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v x = u , v z = v 0 = - P x 0 = - P z + μ 2 v x 2 u x + v z = 0 , t > 0 z > 0 0 < x < h ( z , t ) z h h ( z , 0 ) = h i t = 0 v = 0 z = 0 , t > 0 u = v = 0 x = 0 u = d x d t = h t + v h z τ = - μ v x = 0 x = h ( z , t )

The first equation states that there is zero potential gradient over the thickness of the film. Because the surrounding fluid has zero density and the surface tension is neglected, the pressure in the film is equal to that of the surrounding fluid. Thus,

P = p - ρ g z = p o - ρ g z P z = - ρ g

The velocity profile across the thickness of the film can be determined by integrating the second equation.

v = ρ g h 2 μ x h - 1 2 x h 2

The flux or flow rate per unit width of the film can be determined by integrating the velocity profile over the thickness of the film.

0 h v d x = ρ g h 3 3 μ

The continuity equation can be integrated over the thickness of the film.

0 h u x + v z d x = h t + v h z + 0 h v z d x

The derivative can be taken outside of the integral with the addition of another term that cancels the term in the previous equation.

0 h v z d x = z 0 h v d x - v h z x = h

Thus the differential equation for the film thickness is

h t = - ρ g 3 μ h 3 z = - ρ g h 2 μ h z h ( z , 0 ) = h i , t = 0 , z > 0 h ( 0 , t ) = 0 , z = 0 , t > 0

This is a first order, hyperbolic partial differential equation with constant initial and boundary conditions. Time and distance can be combined into a single similarity variable. The trajectories of constant values of the dependent variable can be calculated from the PDE.

d h = 0 = h t d t + h z d z d z d t d h = 0 = - h t h z = ρ g h 2 μ , h h i

Since the origin of all changes in thickness occur at the origin, the equation can be integrated as straight-line trajectories for each value of thickness between zero and the initial condition.

z t d h = 0 = ρ g h 2 μ , 0 < h < h i z t h i = ρ g h i 2 μ h ( z , t ) = μ z ρ g t , z t < ρ g h i 2 μ h ( z , t ) = h i , z t > ρ g h i 2 μ

This is the classical solution for transient film drainage. Thick films initially drain very rapidly but the rate of drainage slows as the film thins.

Notice that where the thickness has thinned below the initial thickness, the thickness is independent of the value of the initial thickness. Also, notice that the solution does not have a characteristic time, length, or thickness. This suggests that the thickness, time and distance are self-similar. In fact these variables can be combined into a single variable.

ρ g t h 2 μ z = 1 or t h 2 z = μ ρ g for h < h i

The thickness normalized with respect to the initial thickness can be expressed as a function of a single similarity variable or if a system length is specified, it can be expressed as a function of the dimensionless distance and time.

h * = h h i = μ z h i 2 ρ g t , μ z h i 2 ρ g t < 1 1 , μ z h i 2 ρ g t > 1 = η , η < 1 , η = μ z h i 2 ρ g t 1 , η > 1 = z * t * , , η < 1 , z * = z h i , t = ρ g h i t μ 1 , η > 1

One may be interested in the volume of liquid that remains on a vertical wall of length L after the film is everywhere less than the initial thickness. This can be determined by integrating the film thickness profile over the length of the wall.

0 L h d z = 4 μ L 3 9 ρ g t , for t > μ L ρ g h i 2 0 1 h d z * = 4 μ L 9 ρ g t , where z * = z / L

This expression shows that the amount of liquid remaining on a vertical wall is inversely proportional to the square root of time. This solution is valid only after the film has everywhere thinned below the initial thickness.

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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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