Abstract
The two-dimensional spreading under gravity of a thin fluid film with suction (fluid leak-off) or blowing (fluid injection) at the base is considered. The thin fluid film approximation is imposed. The height of the thin film satisfies a nonlinear diffusion equation with a source/sink term. The Lie point symmetries of the nonlinear diffusion equation are derived and exist, which provided the fluid velocity at the base, vn satisfies a first order linear partial differential equation. The general form has algebraic time dependence while a special case has exponential time dependence. The solution in which vn is proportional to the height of the thin film is studied. The width of the base always increases with time even for suction while the height decreases with time for sufficiently weak blowing. The streamlines of the fluid flow inside the thin film are plotted by first solving a cubic equation. For sufficiently weak blowing there is a dividing streamline, emanating from the stagnation point on the centre line which separates the fluid flow into two regions, a lower region consisting of rising fluid and dominated by fluid injection at the base and an upper region consisting of descending fluid and dominated by spreading due to gravity. For sufficiently strong blowing the lower region expands to completely fill the whole thin film.
Highlights
Thin fluid films occur in nature and they play a significant role in many technological processes such as in coating applications and in the spreading of paints
For sufficiently weak blowing there is a dividing streamline, emanating from the stagnation point on the centre line which separates the fluid flow into two regions, a lower region consisting of rising fluid and dominated by fluid injection at the base and an upper region consisting of descending fluid and dominated by spreading due to gravity
The thin fluid film equations were derived and it was shown how the characteristic fluid pressure could be determined from the dimensionless partial differential equations and how the characteristic horizontal fluid velocity could be determined from the two expressions obtained for the characteristic fluid pressure
Summary
Thin fluid films occur in nature and they play a significant role in many technological processes such as in coating applications and in the spreading of paints. The spreading of an axisymmetric liquid drop with fluid injection or suction at the base has been investigated by Mason and Momoniat [12] who later extended their work to include surface tension [13]. Momoniat et al [14] investigated the effect of slot injection or suction on the spreading of a thin film where the surface tension gradient and gravity effects are included They investigated the behaviour of the streamlines and how they evolve with suction or blowing. Nomenclature height of thin fluid film half-width of thin fluid film fluid pressure in thin fluid film Reynolds number of thin fluid film flow fluid injection/leak-off velocity total volume of thin fluid film per unit breadth Lie point symmetry x w(t ) , scaled similarity variable z h (t, 0) , scaled similarity variable invariant solution for h (t, x) invariant solution for vn (t, x) constant of proportionality in equation g (η ) = β f (η )
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