The viscosity of a fluid containing small drops of another fluid

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.

Turbulent drag reduction using fluid spheres

“…The Reynolds-averaged volume fraction in figure 2(a) is computed as φ = 1 − Ψ , where Ψ is a marker function, which is zero in the spheres and one in the suspending liquid. The simulation results in figure 2(c) agree well with the theory of Taylor (1932) and the accuracy increases with increasing grid resolution.…”

“…In the limit of vanishing Reynolds number Re p = U W R 2 /νH 1, vanishing volume fraction Φ 1 and vanishing wall effects R/H 1, Taylor (1932) derived that the shear stress in the suspension is τ = µ(1 + (7/4)Φ) du/dy, where u is the Reynolds-averaged velocity.…”

A rheological investigation of vesicular rhyolite

“…(1) (Taylor, 1932) where ~s is the shear viscosity of the liquid and f is the volume fraction of bubbles.…”

An experimental study of transient effects in the breakup of viscous drops

“…Taylor (1932Taylor ( , 1934Taylor ( , 1964. The problem is of considerable fundamental interest in fluid mechanics as an example of a time-dependent free-boundary problem and as a prototype for flow-induced deformation of a variety of flexible bodies such as red blood cells, macromolecules, floes, elastic particles, etc.…”