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

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, n v 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 n v 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.


Introduction
Thin fluid films occur in nature and they also play a significant role in many technological processes such as in coating applications and in the spreading of paints. There are important applications of thin fluid films to the lung [1] and the eye [2]. Reviews of the research up to the year 2000 on the spreading of thin fluid films and liquid drops have been given by Oron et al. [3] and by Davis [4].
Thin fluid flows usually occur under the action of gravity [5], surface tension gradients [4] or on a rotating substrate [6] [7]. The moving contact line at the boundary of the thin fluid film has been studied by O'Brien and Schwartz [8] and by Hocking [9]. Davis and Hocking [10] [11] have considered the spreading of a liquid drop on a porous base. 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].
A study of the streamlines of the flow in a thin film can illustrate how the thin fluid film spreads. 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 also investigated the behaviour of the streamlines and how they evolve with suction or blowing. They found that a breakup of the streamlines occurs. Mason and Chung [15] considered the spreading of a thin liquid drop with slip at the base but without fluid injection or suction at the base. The streamlines made clear the properties of the flow. It was found that vortices can occur at the base and close to the axis of the liquid drop. The motivation for this research is to obtain an improvement on our understanding of the fluid flow inside a spreading thin fluid film with suction or blowing at the base by analysing the streamlines of the flow.
In general a nonlinear diffusion equation for the height of the thin film occurs in the mathematical formulation of thin fluid film flows. Exact analytical solutions for these equations have been found in the form of similarity solutions [16] [17]. The application of Lie group analysis of differential equations has been successful in solving problems in thin fluid film theory [18] [19] [20] and this method will be used here to reduce the nonlinear diffusion equation obtained to an ordinary differential equation. The half-width of the base is obtained during this reduction.
In this paper we investigate the effect of suction and blowing at the base on the spreading under gravity of a two-dimensional thin fluid film. The general form of the suction/blowing velocity at the base is determined from the condition that the partial differential equation and boundary conditions admit an invariant solution in a Lie symmetry analysis. We consider the case in which the suction/blowing velocity at the base is proportional to the height of the thin fluid film. The streamlines of the fluid flow inside the thin film are derived and analysed.
A nomenclature is provided in Table 1.   [8]. We follow the derivation of Acheson [21].
The thin film is infinite in the y-direction and symmetric in the ( )  where The x and z components of the Navier-Stokes equation and the conservation of mass equation are [21] 2 2 where g is the acceleration due to gravity.
We introduce characteristic quantities and dimensionless variables and write (2.6) to (2.8) in dimensionless form: characteristic velocity in the x-direction U, characteristic velocity in the z-direction characteristic pressure and stress P, The characteristic velocity V is determined from the conservation of mass Equation (2.8) which is not approximated so that the two terms balance while U and P are yet to be specified. The Reynolds number is where ν µ ρ = is the kinematic viscosity. Dimensionless variables are defined by x The x-component of the Navier-Stokes Equation (2.6), becomes Journal of Applied Mathematics and Physics Re .
For the thin fluid film to spread in the x-direction the magnitude of the pressure gradient in the x-direction should be sufficiently large to balance the viscous term. Thus 2 .
The z-component of the Navier-Stokes Equation (2.7), in dimensionless form where (2.12) was used for P in all terms except the body force term. For the thin fluid film to spread the body force term must be sufficiently large to balance the pressure gradient term in the z-direction. Thus we obtain the second condition on P, Equating (2.12) and (2.14) for P gives for the characteristic velocity U, 3 .
The thin fluid approximation is [21] Before stating the boundary conditions we derive two preliminary results. From the Navier-Poisson law for an incompressible fluid [21] 2 , , where zz τ and zx τ are components of the Cauchy stress tensor. Expressed in and neglecting terms of order , .
Also, a fluid particle on the free surface of the thin film remains on the free surface as the fluid evolves. Thus and expressed in terms of dimensionless variables, We now state the boundary conditions and then comment on them briefly. The boundary condition (2.28) is the balance of the normal stress at the surface of the thin film where 0 p is the atmospheric pressure and the boundary condition (2.29) is that the tangential stress vanishes at the surface.
Consider now the partial differential equation for the surface ( ) But using the formula for differentiation under the integral sign [22], and therefore (2.32) becomes , . x Integrating (2.37) twice with respect to z and imposing the no slip boundary condition (2.26) and the no tangential stress boundary condition (2.29) we obtain Consider now the balance law for fluid volume. The total volume of the thin fluid film per unit length in the y-direction is where the characteristic volume is 0 Hw . Applying the formula for differentiation under the integral sign [22] ( ) At the moving contact lines, ( ) , we take as boundary condition that the volume flux of fluid in the x-direction vanishes, that is The balance law for fluid volume reduces to The problem is to solve the nonlinear diffusion Equation In the remainder of the paper the overhead bars are suppressed to keep the notation simple, it being understood that dimensionless quantities are used.

Lie Point Symmetries and Group Invariant Solutions
When deriving the Lie point symmetries of the nonlinear diffusion Equation The Lie point symmetry generators of (2.39) are of the form [23] ( They are derived from the determining equation with summation over the repeated index k from 1 to 2. The total derivative operators with respect to t and x are The determining Equation (3.4) is separated by powers and products of the partial derivatives of h. It is found that provided the velocity ( ) , n v t x satisfies the first order linear partial differential The velocity ( ) , n v t x therefore cannot be arbitrary for the Lie point symme- satisfies the first order linear partial differential equation We first consider the general case in which 2 0 c ≠ , 4 0 c ≠ and

General Case c c c c
The differential equations of the characteristic curves of (3.14) are ( ) Integrating of the first pair of terms in (3.15) gives where 1 a is a constant while integration of the first and last pair of terms gives where 2 a is a constant. The general solution of (3.14) is where F is an arbitrary function. Hence, since h = Φ , the form of the invariant solution is Consider next the partial differential Equation . The differential equations of the characteristic curves of (3.12) are ( ) Integration of the first pair of terms gives again (3.16) and integrating the first and last pair of terms gives where 3 a is a constant. The general solution of (3.12) is where G is an arbitrary function. Hence the form of the invariant solution is where ξ is given by (3.20). Substituting (3.19) and (3.24) into the nonlinear diffusion Equation (2.39) reduces (2.39) to the ordinary differential equation The half-width of the base, ( ) w t , is determined from the boundary condi- Adding we obtain and differentiating (3.31) with respect to t, we find that and therefore since from (2.10), ( ) The boundary condition (3.26) becomes .
The total volume of the thin fluid film per unit length in the y-direction is By using (3.19) and (3.36) it can be expressed as ( ) The balance law for fluid volume is given by (2.47) and using (3.24) for In order to simplify the ordinary differential Equation (3.25) and the boundary conditions (3.35) we make the change of variables Equations (3.25) and (3.35) become ( ) The other variables transform to ( ) ( ) There is one condition still to be imposed. Since the characteristic length in the z-direction is the initial height of the thin film at the central line The problem expressed in terms of the transformed variable η and the parameter α is to solve the differential Equation (3.43), subject to the boundary conditions (3.44) and (2.46), The remaining quantities are given by and η is given by (3.49). The Lie point symmetry (3.10) which generates the invariant solution is ( )

Special
is a group invariant solution of (2.39) provided (3.13) is satisfied, that is, provided ( ) The differential equations of the characteristic curves of (3.65) are The first pair of terms gives which has the same form as (3.65). Hence where ( ) G ξ is an arbitrary function of ξ .
Substituting (3.70) and (3.73) into the partial differential Equation (2.39) reduces it to the ordinary differential equation ( ) The half-width of the base, ( ) w t , is again obtained from the boundary conditions (2.40) which can be written as (3.26) with , .
exp exp The total volume (3.37), expressed in terms of the invariant solution, is ( ) The balance law (2.47) becomes ( ) ( ) The remaining condition, ( ) In order to treat the special case 2 0 c = as the limit 2 0 c → of the general case we transform the special case by making the change of variables (3.41) where now Expressed in terms of the transformed variables the problem is to solve the differential equation 0 The remaining quantities are and using the expansion and therefore ( ) ( )

Normal Velocity at Base Proportional to the Thin Fluid Film Height
An assumption on ( ) , n v t x needs to be made to close the system of equations.
In this section we will investigate the solution for which

Invariant Solution
We will first consider the general case 2 0 c ≠ and then the special case 2 0 c = .
where β is a constant. Then from (3.58) and (3.59) , , . The value 5 3 β = and its relation to the special case 2 0 c = will be considered later. The differential Equation (3.54) reduces to The solution of (4.5) which satisfies the boundary conditions (3.55) and (3.56) is Hence using (3.58) to (3.62), is the Gamma function [22] and ( ) ( ) The Lie point symmetry (3.63) which generates the invariant solution becomes ( ) ( ) Consider now the special case 2 0 c = and that (4.1) is satisfied. Then from Journal of Applied Mathematics and Physics where 0 V is given by (4.11) and ( ) The Lie point symmetry which generates the invariant solution, (3.93), becomes In the same way as (4.17) was derived it can be shown that ( ) and it can be verified that (4.16) to (4.20) can be obtained from the general results (4.7) to (4.12) as 5 3 β → .

Time Evolution of the Invariant Solution
We now investigate the evolution of the thin fluid film with time. The behaviour of the fluid variables depends on the value of the constant β where β −∞ < < ∞ .
We first consider the second condition in the thin fluid film approximation (2.16) and determine the range of values of β for which it is satisfied. Return to dimensional variables and define We assume that the thin fluid film approximation is satisfied initially so that Hence the thin fluid film condition will be satisfied as the fluid evolves if either the thin fluid film approximation will break down.
Return to dimensional variables. From (4.7) and (4.9) for 5 3 β ≠ , ( ) and from (4.16) and (4.18) for Hence in the vicinity of the moving contact lines, ( ) , the thin fluid film approximation is no longer satisfied. The inclusion of blowing or suction does not eliminate the singularity at ( ) . Also, for 2 β −∞ < ≤ , Table 2. Time evolution of ( ) and therefore the height of the centre line of the thin fluid film decreases with time even when there is blowing at the base, 0 1 β < < . The rate of decrease in the height due to spreading is greater than the rate of increase due to blowing. When 1 β = the two rates of change balance and when 1 β > the rate of increase due to blowing is greater than the rate of decrease due to spreading. For  Table 3.   Table 3. Time evolution of ( ) ,0 h t for 2 β −∞ < ≤ . The finite time 1 t is defined by  Table 4.
In Figure 2 the evolution of ( )   , , and therefore on the centre line 0 and on the centre line, . The roots of (4.48) satisfy the following general properties [24].
and substituting (4.66) and (4.69) into (4.68) we obtain The root * 1 z is negative and lies outside the thin fluid film.

Strong Blowing
which can be derived directly from (4.45) with 2 β = . There are no roots inside the thin fluid film for 1 2 The stagnation point * 2 z on the centre line plays a significant part in the pattern of the streamlines in the thin film which we now consider.

Streamlines
In order to study further the fluid flow in the thin film we investigate the streamlines of the flow. The tangent vector to a streamline is everywhere parallel to the fluid velocity vector instantaneously. In two-dimensional incompressible flow the streamlines can be obtained from the stream function ( )  Table 2 and Table 3 for the evolution of ( ) In Figure 3, 0 β −∞ < ≤ and ( ) w t → ∞ and ( ) ,0 0 h t → as t → ∞ . The base will steadily increase and the maximum height will steadily decrease for 0 t < < ∞ .
In Figure 4

Conclusions
We first list the findings of the paper and then comment on the findings.
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.
It was found that the height of the thin fluid film satisfied a nonlinear diffusion equation with a source/sink term consisting of the fluid injection/leak-off velocity, The general form of the Lie point symmetry generates an invariant solution with height and half-width which evolve algebraically with time. A special form of the Lie point symmetry was also found which generates a solution for which the height and half-width evolve exponentially with time.
To close the system of equations, we found that one further condition is required. We assumed that The stagnation point on the centre line was found by solving a cubic equation. At this point the upward flow due to fluid injection at the base is balanced by the downward flow due to spreading by gravity.
The fluid flow inside the thin fluid film was investigated by plotting the streamlines which were obtained from the stream function by solving numerically a second cubic equation. Streamlines for both fluid leak-off and fluid injection were plotted. For fluid injection, it was found that, provided the injection velocity is not too strong, there is a dividing streamline which separates the flow into two regions, an upper region with downflow due to gravity and a lower region with upflow due to fluid injection at the base.
We make the following comments on the findings. When the fluid velocity at the base is proportional to the height of the thin fluid film, a complete analytical solution can be derived and the fluid variables and streamlines can be fully analysed. The thin fluid film approximation is satisfied for all values of time for the whole range of suction and significantly it is also satisfied for blowing at the base which provided the blowing that is not too strong.
There was unexpected behaviour of fluid variables due to the relative importance of spreading by gravity compared with blowing or suction at the base. The base half-width increased with time for all values of β , even for suction when 0 β −∞ < < . The maximum height of the thin film decreased for all time for suction and even for blowing provided the blowing was not too strong ( 0 1 β < < ). Some limits were attained in a finite time and in such a way that the thin fluid film approximation remained satisfied. For sufficiently strong blowing ( 5 2 3 β < ≤ ) the base half-width and the maximum height of the thin fluid film tended to infinity in time 1 t .
Much of the literature on thin fluid films is concerned with how the surface profile evolves with time. By plotting the streamlines numerically we were able to investigate the fluid flow inside the thin film. The streamline pattern for blowing was much richer than that for suction which did not show any unexpected features. The depth of penetration of the injected fluid at the base and its effect on the fluid flow inside the thin film could be investigated. The streamline pattern showed the existence of a dividing streamline which separated the flow