Shear-induced rupturing of a viscous drop in a Bingham liquid

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1 J. Non-Newtonian Fluid Mech. 95 (2000) Shear-induced rupturing of a viscous drop in a Bingham liquid Jie Li, Yuriko Y. Renardy Department of Mathematics and ICAM, Virginia Polytechnic Institute and State University, Blacksburg, VA , USA Received 6 April 2000; received in revised form 16 August 2000 Abstract Deformation and breakup of a viscous drop in a Bingham liquid is investigated numerically with a volume-of-fluid scheme. Initially, a spherical drop is placed between two moving parallel plates. For our parameters, the matrix liquid has yielded. The competing effects driving the motion are the shear and interfacial tension. When interfacial tension effects dominate, the drop evolves to a steady shape which is elongated compared with the case when the outer liquid is Newtonian. Prior to breakup, stress levels are highest at the ends of the elongated drop. When shearing effects dominate, the drop breaks up, again with features that are elongated compared with the Newtonian counterpart. After the initial breakup, the daughter drops assume steady shapes Elsevier Science B.V. All rights reserved. Keywords: Bingham liquid; Newtonian viscous drop; Shear 1. Introduction Emulsions arise in a wide range of industrial applications, in materials processing, waste treatment, and pharmaceuticals [1]. In order to apply the emulsion technology, we need to control and manipulate the rheology of an emulsion and its microstructure [2 4]. There are many ways to make emulsions; one broad category is shear mixing while adding one fluid to another, as found in common kitchen recipes [5]. Experimental studies [5] show that the composition of the complex matrix fluid used in emulsification can lead to a dramatic alteration of the rupturing phenomenon. In this paper, we address the initial stage of the drop elongation, with shear that is below and slightly above the critical value. The outer liquid is a Bingham liquid, which ideally has a constant shear stress τ y over all shear rates [6 10]. The constitutive model is described in Section 2. The goal of this paper is to examine the idealized problem of an isolated drop subjected to simple shear. Fig. 1 shows the initial condition for our numerical investigation. The main result of this paper is that the drop shapes which are produced are dramatically altered from the Newtonian case. Corresponding author. address: renardyy@math.vt.edu (Y.Y. Renardy) /00/$ see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S (00)

2 236 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) Fig. 1. A single drop of silicone oil surrounded by a yield stress liquid is subjected to steady shear. The computational domain is spatially periodic in the horizontal x and y directions. The deformation of drops in shear flow has been examined for Newtonian liquids both theoretically and experimentally [11 20]. The breakup of a drop was investigated numerically in [21] with a volume-offluid continuous-surface-stress formulation. We describe the code SURFER++ and its performance in Section 3. An important parameter is the capillary number which measures the relative effects of viscous stresses that deform a bubble versus the restoring effect of interfacial tension: Ca = a γµ m /σ, where γ is an average shear rate, a is the initial radius of the drop, and µ m is the viscosity (or effective viscosity) of the matrix liquid. In addition, the other dimensionless parameters are: the viscosity ratio λ = µ d /µ m where µ d is the drop viscosity, a Reynolds number Re = ρ m γa 2 /µ m, the vertical plate separation d versus drop radius, and horizontal spatial periodicities. In the simulations, the Reynolds number is always kept low, and the plate separation and periodicity are kept sufficiently large to mimic the situation of a single drop surrounded by an infinite liquid. We present results limited to two Bingham liquids, a surfactant solution at high concentration and an emulsified mixture. The physical parameters for these are modeled in Section 4. The emphasis here is to investigate a reasonable parameter set, and to draw qualitative differences in the gross features between the Newtonian and Bingham liquid results. Numerical results are described in Section 5. Quantitative data on three-dimensional drop shape in this type of flow are lacking. An extension of our computations to higher capillary numbers appears entirely practical, with the future development of front-tracking algorithms and adaptive mesh. It may be premature, however, since it is clear the computational results have forged ahead of experiments. The most urgent need is thus for experiments, in which each stage of drop deformation is observed. 2. Mathematical modeling The governing equations are incompressibility u = 0, and momentum transport ( ) u ρ t + u u = ( pi + τ)+ F. (2) The interfacial tension force is F = σκn s δ s, where σ is the interfacial tension, κ is the mean curvature and n s is the normal to the interface. The constitutive equation that relates τ to the deformation is a (1)

3 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) Fig. 2. (a) Shear stress vs. shear rate according to the modified Bingham fluid of Eq. (3) for several values of the exponent m = 0.5, 1.0 and 10, µ p = 0.05 and τ y = 1; (b) effective viscosity vs. shear rate. modified Bingham equation [6]: { τ = µ p + τ } y [1 exp( m γ )] γ. (3) γ Here, µ p is a constant plastic viscosity, τ y the apparent yield stress, and m a stress growth exponent. The magnitude γ of the rate-of-strain tensor γ = u + ( u) T is given by γ = { γ : γ }. (4) 1 2 Eq. (3) approximates the von Mises criterion for large exponents m, and holds uniformly in yielded and unyielded regions. The analogue of Eq. (3) for unidirectional shear flow gives the relevant component of the stress tensor as { τ 12 = µ p + τ } y γ 12 [1 exp( m γ 12 )] γ 12, (5) where τ 12 and γ 12 are the shear stress and shear rate, respectively. For unidirectional steady flow, the momentum Eq. (2) reduces to a single scalar equation τ 12 = p x y + c, (6) where p/ x is a constant pressure gradient, c is a constant of integration, and the stress component τ 12 is given by Eq. (5). For Couette flow, the pressure gradient is zero, and the stress component is determined from the known shear rate γ 12 = U/d, substituted into Eq. (5). Fig. 2 shows a graphical representation of Eq. (5) for different values of the exponent m. The vertical axis is the magnitude of the stress tensor and the horizontal axis is the magnitude of the shear rate. Fig. 2(b) shows the effective viscosity versus shear rate. The flow is shear-thinning. The idealized Bingham liquid would display a constant shear stress, equal to the yield stress, rather than the sloped curves of Fig. 2(a). The viscosity τ y / γ would be a hyperbola approaching zero rather

4 238 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) than µ p for large shear rates and the fluid would be more shear-thinning than in Fig. 2(b). The more realistic model of Eq. (3) introduces the smoothing toward the origin, through the exponent m. For large shear rates, it introduces a monotonically increasing shear stress through the plastic viscosity µ p, and a limiting nonzero viscosity µ p. For Newtonian liquids, τ y = 0 and Fig. 2(a) would be replaced by a line of constant slope. To emphasize the Bingham nature of the liquid, µ p will be chosen small compared with the τ y term in Eq. (3), so that we have almost the same shear stress over all shear rates. We have set m = 100 in our computations to approximate a strongly Bingham liquid. An effective viscosity is defined by µ = τ / γ. For materials with yield stress, it is appropriate to define a dimensionless yield stress, called the Bingham number: Bi = τ yd (7) µ p V B where d is a characteristic length, µ p the plasitic viscosity, and V B the average velocity of the Bingham fluid. The Bingham number measures the fraction of the stress which is due to the yield stress. The Newtonian fluid corresponds to Bi = SURFER++ Our code SURFER++ is composed of three parts: a second order volume-of-fluid (VOF) method to track the interface [22], a projection method to solve the Navier Stokes equations on the MAC grid, and finally, a continuum method for modeling the interfacial tension. The details of the original code SURFER are given in [22 25], and new capabilities of SURFER++ are described in [21,26 29]: a summary is provided below. The density ρ and the viscosity µ of each fluid is a constant in each fluid. A concentration (or color) function C is used to track the interface: { 1 fluid 1, C(x) = 0 fluid 2. This concentration function is transported by the velocity field u. The fluids are incompressible: u = 0, and governed by the Navier Stokes equation: ( ) u ρ t + u u = p + µs + F where S is the viscous stress tensor. The body force F includes the interfacial tension force. In the VOF method, the interfacial tension condition across the interface is a body force over the cells which contain the interface. Two such formulations have been implemented in SURFER. The first is the continuous surface force (CSF) formulation [30], in which f s = σκn s, and F s = f s C, where σ is the interfacial tension, κ is the mean curvature and n s is the normal to the interface. The second is the continuous surface stress (CSS) formulation [24], in which F s = T T = =σδ s κn s and T = [(1 n s n s )σ δ s ]. Both methods are equivalent at the continuum level. The concentration function is discontinuous across the interface, but in SURFER, this is replaced by a smoothed color function C(x) which varies from 0 to 1 over a distance O(ɛ), where ɛ is of the order of

5 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) the mesh. In SURFER, ɛ= 2h, where h is the mesh size. The mollified color function C is obtained by convolving C with a kernel K(x,ɛ): C(x) = C(x )K(x x,ɛ)dx. Ω The temporal discretization is based on Chorin s projection method, which decouples the pressure equation. A semi-implicit Stokes solver has been incorporated. The spatial discretization is a Cartesian mesh of rectangular cells. It is a finite difference mesh known as the MAC grid. The density and viscosity for each cell are given by ρ = Cρ 1 + (1 C)ρ 2 and µ = Cµ 1 + (1 C)µ 2, where subscripts refer to fluids 1 and 2. A piecewise linear interface calculation (PLIC) method is used to reconstruct the interface position. The approximate normal n to the interface in each cell is equal to the discrete gradient of the volume fraction field: n = h C/ h C. The final step of the VOF method is to evolve the volume fraction field C. Atthenth timestep, the interface is reconstructed, the velocity at the interface is interpolated linearly and then the new interface position for the (n + 1)th timestep is calculated via a Lagrangian method: x n+1 = x n + u( t). The entire code has been parallelized; data on the scalability and timings are given in [29,31]. The implementation for the Bingham liquid case involves a straightforward modification to the stress tensor. The performance of SURFER++ for the Newtonian case is given in the aforementioned references. Specifically, below the critical capillary number, the evolution of a spherical drop to a steady elongated egg-shaped drop predicted by past works have been reproduced [21]. The Cox angle and the Taylor deformation parameter for several capillary numbers have been reproduced. The dependence of the critical capillary number on viscosity ratio has been compared with previous simulations based on the boundary integral method [32]. Convergence tests for near-critical situations and the effect of inertia are given in [33,31]. Above the critical capillary number, the drop continues to evolve and break up. The transition to just before breakup is well known from from previous works based on the boundary integral method. The simulations in [21] above the critical capillary number show the end pinching mechanism, where the ends of the drop develop into dumbbell shapes. Just after the initial elongation and development of dumbbells at the ends, there is a slight retraction, followed by the breakup at the ends of the dumbbells. During this breakup procedure, the middle portion feeds some fluid back into the dumbbells. Upon breakup, the middle filament undergoes its own instability. Daughter drops evolve to state, each below their respective critical capillary numbers. At higher capillary numbers, a sequence of large and small drops are produced upon breakup and resemble the experimental photographs of [15]. 4. Physical parameters We have chosen to work with physical parameters similar to the matrix liquid in the experiments of [5]. This is an aqueous phase having a mass fraction C of the surfactant NP7. The drops are silicone oil, a Newtonian fluid, which is immiscible. We denote dimensional quantities with asterisks. The silicone oil has viscosity µ d = 350 cp. The interfacial tension depends on the concentration C, and can vary over 1 20 dyn/cm (private communication, T.G. Mason). We use the following scalings in our computations. In Fig. 1, the plate separation is d and we code the flow so that the upper plate moves with speed U /2 and the lower plate with speed U /2. Our length

6 240 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) scale is d, the velocity scale is the difference in plate speeds, U, and the scale for stress is µ d (U /d ). The average shear rate is defined as U /d. Interfacial tension has the dimension of stress length, and therefore is scaled with µ d U. Over the range of interfacial tensions, and U = 1/3to6cms 1,wemay set our dimensionless σ (in Eq. (3)) to be If the dimensional shear rate γ = U /d is roughly 100 s 1, then over the range of U, d ranges from 30 to 600 m. This range covers the dimension of the cylinder separation in the Taylor Couette device of [5], where d 200 m. In our computations, we have set the dimensionless shear rate γ = 1, and scaled time with shear rate, so that t represents real time. From the viscosity versus shear rate curves of Fig. 2 of [5] for the NP7 water mixture, we see that at concentration C = 0.54, the curve may be extended by eye to large shear rates and yield a plastic viscosity of 5 P. The dimensionless µ p = µ p /µ d = 1.4. The Reynolds number Re = ρ U d /µ, with µ representing the viscosity of either liquid, is generally much less than 1. Our dimensionless yield stress τ y = τy /(µ d γ ), where the effective viscosity is µ e = τ y / γ, so that τy = µ e /µ d. From the curve for C = 0.54 at shear rate 100 s 1, the effective viscosity ranges over 5 10, so that τ y is roughly in the range 1 3. In our computations, one parameter set is: a dimensionless viscosity µ d = 1, τ = 2, fluid densities to 1, and dimensionless shear rate 1. Our other parameter values are: U = 1, d = 1. The viscosity ratio is λ = µ d /µ e = The initial radius of the drop a = a /d is in the range The larger the radius, the more effect the walls will have on the breakup sequence. The capillary number is Ca = a/(σ λ). In this modeling, the value of the plastic viscosity is of the order of the yield stress, and the liquid is then more Newtonian than Bingham. In order to emphasize the Bingham nature of the liquid, and because the value of the plastic viscosity is disputable, we choose a small but non-zero value, µ p = 0.05, to investigate drop breakup away from the Newtonian case. A second yield stress liquid, the properties of which are described in Fig. 12 of [5], is the emulsified mixture with closely packed droplets. The plastic behavior of this liquid arises from the composite nature of the compressed emulsion. It is not impossible that close to the exit of the Taylor Couette device, drop breakup is occurring within this closely packed mixture. The volume fraction of the emulsion is defined as φ = V d /V c, where V d is the entire volume of oil and V c continous phase volume. For example, at surfactant concentration C = 0.4, the continuos phase alone remains essentially Newtonian, while the effective viscosity is strongly modified by the volume fraction φ. At volume fraction φ = 0.7. The effective viscosity follows an empirical power law behavior of µ eff 210 γ 1/2 P. The emulsion is shear-thinning, exhibiting a smaller effective viscosity at higher shear rates. At large shear rates, the square root dependence on the shear rate means that this is not strictly a Bingham liquid. A Bingham liquid would display a line of constant but small slope. On the other hand, the square root growth can be approximated by the Bingham model over a finite range of shear rates. A reasonable shear rate for a computation is γ = 10 3 s 1. According to the above empirical law, the µ eff 6.64 P. The droplet radius observed in the emulsion at this shear rate is about 1.8 m = cm. Thus, we can calculate the two important dimensionless parameters: λ , Ca The other parameters are chosen as follows. The numerical simulation will be done in a domain with the initial droplet radius a = This dimension should allow sufficient space horizontally for the drop to deform and breakup just above the critical capillary number. The effect of changing this domain is left for discussion a posteriori. As in the preceding paragraph, the droplet viscosity µ d = 1.0, the shear

7 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) rate γ = 1, µ eff = τ y / γ and λ = µ d /µ eff, and τ y = 2. Finally, the surface tension is calculated from the capillary number σ = τ ya Ca = = The choice µ p = 0.05 yields an almost ideal Bingham liquid, with Bingham number Bi = 40. We will conduct two series of simulations, so that we examine the onset of the breakup regime where the drop is stretched to a much longer filament before breaking up into small drops. The first series is to vary the the yield stress τ y, which corresponds to varying the shear rate and, therefore, the shear stress in experiments. The second series is to vary the capillary number, which corresponds to studying the dependence on the droplet radius in experiments. 5. Numerical results 5.1. Evolution to steady solution For low capillary numbers, the restoring force of interfacial tension overcomes the viscous shearing stress and an initially spherical drop evolves to a steady shape. For Newtonian liquids, two parameters have been used to measure the deformation attained by the drop in its final stage. The first is the Taylor deformation parameter, D = (L B)/(L + B), where L and B are the half-length and half-breadth of the drop, respectively. The second parameter is the Cox angle θ of orientation of the drop with the axis of shear strain (see Fig. 4 of [21]). Fig. 3 is a steady-state solution for the aqueous solution. The capillary Fig. 3. Steady drop shape for effective viscosity ratio λ = 0.29, Ca = 0.35, Bi 1 = 0, Bi 2 = 1.4, t = ,a = 0.125,µ d = 1, γ = 1, µ p = 1.4,τ y = 2,U = 1,d = 1, densities mesh on computational box. Re = Velocity vector plot shows the central x z cross-section of the drop.

8 242 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) Fig. 4. Cross-section in x z plane along middle of drop, with velocity vectors. Initial condition is a spherical drop of radius a = 025,µ d = 1, equal viscosities, Ca = µ m γa/σ = 0.40, equal densities ρ = 1, mesh size and domain of calculation Newtonian Stokes flow; steady-state is achieved. number is below critical and the solution is reminiscent of the Newtonian solutions displayed in Figs. 5 7 of [21]. For comparison, Fig. 4 shows a steady-state achieved for Newtonian flow, with equal viscosities, shear rate 1.0, and capillary number This configuration is just below the critical capillary number of 0.41, at which the drop will continue to deform. A computation at 0.42 leads to drop breakup. The diagnostic tools applied to the NP7 solution give L = 0.23,B = 0.08,D = 0.48,θ = 26. Far away from the droplet, we see the basic simple shear flow pattern, with the flow moving toward the left at the top and toward the right at the bottom. Near the droplet, the velocity is tangential to the interface and the flow moves along the interface, which is consistent with the conditions at a free surface. The competition between the externally imposed shear flow and the surface tension driven flow produces a closed vertical motion interior to the drop. Next, the Bingham behavior of the matrix liquid is emphasized with the choice of the plastic viscosity µ p = 0.05, which is much less than the choice for the yield stresses. Numerical simulations for yield stresses τ y = 1.0,1.5,1.75, 2.0, and 2.5 lead to steady shapes. Fig. 5 shows the view from the side of the computational box. We define an effective capillary number by Ca = τ y /(σ/r), and the figure corresponds to Ca = 0.3, 0.45, 0.525, 0.6, These steady forms are different from the steady shapes achieved in Newtonian flow, in that these are more elongated. Fig. 6 shows the velocity vector plot for τ y = 1.0. The viscosities of the drop and matrix fluid are roughly equal. There is a vertical motion inside the drop, and the shear flow is achieved a short distance away from the drop. The vertical motion inside the drop also occurs for the Newtonian matrix liquid case. The Bingham liquid has yielded everywhere. In our computations, we did not find any unyielded regions. A spatial convergence test between and for Ca = 0.75 is shown in Fig. 7, and shows better convergence along the filament than at the ends. This comparison shows that the more refined mesh yields a drop which is slightly more stretched at the ends. The error in the coarse mesh accumulates with time, and will affect the breakup dynamics. The areas of highest stress are found to be the end points.

9 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) Fig. 5. Cross-sectional cut in x z plane along middle of drop, for steady solutions. Initial condition is a spherical drop of radius a = 0.25, µ d = 1,µ p = 0.05,m = 100,σ = Equal densities ρ = 1. Computational domain is for τ y = 2.5 (Bi 2 = 50). From top to bottom; Ca = τ y a/σ = 0.3, 0.45, 0.53, 0.60, Breakup The breakup sequence for a viscous drop in a strongly Bingham matrix liquid is qualitatively different from the Newtonian case. In the Newtonian case, just above the critical capillary number, there is an initial deformation to a dumbbell shape, in contrast to the worm shape of the Bingham liquid case. This is followed by breakup into daughter drops at the ends, with the daughter drops evolving to ellipsoidal shapes typical for the Newtonian liquids (see Fig. 4). Figs. 8 and 9 exemplify the Bingham case. The yield stress is 3.0, the effective viscosity ratio is λ = 0.33 and Ca = The sequence shown in

10 244 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) Fig. 6. Cross-sectional cut in x z plane along middle of drop. Steady-state solution with velocity vectors. µ d = 1,µ p = 0.05,m = 100,σ = 0.83, γ = 1.0, τ y = 1.0, λ = 0.95, Ca = Mesh size and domain of calculation Fig. 9 is the view from above, and corresponds to what would be seen in an experimental study of a drop sheared between concentric cylinders, as in the case of [15]. The initially spherical drop elongates, then develops large oblong ends, in contrast to the Newtonian dumbbells. The oblong ends at time 40 are on the threshhold of breaking. There is a slight retraction after the initial elongation. By time 50, the thin filament in the middle develops nodules, with the outer ones feeding liquid outwards into the Fig. 7. Comparison of calculations and mesh. Initial condition is a spherical drop of radius a = 0.25,µ d = 1,µ p = 0.05,m = 100,σ = 0.83,τ y = 2.5, Ca = 0.75, effective viscosity ratio λ = 0.4. Computational domain is Steady-state is achieved.

11 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) Fig. 8. Ca = 0.915, effective viscosity ratio λ = 0.33,τ y = 3.0. Side view: domain, mesh. ends. This feeding of liquid is much less pronounced in the Newtonian case. The picture at time 50.5 clearly demonstrates this migration from the nodules in the filament. Upon breakage, the picture at time 60 shows the oblong daughter drops at the ends, the nodule at the center, and smaller drops, all of which are at lower capillary numbers than the initial drop. Each of these drops assumes a steady shape much as in Fig. 5. The sequence of capillary numbers in Fig. 5 is obtained by progressively increasing the yield stress. This, however, amounts to increasing the effective viscosity of the Bingham liquid for each plot, and the viscosity ratio λ = µ d /µ m changes over 1.0, 0.6, 0.57, 0.5, 0.4. At τ y = 3.0, the viscosity ratio is roughly 1/3. The outer liquid is much more viscous and much of the flow dynamics occurs in the central region composed of the interface and drop. The velocity vector plot of Fig. 10 shows the drop just before breaking. This is a cross section of the flowfield in the x z plane at t = 35 s. There is a vertical motion

12 246 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) Fig. 9. Ca = 0.915,τ y = 3.0. Top view of computational box. Fig. 10. Cross-section in x z plane. Ca = 0.915,τ y = 3.0,λ = Side view: t = 35 s, mesh, domain.

13 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) Fig. 11. Ca = 0.915, effective viscosity ratio λ = 0.33,τ y = 3.0, domain, mesh. Contour levels of stress τ in two-dimensional cross sections along the drop. x = 0.91 is away from the drop, the center of the neck lies at x = 4. x = 2.38, 2.40 are near the end of the drop.

14 248 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) Fig. 12. Ca = 0.915, effective viscosity ratio λ = 0.33,τ y = 3.0, domain, mesh. Contour surfaces of stress for τ = 3.0 and 3.2, shown below the plot of the interface. inside the head of the drop. Fluid is flowing into the head region from the neck. Outside the drop, the flowfield quickly adjusts to an almost constant velocity, flowing to the right above the drop, and flowing to the left below the drop. The velocity changes in the middle region 0.4 <z<0.6, where the drop is elongated. In most of the flowfleld, the magnitude of the stress τ is approximately equal to the yield stress τ y. At the tip of the head, the magnitude τ achieves a local maximum. The stress field is shown is Fig. 11. In the absence of the drop, the undisturbed shear rate is γ = 1.0 and the corresponding undisturbed stress τ 12 is 3.05 (Eq. (5)). Two-dimensional cross-sections in the x y plane are displayed for contours of the absolute value of stress at time t = 20. The computations extend over 0 x 8 with symmetry across x = 4, 50 that x = 0.91, 2.38, 2.40 and 4.00 are shown. The cross-section at x = 0.91 is far away from the liquid drop and the stress is around the undisturbed level The drop center is at x = 4.0 and the contour curves hug the interface. The transition of the contour levels from the lines at x = 0.91 to the closed curves at x = 4 occurs at x = 2.38 and 2.40, where we are close to one head of the elongated drop. Fig. 12 shows the three-dimensional contour surface for the stress. The contour surface at τ = 3.0 hugs the interface while the level τ = 3.2 are two tiny surfaces close to the ends of the elongated drop. In our final simulation, the deformation and breakup is tracked by keeping the yield stress at 1.0 and changing the interfacial tension. The viscosities of the drop and matrix liquids are then roughly equal. Just above the critical capillary number, at Ca = 0.75, the sequence of pictures depicting breakup is shown in Fig. 13. The shapes involved are similar to the breakup sequence of Figs. 8 9, with the initially spherical drop elongating into a worm-like shape, then breaking off at the ends into oblong daughter

15 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) Fig. 13. Ca = 0.75 effective viscosity ratio λ = 0.95,τ y = 1.0, computational domain 8 1 1, mesh. drops, together with a main nodule at the center and smaller drops. The central filament then undergoes its own instability to form a central nodule surrounded by smaller drops, as does the Bingham liquid case in Fig. 13. The dramatic difference between these and the Newtonian case is the elongated shapes observed in the Bingham case. 6. Conclusion We have examined the dynamics of a Newtonian viscous drop in a Bingham liquid, sheared between two plates. For low capillary numbers, the drop evolves to a steady shape, but unlike the case where the outer liquid is also Newtonian, the steady shapes are elongated. Above the critical capillary number, the

16 250 J. Li, Y.Y. Renardy / J. Non-Newtonian Fluid Mech. 95 (2000) drop continues to elongate, retracts slightly, and breaks up. The daughter drops first break off at the ends, and are oblong compared with the Newtonian case. During breakup, the middle filament undergoes its formation of nodules, and the feeds liquid into the ends. After breakup, the respective drops have lower capillary numbers and attain steady-states. The effective capillary number which measures the relative strengths of interfacial tension versus shear stress, is roughly independent of the shear rate for the idealized Bingham liquid. The capillary number then depends just on the interfacial tension and initial drop radius. When the outer liquid is Newtonian, on the other hand, the capillary number also depends on the shear rate. This distinction allows for the Bingham liquid case to produce monodisperse drops over different shear rates present in a flow. The breakup at higher capillary numbers, where a large initial extension is followed by capillary waves, is likely the cause of the production of monodisperse drops. The numerical simulation of this is not possible without further refinement especially around the interface; this would include the use of an adaptive mesh which is planned for future work. Acknowledgements This research was sponsored by the National Science Foundation under Grant Nos. CTS , NSF-INT , and DMS This work was supported by National Computational Science Alliance under Grant Nos. CTS990010N, CTS990059N, and CTS990063N, and utilized the NCSA SGI Origin We are grateful to ICAM for the use of their Origin We thank Gareth McKinley for suggesting this problem. References [1] K.J. Lissant, Emulsions and Emulsion Technology, Part III, Marcel Dekker, New York, [2] H.M. Princen, A.D. Kiss, Rheology of foams and highly concentrated emulsions. Part IV, J. Coil. Interface Sci. 128 (1989) [3] T.A. Witten, J. Bibette, D.C. Morse, D.A. Weitz, Stability criteria for emulsions, Phys. Rev. Lett. 69 (1992) [4] J. Mason, T.G. Bibette, D.A. Weitz, Yielding flow of monodisperse emulsions, J. Coll. Interface Sci. 179 (1996) [5] T.G. Mason, J. Bibette, Shear rupturing of droplets in complex fluids, Langmuir 13 (1997) [6] T.C. Papanastasiou, Flows of materials with yield, J. Rheol. 31 (1987) [7] E. Mitsoulis, S.S. Abdali, N.C. Markatos, Entry and exit flows of Bingham fluids, J. Rheol. 36 (1992) [8] K. Sekimoto, Motion of the yield surface in a Bingham fluid with simple-shear flow geometry, J. Non-Newtonian Fluid Mech. 46 (1993) [9] S.L. Burgess, S.D.R. Wilson, Unsteady shear flow of a viscoplastic material, J. Non-Newtonian Fluid Mech. 72 (1997) [10] H.A. Barnes, The yield stress: a review everything flows? J. Non-Newtonian Fluid Mech. 81 (1999) [11] G.I. Taylor, The viscosity of a fluid containing small drops of another fluid, Proc. Soc. A 138 (1932) [12] G.I. Taylor, The formation of emulsions in definable fields of flow, Proc. Soc. A 146 (1934) [13] E.J. Hinch, A. Acrivos, Steady long slender droplets in two-dimensional streaming motion, J. Fluid Mech. 91 (1979) [14] H.P. Grace, Dispersion phenomena in high viscosity immiscible fluid systems and application of static mixers as dispersion devices in such systems, Chem. Eng. Commun. 14 (1982) [15] C.R. Marks, Drop breakup and deformation in sudden onset strong flows, Ph.D. thesis, University of Maryland at College Park, [16] S. Kwak, C. Pozrikidis, Adaptive triangulation of evolving closed, or open surfaces by the advancing-front method, J. Comp. Phys. 145 (1998)

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