Effect of confinement on the steady-state behavior of single droplets during shear flow

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1 Effect of confinement on the steady-state behavior of single droplets during shear flow Anja Vananroye, Peter Van Puyvelde a, and Paula Moldenaers Katholieke Universiteit Leuven, Department of Chemical Engineering W. De Croylaan 46, B-3001 (Heverlee) Leuven, Belgium The effect of geometrical confinement on the deformation and orientation of single droplets during steady-state shear flow is investigated microscopically in a counterrotating device. The model system consists of poly(dimethyl siloxane) droplets of varying sizes and viscosities in a poly(isobutylene) matrix. The experimental results are first compared with the predictions of the model by Maffettone and Minale (1998) for bulk flow. For all viscosity ratios, deviations from the Maffettone and Minale model start to occur at a droplet diameter to gap spacing ratio of the order of 0.4. The droplet deformation increases and the droplets orient more towards the flow direction as a consequence of confinement. At low viscosity ratios, the deviations remain small, whereas at high viscosity ratios, larger deviations from bulk behavior are observed. The observations are also compared with the theory of Shapira and Haber (1990) which a Author to whom correspondence should be addressed; Peter.Vanpuyvelde@cit.kuleuven.be 1

2 includes the influence of wall effects on deformation. The Shapira and Haber model is modified by replacing the Taylor model as bulk reference by the Maffettone and Minale model. Good agreement between theory and experimental results is found for a wide range of viscosity ratios. Keywords: droplet deformation, droplet orientation, confinement, steady-state shear flow I. Introduction Over the years, the dynamics of immiscible blends has been thoroughly explored. In particular, the rheology of dilute Newtonian emulsions and the deformation and relaxation behavior of Newtonian droplets immersed in a Newtonian matrix are quite well understood. In this case, two dimensionless numbers govern the behavior of a η droplet, subjected to a shear flow field: the capillary number Ca ( = m Rγ, where ηm, R, Γ γ, and Γ denote respectively the matrix viscosity, the droplet radius, the shear rate and ηd the interfacial tension) and the viscosity ratio p ( =, in which η d is the droplet η viscosity). Droplet deformation increases with increasing Ca and droplet breakup is not observed until Ca reaches a critical value Ca crit. For Newtonian systems, the latter only depends on the type of flow and the viscosity ratio p [Grace (1982)]. Various analyses have been proposed to express the deformation of Newtonian droplets in a bulk flow field, and these models have been summarized in several reviews [Rallison (1984), Stone m 2

3 (1994), Ottino et al. (1999), Tucker and Moldenaers (2002), Guido and Greco (2004)]. For instance, Maffettone and Minale (1998 and 1999) developed a simple phenomenological model (MM-model) which is capable of describing the time dependent deformation and orientation of ellipsoidal Newtonian droplets. Direct microscopic observations have been performed in special flow cells by several research teams [Grizzuti and Bifulco (1997), Guido and Simeone (1998), Yamane et al. (1998), Hayashi et al. (2001) and Cristini et al. (2003)]. Typically, there is good agreement between the MM-model and the experimental data. At present, an important trend in the process industries is the rapid development of microfluidic devices [Stone et al. (2004)]. For instance, Link et al. (2004) were capable of designing dispersions with predetermined droplet sizes and volume fractions using microfluidic technologies. Utada et al. (2005) reported the possibility to generate monodisperse double emulsions in a micro-capillary fluidic device. However, processing two-phase materials with a droplet-matrix structure on a microscale implies that the size of the droplets could become comparable to the dimensions of the flow geometry. Therefore, droplet-wall interactions could play an important role and affect the dynamics of the droplets. Some interesting studies concerning the behavior of confined blends have already been reported [for a recent review, see Vananroye et al. (2006c)]. For instance, Migler (2001) reported the organization of confined droplets in superstructures such as pearl necklaces and strings. Pathak et al. (2002) mapped out a morphology diagram describing the transition of confined droplets to pearl necklaces and strings, as a function of concentration and shear rate for blends of poly(isobutylene) (PIB) and poly(dimethyl 3

4 siloxane) (PDMS) with p = 1. Pathak and Migler (2003) also studied the deformation of confined droplets of a 10% blend with p = 1. They showed that highly confined droplets remain stable during flow when the capillary number exceeds the critical value for breakup: the droplets arrange in pearl necklaces and coalesce in the flow direction, giving rise to string-like structures with high aspect ratios. Vananroye et al. (2006a) showed that individual droplets in confined dilute PIB/PDMS-emulsions behaved according to the bulk theories, as long as the degree of confinement remained moderate. In the case of single droplets, Vananroye et al. (2006b) also demonstrated a substantial effect of confinement on the breakup criterion: for blends with p < 1, confinement was observed to suppress breakup whereas for blends with p > 1, breakup was enhanced. Vananroye et al. (2006b) also reported that droplets with p > 4 can be broken in simple shear flow by increasing the degree of confinement, a phenomenon that is impossible to achieve in bulk conditions. The ratio of droplet diameter to gap spacing at which deviations from bulk behavior started to occur was reported to be approximately 0.4. Recently, Sibillo et al. (2006) investigated confined droplet deformation for p = 1. They observed complex oscillatory transients and elongated droplet shapes in microconfined shear flow. In this work, a systematical investigation of the effects of viscosity ratio and degree of confinement on the dynamics of single Newtonian droplets in a Newtonian matrix has been performed. The deformation and orientation of confined and non-confined single droplets in shear flow are analyzed microscopically in a counterrotating flow cell. The data are compared with the predictions of the phenomenological model by Maffettone 4

5 and Minale (1998) for ellipsoidal droplets in bulk flow and with the theoretical analysis of Shapira and Haber (1990) for droplets confined between two parallel walls. II. Materials and methods A. Materials The materials used in this work are PIB (Parapol, from ExxonMobil Chemical, USA) and PDMS (Rhodorsil and Silbione, from Rhodia Chemicals, France). Guido and Simeone (1998) reported that PIB is slightly soluble in PDMS. Therefore, in the present experiments, PIB is chosen as the matrix phase and PDMS as the droplet phase. Elasticity effects are negligible and the pure components behave as Newtonian fluids under the measurement conditions [Vinckier et al. (1996)]. The density difference between PIB (ρ PIB = 890 kg/m 3 at 20 C) and PDMS (ρ PDMS = 970 kg/m 3 at 20 C) [Minale et al. (1997)] is small enough to neglect gravitational effects. In order to vary the viscosity ratio, PDMS grades with different molecular weights are selected. Table I gives the zeroshear viscosities η 0 at 24 C and the measured activation energies E a of the components together with the viscosity ratios p and critical capillary numbers Ca crit [de Bruijn (1989)] of the blends. The interfacial tension Γ of the PDMS/PIB system is 2.8 mn/m [Sigillo et al. (1997)], and is independent of the molecular weight of PDMS for grades with relatively high molecular weight, as is the case here [Kobayashi and Owen (1995)]. 5

6 Table I. Zero-shear viscosities at 24 C and activation energies of the model components; viscosity ratios and critical capillary numbers of the blends at 24 C. Grade η 0 (24 C) (Pa.s) E a (kj/mole) η p = ηpib (24 C) PDMS Ca crit (p) PIB Parapol matrix matrix PDMS Silbione 70047V PDMS Rhodorsil 47V PDMS Rhodorsil 47V PDMS Rhodorsil 47V unbreakable by bulk shear flow B. Methods The experiments are performed using a counterrotating parallel plate flow cell (Paar- Physica) in which it is possible to keep a droplet in a stagnation plane during shear flow. 6

7 A detailed discussion of the device has been given by Vananroye et al. (2006b). The gap spacing between the plates is set at 1000 µm and droplets with diameters ranging from 180 to 800 µm are injected in the matrix material using a home-made injection device. To avoid droplet contact with the plates, no larger droplets are investigated. To obtain specific capillary numbers, shear rates are calculated for each droplet individually, based CaΓ on its size ( γ = ). All experiments are performed at room temperature. Therefore, η R m the actual temperatures are always measured and the corresponding viscosities are calculated on the basis of the data in Table I. Optical microscopy (Wild M5A stereo microscope) is used to visualize the droplets. The microscope and camera (Basler A301fc) are mounted on translating stages so that images can be captured in the velocityvorticity plane as well as in the velocity-velocity gradient plane [Figs. 1(a) and 1(b)] using Streampix Digital Video Recording Software (Norpix). Image resolutions are varying from 0.63 pixels/µm for the smallest to pixels/µm for the largest droplet. From the analysis of these images using ScionImage Software, it is possible to extract all necessary information about the deformation and orientation of the droplets. The degree of confinement is expressed as the ratio of the droplet diameter 2R to the gap spacing d of the flow cell. 7

8 Fig. 1. Schematic of a deformed droplet with the geometrical parameters in shear flow: (a) velocity-vorticity plane; (b) velocity-velocity gradient plane. The experimental results will first be compared with the predictions of the Maffettone and Minale model [MM-model] (1998 and 1999) for bulk conditions. This phenomenological model assumes that during flow, the shape of a droplet is ellipsoidal and can be described by a symmetric, positive-definite, second rank tensor S with eigenvalues representing the square semi-axes of the ellipsoid. The evolution equation of S, which results from the competing actions of the interfacial tension and the hydrodynamic forces, is given by [Maffettone and Minale (1998)]: 8

9 ds f1 Ω S + S Ω = 2 dt τ [ S g( S) I] + f ( E S + S E) (1) where t is the time, τ = η m R /Γ is a characteristic emulsion time, I is the second rank unit tensor, and E and Ω are the deformation rate and vorticity tensors of the flow field. The parameters f 1 and f 2 are dimensionless, non-negative functions of p and Ca, defined as: 1 40( p + 1) ( 2 p + 3) ( 19 p + 16) f = (2) f Ca = +. (3) 2 2 p Ca g(s) is a function to preserve the volume of the droplet. The model predicts quite accurately the evolution of the three main axes of a droplet and its orientation in an arbitrary bulk flow field up to the critical capillary number for viscosity ratios both above and below one [Maffettone and Minale (1998)]. For simple shear, the steady-state dimensionless axes of a droplet and its orientation angle θ are given by: L 2R = (4) ( ) f + Ca ( f + Ca f Ca ) 3 1 f + Ca + 1 f Ca f + Ca 2 9

10 B 2R W 2R = (5) ( ) f + Ca ( f + Ca f Ca ) 3 1 f + Ca 1 f Ca f + Ca ( ) f + Ca ( f + Ca f Ca ) f1 + Ca f 2 Ca = (6) 2 f 1 1 θ = arctan (7) 2 Ca From Eqs. (4) and (5), an expression for the droplet deformation parameter D MM, defined as the ratio of (L-B) over (L+B), can be calculated. Despite good agreement between the MM-model and the experimental results in bulk flows, one cannot expect the MM-model to be valid in confined flows. Under the latter conditions, models that take into account wall effects, are required. However, such models are scarce, and moreover, they are based on the existing bulk theories. For instance, the theoretical model of Shapira and Haber (1990) [SH-model] adds an additional deformation to the Taylor deformation [Taylor (1932)] for droplets confined between two parallel walls: p R 3 DSH = DTaylor 1 + Cs ( ), (8) 1+ p d 10

11 with D Taylor the deformation parameter as observed by Taylor (1932): p D Taylor = Ca sinθ cosθ. (9) 8( 1+ p) In Eq. (8), C s represents a shape factor that depends on the position of the droplet relative to the walls. In the case of a droplet positioned exactly in the middle between the confining walls (h/d = 0.5 with h the distance from the mass centre of the droplet to the closer wall), a value of is reported for C s. When the droplet is closer to one of the two walls, the shape factor evolves to a value of, for example, 193 at h/d = [Shapira and Haber (1990)]. Although the shape factor grows considerably with decreasing h/d, Shapira and Haber (1990) stated that wall effects do not alter the shape of the droplets, but only change the magnitude of the deformation. In the present experiments, the droplets are carefully positioned at the centre plane between the confining walls and therefore a value of will be used for C s. The model states that the correction term is a function of the viscosity ratio and of the degree of confinement to the power three. As similar dependence (L/2R ~ R/d 2.93 ) was observed for confined string-like structures by Pathak and Migler (2003). Yet, in their results, 2R had to be substantionally larger than the gap spacing to achieve this dependence. For 2R/d below 1, as is the case here, they only found a linear dependence on the degree of confinement. In Eq. (9), θ is assumed to be π/4, as in the original theory of Taylor (1932). However, this assumption is incorrect since it is known that droplets orient more towards the flow direction with increasing shear rate [see, for instance, Maffettone and Minale (1998)]. 11

12 This could already cause large deviations between the theoretical and the experimental results. Therefore, it might be necessary to adapt the bulk deformation term in the SHmodel with a more appropriate one (see further). Since both deformation and orientation are combined in the model, no independent calculation of the droplet orientation can be made. III. Results and discussion A. Droplet deformation The droplet deformation is studied as a function of Ca by applying shear rates that correspond to capillary numbers ranging from 0.05 to 0.4. When a droplet reaches steadystate, images are captured both in the velocity-vorticity plane and in the velocity-velocity gradient plane. Then, the flow is temporarily stopped, allowing the droplet to relax to its spherical shape. Subsequently, the flow rate is increased, corresponding to an increase in Ca with 0.05, and the procedure is repeated. Droplets with varying degrees of confinement and viscosity ratios ranging from 0.3 to 5.2 are analyzed in this manner. Figure 2 illustrates the steady-state deformation of two droplets with a viscosity ratio of 5.2 as a function of capillary number. One droplet is only weakly confined (2R/d = 0.3) whereas the other has a relatively high degree of geometrical confinement (2R/d = 0.77). The lines are the dimensionless axes from the MM-model in bulk shear flow for p = 5.2, 12

13 calculated using Eqs. (4), (5), and (6). As can be seen in Fig. 2(a), the deformation of the unconfined droplet (circles) is well predicted by the model for all capillary numbers shown here. For the more confined droplet, it is shown in Fig. 2(a) (triangles) that large deviations from bulk behavior are present. This droplet deforms noticeably more than predicted by the MM-model, even at very low capillary numbers. The increased deformation is also illustrated in Fig. 2(b), where the microscopic images, captured in the velocity-vorticity plane at a capillary number of 0.3, are depicted for the two droplets. The deformation of the confined droplet is clearly more pronounced compared to the deformation of the less confined one. 13

14 Fig. 2. (a) Comparison of experimentally determined dimensionless axes of two droplets (2R/d = 0.3 and 2R/d = 0.77) with the MM-model as a function of Ca for p = 5.2; (b) Microscopic images for 2R/d = 0.3 and 2R/d = 0.77 at Ca = 0.3. These results could help to understand the breakup results presented by Vananroye et al. (2006b). They showed that for p > 1, confined breakup occurs at lower capillary numbers as compared to the bulk situation. Due to the pronounced increase in deformation at relatively low capillary numbers [see Fig. 2(a)] and the suppression of droplet rotation, breakup can be promoted in confined conditions. For the confined droplet in Fig. 2, breakup was already observed at Ca = 0.43, were normally totally no breakup is expected in bulk flow. Figure 3 shows the deformation of unconfined droplets for three viscosity ratios as a function of Ca. It should be mentioned that the droplets are situated near the middle plane between the confining walls, and the data shown here serve as a reference for the confined conditions (see further). The lines represent the dimensionless axes from the MM-model corresponding to the viscosity ratios under investigation. For the three viscosity ratios used here, the experimental results are in good agreement with the predictions of the MM-model. From these data, it can be concluded that the MM-model is able to predict the shape of a deformed droplet up to capillary numbers close to the critical value, for p-values both below and above 1, as long as the degree of confinement is fairly low. 14

15 Fig. 3. Comparison of experimentally determined dimensionless droplet axes with the MM-model as a function of Ca for p = 0.31, 1.05 and 2.03, and for 2R/d < 0.2. In Fig. 4, the experimental results of three confined droplets with approximately the same viscosity ratios as in Fig. 3 are depicted. The degree of confinement is now relatively high (2R/d ~ 0.8). At low Ca, the uncertainty on the data is somewhat larger than at high Ca, due to the inherent difficulty of determining θ for only slightly deformed droplets. However, the experimentally obtained deformation is clearly larger than the one predicted by bulk theories. The deviations from bulk behavior are not only determined by the degree of confinement, but also depend on the viscosity ratio. For p = 0.31 (full lines and circles in Fig. 4), the differences between the experimental results and the bulk predictions are quite small and do not seem to depend significantly on Ca. Hence, confinement seems to have a limited effect on the deformation of single droplets for p < 1 up to Ca = 0.35, although it has been reported that breakup of highly confined droplets 15

16 with p < 1 is shifted to higher capillary numbers [Vananroye et al. (2006b)]. For p ~ 1, the confined droplet deforms more as compared with the model with increasing Ca (dashed lines and diamonds). For p > 1 (dashed-dotted lines and triangles in Fig. 4), even larger deviations between the bulk model predictions and the experimental values are present for the highly confined droplet. This indicates that the deviations between the bulk model and the behavior of confined droplets seem to grow with increasing viscosity ratio. The latter is in line with the enhanced breakup of highly confined and high viscosity ratio droplets [Vananroye et al. (2006b)]. Fig. 4. Comparison of experimentally determined dimensionless droplet axes with the MM-model as a function of Ca for p = 0.31, 1.07 and 2, and for 2R/d ~ 0.8. B. Droplet orientation 16

17 The orientation angle of confined and non-confined single droplets is also studied here as a function of capillary number and of viscosity ratio. The MM-model provides a simple expression [Eq. (7)] to calculate the orientation of single droplets in bulk shear flow and is capable of predicting qualitatively the orientation angle for p-values both above and below 1 over the whole range of capillary numbers. However, the MM-model slightly predicts more orientation towards the flow direction for p << 1 and less orientation towards the flow direction for p > 1, when compared to experimental results [Maffettone and Minale (1998)]. In Fig. 5, the steady-state orientation angle of two droplets is shown as a function of Ca for a viscosity ratio of The full line in Fig. 5(a) represents the prediction of the MM-model. This model reasonably predicts the orientation angle of the less confined droplet (2R/d = 0.18) up to Ca = 0.3. The highly confined droplet (2R/d = 0.77), however, orients more in the flow direction with increasing Ca. This is also illustrated in the microscopic images in Fig. 5(b). It can also be seen in Fig. 5(b), that the shape of the highly confined droplet starts to deviate from ellipsoidal. The droplet obtains a more sigmoidal shape with increasing Ca and large 2R/d. 17

18 Fig. 5. (a) Comparison of the experimentally determined orientation angle of two droplets (2R/d = 0.18 and 2R/d = 0.77) with the MM-model as a function of Ca for p = (b) Microscopic images for 2R/d = 0.18 and for 2R/d = 0.77 at Ca = The orientation angle of two droplets with a viscosity ratio above unity is shown in Fig. 6. The MM-model slightly predicts more orientation towards the flow direction for the unconfined droplet with 2R/d = 0.16, as was also the case in their own study [Maffettone and Minale (1998)]. For p = 2, the confined droplet is also oriented more towards the flow direction than the non-confined one. In summary, at low Ca, the differences between non-confined and confined droplets are still small, but with increasing Ca, the confined droplets orient more towards the flow direction than the non-confined ones. The 18

19 influence of the viscosity ratio on the deviations in orientation angle between confined and non-confined droplets is however less pronounced as compared to its effect on the droplet deformation. Fig. 6. Comparison of the experimentally determined orientation angle of two droplets (2R/d = 0.16 and 2R/d = 0.79) with the MM-model as a function of Ca for p = 2. C. Comparison with the model of Shapira and Haber As shown in the previous paragraphs, the model of Maffettone and Minale is capable of predicting quite accurately the shape and orientation of droplets in shear flow for any p up to 2R/d ~ 0.3. For higher 2R/d-values, the effects of confinement have to be taken into 19

20 account. Therefore, the results are compared with the analysis of Shapira and Haber (1990). Their theoretical analysis has the same basics as the theory of Taylor derived in the 1930 s for nearly spherical droplets at p ~ 1 [Taylor (1932)]. A correction term is added to the Taylor deformation to include wall and viscosity ratio effects [see Eq. (8)]. Figure 7 shows the deformation parameter of both an unconfined (2R/d = 0.18) and a confined droplet (2R/d = 0.69) with viscosity ratios near unity as a function of Ca. In this figure, the lines represent the predictions of the SH-model with θ = π/4. In the nonconfined case, the model is capable of predicting the experimental droplet deformation for all Ca. In the confined case, it is seen that the model predicts an increase in deformation with respect to the non-confined situation over the whole range of capillary numbers, which is consistent with the experimental observations. Similar observations for p = 1 were reported by Sibillo et al. (2006). However, because the SH-model is based on the deformation theory of Taylor (1932), it has two important drawbacks. Firstly, Taylor s theory is derived for small deformations with θ = π/4. However, it was shown that confinement does not only influence the droplet deformation but also affects the orientation angle. Therefore, the assumption of θ = π/4 could cause large mistakes, especially a high capillary numbers. Secondly, the deformation model of Taylor is derived for viscosity ratios near unity, where in the present experiments, viscosity ratios ranging from 0.3 to 5.2 are investigated. 20

21 Fig. 7. Comparison of the experimentally determined droplet deformation parameter for two degrees of confinement (2R/d = 0.18 and 2R/d = 0.69) with the SH-model as a function of Ca for p = 1. A possibility to partially overcome these drawbacks is to scale both the experimental and the theoretical deformation parameter D SH with sinθ. cosθ. With this scaling relation, the effects of the orientation angle will disappear from Eq. (8). Since the orientation angle θ is directly determined in each experiment, the experimentally obtained droplet deformation can easily be scaled. The experimentally determined scaled droplet deformation parameter is compared with the predictions of the SH-model as a function of the degree of confinement in Figs. 8 and 9. The results are shown for three different capillary numbers (Ca = 0.1, 0.2, and 0.3) and two viscosity ratios. In Fig. 8 (p = 0.3), it is seen that for capillary numbers of 0.1 and 21

22 0.2, the model is capable of describing the correct deformation over the whole confinement range. For Ca = 0.3, some small deviations are seen, which are already present at low 2R/d. The model underestimates the deformation somewhat at high capillary numbers over the whole confinement range. These deviations are probably due to the large deformation at Ca = 0.3 for low viscosity ratios, where the Taylor model is no longer valid. Figure 9 contains similar data for a system with a viscosity ratio of 5.2. Here, the experimentally obtained data nicely coincide with the predictions of the SHmodel for all degrees of confinement and all capillary numbers under investigation. When comparing the model predictions in Figs. 8 and 9, it can be concluded that, at low viscosity ratios, the model predicts less effect of confinement on the deformation for a specific Ca than at high viscosity ratios. This is in agreement with the results shown in Fig. 4, where a higher viscosity ratio resulted in larger deviations from bulk when the droplets are confined. 22

23 Fig. 8. Comparison of the experimentally determined scaled droplet deformation parameter with the scaled SH-model as a function of 2R/d for p = 0.3. Fig. 9. Comparison of the experimentally determined scaled droplet deformation parameter with the scaled SH-model as a function of 2R/d for p = 5.2. An alternative possibility to overcome the limitations of Taylor s theory for bulk flow, would be to replace the bulk deformation parameter from Taylor D Taylor [Eq. (9)] by a deformation parameter that is valid for all viscosity ratios and up to high capillary numbers. In this case, the deformation parameter from the MM-model could be used, resulting in a combined MMSH-model: 23

24 p R 3 DMMSH = DMM ( ), (10) 1+ p d in which D MM is calculated from Eqs. (4) and (5). In this approach, the dependence on the orientation angle again disappears, and a straightforward comparison with the experimentally obtained droplet deformation can be made. In Figs. 10 and 11, the experimentally determined droplet deformation is compared with the predictions of the combined MMSH-model as a function of the degree of confinement for two viscosity ratios. The dotted lines represent the results of the unconfined MMmodel for the appropriate Ca. For both viscosity ratios, the MMSH-model starts to deviate substantially from the MM-model for 2R/d around 0.4. For p = 0.3 (Fig. 10) it is seen that this approach is also capable of predicting the effect of confinement on the deformation. For larger capillary numbers (Ca > 0.2), the model however, overestimates the effect of confinement on the deformation. In Fig. 11 (p = 5.2), good agreement between the experimentally determined deformation and the MMSH-model is seen for all capillary numbers. In conclusion, it can be stated that confined droplet behavior can be reasonably predicted by the model of Shapira and Haber for a wide range of viscosity ratios and degrees of confinement, and for Ca

25 Fig. 10. Comparison of the experimentally determined droplet deformation parameter with the MMSH-model as a function of 2R/d for p = 0.3. Fig. 11. Comparison of the experimentally determined droplet deformation parameter with the MMSH-model as a function of 2R/d for p =

26 IV. Conclusions The effect of geometrical confinement on the dynamics of single droplets was investigated using a counterrotating parallel plate device. Individual PDMS droplets with specific sizes and viscosities were injected in a PIB matrix confined between the two plates. The dimensionless ratio of the droplet diameter to the gap spacing (2R/d) was used to quantify the degree of confinement. The deformation and the orientation of confined and non-confined droplets were microscopically determined during steady-state shearing. It was shown that for all p, both the deformation and the orientation of droplets can be nicely predicted by the phenomenological model of Maffettone and Minale as long as 2R/d is below 0.3. For higher 2R/d, the experimentally obtained droplet deformation exceeded the predictions of the MM-model for all viscosity ratios. The deviations from the model at high 2R/d are increasing with increasing viscosity ratio. It was also seen that the confined droplets were clearly more oriented towards the flow direction than their non-confined counterparts. The experimentally obtained droplet deformation was compared with the theoretical analysis of Shapira and Haber, which basically consists of the Taylor deformation with an additional deformation expressing the effects of confinement. To overcome the drawbacks of the Taylor theory (θ = π/4) for bulk flow, the bulk deformation part of the model was adapted in two ways. In a first approach, a scaled SH-model was proposed where the influence of the orientation angle disappeared from the model. A second approach was to replace the Taylor deformation parameter by the deformation parameter of Maffettone and Minale (1998) which is valid for all viscosity ratios. This resulted in a combined MMSH-model for confined droplets. Both 26

27 the scaled SH-model as well as the combined MMSH-model were capable of predicting the effects of confinement on the deformation for p-values both below and above one. The experimental as well as the theoretical results indicate that the differences between the non-confined and the confined deformation increase with increasing degree of confinement, increasing viscosity ratio, and increasing capillary number. Similar observations were made for the orientation angle: the differences between the nonconfined and the confined orientation angle increase with increasing degree of confinement and increasing capillary number. The viscosity ratio only has a limited effect on the orientation angle under confined conditions. Acknowledgments FWO Vlaanderen (project G ) and onderzoeksfonds K.U.Leuven (GOA 03/06) are gratefully acknowledged for financial support. References Cristini, V., S. Guido, A. Alfani, J. Blawzdziewicz, and M. Loewenberg, Drop breakup and fragment size distribution in shear flow, J. Rheol. 47, (2003). 27

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32 List of figure captions Fig. 1. Schematic of a deformed droplet with the geometrical parameters in shear flow: (a) velocity-vorticity plane; (b) velocity-velocity gradient plane. Fig. 2. (a) Comparison of experimentally determined dimensionless axes of two droplets (2R/d = 0.3 and 2R/d = 0.77) with the MM-model as a function of Ca for p = 5.2; (b) Microscopic images for 2R/d = 0.3 and 2R/d = 0.77 at Ca = 0.3. Fig. 3. Comparison of experimentally determined dimensionless droplet axes with the MM-model as a function of Ca for p = 0.31, 1.05 and 2.03, and for 2R/d < 0.2. Fig. 4. Comparison of experimentally determined dimensionless droplet axes with the MM-model as a function of Ca for p = 0.31, 1.07 and 2, and for 2R/d ~ 0.8. Fig. 5. (a) Comparison of the experimentally determined orientation angle of two droplets (2R/d = 0.18 and 2R/d = 0.77) with the MM-model as a function of Ca for p = (b) Microscopic images for 2R/d = 0.18 and for 2R/d = 0.77 at Ca = Fig. 6. Comparison of the experimentally determined orientation angle of two droplets (2R/d = 0.16 and 2R/d = 0.79) with the MM-model as a function of Ca for p = 2. 32

33 Fig. 7. Comparison of the experimentally determined droplet deformation parameter for two degrees of confinement (2R/d = 0.18 and 2R/d = 0.69) with the SH-model as a function of Ca for p = 1. Fig. 8. Comparison of the experimentally determined scaled droplet deformation parameter with the scaled SH-model as a function of 2R/d for p = 0.3. Fig. 9. Comparison of the experimentally determined scaled droplet deformation parameter with the scaled SH-model as a function of 2R/d for p = 5.2. Fig. 10. Comparison of the experimentally determined droplet deformation parameter with the MMSH-model as a function of 2R/d for p = 0.3. Fig. 11. Comparison of the experimentally determined droplet deformation parameter with the MMSH-model as a function of 2R/d for p =