Drops Hadamard-Rybczynski solution

Figure 7-15. Streamlines for the Hadamard-Rybczynski solution for translation of a spherical drop through a quiescent fluid (plotted in a frame of reference that is fixed to the drop).

This result is known as the Hadamard-Rybczynski solution. It describes the velocity field for a spherical drop that is translating through an otherwise motionless fluid. A plot of the streamlines is shown in Fig. 7-15 for several different ratios of the internal to external viscosity. Obviously, for A. = 10, the interior fluid is moving slowly, and this is reflected in the small number of streamlines inside the drop. In addition, the exterior fluid is required to come almost to a stop on the drop surface, and this results in rather large velocity gradients (the streamlines are close together). As k is decreased, on the other hand, the interior fluid moves more freely, and the velocity gradients in the exterior fluid are reduced. 

Flow separation in the case of a drop is delayed compared with the case of a solid particle, and the vorticity region (wake) is considerably narrower. While in the case of a solid sphere, the flow separates and the rear wake occurs at Re 10 (the number Re is determined by the sphere radius), in the case of a drop there may be no separation until Re = 50. For 1 < Re < 50, numerical methods are widely used. The results of numerical calculations are discussed in [94], For these Reynolds numbers, the internal circulation is more intensive than is predicted by the Hadamard-Rybczynski solution. The velocity at the drop boundary increases rapidly with the Reynolds number even for highly viscous drops, In the limit case of small viscosity of the disperse phase, /3 —> 0 (this corresponds to the case of a gas bubble), one can use the approximation of ideal fluid for the outer flow at Re > 1. 

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)... 

Let us consider a transient solute concentration field in a liquid outside and inside a spherical drop of radius a moving at a constant velocity U in an infinite fluid medium. We assume that the fluid velocity fields for the continuous and disperse phases are determined by the Hadamard-Rybczynski solution [177, 420], obtained for low Reynolds numbers. The concentration far from the drop is maintained constant and equal to C,. At the initial time f = 0, the concentration outside the drop is everywhere uniform and is equal to C inside the drop, it is also uniform, but is equal to Co-... 

Internal circulation patterns have been observed experimentally for drops by observing striae caused by the shearing of viscous solutions (S7) or by photographing non-surface-active aluminum particles or dyes dispersed in the drop fluid [e.g. (G2, G3, J2, L5, Ml, SI)]. A photograph of a fully circulating falling drop is shown in Fig. 3.5a. Since the internal flow pattern for the Hadamard-Rybczynski analysis satisfies the complete Navier-Stokes equation... 

One of the most important analytic solutions in the study of bubbles, drops, and particles was derived independently by Hadamard (HI) and Rybczynski (R5). A fluid sphere is considered, with its interface assumed to be completely free from surface-active contaminants, so that the interfacial tension is constant. It is assumed that both Re and Rep are small so that Eq. (1-36) can be applied to both fluids, i.e.,... 

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