Hadamard-Rybczynski problem

An approximate solution to this problem can also be obtained with the particularization of the Hadamard-Rybczynski problem [6.30, 6.31] from which it is found that C = 1/18 so that ... 

One point that has not been emphasized is that all of the preceding analysis and discussion pertains only to the steady-state problem. From this type of analysis, we cannot deduce anything about the stability of the spherical (Hadamard Rybczynski) shape. In particular, if a drop or bubble is initially nonspherical or is perturbed to a nonspherical shape, we cannot ascertain whether the drop will evolve toward a steady, spherical shape. The answer to this question requires additional analysis that is not given here. The result of this analysis26 is that the spherical shape is stable to infinitesimal perturbations of shape for all finite capillary numbers but is unstable in the limit Ca = oo (y = 0). In the latter case, a drop that is initially elongated in the direction of motion is predicted to develop a tail. A drop that is initially flattened in the direction of motion, on the other hand, is predicted to develop an indentation at the rear. Further analysis is required to determine whether the magnitude of the shape perturbation is a factor in the stability of the spherical shape for arbitrary, finite Ca.21 Again, the details are not presented here. The result is that finite deformation can lead to instability even for finite Ca. Once unstable, the drop behavior for finite Ca is qualitatively similar to that predicted for infinitesimal perturbations of shape at Ca = oo that is, oblate drops form an indentation at the rear, and prolate drops form a tail. 

The mathematical statement of the problem on the concentration distribution outside a drop is described by Eq. (4.4.3) and the boundary conditions (4.4.4) and (4.4.5), in which the dimensionless stream function satisfies the Hadamard-Rybczynski solution (see Section 2.2)... 

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),... 

A theoretical analysis of the Stokes flow problem for a noimeutraUy buoyant droplet is clearly called for. Germane to this problem is the theoretical analysis of Haberman (H3), dealing with axially symmetric Stokes flow relative to a liquid droplet at the axis of a circular tube, and Taylor and Acrivos (T2c) extension of the classical Hadamard-Rybczynski liquid droplet problem to the case of nonzero Reynolds numbers. In particular, Haberman shows that the assumption of a spherical shape for the droplet in a tube is incompatible with the differential equations and boundary conditions. Taylor and Acrivos (T2c) point out that, though Hadamard (H3a) and Rybczynski (RIO) were able to solve the Stokes flow problem by assuming a spherical shape for a liquid droplet, irrespective of the magnitude of the interfacial tension, the correctness of their a priori assumption was, to some extent, fortuitous. These remarks are undoubtedly pertinent to the resolution of Haberman s paradox and, ultimately, to the solution of the nonaxially symmetric droplet problem. 

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) ... 

Fluid sphere Studies by Hadamard [4] and Rybczynski [5] have addressed the problems of steady creeping flow past a fluid sphere analytically. The stream functions representing the motion are given by... 

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