Rheol Acta (2007) 46:521–529 DOI 10.1007/s00397-006-0150-y ORIGINAL CONTRIBUTION The effects of confinement and inertia on the production of droplets Y. Renardy Received: 16 May 2006 / Accepted: 27 September 2006 / Published online: 2 December 2006 © Springer-Verlag 2006 Abstract Recent experiments of Sibillo et al., Phys. Rev. Lett. 97:054502, (2006) investigate the effect of walls on flow-induced drop deformation for Stokes flow. The drop and the fluid in which it is suspended have the same viscosities. The capillary numbers vary from 0.4 to 0.46. They find that complex start-up tran- sients are observed with overshoots and undershoots in drop deformation. Drop breakup is inhibited by lowering the gap. The ratio of initial drop radius to wall separation is fixed at 0.34. We show that inertia can enhance elongation to break the drop by exam- ining Reynolds numbers in the range of 1 to 10. The volumes of the daughter drops can be larger than in the unbounded case, and even result in the production of monodisperse droplets. Keywords Volume-of-fluid methods · Drop breakup · Drop deformation PACS 47.11.-j · 47.55D- · 47.55.df Introduction The effect of walls at close proximity to drops in processing situations is an important issue in micro- fluidics industries (Tan et al. 2004; Cristini and Tan 2004; Tan et al. 2006). Pathak et al. (2002) investigates Y. Renardy (B ) Department of Mathematics and Interdisciplinary Center for Applied Mathematics (ICAM), Virginia Tech, 460 McBryde Hall, Blacksburg, VA 24061-0123, USA e-mail: renardyy@math.vt.edu the influence of confinement on the steady state mi- crostructure of emulsions sheared between parallel plates, in a regime where the average droplet dimension is comparable to the gap width between the confining walls. It is found that droplets can organize into layers, depending on the migration to the centerline due to wall effects and coalescence. In this paper, we focus on a single drop and examine the effect of inertia on drop deformation when the drop dimension is of the order of the gap width. The initial condition for our numerical simulation is a spherical drop of viscosity μ d and density ρ , sus- pended in another liquid of viscosity μ m and the same density. The drop is placed at the center of the channel, which induces flow symmetry in y and antisymmetry in (x, z) and the drop does not migrate. Figure 1 is a schematic of the initial condition. There is an imposed constant shear rate ˙ γ . Time is nondimensionalized with ˙ γ . The top wall moves in the x-direction, and the bot- tom wall in the opposite direction. The initial velocity field is simple shear in both liquids. This adjusts quickly to the flow solution during the simulation. Distances are nondimensionalized with respect to the plate separa- tion. The initial dimensionless drop radius is denoted as a, fixed at 0.34. The computational box has periodic boundary conditions in the x and y directions. The fluids are incompressible and satisfy the Navier-Stokes equation. At the fluid interface, velocity and shear stress are continuous, and the jump in the normal stress is balanced by interfacial tension force (Renardy 2003). The dimensionless parameters are the viscosity ratio of the drop to matrix liquids λ = μ d /μ m , (1)