Unsteady Flow near the Junction Zone of Three Liquids

119 Unsteady Flow near the Junction Zone of Three Liquids Joseph G. Abdulahad Rutayna J. Eisa College of Education College of Computer Sciences University of Dohuk and Mathematics University of Mosul Received on: 13/9/2009 Accepted on: 16/9/2010 ABSTRACT In this paper we consider the motion near the junction zone of three approximately plane liquid films of semi–infinite extent in two dimensional polar coordinate system with negligible inertia. We use Stokes equation to describe such flow. The pressure in the region of large curvature is less than that on the surface of bulk fluid and this pressure gradient ensures that this problem is unsteady state case. The equation that governs such flow is solved analytically, the shape and the thickness are determined for some liquids.


Introduction:
Thin liquid films appear in many contexts such as the cooling of gas turbine blade tips, rocket engines. Apart from these direct cooling applications of thin liquid layers, thin films form a crucial element in many other applications such as in industrial coating and spinning processes. Homsy (2000), studied the slow motion of a thin viscous film flowing over a topographical feature under the action of external forces, using the lubrication approximation and he obtained an equation of the free surface in time and space. Breward The main object of this paper is to study the mechanics of the junction zone of three plane films as shown in Figure (1). An effect of surface tension is necessarily to cause a continual thickening of the films in the junction zone, with the additional liquid being supplied symmetrically by all the three films.
The hydrostatic pressure of the junction zone (called border which is a region of large curvature) is less than that on the surface of bulk fluid, further up the liquid, the pressure is much higher than that at the border and so the pressure gradient inside the liquid film which forces the flow of liquid towards the border and this pressure gradient ensures that there must be some flow and this cannot represent a steady state situation.

Formulation and governing equations:
We consider the flow of viscous liquid within the film in two dimensions and we suppose that there is no inertia the uses the stokes equations to describe the fluid motion in junction zone of three films in vector form as where  , , P q and  are the velocity, pressure, density and viscosity of fluid respectively,  and  are the gradient and the laplacian operator.
The solution of equation (1) is equivalent to the solution of the following biharmonic equation where the stream function is related to the velocity components by Equation (3) can be written in polar coordinates and after simplifications to give: Now, using (5) and (6) into equation (4), and after simplifications, we get By using equation (6), equation (7) gives The constants in (21) can be determined by using the initial and boundary condition. The boundedness at the origin requires that the constants 0 D and 0 C must be vanished, that is 0 , and the solution now is given by where  is the surface tension and k is the curvature, the unit tangent and unit normal vectors at the free surface is give by r denote the natural orthonormal basis of the coordinate system, the curvature k is given by  Now the stream tensor given by (28) and by using the condition (29) and (30) is then become

Conclusions:
The flow of a liquid in the junction zone of these approximately plane liquid films is essentially unsteady initial value problem and the solution controls the life span of foam. The shape of three surfaces is determined for some liquids, namely for water, mercury and glycerin and it is seen from figure (2) and (3) that at time increases, the thickness of the liquid film also increases and when the supply of liquid is exhausted, that component ruptures. Furthermore, the thickness of liquid film in water is less than that of glycerin and the reason may be related to viscosity of the liquid.