Tip streaming from a liquid drop forming from a tube in a co-flowing outer fluid

Abstract

Dynamics of formation of a drop of an incompressible Newtonian fluid of viscosity μ1 and density ρ1 from the tip of a tube of radius R1 into a co-flowing immiscible, incompressible Newtonian fluid of viscosity μ2 and density ρ2 that is enclosed in a concentric cylindrical tube of radius R2 are investigated under creeping flow conditions. Transient drop shapes, and fluid velocities and pressures, are calculated numerically by solving the governing Stokes equations with the appropriate boundary and initial conditions using the Galerkin/finite element method for spatial discretization and an adaptive finite difference method for time integration. In accord with previous studies, the primary effect of increasing the ratio of the volumetric injection rate Q̃2 of the outer fluid to that of the inner fluid Q̃1, Qr≡Q̃2∕Q̃1, is shown to be a reduction in the volume of primary drops that are formed. When Qr is small, calculations show that drop formation occurs in a slug flow regime where the primary drops that are about to be formed are elongated axially and occupy virtually the entire cross section of the outer tube. In this slug flow regime, the primary drops at breakup resemble cylinders that are terminated by hemispherical caps and their aspect ratios Lp∕Dp≫1, where Lp denotes their axial lengths and Dp≈2R2 their maximum diameters. As Qr increases, the dynamics are shown to transition to the dripping regime, where the primary drops are more globular, Lp∕Dp∼1, and their radii are of the order of or smaller than R1. As Qr increases, the importance of viscous stress exerted by the outer fluid relative to the surface tension or capillary pressure increases. Thus, the drop length measured from the tube exit to the drop tip at breakup increases while the primary drop volume decreases as Qr increases. When Qr is sufficiently large, viscous stress exerted by the outer fluid induces a recirculating flow within a forming drop. Once Qr exceeds a critical value Qrt, viscous stress exerted by the outer fluid becomes so large that the growing drop takes on a conical shape and a thin fluid jet with a radius that is a few orders of magnitude smaller than that of the radius of the inner tube emanates from its tip. This latter regime of drop breakup, which is henceforward referred to as tip streaming, is remarkably similar to electrohydrodynamic jetting that is seen from the tips of conical drops in electric fields and tip streaming that occurs from the pointed ends of surfactant-covered free drops subjected to linear extensional or shear flows, but takes place here in the absence of electric fields or surfactants. Scaling arguments for fixed a≡R2∕R1 show that for exterior viscous stress to overcome capillary pressure and cause tip streaming, Ca−1<mQr, where Ca≡μ1Q̃1∕γπR12 is the capillary number, γ is the interfacial tension, and m≡μ2∕μ1 is the viscosity ratio. In accordance with the scaling arguments, the computed predictions show that the critical value of the flow rate ratio signaling transition from dripping to tip streaming Qrt varies inversely with m for small to moderate m but becomes independent of both Ca and m as viscosity ratio grows without bound.