IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org ISSN (e): 2250-3021, ISSN (p): 2278-8719 Vol. 04, Issue 04 (April. 2014), ||V4|| PP 41-46 International organization of Scientific Research 41 | P a g e Steady Flow of Thin Liquid Films on an Inclined Solid surface Joseph G. Abdulahad 1 , Salih A. Derbaz 2 1,2 Department of Mathematics, Faculty of Science, University of Zakho,Duhok, Kurdistan Region, Iraq Abstract: - In this paper, we consider the thinning process of an inclined thin liquid film over a solid boundary with an inclination angle to the horizontal in gravity driven flow. Throughout this work, we assumed that the fluid thickness is constant far behind the front and we neglect the thickness of the film at the beginning of the motion. The differential equation of the film thickness is obtained analytically and the solution of equation that represents the film thickness is obtained numerically by using Rung-Kutta method. Keywords: - Thin Liquid Films , Navier-Stokes equations , continuity equation . I. INTRODUCTION We present here some of the theoretical aspects of the instability development in an inclined thin liquid films on a solid surface in two dimensional coordinate system with an inclination angle  to the horizontal . There are different types of phenomena that can occur, such as drainage, details of rupture, non –Newtonian surface properties in moving contract lines in thin liquid films [1]. These phenomena can help to describe the physical processes that occur in our real world. [2] have studied the case of contact line instabilities of thin liquid films but with constant flux configuration and also they considered some global models of a moving contact lines. [3] studied the thin liquid films flowing down the inverted substrate in three dimensional flow. [4] investigated the dynamics of an inclined thin liquid films of variable thickness in steady and unsteady cases and when the film is stationary and uniform. [5] considered the stability of thin liquid films and sessile droplets. The stability of the contact line of thin fluid film flows with constant flux configuration is considered by [6]. [7] considered the spreading of thin liquid films with small surface tension in the case when the flow is unsteady. [8] have studied the drainage of thin liquid films on an inclined solid surfacefor unsteadynflow by using similarity solution. In this paper we investigate the drainage of the inclined thin liquid films. The solution of the governing equations of the liquid film thickness is obtained numerically. II. GOVERNING EQUATIONS: Let   w u q ,  denotes the fluid velocity, where u and w are the velocity components in x and z directions respectively. Let   t x h z ,  be the equation of the inclined thin liquid films as shown in Figure (1) and the flow is in x direction. The continuity equation is given by: 0       z w x u (1) and from the incompressibility condition, we have x u z w     (2) and this insures that x u   is a function of x only. The Navier-Stokes equations in x and z directions respectively for an inclined thin liquid film are given by:     sin 2 2 2 2 g z u x u x P z u w x u u                               (3) And     cos 2 2 2 2 g z w x w z P z w w x w u                               (4) where  ,  and P are the density, viscosity of fluid and P the pressure. In lubrication theory the inertia terms can be neglected and the Navier-Stokes equations (3) and (4) become    sin 2 2 2 2 g z u x u x P                  (5)