Drop in a Translational Liquid Flow

In [476] the following asymptotic expansion was obtained for the drag coefficient of a drop in a translational flow at small Reynolds numbers  

The spherical form of a drop or a bubble in Stokes flow follows from the fact that the flow is inertia-free. However, even for the case in which the inertia forces dominate viscous forces and the Reynolds number cannot be considered small, the drop remains undeformed if the inertia forces are small compared with the capillary forces. The ratio of inertial to capillary forces is measured by the Weber number We = p U a/a, where cr is the surface tension at the drop boundary. For small We, a deformable drop will conserve the spherical form. 

In Section 2.2 it was already noted that even a small amount of surfactants in any of the adjacent phases may lead to the solidification of the interface, so that the laws of flow around a drop become close to those for a solid particle. This effect often occurs in practice. However, if both phases are carefully purified (do not contain contaminants), the flow around a drop possesses some special features. 

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. 

According to the data presented in [94], to estimate the drag of a spherical drop with high accuracy, one can use the following formula, which approximates numerical results obtain by the Galerkin method  

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