Deformation and breakup of viscoelastic droplets in confined shear flow

The deformation and breakup of Newtonian and viscoelastic droplets are studied in confined shear flow. Our numerical approach is based on a combination of lattice-Boltzmann models and finite difference schemes, the former used to model two immiscible fluids with variable viscosity ratio and the latter used to model the polymer dynamics. The kinetics of the polymers is introduced using constitutive equations for viscoelastic fluids with finitely extensible nonlinear elastic dumbbells with Peterlin's closure. We quantify the droplet response by changing the polymer relaxation time ${\tau }_P$, the maximum extensibility $L$ of the polymers, and the degree of confinement, i.e., the ratio of the droplet diameter to wall separation. In unconfined shear flow, the effects of droplet viscoelasticity on the critical capillary number ${\mathrm{Ca}}_{\mathrm{cr}}$ for breakup are moderate in all cases studied. However, in confined conditions a different behavior is observed: The critical capillary number of a viscoelastic droplet increases or decreases, depending on the maximum elongation of the polymers, the latter affecting the extensional viscosity of the polymeric solution. Force balance is monitored in the numerical simulations to validate the physical picture.

Figure 1

Shear plane ($xz$ plane at $y=L_y/2$) view of the numerical setup for the study of deformation and breakup of confined droplets. A Newtonian droplet (D) (phase $A$) with radius $R$ and shear viscosity ${\eta }_A$ is placed in between two parallel plates at distance $H$ in a Newtonian matrix (M) (phase $B$) with shear viscosity ${\eta }_B$. We then add a polymer phase with shear viscosity ${\eta }_P/{\eta }_P^{\prime }$ in the droplet-matrix (D-M) phase. We work with unitary viscosity ratio, defined in terms of the total (fluid+polymer) shear viscosity: $\lambda =({\eta }_A+{\eta }_P)/{\eta }_B=1$ in the case of droplet viscoelasticity and $\lambda ={\eta }_A/({\eta }_B+{\eta }_P^{\prime })=1$ in the case of matrix viscoelasticity [see Eqs. (1, 2, 3, 4)]. A shear $\dot{\gamma }=2U_{\mathrm{w}}/H$ is applied by moving the two plates in opposite directions with velocities $\pm U_{\mathrm{w}}$. The corresponding capillary number is given in terms of the matrix viscosity and surface tension $\sigma$ at the interface, $\mathrm{Ca}=\dot{\gamma }R{\eta }_M/\sigma$. In order to quantify the deformation of the droplet, we study the deformation parameter $D=(a-b)/(a+b)$, where $a$ and $b$ are the droplet semiaxes in the shear plane, and the orientation angle $\theta$ between the major semiaxis and the flow direction. Droplet deformation is benchmarked in a case of matrix viscoelasticity in Fig. 2. For large $\mathrm{Ca}$ the droplet deformation is increased and the droplet breaks at a critical capillary number ${\mathrm{Ca}}_{\mathrm{cr}}$. Droplet breakup will be analyzed for the case of droplet viscoelasticity.

Figure 2

(a) We report the steady-state deformation parameter $D$ for a Newtonian droplet under steady shear flow as a function of the associated capillary number $\mathrm{Ca}$. The viscosity ratio between the droplet phase and the matrix phase is kept fixed to $\lambda =1$. For small $\mathrm{Ca}$ the linearity of the deformation is captured by the numerical simulations, but the numerical results overestimate Taylor's prediction (unconfined droplet), being well approximated by the theoretical prediction of Shapira and Haber for a confined droplet [10]. (b) Steady-state orientation angle for a Newtonian droplet immersed in a viscoelastic matrix with Deborah numbers $\mathrm{De}=1.44$ and $\mathrm{De}=2.88$ and finite extensibility parameter $L^2={10}^4$. The results for the corresponding Newtonian system ($\mathrm{De}=0$) with the same viscosity ratio are also reported. The reference theory comes from the prediction of “ellipsoidal” models [12, 49], describing the dynamics of a single Newtonian droplet immersed in a viscoelastic matrix, based on the assumption that the droplet deforms into an ellipsoid.

Figure 3

Deformation and breakup after the startup of a shear flow with confinement ratio $2R/H=0.4$. We report the time history of droplet deformation and breakup including three time frames which represent, in the Newtonian case ($\mathrm{De}=0$), initial deformation (left column, $t=25{\tau }_{\mathrm{em}}$); deformation prior to breakup (middle column, $t=75{\tau }_{\mathrm{em}}$); postbreakup frame (right column, $t=100{\tau }_{\mathrm{em}}$). We use the droplet emulsion time ${\tau }_{\mathrm{em}}$ [see Eq. (8)] as a unit of time. The second row of images is related to a weakly viscoelastic droplet ($\mathrm{De}=0.2$), indicating that non-Newtonian properties do not affect the droplet deformation and breakup much. The third row of images is related to a viscoelastic droplet with Deborah number above unity ($\mathrm{De}=2.0$), and indicates that non-Newtonian properties stabilize the droplet deformation and inhibit droplet breakup. Note that the capillary number is kept fixed to the postcritical Newtonian value $\mathrm{Ca}=0.34$, which is the smallest capillary number available for us at which we observe breakup in the Newtonian case. In all cases, the viscosity ratio between the droplet phase and the matrix phase is kept fixed to $\lambda ={\eta }_D/{\eta }_M=1$, independently of the degree of viscoelasticity. The finite extensibility parameter is fixed to $L^2={10}^2$.

Figure 4

Deformation and breakup of a viscoelastic droplet in a Newtonian matrix after the startup of a shear flow with confinement ratio $2R/H=0.4$ and Deborah number $\mathrm{De}=2.0$, at changing the capillary number. The finite extensibility parameter is fixed to $L^2={10}^2$. The first row of images is just the last row of images in Fig. 3, corresponding to $\mathrm{Ca}=0.34$. The droplet deformation increases with increasing $\mathrm{Ca}$, and when $\mathrm{Ca}$ exceeds a critical value ${\mathrm{Ca}}_{\mathrm{cr}}$ between $0.34$ and $0.35$ the droplet breaks into two equally sized droplets (second row of images). The critical capillary number at $\mathrm{De}=2.0$ is close to the Newtonian counterpart ($\mathrm{De}=0$, see Fig. 3). In all cases, the viscosity ratio between the droplet phase and the matrix phase is kept fixed to $\lambda ={\eta }_D/{\eta }_M=1$, independently of the degree of viscoelasticity.

Figure 5

Deformation and breakup after the startup of a shear flow with confinement ratio $2R/H=0.78$. We report the time history of droplet deformation and breakup including three representative time frames, similarly to what is reported in Fig. 3. A distinctive feature of this confined case is the emergence of triple breakup [11]. The second row of images is related to a weekly viscoelastic droplet ($\mathrm{De}=0.2$), indicating that non-Newtonian properties do not affect the droplet deformation and breakup much. The third row of images is related to a viscoelastic droplet with Deborah number above unity ($\mathrm{De}=2.0$) and indicates that such non-Newtonian properties stabilize the droplet deformation and inhibit droplet breakup. Note that the capillary number is kept fixed to the postcritical Newtonian value $\mathrm{Ca}=0.47$, which is the smallest capillary number at which we observe breakup in the Newtonian case. In all cases, the viscosity ratio between the droplet phase and the matrix phase is kept fixed to $\lambda ={\eta }_D/{\eta }_M=1$, independently of the degree of viscoelasticity. The finite extensibility parameter is fixed to $L^2={10}^2$.

Figure 6

Deformation and breakup of a viscoelastic droplet in a Newtonian matrix after the startup of a shear flow with confinement ratio $2R/H=0.78$ and Deborah number above unity ($\mathrm{De}=2.0$). The finite extensibility parameter is fixed to $L^2={10}^2$. The first row of images is just the last row of images in Fig. 5, corresponding to $\mathrm{Ca}=0.47$. The droplet deformation increases with increasing $\mathrm{Ca}$ and when $\mathrm{Ca}$ exceeds a critical value ${\mathrm{Ca}}_{\mathrm{cr}}$ the droplet breaks (second row of images). The critical capillary number is found to be ${\mathrm{Ca}}_{\mathrm{cr}}=0.75$ and increases substantially compared to its Newtonian counterpart ($\mathrm{De}=0$, see Fig. 5). In all cases, the viscosity ratio between the droplet phase and the matrix phase is kept fixed to $\lambda ={\eta }_D/{\eta }_M=1$, independently of the degree of viscoelasticity.

Figure 7

Evolution of the dimensionless droplet length after the startup of a shear flow for various capillary numbers and Deborah numbers for a fixed confinement ratio $2R/H=0.78$ and finite extensibility parameter $L^2={10}^2$. Since the shape of highly deformed droplets may deviate from an ellipsoid, we estimated the droplet elongation from the projection of the droplet length ($L_p$) in the velocity direction. The viscosity ratio between the droplet phase and the matrix phase is kept fixed to $\lambda ={\eta }_D/{\eta }_M=1$, independently of the degree of viscoelasticity. Similarly to Figs. 3, 4, 5, 6, we use the droplet emulsion time ${\tau }_{\mathrm{em}}$ [see Eq. (8)] as a unit of time.

Figure 8

(a) Critical Capillary number for breakup as a function of confinement ratio for systems with finite extensibility parameter $L^2={10}^2$. The viscosity ratio between the polymeric droplet phase and the matrix phase is kept fixed to $\lambda ={\eta }_D/{\eta }_M=1$. Different Deborah numbers are considered, by letting the polymer relaxation time ${\tau }_P$ in Eq. (2) changing in the interval $0\le {\tau }_P\le 7000$ lbu. Black open circles indicate situations where multiple neckings occur. (b) Data analyzed in panel (a) are reported in terms of the dimensionless maximum elongation of the droplet $L_p^{\left(M\right)}/2R$. Panel (c): We plot the dimensionless size of the outer ($R_{\mathrm{out}}/R$) and inner ($R_{\mathrm{in}}/R$) daughter droplets in the triple breakup (see also Fig. 6). Up to $\mathrm{De}\approx 1$, droplets break into roughly equal sized daughter droplets, but for $\mathrm{De}>1$ there is substantial change in the size of these daughter droplets, as the size of inner (outer) daughter droplet starts decreasing (increasing) rapidly

Figure 9

(a) Evolution of the dimensionless droplet length after the startup of a shear flow for various capillary numbers. We fix both the Deborah number ($\mathrm{De}=2.0$), the confinement ratio ($2R/H=0.78$), and the finite extensibility parameter $L^2={10}^2$ [data already shown in Fig. 7]. (b) Same as panel (a) with an increased finite extensibility parameter, $L^2={10}^4$. The increase of $L^2$ determines different elongational properties and is affecting the critical capillary number. (c) The critical capillary number is reported as a function of the Deborah number for the two values of $L^2$ considered in panels (a) and (b). In the inset of panel (c) we also report, for completeness, the critical capillary number as a function of the Deborah number for different confinement ratios and fixed $L^2$. In all cases, the viscosity ratio between the polymeric droplet phase and the matrix phase is kept fixed to $\lambda ={\eta }_D/{\eta }_M=1$.

Figure 10

Influence of the finite extensibility parameter $L^2$ in the breakup after the startup of a shear flow with confinement ratio $2R/H=0.78$ and fixed Deborah number $\mathrm{De}=2.0$. We report the time history of droplet deformation and breakup including three representative snapshots: initial deformation (left column, $t=25{\tau }_{\mathrm{em}}$); deformation prior to breakup (middle column, $t=75{\tau }_{\mathrm{em}}$); postbreakup frame (right column, $t=100{\tau }_{\mathrm{em}}$). We use the droplet emulsion time ${\tau }_{\mathrm{em}}$ [see Eq. (8)] as a unit of time. In all cases, the viscosity ratio between the droplet phase and the matrix phase is kept fixed to $\lambda ={\eta }_D/{\eta }_M=1$, independently of the degree of viscoelasticity.

Figure 11

Feedback stress magnitude in the shear plane ($xz$ plane at $y=L_y/2$) for the viscoelastic data with $\mathrm{De}=2.0$ reported in Figs. 3 and 5. We use the droplet emulsion time ${\tau }_{\mathrm{em}}$ [see Eq. (8)] as a unit of time. In both confinement ratios, the capillary number is such that it corresponds to the postcritical condition for the corresponding Newtonian case ($\mathrm{De}=0$). Gradients in the polymeric stress are modulated in space and more pronounced in the confined case, which qualitatively explains the larger increase in the capillary number at breakup.

Figure 12

We report the forces contributions resulting from Eq. (1) in the shear plane ($xz$ plane at $y=L_y/2$). The capillary number is fixed, $\mathrm{Ca}\approx 0.32$, corresponding to steady states for all the cases studied. We project the viscous forces (${\mathit{F}}_{\nu }$), the pressure forces (${\mathit{F}}_p$), and the viscoelastic forces (${\mathit{F}}_{\mathrm{poly}}$, where applicable) of Eq. (1) in the radial direction at a given distance ($R/10$ lbu) from the interface. The force balance is then studied as a function of the angular position $\theta$ (see Fig 1) from the flow direction. Left panel figures are related to a confinement ratio $2R/H=0.4$, whereas the right panel ones refer to $2R/H=0.78$. Different Deborah numbers are considered. Panels (a) and (b) show the force balance for the Newtonian case ($\mathrm{De}=0$); Panels (c) and (d) show the viscoelastic case with $\mathrm{De}=2.0$ and $L^2={10}^2$. The vertical dashed lines show the orientation angle of the droplet, computed as the one of the equivalent ellipsoid. All forces are reported in lbu (LBM units).

Figure 13

We repeat the analysis of Fig. 12 for different values of the finite extensibility parameter $L^2$, by keeping the Deborah number fixed to $\mathrm{De}=2.0$ and the capillary number fixed to $\mathrm{Ca}=0.32$. The confinement ratio is kept fixed to $2R/H=0.78$. All forces are reported in lbu (LBM units).