Flow of Soap Films on Inclined Plane

يف ةساردل جذومن عضو مت ثحبلا اذه ةيشغلأا ةلئاملا يف ماظنلا يئانثلا دعبلا ذإ تي رثأ ةوـقب نايرجلا ةيبذاجلا . انضرتفا ثيح جذومنلا اذه ةسارد يف تييزتلا ةيرظنو ىذاحملا ليلحتلا مدختسا نأ دنع تبثم ءاشغلا ريفان تلاداعم قيبطت مت دقو هيتياهن مـتو ةـنيعم ةـيدودح طورش عم طاغضنلال لباق ريغلا عئاملل سكوتس زيكرتلاو ءاشغلا كمس لثمت تلاداعم ىلع لوصحلا و اضيأ ءاشغلا حطس ةعرس . صح ل لوـلحلا ىلع ان اذـهل مادختساب تلاداعملا هذه بيرقت متو نايرجلا تايثادحإ نـكمي تلاداعم ىلع لوصحلل رليات ةلسلستمو جناركلا نمزلل ةفلتخم ميقل اهمادختسا .

Thin films have been studied widely in many areas, such as surface coatings in paint, varnish and silver layer on a compact disc. The drainage of thin film is important in understanding foam fabrications for applications like making cushions and these foam are not aqueous. There are a variety of phenomena one can observe such as drainage, details of rapture [1] and these phenomena can help to characterize and describe the physical processes that occur in our real world and such knowledge can be used in surface coatings in paint, protective wax and foam development [2].
The drainage of soap films has been studied by many authors. Mysel [7] gave a comprehensive experimental description of soap films thinning. The theoretical and numerical model for soap film drainage studied by Schwartz [8] which reproduce may of the features of this process that have been observed in experiment. These features include the shape of the film thickness profile when the film is taken to be vertical and supported by wires, and the large differences in drainage time scales for low and high surfactant. Myers [5,6] investigated the driven of thin film flow when the surface tension play an important role. Kondic, L. [3] studied the instabilities in gravity driven flow of thin liquid films. The main purpose of this paper is to study the inclined drainage flow in mobile and immobile films in which the surfactant forces play an important role on the drainage of thin liquid films.

Formulations and Governing Equations:
We consider an inclined two dimensional soap film supported at the ends by guides. The liquid air interface is located at where the film is symmetric with respect to the central line

Flow of Soap Films on Inclined Plane
Which hold within the fluid half layer and where P and  represent density and viscosity of fluid, g is the gravity and t the time.
The boundary conditions to be imposed are as follows: Since the film is symmetric, we have Furthermore at the surface of the film, the kinematics condition is given by: Also the shear stress and normal stress conditions on the free surface of the film are respectively given by: Where s P is the pressure at the liquid side of the free surface. We have to note Since the Reynolds number is taken so small, the inertia term in the Navier Stokes equation can be neglected in comparison with viscous forces per unit volume of the fluid and so equations (2) are reduced to give By using equation (9), equations (7), (8) and (11), become respectively ( ) Scaling and dimensional analysis: If L denotes the characteristic length scale and a typical scale for width is 0 hL = , 1  , then the velocity scale is We introduce the following non-dimensional variables as follows.
We define the Bond number ( ) where  : is the surface tension.
Furthermore, the boundary conditions (4), (5), (13) and (14) become Lubrication model: We expand each of the unknowns v u , 1 and P as a power series in  as follows: The continuity equation (17) gives: Integrating (24) with respect to y we have: and equation (25) reduces to give: Also equation (18) and so on. Integrating both sides of equation (27) From (21), the shear stress condition, gives: …(38) Comparing the similar terms of both sides of (38) in power of , we obtain ( ) Using equation (24) We have to noticed that the term equation (43) is the evolution equation of the film thickness in non-dimensional form.
In dimensional form equation (40) and (43) are respectively The pressure s P in dimensional variables is defined as Surfactant effect: A convectiondiffusion evolution equation for an insoluble surfactant for a film of small slop is given by: is a local surfactant concentration and D is Fickian diffusion constant. The simplest form for small change in surface tension is given by: where 1 k is constant and the zero subscript refer to initial values of these quantities at the start of motion. The surface shear stress is given by: substituting (48) into equation (44), we get we have to noticed that equations (46), (49) and (50) form a complete set.
Sample calculation: Assume that cm L 1 = and 0 h must be less than L , we take 0 0.5 h = . The density of water Integrating equation (52) with respect to x , we get is a function that will be determined by the lower end boundary condition. The derivative following the motion in the Lagrangian system is given by and following the motion, the mass conservation condition requires that Thus equations (56) and (57) gives integrating equation (58), we get (60) is a first order differential equation whose general solution is given by , and then equation (61) gives (57) and (62) gives The solution of equation (63)

Conclusion:
The film drainage reproduces different features that include the shape of the film profile and also in the draining time with or without surfactant effects. The film drains quickly when no surfactant exists and have a hollow ground appearance, but when the surfactant with high concentration exists, the film quickly look into an immobile interface with a parabolic shape that drains in slow manner and retains an appreciable thickness for long time and the film shape will be concaved out if the time drainage increases. x