Abstract
The aim of this investigation is to determine the effect of fluid leak-off (suction) and fluid injection (blowing) at the horizontal base on the two-dimensional spreading under the gravity of a thin film of viscous incompressible fluid by studying the evolution of the streamlines in the thin film. It is assumed that the normal component of the fluid velocity at the base is proportional to the spatial gradient of the height of the film. Lie symmetry methods for partial differential equations are applied. The invariant solution for the surface profile is derived. It is found that the thin fluid film approximation is satisfied for weak to moderate leak-off and for the whole range of fluid injection. The streamlines are derived and plotted by solving a cubic equation numerically. For fluid injection, there is a dividing streamline originating at the stagnation point at the base which separates the flow into two regions, a lower region consisting mainly of rising fluid and an upper region consisting mainly of descending fluid. An approximate analytical solution for the dividing streamline is derived. It generates an approximate V-shaped surface along the length of the two-dimensional film with the vertex of each section the stagnation point. It is concluded that the fluid flow inside the thin film can be visualised by plotting the streamlines. Other models relating the fluid velocity at the base to the height of the thin film can be expected to contain a dividing streamline originating at a stagnation point and dividing the flow into a lower region of rising fluid and an upper region of descending fluid.
Highlights
The aim of this investigation is to determine the effect of fluid leak-off and fluid injection at the horizontal base on the two-dimensional spreading under the gravity of a thin film of viscous incompressible fluid by studying the evolution of the streamlines in the thin film
There is a dividing streamline originating at the stagnation point at the base which separates the flow into two regions, a lower region consisting mainly of rising fluid and an upper region consisting mainly of descending fluid
In a recent paper [1] we investigated the effect of fluid leak-off and fluid injection at the horizontal base on the two-dimensional spreading under the gravity of a thin film of viscous incompressible fluid
Summary
In a recent paper [1] we investigated the effect of fluid leak-off (suction) and fluid injection (blowing) at the horizontal base on the two-dimensional spreading under the gravity of a thin film of viscous incompressible fluid. The system of equations was closed by making the assumption that the normal component of the fluid velocity at the base, vn, is proportional to the height of the thin film at that point With this assumption, an analytical solution for the surface profile could be derived and the streamlines were plotted by solving a cubic equation numerically. Mason and Momoniat [5] made the assumption that vn is proportional to the spatial gradient of the height in the axisymmetric spreading under the gravity of a thin liquid drop with suction and blowing at the base They did not investigate the streamlines in the liquid drop. We will use Lie group analysis of differential equations to reduce the nonlinear diffusion equation to an ordinary differential equation and derive an analytical solution This is a powerful and systematic method that has been applied successfully in other investigations in thin fluid film theory.
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