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

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 of rising fluid and an upper region of descending fluid.


Introduction
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. We derived and analysed the plots of the streamlines to understand the fluid flow in the thin film on which the evolution of the surface profile depended. The system of equations was closed by making the assumption that the normal component of the fluid velocity at the base, v n , 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. In this model, the magnitude of the suction and blowing is greatest near the centre line of the thin film because the height is greatest on the centre line and is least near the moving contact lines where the height vanishes.
In this paper, we will assume that v n is proportional to the spatial gradient of the height. An analytical solution for the surface profile can again be derived. We will plot the streamlines and investigate their properties and compare them with the results obtained in [1]. In this model, the magnitude of the suction and blowing is least near the centre line on which the spatial gradient vanishes and is greatest near the moving contact lines on which the magnitude of the spatial gradient is infinite.
Investigations of the evolution of a thin fluid film with a free surface generally concentrate on the surface profile [2] [3] [4]. In this paper, we investigate the properties and evolution of the thin film by deriving and analysing the streamlines of the fluid flow in the thin film.
Mason and Momoniat [5] made the assumption that v n 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. It can be used to analyse the evolution of a thin fluid film from a point or line source of fluid and from a fluid film with an initial non-zero half-width as considered here. A literature review of the relevant papers of thin fluid film theory and of the application of Lie group analysis to thin fluid films was given in [1].
An outline of the paper is as follows. In Section 2 a summary is given of the

Summary of Results
The fluid is viscous and incompressible. The spreading of the thin fluid film is two-dimensional as shown in Figure The balance law for fluid volume is where ( ) V t is the total volume of the thin film per unit length in the y-direction: An invariant solution is derived using Lie group analysis for differential equa- Using the Lie point symmetry (2.8) the nonlinear diffusion Equation (2.3) is reduced to an ordinary differential equation. There are two cases, the general case, 2 0 c ≠ and the special case, 2 0 c = .
When 2 0 c ≠ the invariant solution is obtained by solving the ordinary dif- The Lie point symmetry which generates the invariant solution is ( ) where 0 V is given by (2.19) and η is given by (2.11). The Lie point symmetry which generates the invariant solution is ( ) and the streamlines at time t are the curves where ( ) k t is given a range of discrete values. In the ( ) , x z -plane the streamlines at time t are obtained by solving numerically for z as a function of x the cubic equation [1] ( ) ( ) ( ) ( ) for a range of discrete values of ( ) , d .
We will also consider the streamlines in the ( ) In order to close the systems of equations and complete the formulation of the problem an assumption needs to be made on ( ) , n v t x . In [1] it was assumed that ( ) , n v t x is proportional to ( ) , h t x . In this paper it will be assumed that The differential Equation (2.10) with α given by (3.4) becomes which can now be integrated subject to the boundary conditions (2.12) and (2.13). We obtain and therefore We introduce the new parameter β defined by The factor 2 5 is a normalisation factor to give 1 β = − for the special case The normal fluid velocity at the base has the same sign as β and From (3.10), (3.11) and (3.14) We see that the physical significance of the solution is that Consider now the special case, 2 0 c = . Assuming (3.1), the balance law (2.22) becomes ( ) and therefore * 5 3

17) Journal of Applied Mathematics and Physics
Integrating and imposing the boundary conditions we obtain Hence from (2.23) to (2.26), where 0 V is given by (3.13).

Analysis of the Solution
We will determine the evolution in time of the fluid variables, especially ( )

Time Evolution of the Fluid Variables
Consider first the thin fluid film approximation [1]. From (3.11) and (3.14) for 1 and from (3.19) and (3.21) for 1 Since it is assumed that the thin fluid film approximation is satisfied initially it will remain satisfied for all time for is satisfied for the whole range of blowing, 0 β < ≤ ∞ , which also has the most interesting streamlines. The spreading due to gravity is the dominant mechanism and is stronger than the effect of blowing on the height. The results are summarised in Table 2.
Consider next the half-width of the base, ( ) w t , for the range in which the solution applies. For this range, ( ) w t is given by (3.11). For  Table 2.
. The thin fluid film approximation is satisfied for all time only for We see that even for blowing when 0 β < ≤ ∞ . The behaviour of ( ) ,0 h t as t evolves for the full range, β −∞ < < ∞ , is given in Table 2.
The evolution of ( ) V t is summarised in Table 2. The time evolution of ( ) , n v t x for a fixed value of η in the range 0 1 η < < is given in Table 3. Also given in Table 3 is the time evolution of ( )

Fluid Velocity on Centre Line
On the centre line, and therefore For points in the thin film on the centre line, The time evolution of ( ) Table 3.

Streamlines
The tangent vector to a streamline is everywhere parallel to the fluid velocity vector instantaneously. The streamlines in the ( ) 1 We will mainly consider the streamlines in the ( ) The coefficients    Table 2 for the evolution of ( ) w t and ( ) ,0 h t . In Figure 2  ,0 h t , will decrease steadily with time and the thin fluid film will disappear as a line sink as t → ∞ . In Figure 2 The half-width of the base, ( ) w t , will increase steadily with time and the maximum height, ( ) ,0 h t , will decrease steadily with time. The thin fluid film will spread over the whole plane 0 z = with decreasing thickness and disappear as t → ∞ . In Figure 3 fluid injection (blowing) is considered and β lies in the range 0 β The half-with of the base will therefore increase steadily with time and the maximum height of the thin film will decrease steadily with time. The angle which the dividing streamline makes with the base at the stagnation point will therefore decrease steadily with time.
The evolution of the streamline pattern with time is illustrated in Figure 4 where the streamlines are plotted in the ( ) , x z -plane at time β − ≤ ≤ ∞ for which the thin fluid film approximation is satisfied for all time. In the next section an approximate analytical solution for the dividing streamline will be derived and its dependence on β and its evolution with time will be determined.

Dividing Streamline
We will derive an approximate solution for the dividing streamline in the ( )   , On the dividing streamline, 0 1 η ≤ < . We perform an expansion in powers of 2 η and derive a solution for blowing, 0 β > , to first order in 2 η . No assumption is placed on β or t. We find that this gives a sufficiently accurate result. We see from Figure 3 that as β increases the dividing streamline moves closer to the centre line and the maximum value 2 ζ on the dividing streamline decreases. The solution should therefore be more accurate as β increases. Now from (6.5) and (6.6), The expansion of the cubic equation in powers of 2 η is therefore and from (6.3), ( ) The discriminant, ∆ , of the cubic Equation (6.9) is [7] ( ) The solution for the dividing streamline is the stagnation point (0, 0).
For the solution to first order in 2 η , 0 ∆ < since 0 β > and the cubic equation has three real and distinct roots [7]. We use the trigonometric method of solution described in [1]. The three roots of (6.9) are ( ) Thus from (6.7) and (6.10), ( ) Thus for first order in 2 η , φ will be small. Using the series expansion for cosine, (6.14) becomes The dividing streamline must pass through the stagnation point (0, 0) and satisfy  As β increases and the strength of the blowing increases, P η decreases, verifying that the dividing streamline moves closer to the centre line. The product 1 2 P β η occurs in the expansions (6.27) and (6.28) which are valid because 1 2 P η β is of order of magnitude unity. The approximation correct to order 2 η is obtained by solving the quadratic Equation (6.32). The positive root of (6.32) can be expressed as  10 tan 1 1 .
The asymptotic solution for large values of β is ( ) where P η is given by (6.35) and ( ) ( )   . We see clearly that as t increases the angle ( ) t θ that the dividing streamline makes with the x-axis steadily decreases because spreading is stronger than blowing near the centre line. In Figure 6 the approximate solution, (6.45) and (6.46), for the dividing streamline is compared with the numerical solution of (5.1) with The approximate solution is more accurate as β increases. The approximate solution (6.43) for ( ) , p P x z is also compared with the numerical solution and its accuracy also increases as β increases. The properties of the fluid flow in this study and in [1] where n v is proportional to ( ) , h t x can be expected to be present in other models for suction and blowing. If more realistic models cannot be solved analytically these properties will be a useful guide in a numerical solution. In both studies, there is a range of suction and blowing for which the thin fluid film approximation is satisfied and outside this range, the solutions are not physically acceptable. In both models a dividing streamline exists, starting at a stagnation point, which separates the flow into two regions, a lower region at the base consisting of rising fluid and an upper region consisting mainly of descending fluid. Both models have fluid variables which tend to zero or infinity in a finite time. The transition solution to an infinite limiting time is an exponential solution.
For the study presented here the thin fluid film approximation was satisfied for 4 5 β − ≤ ≤ ∞ , that is, for moderate to weak suction and the whole range of blowing. The exponential solution, generated by a special case of the Lie point symmetry, and the solutions which tended to zero or infinity in a finite time, lie outside this range. The study complemented the investigation in [1] for which the thin fluid film approximation was satisfied for 2 β −∞ < ≤ , that is, for the whole of suction and for weak to moderate blowing. There was unexpected behaviour in the evolution of ( ) w t and ( ) ,0 h t caused by the relative importance of spreading due to gravity and suction or blowing. The base half-width ( ) w t → ∞ for all blowing and even for weak suction for which decreases as β increases. Because blowing is weakest at the centre line and strongest at the moving contact lines the dividing streamline generated an approximate V-shaped surface along the length of the two-dimensional thin fluid film, with vertex of each section at a stagnation point. This compares with the dividing streamline in [1] which also passed through a stagnation point and generated an approximately horizontal surface that dipped down at the moving contact lines. Journal of Applied Mathematics and Physics Further research can be undertaken. Surface tension was neglected and there was no slip at the fluid/base interface. The streamlines in a thin fluid film with surface tension or slip could be investigated to give a better understanding of the fluid film flow inside the thin film.