную роль в данном процессе играет функциональная зависимость коэффициентов диффузии реагентов от концентрации в растворе последних, что имеет следствием нелинейный вид соответствующих уравнений переноса уже в основном состоянии реакции-диффузии. Как и в случае статического силового поля, вблизи фронта реакции возникает плотностная потенциальная яма, которая определяет нелинейную динамику системы. Получена система нелинейных уравнений, описывающая движение жидкости. Демонстрируются результаты численного моделирования полной нелинейной задачи. Показано, что при некотором соотношении начальных концентраций и значений центробежных чисел Релея в потенциальной яме развивается ячеистая конвекция. При увеличении скорости вращения периодичность структуры нарушается все больше и больше за счет влияния неустойчивости диффузионного слоя, формирующейся у оси вращения, и действия инерционного поля, которое вырывает отдельные ячейки из потенциальной ямы.
The authors study the effect of uniform rotation on the system of two reacting miscible liquids placed in a cylindrical Hele-Shaw cell. The cell performs a rotation with a constant velocity around the axis of symmetry resulting in a radially directed inertial field. The initial configuration of the system is statically stable and consists of two concentric layers of aqueous solutions of acid and base, which are spatially separated. When liquids are brought into contact, a neutralization reaction begins, which is accompanied by the release of salt. In this work, we obtain a system of governing equations and present the results of numerical simulation. We found that reaction-diffusion processes lead to the formation of a non-monotonic density profile with a potential well. If the rotation rate gradually increases, then a cellular convection pattern can develop in the potential well. We found that with further growth of the control parameter, the periodicity of the pattern is violated due to the influence of another convective instability, which independently develops in the domain close to the axis of rotation. The action of the inertial field results in the ejection of some convective vortices from the potential well.
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