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We address the effect of inertia on the deformation and breakup of a drop suspended in a matrix liquid. The drop-to- matrix viscosity ratio is denoted by l5m d /m m . The case of equal viscosities is described in Refs. 1-3. At l51, inertia enhances the angle of tilt, so that the drop's tips experience higher velocity and elongation. In two-dimensions, the elon- gation of a circle ~a circular cross section of a cylinder! in- creases with inertia, 4 and significant elongation is observed at higher viscosity ratios. A circle, however, does not break; it thins indefinitely. The aim of this Brief Communication is to show that in three-dimensions ~3D!, the addition of inertia increases the critical viscosity ratio l c ~beyond which the drop does not break! from the Stokes flow value of 3.1. This, therefore, widens the range of viscosities at which shear- mixing takes place under simple shear. In Stokes flow, the mechanism for achieving a stationary shape is that the drop rotates toward the direction of the flow, away from the axis of elongation which is at 45 degrees to the flow direction. Thus elongation is suppressed, and breakup does not occur.5-9 Does the critical viscosity ratio increase when inertia is included? We denote the Reynolds number by Re5rga2/m m , and the capillary number by Ca 5m mga/s. Here, a denotes the initial radius of the drop, r the density of the matrix liquid, gthe shear rate, and s the surface tension coefficient. When the drop is subjected to simple shear, the capillary number represents the relative im- portance of viscous vs surface tension effects. When the cap- illary number is large enough, then capillary force becomes dominant only with the development of small length scales such as necks. However, until such length scales develop, viscous force dominates over capillary force. We can, there- fore, examine the dependence of initial drop rotation on Re and obtain scalings for the dependence of l c on Re as fol- lows. A drop under simple shear rotates clockwise with an angular velocity g ˙. At the same time, the viscous force elon- gates it at a rate T/m d , where T denotes the magnitude of the stresses acting on a more viscous drop. The stress T is the product of viscosity and deformation rate. The drop cannot break when the rotation rate is larger than the deformation rate

Inertia-induced breakup of highly viscous drops subjected to simple shear