Films thinning velocity

At a quasi-elastic impact the film thinning velocity is much smaller than the particle velocity. Thus dh/dt is very much less than unity and can be neglected in deriving Eq. (11.5). It is easy to obtain the well-known formula of Evans (1954) for the film deformation time from Eq.(11.6),... 

Miles (M10), 1960 Considers stability problem of thin liquid film (linear velocity profile) bounded by a solid wall and a cocurrent gas stream. 

While Equation 3.59 does not apply to viscoelastic films, the velocity and attenuation arising from acoustically thin, viscoelastic films can be determined from Equation 3.58 by inserting complex moduli (e.g., K and G) into the 0 expressions. 

In the case when two plane-parallel ellipsoidal disks of tangentially immobile surfaces are moving against each other under the action of an external force, F,, from Equations 5.255 and 5.256, we can derive the Reynolds equation for the velocity of film thinning ... 

Here and V2 denote the respective velocities of film thinning. After some estimates based on Equations 5.262, 5.273, 5.285, and 5.286, we can express the ratio in Equation 5.291 in the form ... 

The technique itself simply amounts to measuring the sound velocity along a fiber or in a long thin film. This velocity compared with the velocity measured in the same material that is randomly oriented (isotropic) leads to the second moment of the orientation distribution function. 

Figure 11 displays the dependence of the foam dilatational viscosity upon the rate of foam expansion for Na = 1 (21). The foam viscosity, K, increases for foam expansion, as the foam films thin and velocity gradients occurring over the thickness of the film increase in magnitude, whereas, for opposite reasons, the foam viscosity, K, decreases as the foam is compressed. 

For small contact angles, the flow pattern close to the contact line (the wedge region) can be approximated by the so-called lubrication approximation, which compares the flow in the wedge with flow in a thin film. The velocity of fluid molecules is only determined by their distance vertical to the substrate the molecules move preferentially in the direction parallel to the substrate and less in the direction away from the substrate. 

According to Davies and Rideal (6), both types of emulsions (water-in-oil and oil-in-water) are formed during the homogenization process, but only the one with lower coalescence rate siuvives. If the initial drop concentration for the two emulsions (Systems I and II, see Fig. 15) is the same, the corresponding coalescence rates for the two emulsions will be (approximately) proportional to the respective velocities of film thinning, Kj and Fjj (163) ... 

Let us consider first the following situation. A metal pin slides over a ceramic plate at a constant velocity in an electrolyte solution. The metal is polarized anodically in the passive potential region using a potentiostat. Rubbing causes wear of the passive oxide film by abrasion. Under steady state conditions the rate of film growth at the imposed potential is equal to the rate of film thinning due to wear ... 

From Equations 4.270 and 4.273, the ratio between the Reynolds velocity and the velocity of film thinning for a given force is obtained. In Figure 4.42, this ratio is plotted as a function of the film... 

The investigation of the influence of the interfacial viscosity on the rate of film thinning and the shape of the film surfaces is a computationally difficult task, which could be solved only numerically. The results for a symmetrical plane-parallel foam and emulsion films, obeying the classical Boussinesq-Scriven constitutive law (see Sec. III.F) are presented in Refs. 5, 58, 267, 479, and 480. Ivanov and Dimitrov [5,481] showed that to solve this problem, it is necessary to use the boundary conditions on the film ring (at r = jR) for that reason, the calculations given in Refs. 479 and 480 may not be realistic. The only correct way to solve the boundary problem is to include the influence of the Plateau border in the boundary conditions however, this makes the explicit solution much more difficult. As a first approximation, in Ref. 267 an appropriate asymptotic procedure was applied to foam films and the following formula for the velocity of thinning was obtained ... 

In Ref. 429, it was established that for micron-sized nondeformed droplets, the surfactant in the drop phase can slightly influence the velocity of film thinning [in contrast to the case of deformed drops, Fig. 15, described by Eq. (256)]. 

In Eqs. (10.1) and (10.2), it is assumed that the layer of the polymer solution, applied initially, has a uniform thickness of hp and contains volatile substances of density The viscosity of the polymer solution is denoted by ju and e represents the evaporation rate, assumed to be constant. For the spin coating process, the angular velocity of the coater is represented by co. It is assumed that evaporation commences when the rate of film thinning process is reduced below the threshold value of cPa/Pao> where p is the density of the volatile part of the solvent at that instance [26, 27]. In general, the higher the angular velocity of the coater, the thinner is the coating of the polymer film. 

---