The viscosity of a fluid containing small drops of another fluid

TLDR

Einstein’s classic work on the viscosity of suspensions of solid spheres, later extended by Jeffery to ellipsoidal particles, revealed that particle shape requires modification of the theory, but no analogous theory exists for suspensions of liquid drops due to unknown boundary conditions and deformation under viscous forces. The authors aim to extend Einstein’s viscosity analysis to liquid drops by assuming that surface tension keeps small drops nearly spherical when their radius or the fluid distortion velocity is small. They apply Einstein’s theoretical framework to these nearly spherical liquid drops, treating them as effectively solid spheres in a viscous fluid. Experimental observations confirm several of Jeffery’s.

Abstract

The viscosity of a fluid in which small solid spheres are suspended has been studied by Einstein as a problem in theoretical hydrodynamics. Einstein’s paper gave rise to many experimental researches on the viscosity of fluids containing solid particles, and it soon became clear that though complete agreement with the theory might be expected when the particles are true sphered, some modification is necessary when the particles are flattened or elongated. The theory of such systems was developed by G. B. Jeffery, who calculated the motion of ellipsoidal particles in a viscous fluid and their effect on the mean viscosity. Some of his conclusions have been verified by observation. So far no one seems to have extended Einstein’s work to liquids containing small drops of another liquid in suspension. The difficulties in the way of a complete theory when solid particles are replaced by fluid drops are almost insuperable, partly because the correct boundary conditions are not known, and partly because a fluid drop would deform under the combined action of viscous forces and surface tension. Even if the boundary conditions were known to be those commonly used in hydrodynamical theory, the calculation of the shape of the deformed drop would be exceedingly difficult. When the radius of the suspended drops or the velocity of distortion of the fluid are small, surface tension may be expected to keep them nearly spherical, and in that case Einstein’s analysis may be extended so as to include the case of liquid drops.