In this paper, we considerthe unsteady flow within a double-sided symmetric thin liquid film with negligible inertia. We apply the Navier-Stokes equations in two dimensional flows for incompressible fluid.The similarity method is used in which the explicit time dependence can be isolated and thus the shape of the film will depend on one variable only.The differential equation of the film thickness is obtained analytically and the solution of equation thatrepresents the film thickness is obtained numerically by using Rung-Kutta method. This paper is concerned with a dynamical system in such a way that no solid boundaries exist.It is more convened to assume that any solid boundaries which exist are at sufficient distance to produce or have no significant direct mechanical effect on that part of the system which is under consideration that are imposed in this work.The purpose of studying such liquid systems is to determine the global effect of varying the boundary conditions at liquid surfaces, in cases where the physical relevance of these boundary conditions appears to be a matter of doubt. However, in the case of motion in thin films, for instance, it has been suggested (1) that the surfaces of the thin film behave more like inextensible solids rather than ordinary fluid surfaces. (2),considered the thinning process of an inclined thin liquid film over a solid boundary with an inclination angle  to the horizontal in gravity driven flow. They assumed that the fluid thickness is constant far behind the front and they neglected the thickness of the film at the beginning of the motion.The drainage of thin liquid films on an inclined solid surface is considered by (3), and the equation that governs the film thickness is obtained analytically by using the similarity method.The dynamic rupture process of a thin liquid film on a cylinder is investigated numerically; a nonlinear differential equation that describes the long-wave evolution of the interface shape. The tendency of acceleration becomes more explicit in the case of large surface tension (4). The contact line induced instabilities for a thin film of fluid under destabilizing gravitational force in three dimensional setting. The instabilities in the setting vary in the transverse direction. It is argued that the flow pattern strongly depends on the inclination angle and the viscosity gradient (5). A mathematical model is constructed by (6) to describe a two dimensional flow for an inclined thin liquid films with an inclination angle α to the horizontal under the action of gravity. An asymptotic analysis is employed by using the lubrication approximation.

https://iosrjournals.org/iosr-jm/papers/Vol11-issue3/Version-3/F011333846.pdf

Unsteady Flow in a Double-Sided Symmetric Thin Liquid Films

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.

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Steady Flow of Thin Liquid Films on an Inclined Solid surface

filed of the component two dimensi the velocity In differe without ext steady incom Fig. ρ u + and ρ u + The velo u = u(x) v = 0 w = −z p = p(x) which satisf + = since the slo distribution surface of fil 1. Shear τ = μ where μ is gradient an the surface o 2. Norma T = − Now fro have: =, Abdulah fluid, wher in x and z d onal motion and pressure ntial form, t ernal forces pressible flo 1: Crosssecti w = − w = − city distribut ies the incom 0. pe is so sm satisfies th m z = h(x), w stress condit = 0, the viscosity d the subscr f the film. l stress cond p + 2μ . m the cont ad et al / Interna e u and w a irections re s of the liquid distribution he Navier–S in x and z w are given on of a symme + μ + + μ + ion for the fl pressibility all for thin e stress co hich are: ion: of liquid ibe s denote ition: inuity equat tional Journal of re the veloc spectively. , we determ as follow: tokes equatio directions by: tric film , (2.1 . (2.1 ow is given b (2.1 condition, (2.1 liquid films, ndition at (2.1 is the veloc s the values (2.1 ion (2.1.4), (2.1 Advanced and A 103 ity For ine ns for .1) .2) y: .3) .4) the the .5) ity at .6) we .7) sub giv eq als ten sur wh fun wi fro eq giv eq we wh pplied Sciences, 3 stituting it i es: T = −p − The curvatu k = 1 + Since, ≪ uation (2.1.9 k = o, on the su sionσ, the no T = σ . From equat face of the fi p = −2μ ich holds ev ctions of x th respect to = −2μ m the velo uation of mo e: ρ u = − Equations (2 ρ u = σ From the v uation of mo = z ρu Integrating get: p = ρu ich can be w p = p(x) + o (8) 2016, Pages: nto equation 2μ . re of the liqu . 1, then ) reduces to g rface of the rmal compo ions (2.1.8) lm, we have: − σ , erywhere. N only. Differe x, we get: − σ , city distribu tion (2.1.1) fo + μ . .1.12) and (2 + 3μ elocity distr tion (2.1.2), g − ρ − equation (2 − ρ − ritten in the (z ) , |z| : 102‐107 (2.1.4), equ id film is giv can be ne ive: liquid film nent of stres and (2.1.10) otice that p, ntiate equat tions, the r steady flo .1.14), give: . ibutions, th ives, μ . .1.17) with r μ + g(x form: h ation (2.1.6) (2.1.8) en by: (2.1.9) glected and (2.1.10) with surface s is given by, (2.1.11) and on the (2.1.12) u and h are ion (2.1.12) (2.1.13) longitudinal w reduces to (2.1.14) (2.1.16) e transverse (2.1.17) espect to z, ), (2.1.18) Abdulahad et al / International Journal of Advanced and Applied Sciences, 3(8) 2016, Pages: 102‐107 104 This is relevant only to higher order approximation. Since z = h(x) is a free surface of the liquid film, then the conservation of the mass across the film thickness of the film is therefore given by: uh = Q, (2.1.19) where Q is any constant. Thus, equations (2.1.16) and (2.1.19) are the governing equations of and within the liquid film. 3. Flows with negligible inertia The governing equation (2.1.16) with negligible inertia reduces to give: + = 0. (2.2.1) From equation (2.2.1), the only material constant that is relevant is therefore: V = . (2.2.2) We can determine the value of the parameter V for some liquids as shown in the following table. Every solution of equations (2.1.16) and (2.1.19) must be assessed with respect to full equations given in the previous section. In particular, for infinitesimal perturbations as a uniform film and from Table 1, the analysis shows that inertia can never be neglected for mercury; can only marginally be neglected in water and carbon tetrachloride; and can always be neglected in thin films of glycerin, linseed oil and olive oil. From equations (2.2.1) and (2.2.2), we have: + = 0 . (2.2.3) Integrating equation (2.2.3) with respect to x twice, we get: + u = Ax + B (2.2.4) where, A and Bare arbitrary constants and can be found from asymptotic or initial conditions. Table 1: The value of the parameter V for some liquids Liquid density ρ,/cm Surface tension σ, m/sec viscosity μ, g/cm. sec Velocity V, cm/sec Water 0.998 72.97 0.0113 2152.5074 Mercury 13.55 510.76 0.0115 10984.086 Glycerin 1.26 62.75 14.9 1.4038 Carbon Tetrachloride 1.59 26.27 0.00974 899.0418 Linseed oil 0.94 33.57 0.4309 25.9698 Olive oil 0.91 33.56 0.8379 13.3508 From equation (2.1.19), equation (2.2.4) reduces to give: + = Ax + B. (2.2.5) Equation (2.2.5) is related to lubrication theory, but in the absence of surface –active solutes, it seems to be a degenerate relationship; since the shear stress at the edge of the film is then zero, and this ensures that the velocity distribution across the film is uniform, not parabolic. Now equation (2.2.5) gives the following two cases: Case I: If A ≠ 0, we can write Ax + B = Dx and thus equation (2.2.5), then becomes: + = Dx. (2.2.6) Case II: If A = 0, then equation (2.2.5), gives: + = B. (2.2.7) 4. Non‐dimensional analysis For non-trivial solution of equation (2.2.6), we define the following nondimensional parameters for case I as follows: x = η h(x) = f(η) , (2.3.8) and equation (2.2.6), reduces to give: f(η) − ηf(η) = −1. (2.3.9) The following analysis follows from the locus of term of the function of f (η) which is the critical solution of equation (2.3.9). Note that we use ODE45 and plot commands in MATLAB to solve (2.3.9) and all the curves in this paper. The locus of points at which = 0 may be written from equation (2.3.9), as: f(η) = . (2.3.10) Some of the solution curves are obtained for equation (2.3.9) in(η, f(η))-plane. Within the class of the solutions there is a critical solution shown in Fig. Abdulahad et al / International Journal of Advanced and Applied Sciences, 3(8) 2016, Pages: 102‐107 105 2 which divided the region of the definition into two sub regions. The behavior is as follows: f (η) ~ η as η → −∞, (2.3.11) and f (η) ~ as η → ∞. (2.3.12) Now asη → −∞, all the films have constant curvature, and f (η) may be related to the supply of fluid to thin films from a plateau border (a region of large curvature). Note that no solution intersects f (η), and all the solution curves which lie above the critical solution f (η) , and have the asymptotic behavior: f(η) = η as η → ±∞. (2.3.13) The behavior (2.3.13) corresponds to the behavior: h(x) = k x as x → ±∞. (2.3.14) In (x, h(x))-plane, where k is arbitrary constant which labels each solution, so that all of these solutions describe the transition from a film of uniform thickness to one of constant curvature. All the solution curves which lie below the critical solution f (η) in Fig. 2 have the asymptotic behavior: f (η) ~ η as η → ∞, (2.3.15) but f (η ) = 0 with u → ∞ . The behavior of (2.3.18) corresponds to: h(x)~ k x as x → −∞, (2.3.16) but h(x)=0 for all values of x with u(x ) = ∞, and this represents a film which terminates in a sink of fluid at x = x , where the thin film approximation breaks down. In (x, h(x))-plane Fig. 3 shown the thickness of the film for different liquids namely: Mercury, Carbon Tetrachloride, Water, Glycerin, Linseed oil and Olive oil. The comparison of the thickness of films for some liquids is shown in Fig. 4 and 5. For non-trivial solution of equation (2.2.7), we define the following nondimensional parameters for case II as follows: x = η h(x) = f(η) , (2.3.17) and thus equation (2.3.7), reduces to give ( ) + = 1, (2.3.18) or 1 + ( ) = 1. (2.3.19) Fig. 2: Solution curves of equation (2.3.9) in (η, f(η)) plane. Mercury Glycerin Linseed oil Olive oil Fig. 3: Solution curves of equation (2.3.9) in (x, h(x)) plane, for different liquids. Fig. 4: Comparison between the film thickness of carbon tetrachloride and water Again we use ODE45 command in MATLAB to solve (2.3.18). Equation (2.3.19) is the governing equation for case II. Integrating equation (2.3.19) with respect to η, we get: -3 -2 -1 0 1 2 3 0 1 2 3 4 5 6 7 8

Steady flow of horizontal double‐sided symmetric thin liquid films

Unsteady Flow in a Horizontal Double-Sided Symmetric Thin Liquid Films

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 equation of the film thickness is obtained analytically, using the similarity method by which we can isolate the explicit time dependence and then the shape of the film will depend on one variable only. The solution of the governing equations of the film thickness is obtained numerically by the Rung-Kutta method with the aid of Mat lab(ode45). 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. (2,3) 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. (4) studied the thin liquid films flowing down the inverted substrate in three dimensional flow. (5) 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. (6) 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 (7). (8) considered the spreading of thin liquid films with small surface tension in the case when the flow is unsteady. The Non-linear analysis of creeping flow on the inclined permeable substrate plane subjected to an electric field was considered by (9). In this paper we investigate the drainage of the inclined thin liquid films where the gravity and other forces such as viscous and surface tension forces have a significant effect on the flow of the film. We use the similarity method by which we can isolate the explicit time dependence and then the shape of the film will depend on one variable only. The solution of the governing equations of the liquid film thickness is obtained numerically.

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Joseph G. Abdulahad

Papers

The Drainage of Thin Liquid Films on an Inclined Solid Surface

Steady Flow of Thin Liquid Films on an Inclined Solid surface