Thin Films Flow Driven on an Inclined Surface

The flow of unsteady incompressible two dimensional system flow of a thin liquid films with negligible inertia is investigated. Continuity equation and NavierStokes equations are used to obtain the equation that governs this type of flow. Keyword: Flow , Thin liquid films , Navier-Stokes equations لا قفدت حوطس ةقيقرلا لا لئام حطس ىلع ةعوفدم رضخ دمحم رضخ تايضايرلاو بوساحلا مولع ةيلك لصوملا ةعماج :ملاتسلاا خيرات 20 / 11 / 2011 لا خيرات لوبق : 15 / 02 / 2012 صخلملا لالا نايرجلا ةيكيناكيم ةسا رد ىلإ ثحبلا فدهي و رقتسم لالا روصقلا ىوق مادعناب ةقيقرلا ةيشغلأل طغضنم ر اذذن تتداذذعمو ةيرا رمتذذستا ةذذلداعم رمدستذذسا دذذقو ددذذعبلا و،اذذنا ماذذثن وذذ و و اتذذلا وذذتلا ةذذلداعملا داذذجاج لإ وتذذس .نايرجلا نم عونلا اته مكح ر ان تتداعم دةقيقرلا ةيشغتا د نايرجلا: ةيحاتفملا تاملكلا لإ وتس


Introduction:
The flow of thin films of fluids is encountered in many engineering and biological applications. They include; the flow of rainwater on a road, windscreen or other draining problem [3], paint and coating flow [6,1]. The flow of many protective biological fluids [5], and other coating are paint and dry processes [4,7,8]. The fluid film thickness and the average fluid flux are the main characteristics of interest in these applications [9]. Bascom, Cottington, and Singleterry [10] reported experimental observations of contact lines of thin liquid films. Emilia Borsa had studied the flow of a thin layer on a horizontal plate in the lubrication approximation [2]. The objective here is to obtain the equation governing the flow in thin liquid films, and to find the thickness of the film.

Governing Equations:
We consider a two-dimensional thin film flow on an inclined plane at angle α. The x-axis is oriented stream wise along the plane. The y-axis is perpendicular to the plane in the film thickness direction with the origin at the liquid plane interface. The flow is considered to be a laminar incompressible Newtonian fluid with constant density ρ and constant viscosity µ, and governed by the Navier-Stokes equations and continuity equation as: In the thin film lubrication approximation the Navier-Stokes equations read Where g is gravitational acceleration, y=h(x,y) is the free surface, p is the pressure in the fluid , u and p depended on X=(x,y,t), and t is the time.
To complete the problem formulation, the lubrication equations (3), (4) and (5) require the boundary conditions: Where 0 p is the atmospheric pressure in the air face.
Where the boundary conditions (6), (7), (8) and (9) represent the no-slip condition ,the balance of tangential stress, the balance of normal stress and the kinematics condition respectively.
Integrating (4) with respect to y and using the boundary condition (7) we have: Similarly integrating (10) with respect to y and using boundary condition (6) we get : Now, integrating (5) with respect to y and using the boundary condition (8) we obtain : …(12) Drive(12) with respect to x and substitution in to(11) we have: By using the continuity equation in the thin film approximation and the kinematics boundary condition leads the evolution equation for We introduce the following nondimensional variables defined by: Where the velocity U and the length scale L are characteristic quantities of the problem, assume that δ=ho/L, where ho is the characteristic length for the film thickness.
Then, convert the equation (14) into no dimensional form in terms of the no dimensional variables hˉ, xˉ, tˉ, τˉ and the equation (14) becomes For the sake of simpler notation, we drop the "dash" from the non dimensional variables hˉ, xˉ , tˉ , τˉ in the equation (16) By giving different value to the constant f, and angle α, we get the thick the film.
The second case when A≠0 and τ=0, the equation (18) becomes: Now, to get the initial condition for the equation (19) Similarly by taking different value for A and α we get the thick film in another case.  Through our studies to the motion equation for viscous incompressible liquid, we conclude from equation (20)a that the thickness of the film increases when we approach the negative values of x when α=30,49 as shown in Figures (1.1) and (1.3) and it will decrease towards the positive values of x, when α=41, 60 as shown in Figures  (1.2) and (1.4) and the table (1.1), this implies that the value of the angle α will affect the thickness of the film. From equation (21), we note that the thickness of the film will be parallel to the x-axis and it will increase according to the value of α as shown in Figures (1.5), (1.6), (1.7) and (1.8) and the table (1.2). .