Rybczynski-Hadamard formula

For small core radius (e -> 0), this formula tends to the Hadamard-Rybczynski formula (2.2.15) for a drop if the membrane is thin (e - 1), then we obtain the Stokes formula (2.2.5) for a solid sphere. 

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. 

The formula of Hadamard and Rybczynski are also valid for the "moving bubble" problem with Ti Ti. Using the Hadamard-Rybczynski velocity field, it is easy to show that the difference between the tangential component of the velocity in the diffusion boundary layer and the surface velocity field is negligible. This is the reason why the reduction of equation (8.8) to variables 0, P leads to a coefficient on the right hand side which independent of T = xsin 0,... 

The main distinction of the theory of a dynamic adsorption layer formed under weak and strong retardation arises when formulating the convective diffusion equation. At weak retardation the Hadamard-Rybczynski hydrodynamic velocity field is used while at strong retardation the Stokes velocity field. Different formulas for the dependence of the diffusion layer thickness on Peclet numbers are obtained. The problem of convective diffusion in the neighbourhood of a spherical particle with an immobile surface at small Reynolds numbers and condition (8.74) is solved, so that the well-known expression for the density distribution of the diffusion flow along the surface can be used. As a result, Eq. (8.10) takes the form (Dukhin, 1982),... 

At y fi + fii, the formula (17.103) turns into the Stokes formula, which corresponds to a completely retarded surface of the drop, and at y fi + fii it transforms into the Hadamard-Rybczynski formula. In the general case, accounting for the surfactant leads to a decrease in the velocity of the drop s descent. 

For NRt < 1, the problem of bubble motion is closely related to that of the motion of a liquid drop in a liquid medium, and can consequently be derived from the Rybczynski-Hadamard formula (H2, R13) ... 

Redlich-Kwong equation, 181 Rybczynski-Hadamard formula, 318, 332, 348... 

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