Boussinesq formula

The differences between Goldbeck s results and those of Kogler and Scheidig emphasize the difficulties of determining pressure distributions accurately in non-cohesive packings. Unquestionably, if the packings were wetted, so that particulate friction alone were not responsible for transmitting pressure, the Boussinesq formulas would prove satisfactory. 

Any mobility of the surface decreases the velocity difference and the viscous stresses. The result is that the hydrodynamic resistance becomes smaller and the floating velocity of a bubble according to (8.6) increases by a factor of 3/2 as compared to Stokes Eq. (8.5). In early experiments, under the condition of Re < 1, it was found (Lebedev 1916) that small bubbles of a diameters less than 0.01 cm behave like rigid spheres since their velocity is described by Stokes formula (8.5). At the same time. Bond (1927) has found that drops of a sufficiently large size fall at velocities described by Eq. (8.6). To overcome contradictions with the Hadamard-Rybczynski theory, Boussinesq (1913) considered the hypothetical influence of the surfaee viscosity and derived the following relation. 

Boussinesq, J., Existence of surface viscosity in the thin, transition layer separating a liquid from a contiguous fluid. Application of surface viscosity formulas to the surface of a spherical drop falling with uniform velocity into a fluid of less specific gravity. Velocity of the fall of a spherical drop into a viscous fluid of less specific gravity, Ann. Chim., 29, 349, 357, 364, 1914. 

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 ... 

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