Streamlines in the Two-Dimensional Spreading of a Thin Fluid Film: Blowing and Suction Velocity Proportional to the Spatial Gradient of the Height

摘要

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.