Stability Analysis of a Film Flow Down an Incline in the Presence of a Floating Flexible Membrane
19 Jul 2018-Vol. 308, pp 253-263
TL;DR: In this article, the effects of floating flexible membrane on the instability of a gravity-driven flow down an incline was explored using normal-mode analysis, where the influence of membrane tension was taken into account in terms of stress jump at the free surface.
Abstract: The present study deals with the effects of floating flexible membrane on the instability of a gravity-driven flow down an incline. Linear stability of the flow system is explored using normal-mode analysis. Free surface gravity-driven flow is unstable at much lower Reynolds numbers. Instability of such a flow can be controlled either by changing behavioral of the lower wall or by altering the surface waves at the free surface which is done here by including a floating flexible membrane at the top of the liquid layer. Influence of membrane tension is taken into account in terms of stress jump at the free surface. The Orr-Sommerfeld system of the flow is solved numerically using spectral collocation method. The results displays a destabilizing role of membrane tension for a wide range of parameters. The growth rate of the perturbation waves increases with an increase of membrane tension and the critical Reynolds number becomes smaller. Therefore, it is possible to enhance the instability of the flow system with help of membrane properties, which may be useful in Ocean engineering and coating industries.
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...[31] analyzed the role of a floating membrane in the linear stability of a film flow down an inclined plane....
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TL;DR: In this article , two dimensional Navier-Stokes have been applied to an incompressible viscous fluid motion down an inclined plane with net flow and boundary layer thickness analysis.
Abstract: Motion of uid elements can be described by the Navier-Stokes equations. They arise from the use of Law of motion to a uid. In this investigation, two dimensional Navier-Stokes have been engaged. Then they are applied to an incompressible viscous uid movement down an inclined plane with net ow. These leads to examining the effects to the velocity of the motion at various angles of inclination and finding the boundary layer thickness. Viscous laminar incompressible fluid ow also ow on an inclined position which makes it necessary to investigate the ow on an inclined plane. Results that have been achieved are of the ow over horizontal at plate. Solution that has been obtained involves a at photographic film being pulled up by a processing bath by rollers at an angle \(\theta\) to the horizontal. Quadratic polynomial function approximate velocity profile has been obtained under initial boundary layer conditions. This velocity profile has been used in momentum integral equation for ow over an inclined plane to get the boundary layer thickness. Boundary layer thickness is one of the parameters that is used to obtain the ow velocity down inclined plane.
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TL;DR: In this paper, the effect of a temperature dependent variable viscosity fluid flow down an inclined plane with a free surface is investigated, and the full solutions for the temperature and velocity profiles are derived using the Runge-Kutta numerical method.
Abstract: The effect of a temperature dependent variable viscosity fluid flow down an inclined plane with a free surface is investigated. The fluid film is thin, so that lubrication approximation may be applied. Convective heating effects are included, and the fluid viscosity decreases exponentially with temperature. In general, the flow equations resulting from the variable viscosity model must be solved numerically. However, when the viscosity variation is small, then an asymptotic approximation is possible. The full solutions for the temperature and velocity profiles are derived using the Runge-Kutta numerical method. The flow controlling parameters such as the nondimensional viscosity variation parameter, the Biot and the Brinkman numbers, are found to have a profound effect on the resulting flow profiles.
12 citations
TL;DR: Far from the instability threshold, the current reduced models continue to follow the OS solution up to moderate Reynolds numbers, while the averaging model diverges rapidly, and the model SRM gives better results than FRM beyond sufficiently high Reynolds numbers.
Abstract: This paper deals with the long wave instability of an electroconductor fluid film, flowing down an inclined plane at small to moderate Reynolds numbers, under the action of electromagnetic fields. A coherent second order long wave model and two simplified versions of it, referred to as first and second reduced models (FRM and SRM), are proposed to describe the nonlinear behavior of the flow. The modeling procedure consists of a combination of the lubrication theory and the weighted residual approach using an appropriate projection basis. A suitable choice of weighting functions allows a significant reduction of the dimension of the problem. The full model is naturally unique, i.e., independent of the particular form of the trial functions. The linear stability of the problem is investigated, and the influence of electromagnetic field on the flow stability is analyzed. Two cases are considered: the applied magnetic field is either normal or parallel to the fluid flow direction, while the electric field is transversal. The numerical solution of the Orr-Sommerfeld (OS) eigenvalue problem and those of the depth averaging model are used to assess the accuracy of the reduced models. It is found that the current models have the advantage of the Benney-like model, which is known to asymptote the exact solution near criticality. Moreover, far from the instability threshold, the current reduced models continue to follow the OS solution up to moderate Reynolds numbers, while the averaging model diverges rapidly. The model SRM gives better results than FRM beyond sufficiently high Reynolds numbers.
9 citations
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TL;DR: In this paper, a mathematically rigorous justification of a thin-film approximation was given by establishing an error estimate between the solution of the Navier-Stokes equations and those of approximate equations.
Abstract: We consider a two-dimensional motion of a thin film flowing down an inclined plane under the influence of the gravity and the surface tension. In order to investigate the stability of such flow, we often use a thin film approximation, which is an approximation obtained by the perturbation expansion with respect to the aspect ratio of the film. The famous example of the approximate equations are the Burgers equation, Kuramoto--Sivashinsky equation, KdV--Burgers equation, KdV--Kuramoto--Sivashinsky equation, and so on. In this paper, we give a mathematically rigorous justification of a thin film approximation by establishing an error estimate between the solution of the Navier--Stokes equations and those of approximate equations.
4 citations
TL;DR: In this article, a mathematically rigorous justification of a thin-film approximation is given by establishing an error estimate between the solution of the Navier-Stokes equations and those of approximate equations.
2 citations