Nonlinear waves in a liquid film on a near-horizontal surface

Abstract

Nonlinear waves in a liquid film on a slightly inclined rigid plane are studied. A mathematical model is reduced to a system of two evolutionary equations for the layer thickness and the local fluid mass flow. In addition to viscous forces, gravity, and surface tension, the pressure difference over the layer thickness, induced by the gravity force projection on the normal to the underlying surface, is also taken into account. Spatially periodic solutions developing with time from small initial disturbances into regular nonlinear waves are considered. A spectral representation of the solution, the Galerkin method with respect to the uniform coordinate, and subsequent numerical calculation of the corresponding dynamic system on large time intervals are employed. Different variants in the space of the three governing parameters are calculated and some basic mechanisms of nonlinear dynamics of the two-dimensional waves are detected. The calculation results are compared with the existing experimental data. It is shown that the theoretical conclusions can be used to interpret and predict experiments.Comprehensive measurements of nonlinear waves in a film falling down an inclined surface at small angles of inclination to the horizontal (θ = 4 • -10 • ) were presented in [1-3]. This measurement series was carried out to extend and develop a pioneering study [4], in which the waves on a vertical rigid wall were investigated experimentally for the first time. The goal was to perform a more detailed study of the motion and interaction of nonlinear waves developing from small initial disturbances on the basis of modern experimental techniques, including laser probing and fluorescent imaging of the waves. The authors of [1-3] noted that they had no appropriate wave motion theory for a concise and regular representation of the experimental results. They mentioned only certain analogies with the case of waves on a vertical wall for which such a theory exists. The measurement results were given as sets of particular variants with explanations and discussions of possible mechanisms of the observed scenarios of wave development.In this paper, we present a systematic investigation of nonlinear-wave development scenarios in a film on the basis of a numerical solution of the equations of a mathematical model. In the calculations, we model the experimental conditions [1-3] and compare the numerical and experimental results.

Вплив Гідродинамічних Факторів На Рівномірність Товщини Рідинного Шару При Екструзійному Нанесенні

“…Вивчення хвильових режимів плівок рідини, що стікають, представлені в роботах [7,8]. Авторами виведено систему рівнянь для течії двошарової плівки.…”

“…Оскільки об'єм V є функцією перепаду тисків ∆p 1 d∆p = f(t) і може змінюватися у часі, то V= F(∆p)=F[f(t)]. Похідна від об'єму за часом, що входить до рівняння (6) та(7): 𝑝𝑝 ск + ∆𝑝𝑝) 4 �2(𝑝𝑝 ск + ∆𝑝𝑝) �ln ��2(𝜎𝜎 2 − 𝜎𝜎) шару зменшується, і, отже, абсолютні відхилення витрати ∆q ∆р і ∆h ∆p товщина шару негативні. У момент зменшення перепаду товщина шару збільшується і ∆q ∆р і ∆h ∆p стають позитивними.…”

“…У момент зменшення перепаду товщина шару збільшується і ∆q ∆р і ∆h ∆p стають позитивними. Тому в загальному вигляді вирази (6) та(7) матимуть такий вигляд: d∆V ∆р /dt визначається з виразу (10).Висновки.1. Проведені дослідження реологічних характеристик фотоемульсії методом ротаційної віскозиметрії.…”

Stability of Thin Liquid Films Falling Down Isothermal and Nonisothermal Walls

“…The series of wave bifurcations occurring in the solutions of this model is investigated numerically by Shkadov and Sisoev (2005). A comparison with experimental results is done by Tushkanov and Shkadov (2006) for the case of a slightly inclined plane. They show that the model reproduces correctly single-crested waves and solitary waves and all between this range of possibilities.…”

Inertial effects on thin-film wave structures with imposed surface shear on an inclined plane

“…Numerous studies involving flow on an inclined plane range from near-horizontal flows [24] to vertical falling films [4,5,14]. Previous studies, for example [17,23], for shear driven flow on an inclined plane describe wave structures depending on their initial profiles as well as uniform upstream and downstream boundary conditions.…”