THE BREAKUP OF SMALL DROPS AND BUBBLES IN SHEAR FLOWS *

The problem of determining the deformation and burst of a single drop freely suspended in another fluid undergoing shear is of fundamental importance in a variety of physical processes of practical significance; for example, the rheology of emulsions and the dispersion of one fluid phase into another. It was studied both theoretically and experimentally by G. I . Taylor,’.’ who, as he has with many of the other topics in fluid mechanics, obtained quantitative results that were not only the first on the subject but which remain among the most important and fundamental in this field. The systems considered experimentally by Taylor’ are depicted in FIGURES l a and I b. An initially spherical liquid drop of radius a and viscosity Ap was placed in a fluid, with which it was immiscible, of equal density and of viscosity p. A steady shear of strength G was then applied and the drop was found to deform into a steady shape if G was maintained below a critical value G,, but the drop broke when G exceeded G,. The two shear flows set up by Taylor’ were: ( I ) the “hyperbolic” flow, u, = Gx, ug = Gy3 with u , and u , being the corresponding velocity components along the x and y directions, respectively, which is a pure straining motion without vorticity, and (2) the simple shear flow, u, = Gy3 ug = 0, which, as is well known, consists of a pure straining motion, with its principal axis of extension along the diagonal in the xy plane, plus a solid body rotation about the origin. In the absence of inertial erects, which were indeed negligible in Taylor’s experiments,’ the independent parameters, in addition to the type of shear being impressed, are: C , the strength of the shear flow; a, the radius of the initially spherical drop, p, the viscosity of the ambient fluid; A, the viscosity ratio; and y. the interfacial tension. Hence, the deformation, D = ( L B ) / ( L + B ) , where L and B are the half-length and the half-breadth of the drop, respectively, becomes a function of only two dimensionless groups, i.e., the capillary number ( k ’ = Gwa/y) and the viscosity ratio, A. Taylor’ found that, for fixed A, D was linear in Gpa/y for small values of the capillary number k ’ , but that, beyond a certain range, the slope of the D versus k ’ curve increased rapidly in many cases until a point was reached where a steady drop shape could no longer be maintained and the drop burst. There were conditions, however-most notably with high viscosity drops (A >> 1) in a simple shear flow-for which a limiting deformation was attained and drop breakup did not occur. Examples of these two types of deformation curves are sketched in FIGURE 2. From a practical point of view, the quantity of primary interest is the critical shear