Numerical simulation of secondary breakup of shear-thinning droplets

A schematic diagram of the simulation is shown in Fig. 1. An axisymmetric domain is adopted to reduce the computation cost, similar to many previous studies Cao et al. (2022); Feng (2010); Jain et al. (2019); Marcotte and Zaleski (2019). The axisymmetric setting can allow us to consider the initial spherical shape of the droplet and the axisymmetric deformation. However, once the droplet breaks up, the process is three-dimensional, and 2D axisymmetric simulation cannot fully capture the process. Therefore, with the axisymmetric setting, we mainly consider the deformation of the droplet before the breakup. Even though three-dimensional (3D) simulation is possible and adopted by some studies Yang et al. (2017, 2016); Chu et al. (2020); Tavangar, Hashemabadi, and Saberimoghadam (2015), 2D simulation allows us to use a very fine mesh to capture the details of the flow and droplet deformation, which is important for us to unveil the effects of the shear-thinning rheology of the droplet.

The left boundary of the domain is set as uniform velocity inflow ${U_{{r}}}$. The right and the top boundaries are set free outflow, i.e., a Neumann condition for velocity ${{\partial{\bf{u}}}}/{{\partial{\bf{n}}}}=0$ and a Dirichlet condition for pressure $p=0$. The droplet has an initial diameter of $D=4$ mm, and it is initially located on the axis 10$D$ away from the left boundary (see the Supplementary Material for the effect of the distance from the left boundary). At the beginning of the simulation, the velocity in the whole domain is set to zero. Then the droplet gradually deforms in the air stream. To quantify the droplet deformation, we use the cross-stream diameter ${D_{{\rm{cro}}}}$ (i.e., the droplet diameter in the cross direction). This parameter is normalized by the initial droplet diameter as ${D_{{\rm{cro}}}}/D$.

The system contains a droplet of a shear-thinning fluid exposed to the airflow. The droplet and air are considered as incompressible fluids with constant densities. Similar to many previous studies Cao et al. (2022); Meng and Colonius (2017), mass transfer, heat transfer, and gravity are neglected in the simulation because it is very weak in most applications of droplet deformation and breakup processes. With the VOF method, the flow field can be modeled by a single continuum Hirt and Nichols (1981). The continuity equation, the momentum equation, and the transport equation of the volume fraction Berberovic (2010) are solved

$$\nabla\cdot{\bf{u}}=0$$

(1)

$$\rho\left({\frac{{\partial{\bf{u}}}}{{\partial t}}+\nabla\cdot{\bf{uu}}}\right)=-\nabla p+\nabla\cdot(2\mu{\bf{D}})+\sigma\kappa{\bf{n}}{\delta_{S}}\$$

(2)

$$\frac{{\partial f}}{{\partial t}}+\nabla\cdot({\bf{u}}f)=0\$$

(3)

where ${\bf{u}}$ is the velocity vector; $\mu$ is the dynamic viscosity; ${\bf{D}}$ is the deformation rate tensor given by ${\bf{D}}=\left[{\nabla{\bf{u}}+{{(\nabla{\bf{u}})}^{\bf{T}}}}\right]/2$, and $\sigma$ is the surface tension coefficient. In the last item of Eq. (2), $\sigma\kappa{\bf{n}}{\delta_{S}}$ represents the surface tension force of the gas-liquid interface. In Eq. (3), $f$ is the volume fraction of the liquid in the control volume. It is $f$ = 1 if the control volume is full of liquid, and $f$ = 0 if full of gas. $0<f<1$ indicates the existence of an interface in the control volume.

The density and the viscosity of the fluid are expressed as:

$$\rho={\rho_{{L}}}f+{\rho_{{G}}}(1-f)$$

(4)

$$\mu={\mu_{{L}}}f+{\mu_{{G}}}(1-f)$$

(5)

where the subscripts $G$ and $L$ denote the gas (air) and the liquid (shear-thinning or Newtonian droplet) phases, respectively.

The power-law fluid model is adopted to describe the rheology of the shear-thinning fluid Arnold et al. (2010):

$${\mu_{{\rm{eff}}}}=K{\dot{\gamma}^{n-1}}$$

(6)

where ${\mu_{{\rm{eff}}}}$ is the effective viscosity, $\dot{\gamma}$ is the shear rate, $K$ is the consistency index, and $n$ is the flow index of the power-law fluid. When $n=1$, Eq. (6) describes a Newtonian fluid. When $0<n<1$ the fluid is shear-thinning, and when $n>1$ the fluid is shear-thickening. In this study, the flow index $n$ and the consistency index $K$ are varied in the range of $0.4<n<1$ and $0.00569<K<26.785$, respectively. The other properties of shear-thinning fluids, Newtonian fluids, and air used in the simulation are listed in Table 1, except otherwise explicitly specified. The density and the surface tension of the droplet are set according to the properties of water. The viscosity of the Newtonian fluid is calculated according to the Ohnesorge number defined in Eq. (12). The governing equations are solved by the open-source multiphase flow solver Basilisk Popinet . Quadtree space discretization is adopted for adaptive mesh refinement (AMR) Lagrée, Staron, and Popinet (2011); Popinet (2003, 2009), which dynamically changes the grid structure according to the local interface shape and the velocity field. Hence, it can increase the simulation resolution without significantly increasing the computational demand.

To quantify the influence of shear-thinning rheological properties on the deformation and breakup process, several dimensionless numbers are used, including the Ohnesorge number, the Weber number, and the dimensionless time. The Ohnesorge number $Oh$ and the Weber number We can be obtained by nondimensionalize the momentum equation (2). By introducing the following variables, ${\bf{x}}^{*}={\bf{x}}/D$, ${\bf{u}}^{*}={\bf{u}}/{U_{{r}}}$, $t^{*}=t{U_{{r}}}/D$, $p^{*}=p/({\rho_{{L}}}{U_{{r}}}^{2})$, $\kappa^{*}=D\kappa$, ${\delta^{*}_{S}}=D{\delta_{S}}$, $\mu^{*}=\mu/{\mu_{\text{eff}}}$, $\rho^{*}=\rho/{\rho_{{L}}}$, the momentum equation can be nondimensionalized,

		$\displaystyle{\frac{{\partial{\bf{u}}^{*}}}{{\partial t^{*}}}+{\bf{u}}^{*}\cdot\nabla{\bf{u}}^{*}}=-\nabla p^{*}$		(7)
	$\displaystyle+$	$\displaystyle\frac{Oh}{\sqrt{\mbox{{We}}}}\nabla\cdot\mu^{*}\left({\nabla{\bf{u}}^{*}+{{\nabla{\bf{u}}^{*}}^{\bf{T}}}}\right)+\frac{1}{\mbox{{We}}}\kappa^{*}{\bf{n}}{\delta_{S}^{*}}.$

It should be noted that in the definition of the Ohnesorge number, the viscosity of the fluid is involved, which is not a constant for the shear-thinning droplet and depends on the shear rate. Hence, the Ohnesorge number $Oh$ in Eq. (7) is defined in Eq. (12), which can consider the characteristic effective viscosity at the characteristic shear rate in the shear thinning droplet, and the detail will be discussed in Sec. III.1. The Weber number We in Eq. (7) represents the ratio of inertia to surface tension force,

$$\mbox{{We}}=\frac{{{\rho_{{G}}}{U_{{r}}}^{2}D}}{\sigma}.$$

(8)

The Weber number in the simulations is varied by changing the initial relative velocity between gas and liquid phases (${U_{{r}}}$). Instead of using $t^{*}$ directly, the dimensionless time is defined following Nicholls and Ranger Nicholls and Ranger (1969):

$$T=\sqrt{\frac{{{\rho_{{G}}}}}{{{\rho_{{L}}}}}}\frac{{{U_{{r}}}}}{D}t.$$

(9)

Hence, $T=\sqrt{{\rho_{G}}/{\rho_{L}}}t^{*}$.

In the simulation, the size of the computational domain needs to be large enough that the selection of the boundary does not affect the simulation results. To select the proper size of the computational domain, four sizes were tested and the results are shown in Fig. 2. The comparison shows that the difference in the early stage of droplet deformation is indistinguishable. As the droplet deforms, the differences gradually become noticeable ($T=1.28$). The droplet morphology after the breakup ($T=1.46$) shows that the domain size of $L=20D$ is sufficient to capture the whole process of droplet deformation and breakup, therefore, $L=20D$ is selected as the default domain size for further simulations.

The mesh size in the simulation determines the accuracy and numerical stability, which is crucial for capturing the liquid-gas interface during the droplet deformation and breakup. The adaptive mesh refinement is performed as shown in Fig. 3(a). Compared with static global mesh refinement, the computation time and storage cost of adaptive mesh refinement can be greatly reduced. We progressively increase the maximum level of refinement and compare the results. Fig. 3(b) shows the contours of droplets at four mesh resolutions of $D/\Delta x$ = 51, 102, 204, and 408, where $\Delta x$ is the minimum mesh size. The results show that when $D/\Delta x$ is greater than 204, the droplet deformation and breakup are almost independent of the mesh size. Therefore, $D/\Delta x=204$ is used in the following simulations to study the deformation and breakup of droplets.

The numerical results are validated against experimental data in Ref. 27, as shown in Fig. 4. The droplet used in Ref. 27 is a 0.2% CMC solution, which is a shear-thinning fluid. The deformation and breakup process of a droplet with $\mbox{{We}}=16$ from the experiment and simulation is compared in Fig. 4(a). The numerical result is consistent with the experimental result and can reproduce the bag breakup process. In addition, for the multimode breakup, the evolution of the cross-stream diameter ${D_{{\rm{cro}}}}/D$ of two droplets ($Oh=0.065$ and 0.11) at We = 33 from the experiment and simulations is shown in Fig. 4(b). It can be seen that this numerical model can capture the morphological changes of the droplets reasonably well.

To compare the breakup between Newtonian and shear-thinning droplets, two approaches are widely used. One is to simply change the flow index $n$. If $n=1$ the fluid is Newtonian, and if $n<1$ the fluid is shear-thinning. This approach has been widely used in many studies Cao et al. (2022); Kant and Banerjee (2022); Qian et al. (2021). The breakup process of two droplets using this approach is compared in Fig. 5(a). We can see that as the fluid changes from the Newtonian to shear-thinning ($n=0.4$), there is a significant change in the droplet breakup process. The two breakup processes are in different modes at the same We, i.e., the Newtonian droplet is in the bag breakup regime while the shear-thinning droplet is in the multimode breakup regime. This is because in this approach, the magnitudes of viscosities of the two droplets are significantly different. When comparing Newtonian and shear-thinning fluids, it is important to have the same characteristic viscosity. To keep the relative importance of the viscosity the same, a second approach is widely used. Because of the shear-thinning property, the effective viscosity in the droplet is different everywhere. Therefore, a characteristic viscosity should be selected, corresponding to a characteristic shear rate. Many studies used ${U_{{r}}}/D$ as the characteristic shear rate Chu et al. (2020); Kant and Banerjee (2022). By using this, we can maintain the same effective viscosity by adjusting $K$ when we change $n$. The comparison between a Newtonian droplet and a shear-thinning droplet by this approach is shown in Fig. 5(b). We can see that the two breakup processes are also in different modes, i.e., the Newtonian droplet is in the bag breakup regime while the shear-thinning droplet is in the vibrational regime. Moreover, the minimum effective viscosity ${\mu_{\min}}$ of the shear-thinning droplet is actually always larger than that of the Newtonian droplet as shown in Fig. 5(c). This indicates that the shear-thinning droplet is too more viscous than the Newtonian droplet, and it is not reasonable to compare these two cases by this approach.

To make a reasonable comparison between Newtonian and shear-thinning droplets, we consider this process by examining the characteristic shear rate. For droplet breakup, the characteristic velocity that determines the characteristic shear rate should be the characteristic velocity in the droplet ${U_{{L}}}$, not the characteristic velocity in the air ${U_{{G}}}$. Therefore, the characteristic shear rate ($\dot{\gamma}$) should be $\dot{\gamma}={U_{{L}}}/D$. Since the flow velocity in the droplet is induced by the external airflow, we can estimate the characteristic internal velocity of the droplet by considering the inertia of the air and the droplet fluid. Compared with other forces (such as the viscous force), the kinetic energy of the air is the main contribution to the deformation and the breakup of the droplet. Therefore, the inertia of the air should be at the same scale as the inertia of the droplet fluid, ${\rho_{{G}}}{U_{{G}}}^{2}/2\sim{\rho_{{L}}}{U_{{L}}}^{2}/2$, where ${U_{{G}}}$ and ${U_{{L}}}$ are the characteristic velocities in the air and the liquid. As the airflow velocity ${U_{{G}}}$ can be characterized by the relative airflow velocity ${U_{{r}}}$, we can then have ${U_{L}}={U_{{r}}}\sqrt{{{{\rho_{{G}}}}}/{{{\rho_{{L}}}}}}$. Therefore, the characteristic shear rate can be estimated as

$$\dot{\gamma}=\sqrt{\frac{{{\rho_{{G}}}}}{{{\rho_{{L}}}}}}\frac{{{U_{{r}}}}}{D}$$

(10)

Then, the characteristic effective viscosity of the droplets in the deformation and breakup process can be written as

$${\mu_{{\rm{eff}}}}=K{\left({\sqrt{\frac{{{\rho_{{G}}}}}{{{\rho_{{L}}}}}}\frac{{{U_{{r}}}}}{D}}\right)^{n-1}}$$

(11)

Hence, the Ohnesorge number can be expressed as:

$$Oh=\frac{{{\mu_{{\rm{eff}}}}}}{{\sqrt{{\rho_{{L}}}D\sigma}}}=\frac{K}{{\sqrt{{\rho_{{L}}}\sigma{D^{2n-1}}U_{{r}}^{2-2n}{{({\rho_{{G}}}/{\rho_{{L}}})}^{1-n}}}}}$$

(12)

For Newtonian fluids (i.e., $n=1$ and $K={\mu_{{L}}}$), the Ohnesorge number recovers the original definition. Since both $K$ and $n$ affect the Ohnesorge number, in some simulations, to maintain the same $Oh$ while changing $n$ or ${U_{{r}}}$, we change $K$ accordingly.

Based on the above selection of the characteristic shear rate and the definition of the Ohnesorge number, the deformation and breakup processes of Newtonian droplets and shear-thinning droplets are considered under the same We number and same $Oh$ number, i.e., the same characteristic shear rate and the same effective viscosity. Two typical cases are considered first, one in the bag breakup regime ($\mbox{{We}}=20$, $Oh=0.10746$, Fig. 6), and the other in multimode breakup regime ($\mbox{{We}}=30$, $Oh=0.010746$, Fig. 8). The comparison can show that, using this proposed approach, the processes for Newtonian and shear-thinning droplets are comparable. Therefore, the current selection of the characteristic shear rate and the definition of the Ohnesorge number is better than the previous approaches.

For the bag breakup ($\mbox{{We}}=20$, $Oh=0.10746$), in the early stage, the difference in the deformation between the two droplets is small as shown in Fig. 6(a) (Multimedia view). Comparing the velocity fields between the two droplets, we can see that the Newtonian droplet has a high-velocity region on the windward side near the rim, and the internal velocity field is more uniform for the shear-thinning droplets as shown in Fig. 6(b). This will lead to differences in the later deformation of the droplet. In addition, the change rates of ${D_{{\rm{cro}}}}/D$, and ${U_{{\rm{droplet}}}}/{U_{{r}}}$ in these two droplets are almost the same as shown in Fig. 6. While in the later stage, compared with Newtonian fluids, the deformation of the shear-thinning droplet near the center increases, and the center of droplets becomes thinner. The change rate of the cross-stream diameter and the centroid velocity of the shear-thinning droplet is also slower than that of the Newtonian droplet, as shown in Figs. 6(c) and 6(d).

To analyze the origin of the difference between the Newtonian droplet and the shear-thinning droplet in the bag breakup process, the effective viscosity at the droplet center (i.e., the center point on the axis) ${\mu_{{\rm{center}}}}$ and the minimum effective viscosity ${\mu_{\min}}$ of the shear-thinning droplet are shown in Fig. 7(a). In the later stage, due to the shear-thinning property, ${\mu_{{\rm{center}}}}$ and ${\mu_{\min}}$ are much smaller than the viscosity of the Newtonian droplet. The distribution of the effective viscosity in the shear-thinning droplet is shown in Fig. 7(b). Even though the characteristic viscosity of the shear-thinning droplet (the effective viscosity at the characteristic shear rate $\dot{\gamma}$ in Eq. (10)) is the same as that of the Newtonian droplet, the effective viscosity is varied within the shear-thinning droplet, which affects the deformation of the droplet. There is a low-viscosity region near the center of the shear-thinning droplet, which enhances the deformation of the droplet center. The position of the minimum effective viscosity is mainly near the center of droplets as shown in Fig. 7(b). Compared with the Newtonian droplet, the effective viscosity around the center of the shear-thinning droplet decreases, and the deformation of the droplet increases.

For the multimode breakup ($\mbox{{We}}=30$, $Oh=0.010746$), the difference in the droplet shape between the Newtonian droplet and the shear-thinning droplet in the early stage is also small, as shown in Fig. 8(a) (Multimedia view). By comparing the velocity fields of the two droplets, we can see that the high-speed region is distributed near the rim of the shear-thinning droplet, while it is mainly on the windward side of the rim of the Newtonian fluid, as shown in Fig. 8(b). Such difference will lead to different extent of deformation near the rim of the droplets. In addition, this can also be shown quantitatively by ${D_{{\rm{cro}}}}/D$, and ${U_{{\rm{droplet}}}}/{U_{{r}}}$ in Figs. 8(c) and 8(d). In the later deformation stage, the radial stretching of the shear-thinning droplet near the rim is stronger than that of the Newtonian droplet, while the Newtonian droplet shows stronger forward stretching near the rim and higher overall acceleration. Hence, the change rates of ${{D}_{\rm{cro}}}/D$ and ${{U}_{\rm{droplet}}}/{{U}_{{r}}}$ of the Newtonian droplet are larger than that of the shear-thinning droplet.

To explain the differences between the two fluids in the multimode breakup process, the effective viscosity at the droplet center ${{\mu}_{\rm{center}}}$ and the minimum effective viscosity ${{\mu}_{\min}}$ of the shear-thinning droplet are shown in Fig. 9(a). The minimum effective viscosity ${{\mu}_{\min}}$ inside the shear-thinning droplet is much smaller than that of the Newtonian fluid and ${{\mu}_{\rm{center}}}$ is close to the viscosity of the Newtonian droplet. In addition, the distribution of the local viscosity within the shear-thinning droplet is shown in Fig. 9(b), which shows that the viscosity distribution in the shear-thinning droplet is very different from the Newtonian droplet. In the later stage, due to the shear-thinning property, the local viscosity in the outer region is much smaller than that in the inner region, as shown in Fig. 9(b). Compared with the bag breakup case, the effective viscosity is smaller and the aerodynamic effect is stronger in this case. There is also a low-viscosity region in this case but it has transferred from the center to the rim of the shear-thinning droplet, as shown in Fig. 9(b). The low-viscosity region near the rim of the shear-thinning droplet enhances the deformation of the droplet rim. Therefore, the shear-thinning droplet has a strong radial stretching around the rim than the Newtonian droplet.

The influence of the rheological parameters on the deformation and breakup mode in the process of droplet breakup is analyzed by changing $n$. The value of $Oh$ is fixed to ensure that the effective viscosity in the simulation is comparable. The effect of $n$ on the droplet deformation in the bag breakup mode is analyzed by decreasing $n$ from 1 to 0.4, while keeping $\mbox{{We}}=20$ and $Oh=0.10746$. As the value of $n$ decreases, the change rates of ${{D}_{\rm{cro}}}/D$ and ${{U}_{\rm{droplet}}}/{{U}_{{r}}}$ first increase slightly and then decrease, and the maximum change rates of these two parameters are at $n=0.9$ as shown in Figs. 10(a) and 10(b).

This effect can be explained by the shear-thinning property of the droplet. The smaller the value of $n$ is, the stronger the shear-thinning effect is. Hence, as $n$ decreases, the local effective viscosity inside droplets has a significant change, and the minimum effective viscosity ${{\mu}_{\min}}$ decreases as shown in Fig. 10(c). In addition, the difference in the viscosity within the droplet is more obvious as shown in Fig. 10(d). Because the low-viscosity region is mainly located near the droplet center, the deformation near the droplet center is gradually enhanced. In contrast, the high viscosity around the rim of the droplet hinders the deformation of the droplet. In addition, when $n$ approaches 1, the viscosity distribution is almost uniform, and there is no easy-to-deform region. Hence, without strong local deformation, the deformation of the droplet is also relatively slow when $n$ approaches 1. Therefore, the shear-thinning property causes the transition of the change rates of ${{D}_{\rm{cro}}}/D$ and ${{U}_{\rm{droplet}}}/{{U}_{{r}}}$.

The effect of $n$ on the multimode breakup process (We = 30, $Oh$ = 0.010746) is also analyzed. As $n$ increases, the change rates of ${{D}_{\rm{cro}}}/D$ and ${{U}_{\rm{droplet}}}/{{U}_{{r}}}$ also first increase and then decrease, and the maximum change rates of these two parameters are at $n=0.6$ as shown in Figs. 11(a) and 11(b). For this case, as $n$ decreases, the shear-thinning effect increases, and the minimum effective viscosity ${{\mu}_{\min}}$ also decreases as shown in Fig. 11(c). In addition, the low-viscosity region is mainly located near the droplet rim and the differences between high-viscosity and low-viscosity regions are significant as shown in Fig. 11(d). As $n$ decreases, the radial stretching of the droplet at the rim is gradually enhanced. While the high viscosity region near the center of droplets also hinders the deformation. In addition, when $n$ approaches 1, the deformation is also hindered because the viscosity distribution is almost uniform and there is no easy-to-deform region. Therefore, as $n$ decreases, there is a change for the rim, i.e., from the forward stretching to the radial stretching. This effect can also be explained by the shear-thinning property. When $n$ decreases, the local effective viscosity decreases due to high local velocity and high local shear rate. Therefore, with the decrease of the local effective viscosity, the rim of the droplet is more likely to be accelerated and make stronger radial stretching at a smaller $n$ value.

The effect of the Weber number We on the deformation and breakup mode of shear-thinning droplets is analyzed in this section. To consider the effect of We, we fixed other parameters ($Oh$ and $n$) and changed the airflow velocity ${{U}_{{r}}}$ from 8.54 to 22.59 m/s (that is, We from 5 to 35) to explore its influence. Meanwhile, to maintain the constant $Oh$ as ${{U}_{{r}}}$ changes, we change $K$ accordingly. Then, we change the Weber number We and the power-law index $n$, and produce a regime map for the breakup mode in the We–$n$ space, as shown in Fig. 12. We can see that the increase of We alters the breakup mode: as the We increases, the droplet changes from the vibrational to the bag breakup, and finally to the multimode breakup. This could be attributed to the higher inertia at large We, which accelerates the droplet deformation remarkably. In addition, the results show that $n$ does not influence the breakup mode. This is mainly because we have considered the effect of $n$ on the viscosity by using a modified definition of the Ohnesorge number by Eq. (12). Hence, when we change $n$, the overall effect of the viscosity does not change because $Oh$ is fixed at a low value and the effect of $n$ is mainly on the local viscosity distribution and the local deformation of the droplet. We will see that the change of $n$ does affect the breakup mode when the $Oh$ number is large in Section III.4.

To further see the effect of the Weber number on the droplet deformation and breakup quantitatively, we vary the droplet Weber number and analyze the cross-stream diameter and the centroid velocity of the droplet, as shown in Fig. 13. The change rates of ${{D}_{\rm{cro}}}/D$ and ${{U}_{\rm{droplet}}}/{{U}_{{r}}}$ increase significantly as We increases, as shown in Figs. 13(a) and 13(b). This is because of the stronger effect of the inertia, the differences between high-viscosity and low-viscosity regions are significant and ${{\mu}_{\min}}$ decreases with the increase of We, as shown in Fig. 13(c), which accelerates the droplet deformation. The distribution of the effective viscosity is shown in Fig. 13(d). We can see that, as the Weber number increases, the low-viscosity region gradually transits from the center region to the rim region of the droplet. As the low-viscosity region moves, the deformation of the droplet is also affected.

The effects of the Ohnesorge number $Oh$ on the deformation process and breakup mode of shear-thinning droplets are analyzed in this section. To consider the effect of $Oh$, we fix other parameters (We and $n$) and change the consistency index $K$ to explore its influence. Then, we vary $Oh$ and $n$ to produce a regime map in the $Oh$–$n$ space, as shown in Fig. 14. Three modes occur as $n$ and $Oh$ vary, namely multimode breakup, bag breakup, and vibrational mode. When $Oh$ is small and $n$ is large, the breakup is a multimode breakup, which is mainly because the effective viscosity is small and the droplet deforms easily. As $Oh$ increases, the low-viscosity region transfers from the rim to the center of the droplet, and the breakup is in the bag breakup mode. When $Oh$ is large and $n$ is small, the effective viscosity is very large. Hence, the droplet only vibrates instead of breakup. From Fig. 14, we can also see that as $n$ increases, the transitional $Oh$ for the change in the breakup mode increases. This is mainly because when $n$ is large, the value of $K$ needs to be increased to reach the same breakup mode, then the $Oh$ number increases accordingly.

To study the effect of the Ohnesorge number on droplet deformation and breakup quantitatively, we vary the Ohnesorge number and analyze the cross-stream diameter and the centroid velocity of the droplet. As $Oh$ increases, the change rates of ${{D}_{\rm{cro}}}/D$ and ${{U}_{\rm{droplet}}}/{{U}_{{r}}}$ of droplets decrease significantly, as shown in Figs. 15(a) and 15(b). This is because of the higher viscous stress within the droplet at large $Oh$, which inhibits the droplet deformation remarkably. To confirm this, the minimum effective viscosity ${{\mu}_{\min}}$ for different $Oh$ at $T$ = 1.97 is given in Fig. 15(c). When $Oh$ is small and as it increases, the aerodynamic force dominates the deformation of the droplet and the minimum effective viscosity ${{\mu}_{\min}}$ has a small increase. With the further increase of $Oh$, ${{\mu}_{\min}}$ significantly increases as shown in Fig. 15(c). In addition, the distribution of the effective viscosity within the shear-thinning droplets is plotted in Fig. 15(d). With the increases of $Oh$, the effective viscosity of the droplet increases, and the low-viscosity region inside the droplet transits from the rim of the droplet to the center. Hence, a high-viscosity region appears on the leeward side of the droplet and prohibits the deformation of the droplet. Therefore, the deformation and breakup mode of droplets are also affected.