2.1. Experimental method

The schematic in figure 1(a) illustrates the key features of our experimental apparatus. The air flow is supplied by an air compressor through a rectangular nozzle (width $25.4\ \textrm {mm} \times \textrm {length}\ 0.2\ \textrm {mm}$). The flow pressure is controlled by a back-pressure regulator (Wilkerson), while a diffuser and contraction region ensure a low-turbulence flow (i.e. turbulence intensity below 2 %) of nearly uniform velocity at the outlet. In each experimental run, we first deposit a distilled water droplet of volume $V$ on a polished aluminium surface (240 grit SiC); we allow the droplet to reach an equilibrium shape before applying the jet of air. We measure the droplet's static contact angle, $\theta _0$, with the solid substrate by first identifying the water–air interface via image processing and computing its slope close to the contact line; $\theta _0$ is approximately equal to $30^{\circ }$ for all experiments reported herein. The initial droplet half-width is given by $L$, and the distance between the nozzle outlet and the substrate, $H_0$, is fixed at 2 cm, as illustrated in figure 1(b). Then, prior to opening the air compressor valve, we set a desired pressure for the regulator to ensure that the jet speed at the outlet is kept constant throughout the entire experiment. We measure the flow speed, $U_0$, along the centreline at the outlet using a hot-wire anemometer (Testo). For the range of $U_0$ in this study, typical temporal fluctuations in air speed are measured to be 3 %. Similarly, variations in jet speed along the centreline at different lateral positions are less than 5 % for locations at least 5 mm from the sides of the outlet.

The experiments are backlit by a light-emitting diode panel (ASD Lighting) and recorded with a high-speed camera (Photron, 2000 frames per second, $1280\times 1024$ resolution) and a macro lens (Nikon). The camera is placed in front of the droplet (see figure 1a) to capture the side-view images of the evolving droplet. We systematically vary the droplet volume (i.e. $V=20\text {--}200\ \mathrm {\mu }\textrm {l}$) as well as the speed of air flow $U_0$, which can go up to 40 m s$^{-1}$. The characteristic Reynolds number of the air flow corresponds to $Re=U_0L/\nu _{a}=10^{3}$ to $8\times 10^{3}$, where $\nu _{a}=15\times 10^{-6}\ \textrm {m}^{2}\ \textrm {s}^{-1}$ is the kinematic viscosity of air at room temperature.

2.2. Experimental observations

In the first set of experiments, we examine the dynamic response of a partially wetting droplet when a 2-D jet of air is applied along the drop's centreline for varying $V$ and $U_0$. For given $V$, the droplet exhibits two different behaviours – ‘hanging’ versus ‘splitting’ – depending on the value of $U_0$; the transition between the two regimes is marked by the threshold air speed $U_{cr}$. When the air flow first reaches the drop, the droplet deforms into a concave shape at the centre (see figure 2a,ii), which corresponds to the stagnation point of the jet with the maximum external pressure. The pressure difference between the high-pressure centreline and the low-pressure edges drives an internal flow towards the two opposite sides of the droplet, which results in the observed necking in the centre. For $U_0<U_{cr}$ (‘hanging’ regime), the neck continues to thin in the $z$-direction until the droplet reaches an equilibrium state. The droplet continues to oscillate about this equilibrium configuration, which is related to the propagation of capillary waves along the droplet surface (Milne et al. Reference Milne, Defez, Cabrerizo-Vílchez and Amirfazli2014). However, the unsteady oscillations that are observed after the droplet reaches the hanging state are outside the scope of the current study. When $U_0>U_{cr}$ (‘splitting’ regime), the necking along the centreline of the droplet thins further until the droplet splits into two (figure 2a,iv).

Figure 2. (a) Visualising the time evolution of droplet shapes held against a 2-D stagnation-point flow. (i) A partially wetting droplet of 200 $\mathrm {\mu }$l is initially in the equilibrium state ($t=0$). (ii) As the droplet is subject to a 2-D jet of air with velocity $U_0=10.8\ \textrm {m}\ \textrm {s}^{-1}$, it forms a neck along the centreline ($t=0.3\ \textrm {s}$). (iii) The thinning of the meniscus continues until the drop reaches the onset of breakup ($t=0.55\ \textrm {s}$). (iv) The two resultant drops depin and move in opposite directions ($t=0.75\ \textrm {s}$). (b) The plot of the critical jet speed for splitting, $U_{cr}$, as a function of the droplet volume $V$; the value of $U_{cr}$ has been extracted from the experiments.

Figure 2(b) shows the plot of $U_{cr}$ as a function of the droplet volume $V$. As $V$ increases, the droplet splits at increasingly lower jet speeds. This can be qualitatively explained in terms of the capillary pressure. As the droplet size increases, the magnitude of the mean curvature of the drop must decrease, corresponding to an overall decrease in the drop's internal pressure. Therefore, a lower stagnation pressure is required to drive the internal flow from the drop's centre towards the edges and to result in a breakup, compared with smaller droplets.

In the second series of experiments, we examine the effect of systematically varying the stagnation-point position relative to the droplet's centre for given $V$ and $U_0$. Here, $\alpha$ characterises the dimensionless distance between the centre of the jet and that of the drop normalised by $L$ (i.e. $\alpha =1$ at the contact line). Analogous to the $\alpha =0$ case, at low jet speeds, the droplet reaches the ‘hanging’ equilibrium state and does not split, as shown in figure 3(a). However, at intermediate $U_0$ between the hanging and splitting regimes, a new droplet behaviour emerges. As shown in figure 3(c), droplets depin towards the side of the stagnation point with a larger droplet volume, for a range of $U_0$ that depends on $\alpha$ and $V$. In this ‘depinning’ regime, the droplet initially forms the neck at the stagnation position. Once the neck reaches a minimum thickness, the fluid on the smaller side of the drop drains towards the larger side due to relative pressure differences inside the droplet. Then, at higher values of $U_0$, drops split into two, as shown in figure 3(b). Note that $U_{cr}$ marks the onset of the splitting regime and is now a function of $V$ and $\alpha$.

Figure 3. (a–c) Different behaviours of a $50\ \mathrm {\mu }\textrm {l}$ droplet under an off-centred high-speed jet flow. (a) A jet with $U_0=6.5\ \textrm {m}\ \textrm {s}^{-1}$ located at $\alpha =0.2$ deforms the droplet until it reaches the equilibrium. (b) The droplet is subject to a jet with $U_0=17.8\ \textrm {m}\ \textrm {s}^{-1}$ at $\alpha =0.2$. The neck that is formed under the jet thins until it splits into two smaller drops of different sizes. (c) The initial extensional flow inside the droplet caused by $U_0=8.7\ \textrm {m}\ \textrm {s}^{-1}$ at $\alpha =0.2$ transitions to a unidirectional flow from the smaller side to the larger side of the drop, resulting in a drifting behaviour. (d,e) Summary of different droplet responses to a high-speed jet for varying jet velocity $U_0$ and non-dimensional jet position $\alpha$, for (d) $V=50\ \mathrm {\mu } \textrm {l}$ and (e) $V=200\ \mathrm {\mu } \textrm {l}$.

Figure 3(d,e) summarises the three droplet behaviours in an $\alpha$–$U_0$ phase diagram for $V=50$ and $200\ \mathrm {\mu }\textrm {l}$, respectively. For $V=200\ \mathrm {\mu }\textrm {l}$, the ‘depinning’ regime becomes suppressed for small $\alpha$, and the droplet transitions from hanging to splitting regimes directly with increasing $U_0$. When the stagnation point moves further from the initial centre of the drop (i.e. $\alpha >0.2$), the depinning behaviour emerges. By contrast, for $V=50\ \mathrm {\mu }\textrm {l}$, all three droplet behaviours are observed with an increase in $U_0$, for all non-zero $\alpha$ values considered (see figure 3e). Furthermore, for both small and large droplets, an increase in $\alpha$ consistently shifts the onset of the splitting regime to a higher jet speed (i.e. $U_{cr}$ increasing with $\alpha$), which will be explored in a later section.

2.3. Capillary pressure inside the evolving droplet

The internal flows that lead to ‘hanging’, ‘splitting’ or ‘depinning’ regimes are driven by the non-uniform pressure distribution inside the droplet. Hence, we now estimate the capillary pressure inside the evolving drop, by measuring the local in-plane curvature $\kappa (t)$ at the stagnation point, and at the left and right corners of the droplet, respectively (denoted with the subscripts ‘$s$’, ‘$l$’ and ‘$r$’) and applying the Young–Laplace equation. The dimensionless external pressure over the droplet in the $x$–$z$ plane is approximated by a quadratic function, $P^{*}=P^{*}_{s}-(L/H_0)^{2}(x^{*}-\alpha )^{2}$, where $x^{*}=x/L$ and $P^{*}_{s}$ is the dimensionless stagnation pressure at $x^{*}=\alpha$, normalised by $(1/2)\rho _{a}U^{2}_0$. Here, $\rho_a$ corresponds to the density of air. By neglecting the drop's out-of-plane curvature, the non-dimensional internal pressure $p^{*}$ can be approximated as $p^{*}(t)=P^{*}-2\sigma \kappa (t)/(\rho _{a} U^{2}_0)$, where $\sigma$ is the surface tension coefficient. Hence, the values of $p^{*}-P^{*}_{s}$ at a given time are determined by extracting $\kappa _{s}(t)$, $\kappa _{l}(t)$ and $\kappa _{r}(t)$ from the experimental images.

We focus on the case of $\alpha = 0.2$ and $V=50\ \mathrm {\mu }\textrm {l}$, as presented in figure 3. At the initial time $t=0$, $\kappa _{s}(0)\approx \kappa _{r}(0)\approx \kappa _{l}(0)$, so that we obtain $p^{*}_{s}(0)>p^{*}_{r}(0)>p^{*}_{l}(0)$, mirroring $P^{*}_{s}(0)>P^{*}_{r}(0)>P^{*}_{l}(0)$ (see figure 4a). This internal pressure gradient must drive an asymmetric diverging flow from the stagnation point towards both sides of the droplet. As fluid drains from the stagnation point, $\kappa _s$ increases, while $\kappa _{r}$ and $\kappa _{l}$ decrease, as illustrated in figure 4(b). The resultant shape deformations cause time-dependent changes in the internal pressure. Figure 4(c–e) shows the plots of $p^{*}-P^{*}_{s}$ over time at the stagnation point ($x^{*}=\alpha$), and at the left ($x^{*}=-1$) and right ($x^{*}=1$) corners of the drop.

Figure 4. (a) The schematics of a droplet's initial shape at the moment we apply the jet to an arbitrary off-centred point on the droplet interface. (b) The droplet shape at time $t$ can be divided into three zones: the stagnation zone with a concave shape, the smaller convex side with a higher mean curvature, and the larger convex side with a lower mean curvature. (c–e) The relative internal pressure in the different droplet zones is computed from the extracted mean curvature, for (c) the hanging regime in figure 3(a), (d) the splitting regime in figure 3(b), and (e) the depinning regime in figure 3(c).

When $U_0$ is low (i.e. $6.5\ \textrm {m}\ \textrm {s}^{-1}$), the shape deformations continue until the internal pressure becomes uniform, reaching the steady state at time $t_{f}$. This is evidenced by $p^{*}_{s}$, $p^{*}_{l}$ and $p^{*}_{r}$ converging to a single value in figure 4(c) and corresponds to the ‘hanging’ regime (see figure 3a). If $U_0$ is increased to 17.8 m s$^{-1}$, the deformations are not adequate to equalise the internal pressure before the necking yields breakup at $t_{f}$, as evident in figure 4(d). This exemplifies the ‘splitting’ regime, previously shown in figure 3(b).

At $U_0=8.7 \ \textrm {m}\ \textrm {s}^{-1} < U_{cr}$, $p^{*}_{r}-P^{*}_{s}$ gradually increases past $p^{*}_{s}-P^{*}_{s}$. This reverses the direction of the part of the internal flow from the positive $x$-direction (from the stagnation point to the right-hand side) to the negative $x$-direction (from right to left), which leads to droplet depinning. This flow reversal is evident in the time-sequential images in figure 3(c): fluid first drains away from the stagnation point (i), then temporarily stalls as the internal pressure on the right-hand side marginally surpasses that of the stagnation point at time $t_{f}$ (ii), and finally drains from right to left (iii, iv) before depinning (v). Therefore, the ‘depinning’ regime is marked by the gradual pressure buildup above the stagnation pressure on one side of the droplet, which drives the net internal flow in one direction before the internal pressure can equalise. Figure 4(e) demonstrates the evolution of the internal pressure up to the moment when the droplet stalls, as illustrated in figure 3(c,ii). Hence, $t_{f}$ in figure 4(e) marks the transition from the diverging flow to a unidirectional flow inside the droplet.

The direct correlation between $p^{*}-P^{*}_{s}$ and the experimental observation signifies that the external pressure field (modelled as a stagnation-point flow) and the droplet curvature in the $x$–$z$ plane constitute the two dominant physical effects in our current system. Hence, we will construct a theoretical model of the evolving droplet that incorporates these two effects to leading order.

3.1. Lubrication equation

Under lubrication approximations, the linear momentum equations inside the droplet reduce to

(3.1a,b)\begin{equation} -\frac{\partial p_{d}}{\partial x}+\mu_{d}\frac{\partial^{2} u_{d}}{\partial z^{2}}=0,\quad \frac{\partial p_{d}}{\partial z}=-\rho_{d} g, \end{equation}

where $\mu _{d}$ and $\rho _{d}$ are the dynamic viscosity and density of the droplet, respectively, while the pressure inside the drop is given by $p_{d}(x,z,t)$, and the gravitational acceleration by $g$. We solve for the horizontal velocity inside the drop, $u_{d}(x,z,t)$, by integrating (3.1a,b) subject to boundary conditions on the droplet surface, $z=h(x,t)$:

(3.2a,b)\begin{equation} \mu_{d}\frac{\partial u_{d}}{\partial z}=\tau_{s},\quad P(x,h)-p_{d}(x,h)=\sigma \kappa(x,t), \end{equation}

in addition to $u_{d}(z=0)=0$. Here, the first corresponds to the continuity of shear stress at the fluid–air interface, where $\tau _{s}$ denotes the external shear stress acting on the droplet surface. The second condition relates the jump between $p_{d}$ and external pressure $P$ due to surface tension (with coefficient $\sigma$); here $\kappa (x,t)$ denotes the planar curvature of the droplet surface, which reduces to $h''(x)$ in the lubrication limit. We non-dimensionalise the governing equations and boundary conditions using the following rescalings:

(3.3a–d)\begin{gather} u_{d}^{*}=\frac{u_{d}}{U_0},\quad (x^{*},z^{*})=\left(\frac{x}{L},\frac{z}{\varepsilon L}\right),\quad h^{*}=\frac{h}{\varepsilon L},\quad t^{*}=\frac{tU_0}{L}, \end{gather}

(3.4a,b)\begin{gather}P^{*}=\frac{P}{\tfrac{1}{2}\rho_{a} U^{2}_0},\quad \tau^{*}_{s}=\frac{\tau_{s} \varepsilon L}{\mu_{a}U_0}. \end{gather}

Here $\varepsilon =\tan (\theta _0/2)\approx \theta _0/2\ll 1$ is the expansion parameter and a measure of the surface wettability; and $\mu _{a}$ refers to the dynamic viscosity of air. Combining (3.1a,b) and (3.2a,b) with the continuity equation yields the dimensionless evolution equation for $h^{*}(x,t)$,

(3.5)\begin{equation} \frac{\partial h^{*}}{\partial t^{*}}=\frac{\partial}{\partial x^{*}}\left[ Ca^{-1} \left( We\,\varepsilon^{2} \frac{\partial P^{*}}{\partial x^{*}}-\varepsilon^{3} \frac{\partial^{3} h^{*}}{\partial {x^{*}}^{3}}+Bo\, \varepsilon^{3} \frac{\partial h^{*}}{\partial x^{*}} \right)\frac{{h^{*}}^{3}}{3}- \frac{\mu_{a}}{2\mu_{d}}{\tau^{*}_{s}}{h^{*}}^{2} \right], \end{equation}

with capillary number $Ca=\mu _{d}U_0/\sigma$, Weber number $We=\rho _{a}U_0^{2}L/\sigma$ and Bond number $Bo=\rho _{d}gL^{2}/\sigma$. The values of all the relevant dimensionless numbers, including the Reynolds number $Re$, are tabulated in figure 5, based on the experimental parameters. Equation (3.5) requires four boundary conditions, which are given as

(3.6a,b)\begin{equation} h^{*}(\pm 1,t^{*})=0,\quad \frac{\partial^{2} h^{*}}{\partial {x^{*}}^{2}}(\pm 1,t^{*})=\kappa^{*}_0, \end{equation}

where $\kappa ^{*}_0$ is the initial dimensionless curvature of the droplet at the corners. The boundary conditions specify a fixed contact line with a constant curvature at the edge of the droplet, as previously implemented by Constantin et al. (Reference Constantin, Dupont, Goldstein, Kadanoff, Shelley and Zhou1993) to simulate droplet breakup inside a Hele-Shaw cell. The constant-curvature condition leads to an internal flow towards the sides of the drop that could cause splitting (Constantin et al. Reference Constantin, Dupont, Goldstein, Kadanoff, Shelley and Zhou1993; Bertozzi Reference Bertozzi1998).

To solve (3.5), we must model the external flow to find expressions for $P$ and $\tau _{s}$. Consistent with the lubrication approximation, the external flow is modelled as a steady stagnation-point flow over a flat surface to leading order; the presence of the thin droplet is incorporated into the external flow model at $O(\varepsilon )$. Based on the high-Reynolds-number nature of the air flow (i.e. $Re \sim 10^{3}\text {--}10^{4}$), we further divide the external flow into outer and inner regions (see figure 5). To characterise the inner region, we estimate the thickness of the boundary layer, $\delta /L$, as $Re^{-1/2}\sim 10^{-2}$, which is an order of magnitude smaller than $\varepsilon \sim 10^{-1}$, or the dimensionless droplet height. In the current limit of $Re^{-1/2} \ll \varepsilon$, the effects of viscous shear stress $\tau _{s}$ must be an order of magnitude smaller than the external pressure $P$ (Smith et al. Reference Smith, Brighton, Jackson and Hunt1981; Durbin Reference Durbin1988). Therefore, we reasonably assume that the droplet dynamics is governed by $P$ to leading order, which matches our experimental observations in § 2.3. Notably, the current problem is in a completely different physical regime from the well-established work of Fry (Reference Fry2012), in which a droplet or a protrusion is fully submerged inside the inner deck of the viscous boundary layer, with no imposed external pressure gradient (see also Smith et al. Reference Smith, Brighton, Jackson and Hunt1981).

The relative sizes of $P$ and $\tau _{s}$ can also be deduced by directly comparing the first and last terms on the right-hand side of (3.5). The first term corresponds to the flux inside the droplet that is induced by the external pressure gradient, while the last term corresponds to that due to external shear stress. The ratio between the two terms scales as $Re\,\varepsilon ^{2}\sim \varepsilon ^{-1}$, confirming that the shear stress term is indeed at least $O(\varepsilon )$ smaller than the external pressure gradient term. Based on the scaling argument, we calculate the external pressure up to $O(\varepsilon )$, which accounts for the linearised effect of the droplet, and shear stress to $O(1)$. The latter is equivalent to computing shear stress due to a stagnation-point flow over a flat plate only. Our method consists of three steps:

1. (1) Determine the external pressure $P$ in the outer flow and shear stress $\tau _{s}$ in the inner flow along the solid surface to $O(1)$.

2. (2) Given the droplet shape at time $t$, compute $P$ to $O(\varepsilon )$.

3. (3) Evolve the droplet shape by solving (3.5) and return to step 2.

Owing to the importance of the $O(1)$ outer velocity field for resolving $\tau _s$, we first describe our approach for computing $P$ in the irrotational outer flow (steps 1 and 2). Next, we discuss the solution to the Prandtl boundary layer equations (step 1). In § 4, we present our numerical simulations (step 3). Finally, we include the steady-state analysis of the governing equations and the corresponding analytical results to justify experimental observations.

3.2. External flow: outer region

In the outer region sufficiently far from the boundaries, we take the flow as inviscid. As illustrated in figure 6(a), we rescale the outer flow with $[\tilde {x}]= [\tilde {z}]=H_0$, where $H_0$ is the distance between the flow source and the solid surface (i.e. $\tilde {x}=(L/H_0)x^{*}$). Accordingly, the droplet height in the rescaled coordinates is given by $\tilde {z}=\xi h^{*}$, where $\xi =\varepsilon L/H_0$ is a modified small parameter, and $L/H_0=O(1)$. The external velocity $U$ is scaled by $U_0$, such that $\tilde {U}=U/U_0$. In general, the tilde denotes dimensionless variables in the external flow, while the asterisk indicates those from the original lubrication equation.

Figure 6. (a) The schematic of the potential flow past the drop and the flat plate with zero-flux condition at the boundaries. (b) The computed values of the dimensionless pressure gradient over the droplet interface for different droplet shapes, when a jet of $U_0=10\ {\rm m}\ {\rm s}^{-1}$ is applied to a droplet with $L=7.7\ \textrm {mm}$ and $\theta _0=10^{\circ }$.

As the outer flow is irrotational, $\tilde {U} = \tilde {\boldsymbol {\nabla }} \phi$, where $\phi$ is a dimensionless harmonic velocity potential. We then expand $\phi$ as a power series of $\xi$: $\phi =\phi _0+\xi \phi _1+O(\xi ^{2})$. The leading-order velocity potential reduces to the stagnation-point flow over a flat surface and is given by $\phi _{0}=\frac {1}{2}(\tilde {x}^{2}-\tilde {z}^{2} )$. Then at $O(\xi )$, the presence of a thin droplet is incorporated into the outer flow through the modified geometry of the boundaries, which must satisfy the no-flux condition. The linearised velocity potential gradient on the droplet surface corresponds to

(3.7)\begin{equation} ( \tilde{\boldsymbol{\nabla}}\phi )_{(\tilde{x},\xi h^{*})} =(\tilde{\boldsymbol{\nabla}}\phi_{0})_{(\tilde{x},0)} +\xi\left[h^{*}\frac{\partial}{\partial \tilde{z}} (\tilde{\boldsymbol{\nabla}}\phi_{0}) +\tilde{\boldsymbol{\nabla}}\phi_{1}\right]_{(\tilde{x},0)}+O(\xi^{2}). \end{equation}

Then, at $O(\xi )$, the zero-flux condition on the surface becomes

(3.8)\begin{equation} \left[\frac{\partial \phi_{1}}{\partial \tilde{z}}\right]_{\tilde{z}=0}= \frac{\partial h^{*}}{\partial \tilde{x}} \tilde{x}+h^{*}+\frac{\partial h^{*}}{\partial t^{*}}= \frac{\partial}{\partial x^{*}}\left(x^{*}h^{*} \right) +\frac{\partial h^{*}}{\partial t^{*}}. \end{equation}

We note that (3.8) neglects the effect of the kinematic boundary condition transmitted through the inner flow, as such corrections appear at the next order (i.e. boundary layer thickness, $\delta /L\sim O(\varepsilon ^{2}$)). By setting $\phi _{1}=\sum _{k=-\infty }^{\infty} \hat {\phi }_{k}(\tilde {z},t^{*})\,\textrm {e}^{\textrm {i}k{\rm \pi} x^{*}}$, we note that $\tilde {\nabla }^{2}\phi _1=( (\partial ^{2}/\partial \tilde {z}^{2})-k^{2}{\rm \pi} ^{2} ) \hat {\phi }_{k}=0$, which yields $\hat {\phi }_{k}(\tilde {z},t^{*})=A_{k}(t^{*})\,\textrm {e}^{| k | {\rm \pi}\tilde {z}}+B_{k}(t^{*})\,\textrm {e}^{-| k | {\rm \pi}\tilde {z}}$. The boundedness of $\phi _{1}$ requires that $A_{k}=0$. By substituting $\phi _1$ into the left-hand side of (3.8) and through the Fourier expansion of the right-hand side, we obtain

(3.9)\begin{equation} B_{k}=-\frac{\textrm{i}k}{| k |}\mathcal{F}( x^{*}h^{*} ) - \frac{1}{{\rm \pi} | k |}\mathcal{F}(\partial h^{*}/\partial t^{*}), \end{equation}

where $\mathcal {F}(X(\omega ))=\frac {1}{2}\int _{-1}^{+1}X(\omega )\, \textrm {e}^{-\textrm {i}k{\rm \pi} \omega }\,{\textrm {d}}\omega$. In addition, the Bernoulli equation applies throughout the outer region, such that

(3.10)\begin{equation} \left[2\frac{\partial \phi}{\partial t^{*}}+P^{*}+ |{\tilde{U}}|^{2}\right]_{(\tilde{x},\xi h^{*})}=P^{*}_0+1, \end{equation}

where $P^{*}_0$ is the dimensionless pressure at the nozzle. After substituting (3.7) into (3.10), the external pressure on the interface can be written as

(3.11)\begin{align} P^{*}(x^{*},h^{*}) &= P^{*}_0+1-\left(\frac{L^{2}}{{H_0}^{2}}\right){x^{*}}^{2} -\left(\frac{2{\rm \pi} L^{2}}{{H_0}^{2}}\right) \varepsilon x^{*} {\sum_{k=-\infty}^{\infty} | k | \mathcal{F}( x^{*}h^{*}) \textrm{e}^{\textrm{i}k{\rm \pi} x^{*}}} \nonumber\\ &\quad +\left(\frac{2 L^{2}}{{H_0}^{2}}\right) \varepsilon x^{*} {\sum_{k=-\infty}^{\infty} \frac{\textrm{i}k}{| k |} \mathcal{F}( \partial h^{*}/\partial t^{*} ) \textrm{e}^{\textrm{i}k{\rm \pi} x^{*}}} -2\left(\frac{\partial \phi_1}{\partial t^{*}}\right)_{({\tilde{x},0})} +O(\varepsilon^{2}). \end{align}

The corresponding external pressure gradient ${\textrm {d}}P^{*}/{\textrm {d}}x^{*}$ is plotted as a function of $x^{*}$ in figure 6(b) at different times. It demonstrates the extent to which external pressure is modified due to the evolving droplet shape.

3.3. External flow: inner region

In this section, we outline our approach for computing the shear stress $\tau _s$ to leading order, which we later incorporate into (3.5). Thus, we only consider the inner region, which consists of a thin layer surrounding the solid surface where the effects of viscosity come to bear, as shown in figure 7. The Navier–Stokes equations in the boundary layer reduce to Prandtl boundary layer equations (Prandtl Reference Prandtl1904; Schlichting Reference Schlichting1960):

(3.12a,b)\begin{equation} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial z}=0,\quad u\frac{\partial u}{\partial x} +v\frac{\partial u}{\partial z}-\nu_{a} \frac{\partial^{2} u}{\partial z^{2}}=U\frac{\partial U}{\partial x}, \end{equation}

where $(u,v)$ denote velocity components of the external flow in the inner region, while $U$ refers to the fluid velocity in the inviscid outer flow. The Prandtl equations hold within a boundary layer of thickness $\delta (x)$, which has characteristic scale $\bar \delta =L/\sqrt {Re}$. No-slip boundary conditions and matching conditions to the outer flow include

(3.13a,b)\begin{gather} u(x,0) = v(x,0)=0,\quad \frac{\partial^{2} u}{\partial z^{2}}(x,0)=-\frac{1}{\nu_{a}}U\frac{\partial U}{\partial x}, \end{gather}

(3.14)\begin{gather}u(x,z)\rightarrow U(x),\quad \textrm{as}\ z/\delta\rightarrow \infty. \end{gather}

Figure 7. (a) The schematic of the leading-order boundary layer flow on the flat substrate. (b) The inner flow velocity profile as a function of the normal distance from the solid boundary at $x=L$ using the similarity solution by Hiemenz (Reference Hiemenz1911) and the VKP method. (c) The plot of the shear stress on the surface as a function of $x$ based on the similarity solution and the VKP method for $L=7.7\ \textrm {mm}$, $U_0=10\ \textrm {m}\ \textrm {s}^{-1}$ and $\theta _0=10^{\circ }$.

While a similarity solution to (3.12a,b) exists (Hiemenz Reference Hiemenz1911; Howarth Reference Howarth1935), in our simulations, we employ the von Kármán-Pohlhausen (VKP) method to approximate the solution to the Prandtl equations (von Kármán Reference von Kármán1921; Pohlhausen Reference Pohlhausen1921). This method approximates the inner flow with a fourth-order polynomial that satisfies the boundary conditions (3.13a,b). Once we find $\delta (x)$ with the VKP method, we can write the shear stress on the solid surface in this method as

(3.15)\begin{equation} \tau^{*}_s=\varepsilon \sqrt{Re} \left(\frac{L}{H_0} \right)^{{3}/{2}} \left(\frac{12+(\delta/\bar\delta)}{6\sqrt{\delta/\bar\delta}}\right) x^{*}. \end{equation}

Figure 7(b,c) compares the velocity profile and the tangential shear stress over the surface for the similarity solution and the VKP method for sample input parameters (i.e. $U_0=10\ \textrm {m}\ \textrm {s}^{-1}$, $L=7.7\ \textrm {mm}$ and $\theta _0=10^{\circ }$). More details about the VKP calculations can also be found in the related study by Moore et al. (Reference Moore, Ristroph, Childress, Zhang and Shelley2013).