INERTIA-INDUCED BREAKUP OF HIGHLY VISCOUS DROPS IN SHEAR FLOW
We consider a viscous drop suspended in a matrix fluid between two parallel plates (upper and lower walls) moving with equal velocities but in opposite directions. The initial shape of the drop is spherical and the difference between the drop and matrix densities is negligibly small, i.e., rd = rm = r. The geometry of this simple shear flow with zero gravity is shown on the right. The superposed shear is described by the velocity field u = gzi, where g is the imposed shear rate and z the vertical coordinate. The incompressible flow of these immiscible fluids is governed by the Navier-Stokes equations. We solve these equations with periodic boundary conditions in the horizontal (x and y) directions and no-slip conditions at the walls. In our simulations, the computational domain measures Lx x Ly x Lz = 8(2a x 0.5a x a), where a is the initial radius of the drop. Details on the numerical algorithm can be found here.
When the matrix fluid is sheared, the resulting viscous forces distort the drop from a spherical shape. The drop either is deformed to an ellipsoid with the long axis inclined at less than 45 degrees to the flow direction or breaks up into several smaller drops (satellites or daughter drops). The latter occurs if the capillary number Ca = mmga/s, which is the ratio of viscous forces to interfacial tension forces, exceed some critical value Cac. Here s is the interfacial tension. Apart from the capillary number, the drop deformation in shear flow is characterized by the following dimensionless parameters: the drop to matrix viscosity ratio l = md/mm, the Reynolds number Re = rmga2/mm (based on the matrix), and the Weber number We = Re x Ca (ratio of inertial forces to the force due to interfacial tension). Our numerical experiments show that the critical capillary number Cac depends on both the Reynolds number and the viscosity ratio (see figure on the left). If the Reynolds number is negligibly small (Stokes flow), simple shear flow is known to be incapable of rupturing drops for which the viscosity ratio is greater than 3.5 (according to recent numerical results, the critical viscosity ratio lc = 3.1). The critical curve for Stokes flow in the capillary number Ca versus the viscosity ratio l plane has a vertical asymptote for l = lc. It is, however, a mistake to think that the critical capillary number tends to infinity at l = 3.5 (or 3.1), if the Reynolds number is finite. An increase in the Reynolds number leads to significant changes in the critical parameters. In particular, inertia shifts the vertical asymptote to higher values of the viscosity ratio. Hence, very viscous drops can be fragmented by shear mixing provided the shear rate is sufficiently high.
The next figure (on the right) illustrates the inertial mechanism of rotation of the drop away from the direction of the flow, which is responsible for the breakup of very viscous drops. This figure shows plots of the velocity field for the x-z cross-section (deformation plane). Here l = 5 and (a) Re = 1, Ca = 0.46; (b) Re = 10, Ca = 0.139; (c) Re = 50, Ca = 0.05. My explanation of this inertial mechanism is as follows. Simple shear flow produces a constant vorticity field with the vector aligned with the y-axis. This shear vorticity tends to rotate the drop clockwise (for geometry given in the top figure). If the Reynolds number is small and the drop is highly viscous, the drop cannot readjust its shape as quickly as the flow field can cause rotation. This causes the major axis of the drop to be aligned close to the flow axis, where the stretching is weak. This is the reason why highly viscous drops do not burst in Stokes flow. However, curvature vorticity is non-zero (and hence eddies exist) in the flow domain if the Reynolds number is suffuciently large. In particular, eddies are produced at the drop interface due to interfacial tension and discontinuities in viscosity, and at the walls due to the no-slip conditions. They diffuse into the fluid while being advected by the flow. The vortex lines inside the produced circulation zones point in y-direction. Like shear vorticity, curvature vorticity exerts the force that tends to rotate the drop in the x-z plane. But the rotation due to this force is counterclockwise. In the high Reynolds number flows, there appear, therefore, the lift force that prevents rotation of the drop toward the flow direction.
MOVIES
The first set of movies shows the breakup of a high viscosity drop (l = 4) in the low, intermediate and high Reynolds number regimes (Re = 1, 10, and 50, respectively). The capillary number is just above critical. It may be noticed that inertia accelerates the drop breakup.
The second set of movies compares the drop breakup in the low Reynolds number regime (Re = 1) at different viscosity ratios (l = 1 and 4). The capillary number is just above critical. These movies demonstrate that the breakup process is slower for more viscous drops. Just before the breakup, low viscosity drops are less elongated than high viscosity ones.
The last set of movies illustrates the inertial mechanism of rotation of the drop away from the direction of the flow. The subcritical evolution of a high viscosity drop (l = 5) in the low, intermediate and high Reynolds number regimes is shown.