Bubble hydrodynamics and interfacial rheology

The theoretical description of a diffusion process of a surfactant to, or from, the surface of a floating bubble is impossible without information on the floating velocity and the hydrodynamic field around the bubble. The first of these quantities can be found comparatively easily experimentally, whereas the Navier-Stokes equation is used to define the hydrodynamic field around the floating bubble. A solution of the equation must satisfy all boundary conditions at the bubble surface. It should be stated that a general analytical solution of this 

If the surface of a drop or bubble is immobile for any reason and the coordinate system is moving together with the bubble, the floating velocity is the same as that of a solid sphere. In particular, at small Reynolds numbers, the drop movement can be described by Stokes equation. 

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. 

Velocity of bubbles floating up in aqueous (distilled water) solutions of sodium dodecylsulphate of different concentrations (mole/1) 1-0 2-10 3-1.2-10 dotted curve - solid bubble in distilled water 

The dotted curve corresponds to solid bubbles. It was obtained by recalculations from experiments on spherical glass balls with a density of 2.37 g/m. Significant surfactant effects appear for large bubbles. Small bubbles rise as solid spheres, even in thoroughly cleaned liquids from a = 0.03 cm (Re = 36) and at c lO M up to a = 0.065 cm (Re = 182). It is typical that even at such low concentrations as lO M the velocity of bubbles is essentially decelerated. 

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