4.1 Droplet velocity

Figure 2(a) depicts the dependence of the scaled droplet velocity $U_{d}/U^{\infty }$ with the capillary number $Ca^{\infty }$ and confinement $R/H$ . The velocity increases slightly with $R/H$ . This weak dependence is in accordance with the experimental observations of Shen et al. (Reference Shen, Leman, Reyssat and Tabeling2014) for ${\it\lambda}\approx 1.4$ and capillary numbers several orders smaller than ours. The scaled droplet velocity increases with $Ca^{\infty }$ monotonically and surpasses $1$ , in contrast with the predicted velocity of $U_{d}/U^{\infty }=1$ by Gallaire et al. (Reference Gallaire, Meliga, Laure and Baroud2014) for a matching-viscosity pancake droplet modelled by an undeformed cylinder at sufficiently low $Ca^{\infty }$ . The mismatch results from two drawbacks of their model: it neglects the impeding effect of the dynamics of the menisci of the drop at low $Ca^{\infty }$ ; and it does not capture the film thickening at high $Ca^{\infty }$ that enhances the droplet velocity.

Figure 2.

(a) The scaled droplet velocity $U_{d}/U^{\infty }$ as a function of $Ca^{\infty }$ for varying confinement. (b) Stretching the thin-film region of the drop as in figure 1(b) by $7.5$ times in $z$ .

Figure 3.

Contour lines of the scaled film thickness $h/H$ for droplets with $Ca^{\infty }=0.007$ (a), $0.02$ (b) and $0.08$ (c) under confinement $R/H=2$ . The black contour line $h/H=0.5$ indicates the edge of the droplet cut by the $z=0$ plane.

4.2 Shape of the droplet and film thickness

To better visualize the fine-scale geometrical features of the drop shown in figure 1(b), we stretch its top interface by $7.5$ times vertically and the zoomed view is shown in figure 2(b). The interface clearly bulges on the rear half of the rim of the interface, displaying an arc-shaped ridge.

We show in figure 3 the contour lines of constant film thickness $h(x,y)/H$ for droplets with $Ca^{\infty }=0.007$ , $0.02$ and $0.08$ under confinement $R/H=2$ . Note that the height $z(x,y)$ of the droplet interface is inversely correlated to the film thickness $h(x,y)$ , i.e. $z(x,y)+h(x,y)=H/2$ . The black curve $h/H=0.5$ represents the edge of the droplet cut by the $z=0$ plane, which resembles a circle at $Ca^{\infty }=0.007$ but becomes elongated at $Ca^{\infty }=0.08$ . For all $Ca^{\infty }$ investigated, the contour map exhibits three local minima: one at the rear and a symmetric pair on the lateral edges. These minima correspond to the peaks of the interfacial protrusions. The two symmetric lateral protrusions are higher than the rear one. They have been recently observed experimentally for a pancake droplet with ${\it\lambda}=25$ by Huerre et al. (Reference Huerre, Theodoly, Leshansky, Valignat, Cantat and Jullien2015), who noted the resulting ‘catamaran-like shape’ adopted by the droplet. This feature has also been portrayed theoretically by Burgess & Foster (Reference Burgess and Foster1990), performing a multi-region asymptotic analysis of a pancake bubble (see figure 5 of their paper). As far as we know, our study represents the first computational work that identifies this unique interfacial topology.

Burgess & Foster (Reference Burgess and Foster1990) showed in the low capillary number limit that the contour lines of $h/H$ are streamwise parallel in the central film region (excluding the lateral portion) where the viscous forces dominate, resulting in flat film. The contour lines of the $Ca^{\infty }=0.08$ case are indeed parallel in the region $x\in (-1,1),y\in (-0.75,0.75)$ . At a reduced capillary number $Ca^{\infty }=0.007$ , such parallel lines disappear and the three protrusions instead occupy a large portion of the film, pointing to its $3$ D nature.

Figure 4.

The scaled mean $\bar{h}/H$ (a) and minimum $h_{min}/H$ (b) film thickness versus the capillary number $Ca_{d}$ , for a pancake droplet under confinement $R/H=1.5$ (circles), $2$ (squares) and $3$ (diamonds). Its minimum thickness on the middle vertical slice is denoted by $h_{min}^{y=0}$ . The dashed curve corresponds the constant film thickness $h^{EB}/H$ of a $2$ D drop predicted by the EB model (Klaseboer, Gupta & Manica Reference Klaseboer, Gupta and Manica2014) with $P=0.6$ and $Q=1.5$ . The triangles denote the numerical data $h^{sim}|_{2D}/H$ (constant) and $h_{min}^{sim}|_{2D}/H$ (minimum) for a $2$ D drop.

We show in figure 4(a) the dependence of the mean thickness $\bar{h}$ on the capillary number. $Ca_{d}$ is adopted instead of $Ca^{\infty }$ to be consistent with prior studies. We obtain $\bar{h}$ by averaging $h$ over a central circular patch with radius $R_{cen}=0.3R_{xy}$ , where $R_{xy}$ is the effective radius of the nearly circular droplet profile in the $z=0$ plane. The scaled film thickness $\bar{h}/H$ increases with $Ca_{d}$ monotonically and weakly depends on $R/H$ .

For comparison, we use the flow solver Gerris to simulate a $2$ D matching-viscosity droplet in a channel of width $H$ where the droplet length is much larger than its size in the confined direction. The film far away from the dynamic menisci is almost flat with a constant thickness of $h^{sim}|_{2D}$ which is reported in figure 4(a). Additionally, we include the prediction of the extended Bretherton (EB) model proposed by Klaseboer et al. (Reference Klaseboer, Gupta and Manica2014) for a bubble, according to which, apart from the dynamic meniscus regions, the lubrication film has a constant thickness of $h^{EB}$

where $H$ is the tube diameter, and $P=0.643$ and $Q=2.79$ (Bretherton Reference Bretherton1961). This model agrees well with the empirical fit of Aussillous & Quéré (Reference Aussillous and Quéré2000) of Taylor’s (Reference Taylor1961) experimental data. We adopt $P=0.6$ and $Q=1.5$ in (4.1), and the fitted thickness $h^{EB}/H$ almost coincides with the numerical value $h^{sim}|_{2D}/H$ . The mean film thickness $\bar{h}/H$ agrees well with the two values $h^{sim}|_{2D}/H$ and $h^{EB}/H$ of the $2$ D drop at low capillary numbers, but starts deviating when $Ca_{d}$ increases. As the confinement increases, the film thickness $\bar{h}/H$ agrees better with the $2$ D results. The agreement between $\bar{h}/H$ with the thickness $h^{sim}|_{2D}/H\approx h^{EB}/H$ can be attributed to two reasons: first, the central region where $\bar{h}$ is measured is rather flat, as illustrated by the sparsely distributed contour lines in figure 3, implying the mean film thickness $\bar{h}$ adopts the constant thickness $h$ of the vertical slice ( $y=0$ ); second, as we will show in § 4.3, the velocity field of this slice strongly resembles that of a $2$ D matching-viscosity droplet.

Figure 5.

Velocity field in the droplet frame including the vectors and streamlines of the flow (a) on the $y=0$ plane of the drop with $Ca^{\infty }=0.007$ and $R/H=2$ , (b) of a $2$ D droplet with $Ca^{\infty }=0.007$ and ${\it\lambda}=1$ travelling in an infinitely long channel. Red curves denote the droplet interface and black/magenta/tip circles denote the interfacial/axial/tip stagnation points; the contour colour indicates the in-plane velocity magnitude scaled by the droplet velocity $|\mathbf{u}|/U_{d}$ .

We plot in figure 4(b) the scaled minimum film thickness $h_{min}/H$ of the pancake droplet, where $h_{min}^{y=0}/H$ denotes the scaled minimum thickness of its middle vertical slice, and $h_{min}^{sim}|_{2D}/H$ that of the $2$ D drop. For all $R/H$ , $h_{min}^{y=0}/H$ is slightly below $h_{min}^{sim}|_{2D}/H$ and increases with $R/H$ . For the most confined case, $R/H=3$ , $h_{min}^{y=0}/H$ agrees with $h_{min}^{sim}|_{2D}/H$ reasonably well, which is in accordance with the agreement between their mean thickness counterparts, i.e. $\bar{h}/H$ and $h^{sim}|_{2D}/H$ , as discussed previously.

The global minimum $h_{min}/H$ is, however, approximately half of the local $h_{min}^{y=0}/H$ , as can be inferred from the minima of the contour maps (figure 3) that represent the thickness of the film above the lateral and rear interfacial protrusions. The difference between these two minima indicates the $3$ D nature of the droplet interface. Note that, while $\bar{h}/H$ slightly increases with the confinement $R/H$ , $h_{min}/H$ decreases significantly with $R/H$ , especially at large $Ca_{d}$ numbers. This suggests that the $3$ D nature is more pronounced for a more confined drop.

4.3 Flow field in the reference frame of the droplet

In this section, we focus on the flow field, $\boldsymbol{u}_{drop}=\boldsymbol{u}_{lab}-(U_{d},0,0)_{xyz}$ , in the reference frame of the droplet, where $\boldsymbol{u}_{lab}$ indicates that in the lab frame; the disturbance flow field will be discussed in § 4.4. The velocity fields projected on the vertical, horizontal and transverse planes in the reference frame of the drop are depicted. We first show in figure 5(a) that on the middle vertical plane $y=0$ of the drop with $Ca=0.007$ under confinement $R/H=2$ . We compare it to the $2$ D drop with ${\it\lambda}=1$ in figure 5(b). We find the two flow patterns resemble each other closely, supporting the hypothesis made in § 4.2 regarding their film thickness. In the top-half domain, the interior flow consists of three recirculating zones, two clockwise ones appearing beside the front and rear meniscus respectively and a third anti-clockwise one in between; they are clearly distinguished by six stagnation points, two on the interface (black circles), two on the axis (magenta circles) and the other two as the tips (green circles) of the droplet. The front interfacial stagnation point has been predicted for an axisymmetric inviscid bubble in a tube by Taylor (Reference Taylor1961), as also discussed by Hodges, Jensen & Rallison (Reference Hodges, Jensen and Rallison2004). The recirculation has been observed numerically by Westborg & Hassager (Reference Westborg and Hassager1989) and Martinez & Udell (Reference Martinez and Udell1990) for an axisymmetric viscous droplet both near its front and rear meniscus, as well as by Ling et al. (Reference Ling, Fullana, Popinet and Josserand2016) for a $2$ D drop with ${\it\lambda}\approx 1.35$ .

As explained by Martinez & Udell (Reference Martinez and Udell1990), this flow structure appears as a result of the combination of the shear exerted by the wall onto the film and the zero net flux condition inside the drop. The interface tends to follow the moving wall to reduce the viscous dissipation in the film, producing the interior backward flow; the zero net flux condition dictates a compensating forward flow in the near-axis region. This global balance results from the local divergence-free condition $\partial u_{x}^{2D}/\partial x+\partial u_{z}^{2D}/\partial z=0$ .

This $2$ D scenario holds in any vertical slice of a spanwise, infinitely long droplet confined by two plates. But there is no reason why this condition should be satisfied in the middle slice of the ‘pancake’. The symmetry imposes indeed $u_{y}=0$ but not necessarily $\partial u_{y}/\partial y=0$ . The similarity between the two flows shows a posteriori that the in-plane divergence-free condition is approximately verified though, $\partial u_{x}/\partial x+\partial u_{z}/\partial z=-\partial u_{y}/\partial y\approx 0$ . This will be confirmed in the horizontal flow fields investigated next.

In figure 6, we display the velocity fields on the planes located at $z=0$ , $0.1$ , $0.2$ and $0.285$ together with the colour-coded wall-normal velocity $u_{z}$ ; note that the walls are located at $z=\pm 0.347$ . The flow field can be partitioned into three patches depending on the radial position $r_{xy}$ with respect to the origin: first, the inner patch that is circular ( $r_{xy}\lessapprox 1$ ) inside which the flow is mostly in the streamwise direction, i.e. $u_{y}\approx 0$ and $\partial u_{y}/\partial y\approx 0$ ; second, the outer patch ( $r_{xy}\gtrapprox 1.5$ ) that contains the flow passing around the droplet; and third, the annular patch ( $1\lessapprox r_{xy}\lessapprox 1.5$ ) that bridges the other two, where the flow mainly follows the in-plane curvature of the interface (red). The flow inside all the patches varies direction when the horizontal plane shifts from the middle $z=0$ towards the top wall. More specifically, in the inner patch, the flow goes forward at $z=0$ but backward at $z=0.285$ , reflecting the anti-clockwise recirculation on the vertical planes (see figure 5 a). In addition, the low in-plane velocities at $z=0.2$ correspond to the core of this recirculation. The velocity field in the outer patch represents the relative motion of the ambient flow with respect to the drop: near $z=0$ , the flow is faster than the drop and ‘pushes’ it; near the wall, the flow is slower and ‘retards’ it. The annular patch encompasses the droplet interface, and due to the non-penetration condition, the flow mostly follows the motion of the fluid elements along the interface: at $z=0$ , the ambient flow ‘pushes’ the droplet forward, resulting in a clockwise annular flow; near the top wall, the ambient flow ‘drags’ the droplet backward resulting in a counter-clockwise flow. Unlike the middle vertical slice, the in-plane divergence-free condition in the middle horizontal plane is clearly broken, as a source (respectively a sink) emerges on the axis at $x\approx -1.3$ (respectively $x\approx 1.2$ ) which exactly corresponds to the back (respectively the front) axial stagnation point on the middle vertical plane (see figure 5 a).

Figure 6.

Flow on the horizontal planes at (a) $z=0$ , (b) $z=0.1$ , (c)  $z=0.2$ and (d) $z=0.285$ for the same drop as in figure 5(a), shown in half ( $y\geqq 0$ ) of the domain. The top wall is located at $z=0.347$ . The contour colour indicates the wall-normal velocity $u_{z}$ . A reference vector with norm $|\mathbf{u}|=1$ is given. Red curves represent the droplet interface cut by the planes and the black dashed curves indicate the radial position of $r_{xy}=1$ and $r_{xy}=1.5$ . Magenta circles in (a) denote the same axial stagnation points as in figure 5(a).

We then come to the flow in the transverse planes shown in figure 7. Because of symmetry, we focus on the quarter ( $y\geqq 0,z\geqq 0$ ) and we zoom in the lateral interface of the drop. We observe two vortical structures aligned in the streamwise direction: one at the rear, rotating clockwise, and the other in the front, rotating anti-clockwise. The two structures are most intense at approximately $x=-0.85$ and $0.85$ , i.e. where their axis intersects the interface; they both decay in strength away from these maximum swirl regions and are connected at a no-swirl position slightly aft the droplet centre, i.e. between the $x=-0.15$ and $x=0$ plane. At this position, the vorticity switches sign and streamlines change their spiralling direction. These streamwise vortex structures are closely related to the flow in the horizontal planes shown in figure 6: at $x=-0.85$ and $y\approx 1$ , the flow is in the positive (respectively negative) $y$ direction in the annular patch at $z=0$ (respectively $z=0.285$ ), which generates a clockwise vortex; the vortex at $x=0.85$ appears likewise though oppositely oriented, because the flows in the annular patch reverse their spanwise directions.

Figure 7.

Flow on the transverse planes at (a) $x=-0.85$ , (b) $x=-0.4$ , (c) $x=-0.15$ , (d) $x=0$ , (e) $x=0.4$ and (f) $x=0.85$ for the same drop as shown in figure 5(a), illustrated near the droplet interface (red) in the $y\geqq 0,z\geqq 0$ quarter of the domain. The contour colour indicates the streamwise velocity $u_{x}$ . A reference vector with norm $|\mathbf{u}|=0.4$ is given.

Figure 8.

Disturbance flow field on the $y=0$ plane of the droplet analysed in § 4.3.

4.4 Disturbance flow field

We hereby analyse the disturbance flow $\boldsymbol{u}^{\prime }=\boldsymbol{u}_{lab}-\boldsymbol{u}^{\infty }$ induced by the presence of a translating pancake droplet, where $\boldsymbol{u}^{\infty }=U^{\infty }(1.5-6z^{2}/H^{2},0,0)_{xyz}$ . For the same drop as that examined in § 4.3, we depict $\boldsymbol{u}^{\prime }$ on the middle vertical plane in figure 8. In most of the domain, the disturbance flow is parallel, in the direction against the underlying flow. This represents the obstructive effect of the droplet travelling at a velocity $U_{d}$ smaller than the mean flow velocity $U^{\infty }$ ; in other words, the extra pressure drop stemming from the presence of the droplet is positive. Interestingly, the disturbance flow $\boldsymbol{u}^{\prime }$ reverses its direction near the front and rear dynamic meniscus regions that extend from the lubrication film towards the static meniscus regions. As a result, two vortical structures aligned in the positive $y$ direction emerge, akin to those observed in the flow field in the droplet frame $\boldsymbol{u}_{drop}$ projected on the transverse ( $yz$ ) planes as shown in figure 7. In fact, the projections of $\boldsymbol{u}^{\prime }$ , $\boldsymbol{u}_{lab}$ and $\boldsymbol{u}_{drop}$ on the transverse planes are equivalent, because both the droplet velocity and the underlying flow $\boldsymbol{u}^{\infty }$ have only one non-zero component that is the $x$ component.

The disturbance flow field $\boldsymbol{u}^{\prime }$ projected on three horizontal planes is shown in figure 9. On the middle $z=0$ plane, the droplet sucks in/ejects fluid in the front/rear, the interior flow is mostly parallel and opposite to the moving direction of the droplet but reverses the sign near its lateral edge. This resembles a $2$ D dipolar flow field decaying as $1/r^{2}$ (see figure 9 e for a typical sketch), which has been observed experimentally for a pancake droplet by Beatus, Tlusty & Bar-Ziv (Reference Beatus, Tlusty and Bar-Ziv2006). This dipolar field, as an elementary solution of potential flow, was also assumed to predict the velocity of a buoyancy-driven bubble (Maxworthy Reference Maxworthy1986). In figure 9(d), we examine how the disturbance velocity magnitude $U_{xy}^{\prime }=\sqrt{(u_{x}^{\prime })^{2}+(u_{y}^{\prime })^{2}}$ varies with the radial distance $r=\sqrt{x^{2}+y^{2}}$ , along the three paths emitting from the centre of the domain; the angles between these paths and the positive $x$ direction are ${\it\theta}={\rm\pi}/4,{\rm\pi}/2$ and $3{\rm\pi}/4$ . The log–log plot in the inset indicates that the decaying rate does indeed closely follow the $1/r^{2}$ scaling law. The dipolar flow field is also detected on the $z=0.15$ plane with a decreased strength. However, it disappears on the $z=0.285$ plane where the droplet ejects/sucks in fluid near its front/rear meniscus; this reversed disturbance flow has in fact been revealed on the middle vertical plane in figure 8.

Figure 9.

Disturbance flow field $\boldsymbol{u}^{\prime }$ projected on the horizontal planes at (a) $z=0$ , (b) $z=0.15$ , (c) $z=0.285$ for the same drop as in figure 8(a); the contour colour indicates the disturbance velocity magnitude $U_{xy}^{\prime }/U_{d}$ . (d) Spatial variation of $U_{xy}^{\prime }/U_{d}$ on the $z=0$ plane, along three directions; the inset shows the log–log scale. (e) Sketch of a typical dipolar flow pattern.