Levelling of a Horizontal Film

FIGURE 5.4. Profile of a vertical film thinning down under the influence of gravity. 

The first step is to determine whether one of the two mechanisms at play (gravity and capillarity) might dominate over the other. The hydrostatic pressure difference between a crest (or a trough) and the average height of the film is pg6e. The Laplace pressure, on the other hand, is equal to the product of the surface tension 7 of the liquid and the curvature C of the interface. For a weakly curved surface (implying de/dx 1) C —d efdx.  

In the first case (capillary waves), we can specify the evolution law by expressing the liquid flow rate Q due the gradient of the Laplace pressure. Using equations (5.9) and (5.4), we obtain 

We see that a capillary wave in a viscous thin film relaxes exponentially as a function of time. The characteristic time of the relaxation process is a function of the liquid—it varies as the inverse of a characteristic velocity Y = y/rj—and, more important, of the geometrical characteristics of the film (through A and eo). For A = 1 mm, cq = 100 pm, and for an ordinary cooking oil rj 200 mPa-s and 7 20 mN/m, meaning V = 10 cm/s), we find a time constant of the order of 10 s. 

FIGURE 5.6. Relaxation time r of a sinusoidal film as a function of the wavelength of the disturbance. For ripples (A r increases as A. For waves 

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