Reduced model for droplet dynamics in shear flows at finite capillary numbers

We propose an extension of the Maffettone-Minale (MM) model to predict droplet dynamics in shear flow. The parameters of the MM model are traditionally retrieved in the framework of the perturbation theory for small deformations, i.e., small capillary numbers ($\text{Ca}\ll 1$) applied to Stokes equations. In this work, we take a novel route, in that we determine the model parameters at finite capillary numbers ($\text{Ca}\sim \mathcal{O}\left(1\right)$) without relying on perturbation theory results, while retaining a realistic representation in loading time and steady deformation attained by the droplet for different realizations of the viscosity ratio $\lambda$ between the inner and the outer fluids. This extended MM (EMM) model hinges on an independent characterization of the process of droplet deformation via fully three-dimensional numerical simulations of Stokes equations employing the immersed boundary-lattice Boltzmann numerical techniques. Issues on droplet breakup are also addressed and discussed within the EMM model.

Figure 1

Shear plane section of a droplet with viscosity $\lambda \mu$ deforming under the effect of a simple shear flow with intensity $\dot{\gamma }$ applied in an outer fluid with viscosity $\mu$. The droplet has an initial spherical shape with radius $R$ and deforms into a nonspherical shape due to the action of the shear flow. The shape of the droplet is described via a second-order tensor $\mathit{S}\left(t\right)$. The major $\left[L\right(t\left)\right]$ and minor $\left[B\right(t\left)\right]$ axes in the shear plane are used to compute the time dependent Taylor index $D\left(t\right)$, while $\theta \left(t\right)$ measures the inclination angle with respect to the streamflow direction ($x$-axis).

Figure 2

Time evolution of the Taylor index $D$ at increasing $\text{Ca}$ (lighter to darker colors). Time is made dimensionless via droplet time $\tau =\mu R/\sigma$. Numerical simulations data (points) are fitted with a stretched exponential [solid lines, Eq. (23)]. Different viscosity ratios $\lambda$ are considered. (a) $\lambda =0.2$ and $\text{Ca}=0.1$ ($▴$), $\text{Ca}=0.2$ ($▴$), $\text{Ca}=0.3$ ($▴$), $\text{Ca}=0.4$ ($▴$); (b) $\lambda =1$ and $\text{Ca}=0.1$ ($•$), $\text{Ca}=0.2$ ($•$), $\text{Ca}=0.3$ ($•$), $\text{Ca}=0.4$ ($•$); (c) $\lambda =2$ and $\text{Ca}=0.15$ ($\blacksquare$), $\text{Ca}=0.3$ ($\blacksquare$), $\text{Ca}=0.45$ ($\blacksquare$), $\text{Ca}=0.55$ ($\blacksquare$).

Figure 3

Comparison between numerical simulations data (points), a stretched exponential fit [solid lines, Eq. (23) with $\delta \ne 1]$ and pure exponential fit [dashed lines, Eq. (23) with $\delta =1]$ for the time evolution of the Taylor index $D$ for different $\lambda$ and selected values of $\text{Ca}$. Time is made dimensionless via droplet time $\tau =\mu R/\sigma$. (a) $\lambda =0.2, \text{Ca}=0.4$ ($▴$), $\delta =0.742$; (b) $\lambda =1, \text{Ca}=0.4$ ($•$), $\delta =0.723$; (c) $\lambda =2, \text{Ca}=0.55$ ($\blacksquare$), $\delta =0.796$.

Figure 4

(a) Characteristic dimensionless loading frequency $f_1^{\left(\mathrm{EMM}\right)}$ as a function of the capillary number $\text{Ca}$ [see Eq. (23)] for $\lambda =0.2$ ($▴$), $\lambda =1$ ($•$), $\lambda =2$ ($\blacksquare$). The horizontal dashed lines indicate the values of $f_1^{\left(\mathrm{MM}\right)}\left(\lambda \right)$ [see Eq. (7)], while solid lines refer to a polynomial fit for $f_1^{\left(\mathrm{EMM}\right)}$ [see Eq. (24) and Table 1]. (b) Stretching factor $\delta$ [see Eq. (23)] as a function of $\text{Ca}$ for different values of $\lambda$. The horizontal dashed line is drawn in correspondence of $\delta =1$.

Figure 5

Subcritical scaling for the dimensionless characteristic loading time ${\tau }_{\mathrm{load}}={\left[f_1^{\left(\mathrm{EMM}\right)}\right]}^{-1}$ for $\lambda =0.2$ ($▴$), $\lambda =1$ ($•$) and $\lambda =2$ ($\blacksquare$). Dashed lines indicate the scaling prediction with exponent $-1/2$ [70].

Figure 6

Stationary inclination angles for different $\lambda$. We report results of numerical simulations (points) for $\lambda =0.2$ ($▴$, panel (a)), $\lambda =1$ ($•$, panel (b)), $\lambda =2$ ($\blacksquare$, panel (c)), as well as the prediction of Eq. (10) with $f_1=f_1^{\left(\mathrm{MM}\right)}$ (MM, dashed lines) and with $f_1=f_1^{\left(\mathrm{EMM}\right)}$ (EMM, solid lines). The colored regions refer to the evaluated breakup regime.

Figure 7

Stationary droplet deformations for different $\lambda$. We report results of numerical simulations (points) for $\lambda =0.2$ ($▴$, panel (a)), $\lambda =1$ ($•$, panel (b)), $\lambda =2$ ($\blacksquare$, panel (c)), as well as the prediction of Eq. (9) with $f_{1,2}=f_{1,2}^{\left(\mathrm{MM}\right)}$ (MM, dashed lines) and the one with the values $f_1=f_1^{\left(\mathrm{EMM}\right)}$ and the optimal $f_2=f_2^{\left(\mathrm{EMM}\right)}$ that allow to match the numerical results (EMM, solid lines). In panel (b) data from Ref. [46] (Gounley et al., $◆$, Boundary Element Method) and Ref. [47] (Li et al., $⎔$, Volume of Fluid) are considered. The colored regions refer to the evaluated breakup regime.

Figure 8

We report the coefficient $f_2^{\left(\mathrm{EMM}\right)}$ as a function of the capillary number $\text{Ca}$ that allows to match the prediction of the steady deformation in Eq. (9) with numerical simulations data (points) (see Fig. 7) for $\lambda =0.2$ ($▴$), $\lambda =1$ ($•$), $\lambda =2$ ($\blacksquare$). The colored regions refer to a range of $\text{Ca}$ where breakup occurs. The heuristic correction suggested and reported in Eq. (34) of Ref. [16] is also displayed (Heuristic Correction, colored dashed lines). The limit where the affine deformation takes place ($f_2=1$) is indicated with a black dashed line; solid lines refer to a polynomial fit for $f_2^{\left(\mathrm{EMM}\right)}$ as in Table 1.

Figure 9

Time evolution of the Taylor index for different $\lambda$ and $\text{Ca}$. Numerical simulations data (points) are compared with the original MM model as in Eq. (2) with $f_{1,2}=f_{1,2}^{\left(\mathrm{MM}\right)}$ (MM, dashed lines). We also report the predictions of Eq. (2) with $f_{1,2}=f_{1,2}^{\left(\mathrm{EMM}\right)}$ (EMM, solid lines) that we evaluated from the loading times and the steady deformations (see Figs. 4 and 8). (a) $\lambda =0.2$ and $\text{Ca}=0.1$ ($▴$), $\text{Ca}=0.2$ ($▴$),$\text{Ca}=0.3$ ($▴$), $\text{Ca}=0.4$ ($▴$); (b) $\lambda =1$ and $\text{Ca}=0.1$ ($•$), $\text{Ca}=0.2$ ($•$), $\text{Ca}=0.3$ ($•$), $\text{Ca}=0.4$ ($•$); (c) $\lambda =2$ and $\text{Ca}=0.15$ ($\blacksquare$), $\text{Ca}=0.3$ ($\blacksquare$), $\text{Ca}=0.45$ ($\blacksquare$), $\text{Ca}=0.55$ ($\blacksquare$).

Figure 10

Droplet shapes in the shear plane for selected times indicated by arrows in Fig. 9. Numerical results for $\lambda =0.2$ ($▴$), $\lambda =1$ ($•$), $\lambda =2$ ($\blacksquare$) are compared with the ellipsoidal shape (dashed lines) obtained via the major/minor axes $L,B$ and inclination angle extracted from the inertia tensor in numerical simulations (see Sec. 2 for details). Colored areas indicate the ellipsoidal shapes obtained via the use of Eq. (2) with $f_{1,2}=f_{1,2}^{\left(\mathrm{EMM}\right)}$.