4.2.1. Effect of Ca

It is known that in a Newtonian fluid system the droplet would eventually reach a steady shape and slide on the solid wall with a constant velocity at low values of $Ca$ (Ding et al. Reference Ding, Gilani and Spelt2010; Liu et al. Reference Liu, Zhang, Ba, Wang and Wu2020). In N/V and V/N systems, we run the simulations with $Wi=1.0$, $\beta =0.5$ and $m=1.0$ for varying $Ca$. Like in the Newtonian fluid system, the steady-state droplet is achieved in both N/V and V/N systems as $Ca$ is increased up to 0.35. Three dimensionless parameters are introduced to quantify the steady-state droplet deformation, namely the relative wetting area ${{A}_{r}}={(A-{{A}_{0}})}/{{{A}_{0}}}$, relative droplet height ${{h}_{r}}={(h-{{h}_{0}})}/{{{h}_{0}}}$ and relative droplet surface area ${{S}_{r}}={(S-{{S}_{0}})}/{{{S}_{0}}}$, which are plotted against $Ca$ in figure 3. Herein, the variables $A$, $h$ and $S$ are the wetting area, droplet height and droplet surface area in the steady state, and their corresponding variables in the initial state are ${{A}_{0}}$, ${{h}_{0}}$ and ${{S}_{0}}$, respectively. In figure 3(b), an inset is included to show the steady-state droplet shapes for a representative capillary number of $Ca=0.35$ at the x–z mid-plane (upper half) and x–y bottom plane (lower half) in the N/N, N/V and V/N systems, where the droplet shapes are represented by the contours ${{\rho }^{N}}=0$.

Figure 3.

(a) The steady-state relative wetting area ${{A}_{r}}$ and relative droplet height ${{h}_{r}}$ and (b) the relative droplet surface area ${{S}_{r}}$ as functions of $Ca$ in three different systems. The inset in (b) shows the steady-state droplet shapes for a representative capillary number of $Ca=0.35$ at the x–z mid-plane (upper half) and x–y bottom plane (lower half) in the N/N, N/V and V/N systems, represented by solid, dashed and dash-dot-dot lines, respectively. The other simulation parameters are $Wi=1.0$, $\beta =0.5$ and $m=1.0$.

With an increase of $Ca$ in the N/N system, the droplet deforms into a bulge shape in the shear direction (see the inset in figure 3b). The droplet surface area ${{S}_{r}}$ and the wetting area ${{A}_{r}}$ obviously increase, while the droplet height ${{h}_{r}}$ decreases in a small range (figure 3a,b). In the N/V system, ${{S}_{r}}$, ${{A}_{r}}$ and ${{h}_{r}}$ exhibit the same variations in trend with $Ca$ as those in the N/N system. However, the increase in ${{S}_{r}}$ is smaller, ${{A}_{r}}$ spreads more in the $x$ direction and ${{h}_{r}}$ is reduced more rapidly, as compared with the results in the N/N system. In terms of the V/N system, the droplet deforms the least with minimum variations in ${{A}_{r}}$, ${{S}_{r}}$ and ${{h}_{r}}$ among all three systems. In addition, the curvature radius of the contact line appears always larger at the front than at the rear, especially for higher $Ca$ (see the inset in figure 3b) which is consistent with previous findings in a Newtonian system (Ding et al. Reference Ding, Gilani and Spelt2010) and in a surfactant-covered-droplet system (Liu et al. Reference Liu, Zhang, Ba, Wang and Wu2020).

To understand the viscoelastic effect on the steady-state droplet deformation, we compute the traces of the conformation tensor, defined by $\text {tr}{{\boldsymbol{\mathsf{A}}}}={{A}_{xx}}+{{A}_{yy}}+{{A}_{zz}}$, at $Ca=0.15$ in N/V and V/N systems, and the results are displayed in figure 4. Since $\text {tr}{{\boldsymbol{\mathsf{A}}}}$ is directly proportional to the extension of the polymer molecules (Verhulst et al. Reference Verhulst, Cardinaels, Moldenaers, Renardy and Afkhami2009), one can observe two obvious extension regions, i.e. (I) and (II), in the N/V system (see figure 4a). Region (I) is located near and outside the bulge tip of the droplet, similar to that reported by Verhulst et al. (Reference Verhulst, Cardinaels, Moldenaers, Renardy and Afkhami2009) and Afkhami, Yue & Renardy (Reference Afkhami, Yue and Renardy2009) where the deformation of a Newtonian droplet suspended in viscoelastic matrix subject to simple shear flow was considered. The local extensional flow around the droplet tip leads to severe velocity gradients and increases the stretching of polymer molecules. Region (II) is located around the advancing contact line, where larger velocity gradients are induced by the droplet motion on the stationary bottom wall. In addition, the eigenvector $\boldsymbol {n}_p$ corresponding to the primary eigenvalue of the conformation tensor ${{\boldsymbol{\mathsf{A}}}}$ is often used as an indicator of the orientation of polymer molecules (Yue et al. Reference Yue, Feng, Liu and Shen2005; Aggarwal & Sarkar Reference Aggarwal and Sarkar2007). As an example, figure 4(c) shows the distribution of $\boldsymbol {n}_p$ in the N/V system. It is seen that the polymers around the receding contact line are nearly parallel to the wall, while the polymers around the advancing contact line are distributed on the wall with a large inclination angle to the $x$ direction. Therefore, the polymers near the advancing contact line are easier to be stretched, thus leading to the high values of $\text {tr}{{\boldsymbol{\mathsf{A}}}}$.

Figure 4.

Contour plots for $\text {tr}{{\boldsymbol{\mathsf{A}}}}$ at the x–z mid-plane and the x–y bottom plane in (a) N/V and (b) V/N systems. The dominant polymer orientation $\boldsymbol {n}_p$ presented by line segments with equal length at the x–z mid-plane in (c) N/V and (d) V/N systems. The dashed rectangles with labels (I) or (II) in (a,b) are used to highlight the regions with high $\text {tr}{{\boldsymbol{\mathsf{A}}}}$. The simulation parameters are $Ca= 0.15$, $Wi=1.0$, $\beta =0.5$ and $m=1.0$.

In figure 5, we plot the corresponding distributions of the viscous stress component ${{( \boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}} )}_{x}}$ and the elastic stress component ${{( \boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}} )}_{x}}$ on the droplet surface, as well as the vectors of $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}}$ and $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ on the droplet interface in the x–z mid-plane, where the results obtained in the N/N system are also displayed for comparison. As a result of the polymer extension, the elastic stress $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ in the N/V system not only forms a pulling force outside the bulge tip of the droplet, but also expands the bottom wetting area (figure 5b(ii)). In comparison with the N/N system (figure 5a), the viscous stress $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}}$ in the N/V system is generally smaller, especially in the top region (figure 5b(i)). This is because the Newtonian solvent viscosity of the viscoelastic fluid is only half that of the Newtonian matrix at $\beta =0.5$. Moreover, due to the expansion of the droplet on the wall, the droplet height is reduced, which in turn lowers the viscous stress on the top of the droplet. This can be explained as follows. (1) In Couette flow, the velocity of the ambient fluid increases along the height direction, and the sliding velocity of a droplet is relatively small in the creeping flow condition. Accordingly, when the droplet height gets lower, the velocity difference between the droplet and the ambient fluid at the top surface is smaller, which leads to a lower shear rate and thus a decreased viscous stress. (2) The wall confinement still plays a role in the present problem. For a droplet with lower height, the flow passage that allows the ambient fluid to pass through becomes wider, which decreases the velocity of the ambient fluid near the top of the droplet and also contributes to a decreased viscous stress. On the other hand, even though $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ contributes to pull the droplet front, it is weak in pushing the droplet rear, leading to a smaller droplet deformation and thus to a lower ${{S}_{r}}$. It is also noticed that the maximum pulling stress $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ is positioned outside the droplet bulge tip in the N/V system, deviating from the location of the maximum viscous stress $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}}$ in the N/N system. Such a deviation is attributed to the finite relaxation of the elastic polymer molecules, consistent with previous results obtained from the investigation of a suspended droplet subject to a simple shear flow by Yue et al. (Reference Yue, Feng, Liu and Shen2005), Aggarwal & Sarkar (Reference Aggarwal and Sarkar2008), Verhulst, Moldenaers & Minale (Reference Verhulst, Moldenaers and Minale2007) and Gupta & Sbragaglia (Reference Gupta and Sbragaglia2014).

Figure 5.

Contour plots of the normalized ${{( \boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}} )}_{x}}$ on the droplet surface and vector distributions of the normalized $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}}$ on the interface at the x–z mid-plane for (a) N/N, (b(i)) N/V and (c(i)) V/N. Contour plots of the normalized ${{( \boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}} )}_{x}}$ on the droplet surface and vector distributions of the normalized $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ on the interface at the x–z mid-plane for (b(ii)) N/V and (c(ii)) V/N. The stresses and their $x$ components are all normalized by ${{\mu }_{B}}\dot {\gamma }$. The simulation parameters are $Ca=0.15$, $Wi=1.0$, $\beta =0.5$ and $m=1.0$.

We then analyse the viscoelastic effect in the V/N system. It is seen in figure 4(b) that the polymer extension mainly occurs around the receding contact lines (region (I)) where the polymer molecules exhibit an obvious inclination angle to the bottom wall (figure 4d). A similar strong extension region was numerically obtained in the two-dimensional study of Varagnolo et al. (Reference Varagnolo, Filippi, Mistura, Pierno and Sbragaglia2017). The distributions of ${{( \boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}} )}_{x}}$ in the V/N system (figure 5c(i)) resemble those in the N/N system, but with lower values due to the solvent viscosity difference. Compared with the N/V system, the pulling effect of ${{( \boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}} )}_{x}}$ at the droplet front is weaker, and the compressive effect on the droplet rear is almost negligible (figure 5c(ii)). As a consequence, the overall viscous and elastic forces are the lowest and accordingly the droplet deformation is the smallest in the V/N system. With an increase of $Ca$, the droplet deforms more due to an increased stretching effect of $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}}$ on the droplet tip in all the N/N, N/V and V/N systems. However, the normalized elastic stress is found to vary little with $Ca$ for both N/V and V/N systems. This means that in figure 3, the increase of droplet deformation with $Ca$ in the N/V and V/N systems is mainly attributed to the increased viscous stress.

From the elastic stress vectors at the contact lines, as shown in figure 5(b(ii),c(ii)), it is noticeable that $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ in both N/V and V/N systems tends to expand the wetting area. However, compared with the N/N system, an obvious increase in ${{A}_{r}}$ is only observed in the N/V system, and ${{A}_{r}}$ is even lower in the V/N system. What causes a lower ${{A}_{r}}$ in the V/N system? Ramaswamy & Leal (Reference Ramaswamy and Leal1999) pointed out that the viscoelastic effect not only directly acts on the droplet interface through $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$, but also modifies the flow field. Aggarwal & Sarkar (Reference Aggarwal and Sarkar2008) further explained that the flow modification is induced by the viscoelastic stress gradients over the flow domain. To clarify this, we investigate the elastic force $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ at $Ca= 0.15$ and $Wi=1.0$, and plot the results in figure 6. It is clear that in the N/V system, $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ at the x–z mid-plane and x–y bottom plane generates outward stretching forces outside the droplet front and in the front and rear contact-line regions, consistent with the effect of $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ on the droplet deformation. By contrast, in the V/N system, $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ exhibits severe variation across the rear contact-line region, which is first pointed inward and then outward across the diffuse interface from outside to inside of the droplet. Overall, the inward $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ in the rear contact-line region is more dominant, thereby inhibiting the spreading of the viscoelastic droplet in the Newtonian matrix.

Figure 6.

Vectors of $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ at the x–z mid-plane and x–y bottom plane in (a) N/V and (b) V/N systems for $Ca= 0.15$, $Wi=1.0$, $\beta =0.5$ and $m=1.0$. The length of the arrow represents the magnitude of $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ that is normalized by ${{{\mu }_{B}}\dot {\gamma }}/{R}$. In each panel, the solid contour lines correspond to ${{\rho }^{N}}=0.9$, 0 and $-0.9$, respectively, from inside to outside of the droplet. The vectors at ${{\rho }^{N}}=0$ in the bottom plane are highlighted in red.

In addition to droplet deformation, another meaningful indicator characterizing the droplet behaviour is the sliding velocity of the droplet, which depends on the force balance in the moving direction. Following the arguments by Ding et al. (Reference Ding, Gilani and Spelt2010) and Chen et al. (Reference Chen, Shu, Liu and Zhang2021), the driving force exerted on the droplet interface is balanced by the wall resistance in the $x$ direction. In the N/V and V/N viscoelastic systems, the driving force is tentatively expressed by the sum of the shear stresses ${{\alpha }_{{{\mu }_{s}}}}{{\mu }_{Bs}}\dot {\gamma }{{R}^{2}}+{{\alpha }_{{{\mu }_{p}}}}{{\mu }_{Bp}}\dot {\gamma }{{R}^{2}}$ and the form drag force ${{\alpha }_{i}}\rho {{R}^{2}}{{( \dot {\gamma }R-{{V}_{cl}} )}^{2}}$, where ${{V}_{cl}}$ is the steady-state sliding velocity of the moving droplet. Assuming that the contact angle hysteresis is neglected, the wall resistance can be written as ${{{\alpha }'}_{{{\mu }_{s}}}}{{\mu }_{Rs}}{{V}_{cl}}R+{{{\alpha }'}_{{{\mu }_{p}}}}{{\mu }_{Rp}}{{V}_{cl}}R$. Thus, the force balance in the $x$ direction is expressed as

(4.1)\begin{equation} {{\alpha }_{{{\mu }_{s}}}}{{\mu }_{Bs}}\dot{\gamma }{{R}^{2}}+{{\alpha }_{{{\mu }_{p}}}}{{\mu }_{Bp}} \dot{\gamma }{{R}^{2}}+{{\alpha }_{i}}\rho {{R}^{2}}{{\left( \dot{\gamma }R-{{V}_{cl}} \right)}^{2}}= {{{\alpha }'}_{{{\mu }_{s}}}}{{\mu }_{Rs}}{{V}_{cl}}R+{{{\alpha }'}_{{{\mu }_{p}}}}{{\mu }_{Rp}}{{V}_{cl}}R, \end{equation}

where ${{{\alpha }'}_{{{\mu }_{s}}}}$, ${{{\alpha }'}_{{{\mu }_{p}}}}$, ${{\alpha }_{{{\mu }_{s}}}}$, ${{\alpha }_{{{\mu }_{p}}}}$ and ${{\alpha }_{i}}$ are the coefficients to be determined, which may be relevant to fluid properties, droplet shape and flow conditions (Chen et al. Reference Chen, Shu, Liu and Zhang2021).

In the N/V system, the parameter ${{\mu }_{Rp}}$ is zero. Using the definitions of the two-phase fluid viscosity ratio ($m$) and the solvent viscosity ratio of the matrix fluid ($\beta$), (4.1) is further derived as

(4.2)\begin{equation} \text{N/V:}\quad {{\alpha }_{{{\mu }_{s}}}}\beta {{\mu }_{B}}\dot{\gamma }R+{{\alpha }_{{{\mu }_{p}}}} \left( 1-\beta \right){{\mu }_{B}}\dot{\gamma }R+{{\alpha }_{i}} \rho R{{\left( \dot{\gamma }R-{{V}_{cl}} \right)}^{2}}={{{\alpha }'}_{{{\mu }_{s}}}}m{{\mu }_{B}}{{V}_{cl}}. \end{equation}

By dividing (4.2) by the interfacial tension parameter $\sigma$, we have

(4.3)\begin{equation} \text{N/V:} \quad {{\alpha }_{{{\mu }_{s}}}}\beta Ca+{{\alpha }_{{{\mu }_{p}}}}(1-\beta )Ca+{{\alpha }_{i}} ReCa{{\left( 1-\frac{C{{a}_{cl}}}{Ca} \right)}^{2}}={{{\alpha }'}_{{{\mu }_{s}}}}mC{{a}_{cl}}, \end{equation}

where $C{{a}_{cl}}$ is the contact-line capillary number defined with ${{V}_{cl}}$ as (Liu et al. Reference Liu, Zhang, Ba, Wang and Wu2020)

(4.4)\begin{equation} C{{a}_{cl}}=\frac{{{\mu }_{B}}{{V}_{cl}}}{\sigma }. \end{equation}

The sliding velocity ${{V}_{cl}}$ is normally much smaller than ${{u}_{w}}$ and the corresponding ${C{{a}_{cl}}}/{Ca}$ is often of $O(10^{-1})$, which means that $({C{{a}_{cl}}}/{Ca})^2$ is a small quantity of $\textit {O}(10^{-2})$ in (4.3). Therefore, the form drag ${{\alpha }_{i}}ReCa{{( 1-{C{{a}_{cl}}}/{Ca} )}^{2}}$ can be simplified as ${{\alpha }_{i}}Ca{( 1-2{C{{a}_{cl}}}/{Ca} )}$ with $Re=1.0$ in the present study. Then, (4.3) for the N/V system is written as

(4.5)\begin{equation} \text{N/V:}\quad \frac{C{{a}_{cl}}}{Ca}=\frac{\left( {{\alpha }_{{{\mu }_{s}}}}-{{\alpha }_{{{\mu }_{p}}}}\right) \beta +{{\alpha }_{{{\mu }_{p}}}}+{{\alpha }_{i}}}{{{{{\alpha }'}}_{{{\mu }_{s}}}}m+2{{\alpha }_{i}}}. \end{equation}

Following a similar derivation procedure, the prediction for ${C{{a}_{cl}}}/{Ca}$ in the V/N system is obtained as

(4.6)\begin{equation} \text{V/N:}\quad \frac{C{{a}_{cl}}}{Ca}=\frac{\left( {{{\bar{\alpha }}}_{{{\mu }_{s}}}}- {{{\bar{\alpha }}}_{{{\mu }_{p}}}} \right)\beta +{{{\bar{\alpha }}}_{{{\mu }_{p}}}}+ {{{\bar{\alpha }}}_{i}}}{{{{\bar{{\alpha }}'}}_{{{\mu }_{s}}}}\beta m+{{{\bar{{\alpha }}'}}_{{{\mu }_{p}}}}m \left( 1-\beta \right)+2{{{\bar{\alpha }}}_{i}}}, \end{equation}

where ${{{\bar {\alpha }}}_{{{\mu }_{s}}}}$, ${{{\bar {\alpha }}}_{{{\mu }_{p}}}}$, ${{{\bar {\alpha }}}_{i}}$, ${{{\bar {{\alpha }}'}}_{{{\mu }_{s}}}}$, ${{{\bar {{\alpha }}'}}_{{{\mu }_{p}}}}$ are the coefficients to be determined for the V/N system. The wall friction in the denominator of (4.6) consists of both viscous and elastic contributions and it is different from that in the N/V system. Under constant $\beta$ and $m$ conditions, $C{{a}_{cl}}$ is expected to vary linearly with $Ca$ for either N/V or V/N system. In particular, the linear relation between $C{{a}_{cl}}$ and $Ca$ has been demonstrated for each two-phase viscosity ratio in the Newtonian system ($\beta =1.0$) (Ding et al. Reference Ding, Gilani and Spelt2010; Liu et al. Reference Liu, Zhang, Ba, Wang and Wu2020).

To reflect the variation in the sliding velocity, we plot $Ca_{cl}$ versus $Ca$ for the N/N, N/V and V/V systems in figure 7, where $Wi=1.0$, $\beta =0.5$ and $m=1.0$. As anticipated, the linear relation between $Ca_{cl}$ and $Ca$ is obtained not only in the N/N system, but also in the N/V and V/N systems. By using least-squares linear fitting, the fitted slopes are 0.2236 in the N/N system, 0.2269 in the V/N system and 0.2212 in the N/V system. This means that the droplet is slightly faster in the V/N system but slower in the N/V system when compared with the droplet in the N/N system, which is also reflected in the inset of figure 7. As previously shown in figure 5, the normalized elastic driving force is barely affected by $Ca$, while the viscous driving force in the N/V system is not as strong as that in the V/N system or in the N/N system due to the lower droplet height caused by spreading on the wall. Thus, it leads to a lower slope of $C{{a}_{cl}}$ against $Ca$ in the N/V system. Meanwhile, even though the sum of the driving forces in the V/N system is smaller than that in the N/N system, the corresponding velocity is higher due to lower wall resistance arising from smaller wetting area. With the value of $\beta =0.5$ used in this section, the above analysis indicates that the coefficient $0.5( {{\alpha }_{{{\mu }_{s}}}}+{{\alpha }_{{{\mu }_{p}}}} )$ in the numerator of (4.5) related to driving force is less than ${{\alpha }_{{{\mu }_{s}}}}$ in the N/N system, while the coefficient in the denominator of (4.6) related to wall friction in the V/N system, $0.5( {{{\bar {\alpha }}'}}_{{{\mu }_{s}}}+{{{\bar {\alpha }'}}_{{{\mu }_{p}}}} )$, is smaller than ${{{\bar {\alpha }'}}_{{{\mu }_{s}}}}$ in the N/N system. In Appendix B, we further investigate the variation of $Ca_{cl}$ with $Ca$ in both N/V and V/N systems for varying $Wi$, $\beta$ and $m$, and the results are compared with those obtained in the N/N system to show the role of viscoelasticity in the droplet motion.

Figure 7.

The contact-line capillary number $Ca_{cl}$ as a function of the capillary number $Ca$ at $Wi$=1.0, $\beta =0.5$ and $m=1.0$. An inset is included to show a local enlarged view of the $Ca$–$Ca_{cl}$ plot for $Ca\geq 0.24$. The simulated data are shown by discrete symbols and the fitting lines for the N/N, N/V and V/N systems are represented by solid, dashed and dash-dot-dot lines, respectively.

4.2.2. Effect of $Wi$

To investigate the effect of viscoelasticity on droplet deformation, $Wi$ is varied from 0.1, 0.25, 0.5, 1.0, 2.0, 5.0, 8.0 to 10 in the N/V system, and from 0.1, 0.25, 0.5, 1.0, 2.0, 5.0, 10, 20 to 40 in the V/N system. Steady-state droplet deformation with a constant sliding velocity is eventually achieved under various $Wi$ conditions in both N/V and V/N systems. Based on the steady-state results, we plot the contact-line shapes at several representative $Wi$, along with the variations of ${{A}_{r}}$, ${{h}_{r}}$ and ${{S}_{r}}$ with $Wi$ in figure 8.

Figure 8.

(a) The steady-state contact-line shapes for $Wi= 0.1$, 5.0, 10 in the N/V system and for $Wi= 0.1$, 10 in the V/N system. (b) The relative wetting area ${{A}_{r}}$ and relative droplet height ${{h}_{r}}$ and (c) the relative droplet surface area ${{S}_{r}}$ as functions of $Wi$ in the N/V and V/N systems. The other simulation parameters are $Ca=0.15$, $\beta =0.5$ and $m=1.0$.

We first consider the N/V system. As $Wi$ increases, the wetting area ${{A}_{r}}$ increases and the droplet height ${{h}_{r}}$ decreases monotonically. However, the droplet surface area ${{S}_{r}}$ varies little for $Wi \leq 1.0$, and then obviously increases for $Wi$ above 1.0 (see figure 8b,c). This indicates that the droplet deformation is different from the deformation of a spherical droplet suspended in an Oldroyd-B simple shear flow where the droplet deformation first decreases and then increases with $Wi$ (Yue et al. Reference Yue, Feng, Liu and Shen2005; Aggarwal & Sarkar Reference Aggarwal and Sarkar2008). To understand the cause of the difference, we perform a force analysis at two representative $Wi$ values, i.e. $Wi=0.1$ and 5.0, and the results including the distributions of the normalized viscous stress $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}}$ and the elastic stress $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ on the droplet surface, as well as the distributions of the normalized $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ at the x–z mid-plane and x–y bottom plane, are presented in figure 9.

Figure 9.

Vector distributions of $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ at the x–z mid-plane and x–y bottom plane for (a) $Wi=0.1$ and(b) $Wi=5.0$; and (c) contour plots of the normalized ${{( \boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}} )}_{x}}$ and ${{( \boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}} )}_{x}}$ on the droplet surface and vector distributions of the normalized $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}}$ and $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ at the x–z mid-plane for $Wi=5.0$ in the N/V system. In (a,b), the solid contour lines correspond to ${{\rho }^{N}}=0.9$, 0 and $-0.9$, respectively, from inside to outside of the droplet. The vectors at ${{\rho }^{N}}=0$ in the bottom plane are highlighted in red. Various stresses or components are normalized by ${{\mu }_{B}}\dot {\gamma }$, while $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ is normalized by ${{{\mu }_{B}}\dot {\gamma }}/{R}$. The other parameters are chosen as $Ca=0.15$, $\beta =0.5$ and $m=1.0$.

As indicated in Yue et al. (Reference Yue, Feng, Liu and Shen2005) and Aggarwal & Sarkar (Reference Aggarwal and Sarkar2008), the decreased deformation for a suspended droplet at $Wi \leq 1.0$ in the N/V system is mainly caused by the reduced viscous stretching force due to the decrease in droplet orientation angle caused by elasticity. For the attached droplet in the N/V system with weak elasticity, e.g. $Wi=0.1$, the confinement of the bottom wall resists the droplet rotation. A small difference is observed between the angular position of the maximum ${{( \boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}} )}_{x}}$ and that of the maximum ${{( \boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}} )}_{x}}$ (similar to figure 5c(i),c(ii) and thus not shown here), which marginally expands the wetting area and lowers the droplet height. Also, the flow modification caused by $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ is negligible (figure 9a). Both effects lead to a less reduced viscous force $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}}$, so the droplet deformation quantified by ${{S}_{r}}$ is almost a constant for $Wi \leq 1.0$. After further increasing $Wi$, the increased polymer elasticity in the matrix fluid induces a much stronger stretch locally outside the droplet front (see e.g. $Wi=5.0$ in figure 9c). The locally increased $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ with $Wi$ was also observed by Aggarwal & Sarkar (Reference Aggarwal and Sarkar2008) for a suspended droplet in simple shear flow. However, it is noted that the maximum $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ occurs near the droplet tips for a suspended droplet, but around the advancing contact lines for the wall-attached droplet. Such a distribution of $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ promotes the droplet spreading on the wall, which turns out to exhibit increased ${{A}_{r}}$ and ${{S}_{r}}$ and decreased ${{h}_{r}}$.

On the other hand, the flow modification by $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ (figure 9b) acts equally with $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ in promoting droplet spreading in the N/V system. We also surprisingly find that for $Wi \geq 5.0$ the moving contact line eventually develops into a prolate shape with smaller curvature radius at the front than at the rear (see figures 8a and 9b), which is opposite to the curvature radius distribution of the moving contact line at $Wi =1.0$ (see figure 3b). For high $Wi$, i.e. $Wi \geq 5.0$, the smaller curvature radius at the front is attributed to the fact that $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ increases rapidly especially near the leading edge of the contact line (see e.g. figure 9b), as opposed to the relatively even distribution of $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ in the advancing contact line region at $Wi =1.0$ (see e.g. figure 6a).

We then consider the V/N system, and the variations of ${{A}_{r}}$, ${{h}_{r}}$ and ${{S}_{r}}$ with $Wi$ are given in figure 8. It is clear that increasing $Wi$ slightly decreases the droplet surface area ${{S}_{r}}$ and almost has negligible influence on the wetting area ${{A}_{r}}$ and the droplet height ${{h}_{r}}$. This indicates that the droplet deformation is inhibited to some extent by the droplet viscoelasticity, which is consistent with the trend in the observation obtained for a viscoelastic droplet suspended in a Newtonian shear flow (Aggarwal & Sarkar Reference Aggarwal and Sarkar2007). To elucidate the behaviour of the attached droplet, we plot the distributions of $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ for $Wi = 0.1$ and 5.0 and $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}}$, $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ for $Wi = 5.0$ in figure 10. By comparing these forces, we find that the viscous force $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}}$ varies little with $Wi$, and the elastic force $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ mainly acts at the droplet interface near the contact line (see figure 10c). It is also noticed that the increased inward-pointing elastic force $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ becomes dominant at high $Wi$ (see figure 10b), which overcomes the outward-pointing $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ and causes the moving contact line to contract in the $x$ direction. As a result, the viscoelastic droplet is squeezed in the $x$ direction but marginally expands in the $y$ direction for high $Wi$, as observed in figure 8(a).

Figure 10.

Vector distributions of $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ at the x–z mid-plane and x–y bottom plane for (a) $Wi=0.1$ and(b) $Wi=5.0$; and (c) contour plots of the normalized ${{( \boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}} )}_{x}}$ and ${{( \boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}} )}_{x}}$ on the droplet surface and vector distributions of the normalized $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{s}}$ and $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ at the x–z mid-plane for $Wi=5.0$ in the V/N system. In (a,b), the solid contour lines correspond to ${{\rho }^{N}}=0.9$, 0 and $-0.9$, respectively, from inside to outside of the droplet. The vectors at ${{\rho }^{N}}=0$ in the bottom plane are highlighted in red. Various stresses or components are normalized by ${{\mu }_{B}}\dot {\gamma }$, while $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ is normalized by ${{{\mu }_{B}}\dot {\gamma }}/{R}$. The other parameters are chosen as $Ca=0.15$, $\beta =0.5$ and $m=1.0$.

To show how $Wi$ influences the sliding velocity of the droplet in the N/V and V/N systems, we plot the variation of ${C{{a}_{cl}}}/{Ca}$ with $Wi$ in figure 11. It is observed that the variations of ${C{{a}_{cl}}}/{Ca}$ turn out to generally match the variations of ${{A}_{r}}$ displayed in figure 8(b). In both N/V and V/N systems, the wetting area varies little with $Wi$ for $Wi<1.0$, and the sliding velocities are almost a constant. Upon further increasing $Wi$ above unity, ${C{{a}_{cl}}}/{Ca}$ decreases (increases) with an increase (decrease) of ${{A}_{r}}$ in the N/V (V/N) system due to the enhanced (reduced) wall friction.

Figure 11.

Plot of ${C{{a}_{cl}}}/{Ca}$ as a function of $Wi$ in both N/V and V/N systems for $Ca=0.15$, $\beta =0.5$ and $m=1.0$. The simulated results in the N/V and V/N systems are represented by the red triangles and blue circles, respectively. The dashed and dash-dot-dot lines are used to show the variation of ${C{{a}_{cl}}}/{Ca}$ with $Wi$ in the N/V and V/N systems, respectively.

4.2.3. Effect of solvent viscosity ratio $\beta$

In this section, we choose $Ca=0.15$ and $Wi=1.0$, but decrease $\beta$ from 1.0 to 0.1 at three typical two-phase viscosity ratios, i.e. $m=0.5$, 1.0 and 2.0. Note that for each two-phase viscosity ratio, the elastic effect strengthens with decreasing $\beta$ and vice versa. The droplet deformation is quantified by ${{A}_{r}}$, ${{h}_{r}}$ and ${{S}_{r}}$ in the steady state. The corresponding results in the N/V system are presented in figure 12(a(i),a(ii)). It is seen that a decrease of solvent viscosity ratio $\beta$ leads to an increase in the wetting area ${{A}_{r}}$, but to a decrease in the droplet height ${{h}_{r}}$ and the droplet surface area ${{S}_{r}}$. As anticipated, the elastic force is more pronounced at lower $\beta$, which promotes the droplet spreading on the wall. This will reduce the droplet height, so the droplet is exposed to a weaker shear flow, which reduces the droplet surface area ${{S}_{r}}$. We also notice in figure 12(a(i),a(ii)) that the variations of ${{A}_{r}}$, ${{h}_{r}}$ and ${{S}_{r}}$ with $\beta$ are similar at different values of $m$, while the droplet always deforms the most at $m=2.0$ and the least at $m=0.5$ for each $\beta$. The increased droplet deformation with $m$ was also identified in our previous studies, which considered a clean or surfactant-covered droplet attached on the substrate subject to a Newtonian Couette flow (Liu et al. Reference Liu, Zhang, Ba, Wang and Wu2020). The reason is that a higher droplet viscosity leads to an increased wall friction, which slows down the droplet motion and thus increases the viscous stretching acting on the droplet surface.

Figure 12.

Relative wetting area ${{A}_{r}}$, relative droplet height ${{h}_{r}}$ and relative droplet surface area ${{S}_{r}}$ as functions of decreasing $\beta$ for the two-phase viscosity ratios of $m=0.5$, 1.0 and 2.0 in the (a) N/V and (b) V/N systems. The other parameters are fixed at $Ca=0.15$ and $Wi=1.0$.

We then focus on the V/N system, and investigate the effect of $\beta$ on ${{A}_{r}}$, ${{h}_{r}}$ and ${{S}_{r}}$ at different values of $m$, which is shown in figure 12(b(i),b(ii)). It is clear in figure 12(b(i)) that ${{A}_{r}}$ and ${{h}_{r}}$ both tend to converge to zero with decreasing $\beta$ for each $m$. The main reason is that the decrease of $\beta$ enhances the droplet elastic effect, which causes the contact line to contract in the $x$ direction (see figure 10b), thus inhibiting the development of wetting area and droplet height. Like in the N/V system, ${{S}_{r}}$ is found to decrease with decreasing $\beta$ as well in the V/N system. However, ${{S}_{r}}$ varies with a steeper slope in the V/N system. The reason is that, on the one hand, $\boldsymbol {n}\boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ acting as a driving force to promote droplet deformation is smaller and, on the other hand, the inhibiting effect induced by $\boldsymbol {\nabla } \boldsymbol {\cdot } {{\boldsymbol {\tau }}_{p}}$ contributes more at lower $\beta$.

We next investigate how ${C{{a}_{cl}}}/{Ca}$ is influenced by $\beta$ at three different values of two-phase viscosity ratio in both N/V and V/N systems. The obtained results are shown in figure 13. In the N/V system, it is seen that with a decrease of $\beta$, ${C{{a}_{cl}}}/{Ca}$ monotonically increases for $m=0.5$ and 1.0 but monotonically decreases for $m=2.0$. This means that the presence of viscoelascity in the matrix fluid promotes the droplet motion for $m$ below unity, but inhibits the droplet motion for $m$ above unity. The difference in droplet motion as functions of $\beta$ and $m$ can be explained as follows. As $\beta$ decreases, the elastic viscosity of the matrix fluid increases, which brings two competing effects on the droplet. On the one hand, it promotes the droplet motion through the stretching of the droplet interface. On the other hand, the elastic stretching enlarges the wetting area of the droplet which in turn increases the wall friction. At $m=0.5$ and 1.0 where the wall friction is relatively small due to lower wetting area (see figure 12a(i)) and droplet viscosity, the promoting effect dominates and thus the sliding velocity shows an increasing trend with decreasing $\beta$; whereas at $m=2.0$, the wall friction becomes important so that the sliding velocity decreases with decreasing $\beta$. Unlike that in the N/V system, it is seen in figure 13 that the sliding velocities (${C{{a}_{cl}}}/{Ca}$) under three tested $m$ conditions in the V/N system all increase with decreasing $\beta$. As learnt from the droplet deformation trends, it is easier for the droplet to maintain its initial hemispherical shape, droplet height and wetting area at smaller $\beta$. As a result, the droplet would undergo a larger shear force, which promotes the droplet motion and increases the steady-state sliding velocity. In particular, we notice for $\beta <0.4$ that the droplet in the V/N system always slides faster than in the N/V system.

Figure 13.

The variation of ${C{{a}_{cl}}}/{Ca}$ with decreasing $\beta$ for different two-phase viscosity ratios in the N/V and V/N systems. The simulation results are represented by the discrete symbols, i.e. upper-filled symbols for $m=0.5$, open symbols for $m=1.0$ and lower-filled symbols for $m=2.0$, while the predicted curves from (4.5) and (4.6) are represented by dashed, solid and dash-dot-dot lines for $m=0.5$, 1.0 and 2.0, respectively. The other parameters are fixed at $Ca=0.15$ and $Wi=1.0$.

According to (4.5) and (4.6), for each $m$, ${C{{a}_{cl}}}/{Ca}$ is related to $\beta$ by ${C{{a}_{cl}}}/{Ca}={( \beta +{{a}_{1}} )}/{( {{b}_{1}}m + {{c}_{1}} )}$ in the N/V system and by ${C{{a}_{cl}}}/{Ca}={( \beta +{{a}_{2}} )}/{( {{b}_{2}}m+{{c}_{2}}\beta m +{{d}_{2}})}$ in the V/N system, where (${{a}_{1}}$, ${{b}_{1}}$, ${{c}_{1}}$), (${{a}_{2}}$, ${{b}_{2}}$, ${{c}_{2}}$, ${{d}_{2}}$) are the fitting parameters related to (${{\alpha }_{{{\mu }_{s}}}}$, ${{\alpha }_{{{\mu }_{p}}}}$, ${{\alpha }_{i}}$, ${{{\alpha }'}_{{{\mu }_{s}}}}$, ${{{\alpha }'}_{{{\mu }_{p}}}}$) and (${{{\bar {\alpha }}}_{{{\mu }_{s}}}}$, ${{{\bar {\alpha }}}_{{{\mu }_{p}}}}$, ${{{\bar {\alpha }}}_{i}}$, ${{{\bar {{\alpha }}'}}_{{{\mu }_{s}}}}$, ${{{\bar {{\alpha }}'}}_{{{\mu }_{p}}}}$), respectively. By using least-squares fitting on the simulated data in the N/V system, the obtained values of $({{a}_{1}},{{b}_{1}},{{c}_{1}})$ are ($-$11.34, $-$87.42, $2.98\times 10^{-4}$) for $m=0.5$ (N/V-Case 1), ($-$27.11, $-$121.18, $-5.06\times 10^{-6}$) for $m=1.0$ (N/V-Case 2) and (24.54, 68.02, $1.79\times 10^{-4}$) for $m=2.0$ (N/V-Case 3). Similarly, the fitting parameters of $({{a}_{2}},{{b}_{2}},{{c}_{2}},{{d}_{2}})$ in the V/N system are (0.14, 0.90, 8.74, 0) for $m=0.5$ (V/N-Case 1), (0.19, 0.73, 4.81, 0) for $m=1.0$ (V/N-Case 2) and (0.37, 0.87, 2.82, 0) for $m=2.0$ (V/N-Case 3). With these fitting parameters, the predicted curves can be obtained, which are plotted in figure 13. Clearly, the simulated results match well with the predicted ones in both N/V and V/N systems, justifying the effectiveness of the derived relationships. Moreover, one easily finds that both ${{c}_{1}}$ and ${{d}_{2}}$ simplified from the coefficients of the form drag term are nearly zero, indicating that the form drag term makes a negligible contribution to the sliding velocity of the droplet under the creeping flow condition for both N/V and V/N systems. Similarly, the form drag term has been commonly neglected in modelling of the droplet motion for the N/N system. For example, Ding & Spelt (Reference Ding and Spelt2008) found that the droplet shape does not have a strong effect on the force balance; Liu et al. (Reference Liu, Zhang, Ba, Wang and Wu2020) established a force balance without considering the form drag term for a system with negligible inertia, and the resulting linear relationship between $Ca_{cl}$ and $Ca$ was numerically verified.