The Dynamics of Thin Liquid Film

The dynamics of the thin layer which flows steadily between two vertical guide wires was investigated but with zero shear stress at their bounding surfaces where the gravity has no significant effect on the liquid film. We apply the Navier-Stokes equations in two dimensional steady flows for incompressible fluid to a falling liquid curtain and we present the derivation of the differential equation that governs such flow and we obtain a solution for these equations which is valid for this liquid curtain , where we restrict our works to the case where the domain under consideration is long and thin, the solution of the governing equation is obtained by analytical method, and in this case there is a critical solution 2 / 1 ) 2 ( ) (    c g for large  when the parameter  is equal to zero, where 2 2 2 0 H R P    and which is identical to the case when the normalized pressure 0 p is equal to zero. Generally, we solve the equation when 0 p is not equal to zero, and the thickness of the film increases as  increases where H 2     .

and which is identical to the case when the normalized pressure 0 p is equal to zero. Generally, we solve the equation when 0 p is not equal to zero, and the thickness of the film increases as  increases where

Introduction
The dynamic of a thin liquid film flowing steadily between two vertical guide wires, where the effect of surface curvature is taken into account, is investigated. The Navier -stokes equations integrated over the film thickness and an approximate non-linear differential equation is obtained by neglecting the higher order terms with respect the thickness of the thin liquid films and the results are compared with Cyrus (Cyrus,1987) works who neglects the effect of surface curvature and also results are compared with the experimental measurements of Brown (Brown, 1961).
The objective of the present analysis is to apply the Navier-Stokes equation to a falling liquid curtain, and present the derivation of the differential equations that governs the flow of the liquid curtain and to obtain a solution of these equations which is valid for thin liquid film.
In addition to engineering applications, a solution will serve as a first order approximation of the velocity profile. The domain of validity for each equation is established by comparing the numerical solution with the experimental results of Brown (Brown, 1961) Equations of steady motion in films with zero shear-stress at their bounding surface.
To describe the flow of a viscous fluid within a symmetric film in two dimensions, the Cartesian coordinates x and y are taken in which the xaxis is the axis of symmetry, and the flow is predominantly in the xdirection. In figure (1)  Normally in thin liquid films, the film thickness is much smaller than the width as (Rutayna, 2005), and therefore we assume two-dimensional incompressible flow.
The steady two dimensional incompressible fluid flows governed by the following equation of motion:

1-Continuity equation:
We can express the connection between area and velocity in an equation, called the equation of continuity. The continuity equation in a differential form for two dimensional incompressible flows has the form (Stokes,1945) For steady flow from Stoke's (Stokes,1945) the momentum equation has the form: where xx  , xy  and yy  are the components of the total stress tensor given in the standard notation, where the stress tensor and then for incompressible flow, these stresses has the following forms: where  is the coefficient of viscosity of liquid, and u is component velocity, the density  is assumed to be constant throughout the process. Let represent the thickness of the liquid film at a point x (see equation (1)). We define the equation of the free surface of the film by the function over the domain x under consideration. However we do not impose any geometrical constraint on the total variation of ) (x h over this domain. In these respects, the theory retains many of the features of standard theories of fluid flow in thin domains: namely mathematical hydraulics in an experiments, Lubrication theory and boundary layer theory. The appropriate mode of incorporating the condition eq.(7) into the theory appears to be as follows.
We consider the class of steady flows in which the asymptotic conditions as The boundary of the film is a streamline, and therefore the substantial Thus results in the following boundary conditions: From eq. (2), we have For the balance of the surface forces on the boundary, the Cartesian components of the unit normal vector n are needed which are given by: The curvature of the liquid film is given by (Rutayna, 2005) and since we restrict attention to the case given by eq.(7), the curvature eq.(12) can be simplified by since The surface tension  , creates a stress on the free surface of the liquid film, following (Stokes,1945), the balance of the surface forces on the free surface, is given by where  is the surface tension, using equations (6) for incompressible flow (14) and (15) are gives respectively as follows: (17) Now we decompose the velocity u and the normal stress here  is related to . , and the function u and xx  are weakly dependent on y , when two variables have decomposition slices that contain statements in common, the variable are said to be weakly dependent.
With the decomposition equations (16) and (17), the continuity equation (2) can be integrated over the film thickness to give Since the liquid film is symmetric as (Rutayna, 2005), then and then equation (20) gives substituting the boundary condition eq.(9) in the equation (21), we get Now, using the decomposition eq.(18) of u , to obtain where Q is a constant representing the volumetric mass flow and 0 u is, therefore, the average velocity defined by Restricts to the first order, we get    Similarly the integral over the film thickness of the right hand side of equation (3) gives, To the first order approximation, we get ) ( Now from equation (14), Using eq.(11), the following can be obtained . … (32) Therefore the integral of the x-component of the momentum equation over the film thickness can now be written as: ...(34) Using equation (5), we can express the y-component of equation (15) by Now if the magnitude of x and u are assumed to be  (41) for the x-component of the normal stress tensor in equation (34) Using equation (12) into equation (42), the following can be obtained: Thus from continuity equation (2), we get Furthermore equations (39) and (45), gives the pressure ) ... (47) In this section, we obtain the dimensional non-linear equation (43) such governs the thin liquid films on the vertical guide wires.

2-Dimensional analysis:
We now introduce non-dimensional variables as follows: where  is a non-dimensional variable and ) ( f is a non-dimensional function, x and ) (x h are dimensional variables, function respectively and R is Reynolds number. Using equation (45), equation (43), reduces to Now from the transformation eq.(48) and eq.(49), we have with the transformations eq.(48) and eq.(49) become where  is an arbitrary constant and 0 ) (   g , and the differential equation (52), then gives The root m in (58) represents a balance among viscosity, inertia and pressure, the effect of surface tension being negligible. A part from special circumstances, in which the amplitude of the solution representing a viscosity-inertia is zero, the inertia term is almost always not negligible as the film approaches a condition of asymptotic uniform thickness and the importance of inertia, however, small is the Reynolds number When ) ( f departs from its asymptotic value of unity to an appreciable extent, the complete differential equation (52) must be satisfied. Now the variable  does not appear explicitly in equation (52), so that its differential order can be reduced by one by the following transformations: Some of the solution curves of equation (68)

Conclusion
The dynamics of a free-surface liquid film is very useful in industrial coating and spinning processes. The solution of the thickness of the liquid film is determined for different values of the parameter  and from the solution curves it seems that the thickness or the liquid film increases as the parameter  increases for a thin liquid film flowing steadily between two vertical guide wires where we consider two cases.  g In the first case the gravity is taken to be zero and in the second case when gravity is taken into account, the two cases are considered to see what the significant effect of gravity is.