3.1. Impact morphologies

Figure 3 shows the temporal evolution of the morphology of simultaneous droplet-pair impacts for droplets with glycerol volume fraction $\phi _g$ ranging from 0 % to 80 % through time-resolved front-view images and side-view insets. The impact Weber number ( $ \textit{We}$ ) and the non-dimensional inter-droplet spacing ( $\Delta x^* = \Delta x / D_0$ ) for these droplet impacts are fixed at $ \textit{We} = 130 \pm 2$ and $\Delta x^* = 1.80 \pm 0.04$ , respectively. For pure water droplets (i.e. $\phi _g = 0$ %), the interaction between the two droplet lamellae generates stagnation points along their interaction line, where the liquid flows in the lamellae are redirected upward and sideways. At the midpoint of this interaction line (aligned with the $z$ -axis of the test rig coordinate system), the upward flow is maximum and reduces symmetrically about the $z$ -axis, generating a rim-bounded uprising ‘semilunar’ central sheet, as depicted in the inset image for $\tau = 0.75$ and $\phi _g = 0$ % in figure 3. Similar ‘semilunar’ sheets are also observed for glycerol–water droplet pairs with $\phi _g = 40$ % and 60 % (figure 3), which possess higher viscosities than the water droplets ( $\phi _g = 0$ %). However, with increased liquid viscosity (i.e. higher $ \textit{Oh}$ ), the central sheet size reduces and dissipates faster due to the increased shear stress within the droplet liquid, resisting droplet deformation and central sheet growth. Consequently, as $\phi _g$ is increased to 70 % and 80 %, the sheets’ ‘semilunar’ shape does not appear in side-view images; instead, the sheets’ front views resemble relatively ephemeral liquid bumps, similar to those formed during the coalescence of two droplets upon their low-velocity impacts (Batzdorf et al. Reference Batzdorf, Breitenbach, Schlawitschek, Roisman, Tropea, Stephan and Gambaryan-Roisman2017). It is noted that probing the velocity field within thin liquid films, such as lamellae and the central uprising sheet, is experimentally challenging. Kanjirakat et al. (Reference Kanjirakat, Sadr and Alvarado2021) used microparticle tracking velocimetry ( $\mu$ PTV) to measure the near-wall velocities in the impingement zone of a droplet train and found significant differences compared with a jet-stream impact, primarily due to the enhanced mixing induced by the crown propagation on the substrate. The droplet impacts studied here do not exhibit such crown propagation around the impact centres, but rather involve interacting lamella segments that generate an uprising ‘semilunar’ sheet with distinct morphological characteristics. Therefore, although the present study does not measure liquid velocity fields, these observations highlight the need for future measurements using $\mu$ PTV or similar techniques to probe the flow within the central interaction zone and explore further the hydrodynamics of droplet-pair impacts.

Figure 3. Impact morphologies of droplet pairs of different glycerol–water mixtures with glycerol volume fraction $\phi _g$ from 0 % ( $\mu = 1.01$ mPa s, $ \textit{Oh} = 0.002$ ) to 80 % ( $\mu = 91.46$ mPa s, $ \textit{Oh} = 0.177$ ). Related physical properties are summarised in table 1. The 2 mm scale bar indicated in the first front-view image applies to all front-view images and the 2 mm scale bar indicated in the first side-view inset applies to all side-view insets. The Weber number ( $ \textit{We}$ ) and the dimensionless inter-droplet spacing ( $\Delta x^*$ ) are maintained at $ \textit{We} = 130 \pm 2$ and $\Delta x^* = 1.80 \pm 0.04$ , respectively.

Figure 4. Impact morphologies of droplet pairs of different glycerol–water mixtures with glycerol volume fraction $\phi _g$ from 0 % ( $\mu = 1.01$ mPa s, $ \textit{Oh} = 0.002$ ) to 80 % ( $\mu = 91.46$ mPa s, $ \textit{Oh} = 0.177$ ). Related physical properties are summarised in table 1. The 2 mm scale bar indicated in the first front-view image applies to all front-view images and the 2 mm scale bar indicated in the first side-view inset applies to all side-view insets. The Weber number ( $ \textit{We}$ ) and the dimensionless inter-droplet spacing ( $\Delta x^*$ ) are maintained at $ \textit{We} = 104 \pm 1$ and $\Delta x^* = 1.80 \pm 0.03$ , respectively.

Figure 5. Impact morphologies of droplet pairs of different glycerol–water mixtures with glycerol volume fraction $\phi _g$ from 0 % ( $\mu = 1.01$ mPa s, $ \textit{Oh} = 0.002$ ) to 80 % ( $\mu = 91.46$ mPa s, $ \textit{Oh} = 0.177$ ). Related physical properties are summarised in table 1. The 2 mm scale bar indicated in the first front-view image applies to all front-view images and the 2 mm scale bar indicated in the first side-view inset applies to all side-view insets. The Weber number ( $ \textit{We}$ ) and the dimensionless inter-droplet spacing ( $\Delta x^*$ ) are maintained at $ \textit{We} = 81 \pm 2$ and $\Delta x^* = 1.80 \pm 0.03$ , respectively.

Figure 6. Schematic representation of the characteristic quantities measured for the impact processes: the lamella spread radius ( $R$ ), the central sheet height ( $H_S$ ), the sheet length ( $L_S$ ), the lamella edge velocity ( $V_L$ ) and the lamella edge thickness ( $T_L$ ).

Figures 4 and 5 show droplet-pair impact morphologies for different $\phi _g$ values for two other impact Weber numbers, $ \textit{We} = 104 \pm 1$ and $ \textit{We} = 81 \pm 2$ , respectively, while keeping the inter-droplet spacing unchanged at $\Delta x^* \approx 1.80$ . The temporal evolutions of these impact cases are qualitatively similar to those observed for $ \textit{We} \approx 130$ in figure 3. However, for $ \textit{We} \approx 81$ and $\phi _g = 80$ %, the interaction between two lamellae is not strong enough to form a central sheet (or liquid bump). Instead, the lamella fronts gently touch and coalesce during the impact process. This weak interaction directly results from the combined effect of the reduced impact Weber number and increased Ohnesorge number ( $ \textit{Oh}$ ), indicative of the reduced impact inertia and increased viscous dissipation. The next section will provide a quantitative assessment of the liquid viscosity effect on lamella spreading.

Careful observation of the impact processes shown in figure 3–5 reveals several impact characteristics as follows. For a given impact Weber number ( $ \textit{We}$ ) and non-dimensional inter-droplet spacing ( $\Delta x^*$ ), a higher droplet liquid viscosity (i.e. higher $ \textit{Oh}$ or $\phi _g$ ) leads to a delayed lamella interaction and increased lamella-front thickness at their first instant of interaction, i.e. at $\tau = \tau _0$ . This increased lamella thickness for higher liquid viscosities is not surprising, as the thickness of a spreading lamella formed during droplet impact on a solid substrate scales with a viscous boundary layer thickness, $\sqrt {\nu t}$ , with $\nu (= \mu / \rho )$ as the kinematic viscosity of the droplet liquid (Josserand et al. Reference Josserand, Lemoyne, Troeger and Zaleski2005; Roisman, Horvat & Tropea Reference Roisman, Horvat and Tropea2006; Xu, Barcos & Nagel Reference Xu, Barcos and Nagel2007; Bird, Tsai & Stone Reference Bird, Tsai and Stone2009). Consequently, the interaction of thicker lamella fronts leads to thicker central sheet formation for high-viscosity droplet impacts, as revealed in figures 3–5. For droplet impacts with $\phi _g = 0$ %, several concentric ripples (capillary waves) appear on the ascending central sheet surface and propagate toward the outer sheet rim (figures 3–5). The outer rim of these central sheets becomes destabilised with corrugations due to coupled Rayleigh–Taylor and Rayleigh–Plateau instabilities, as identified in our previous work (Goswami & Hardalupas Reference Goswami and Hardalupas2023). These instabilities amplify the sheet rim corrugations, eventually forming protruding fingers during the vertical descent of the central sheet. In contrast, such propagating ripples and subsequent rim corrugations are not observed for the droplet impacts with $\phi _g \gt 0$ %, indicating that the liquid viscosity suppresses the capillary waves. The next sections present a qualitative and quantitative analysis of the spreading dynamics and central sheet evolution for such droplet-pair impacts. Figure 6 provides a schematic overview of the main quantities measured and discussed in these sections.

The evolution of the central sheet height ( $H_S$ ) and the liquid spreading on the impact surface (i.e. spread radius $R$ and lamella edge velocity $V_L$ ) was quantified from predefined batches of filtered post-impact binary front-view images. For the measurement of $H_S$ , the vertical position of the peak of the central sheet relative to the impact substrate, and for the measurement of $R$ and $V_L$ , the horizontal positions of extreme points of the lamellae on the impact substrate were tracked in the processed images. An essential input parameter to these analyses is the position of the impact substrate line (the horizontal red dashed line in figure 6), which was carefully determined from the corresponding touchdown frame (e.g. images for $\tau = 0$ in figures 5–6) based on the intersection of the droplet and its mirror reflection. It is noted that although fingers appear in central sheets for the water droplet ( $\phi _g = 0$ %) impacts with impact $ \textit{We} = 130$ and $104$ , the fingers are formed only during the vertical descent of central sheets, i.e. only after $\tau _{\textit{max}}$ (corresponding to the maximum sheet height, $H_{S,\textit{max}}$ ). This finger formation can lead to relatively increased standard deviations in sheet height ( $H_S$ ) measurements for $\tau \gt \tau _{\textit{max}}$ , as shown in § 3.3 (figure 9). The length of the inclined central sheet, $L_S$ , for non-simultaneous droplet impacts was measured by analysing a predefined batch of images, as detailed in Appendix B. The lamella edge thickness ( $T_L$ ) was measured only for the instant of the onset of lamella–lamella interaction (i.e. for $\tau = \tau _0$ ), using ImageJ software, as discussed in Appendix A.

3.2. Spreading dynamics

In this section, the time-dependent radius of spreading lamellae, denoted by $R(\tau )$ and defined in the inset image of figure 7, for different viscosity cases is explored while compared with existing theoretical models. It is noted that, although numerous theoretical models for predicting the maximum spread radius ( $R_{\textit{max}}$ ) have been proposed in the literature (Josserand & Thoroddsen Reference Josserand and Thoroddsen2016), quite a few studies have proposed theories for predicting the instantaneous spread radius $R(\tau )$ (Roisman et al. Reference Roisman, Prunet-Foch, Tropea and Vignes-Adler2002; Gordillo, Riboux & Quintero Reference Gordillo, Riboux and Quintero2019). Among these, the theory by Roisman et al. (Reference Roisman, Prunet-Foch, Tropea and Vignes-Adler2002) assumes cylindrical droplet spreading beyond $\tau = 1$ , leading to significant overestimation of $R(\tau )$ , as shown in Goswami & Hardalupas (Reference Goswami and Hardalupas2023). In contrast, the spreading theory of Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) reliably predicts the dimensionless spread radius, $R^*(\tau ) = R(\tau ) / D_0$ , for water droplet impacts across different impact Weber numbers (Goswami & Hardalupas Reference Goswami and Hardalupas2023, Reference Goswami and Hardalupas2024). However, this theory depends on some adjustable constants, which have not been verified in the literature for high-viscosity droplet impacts, such as those involving droplets with $\phi _g = 70$ % ( $ \textit{Oh} = 0.067$ ) and $\phi _g = 80$ % ( $ \textit{Oh} = 0.177$ ) in the present study. Therefore, this section initially evaluates how the proposed model of Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) compares with the experimental data for high-viscosity cases of the present study, such as those involving $\phi _g = 70$ % and 80 %.

Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) proposed a set of differential equations derived from mass and momentum balances at the expanding rim as follows:

(3.1) \begin{align} \left . \begin{aligned} \phi \frac {\pi }{4} \frac {{\rm d} w^2}{{\rm d} t^*} &= [u(s,t^*) - v] h(s,t^*) \\[5pt]\frac {{\rm d} r}{{\rm d} t^*} &= v \\[5pt]\phi \frac {\pi }{4} w^2 \frac {{\rm d} v}{{\rm d} t^*} &= [u(s,t^*) - v]^2 h(s,t^*) - (1 + \varepsilon ) \textit{We}_R^{-1} \end{aligned} \right \} , \end{align}

with $h(r,t^*)$ the dimensionless averaged lamella thickness and $u(r,t^*)$ is the dimensionless averaged radial velocity of the lamella. Also, $s(t^*)$ , $w(t^*)$ and $v(t^*)$ are the dimensionless radial position, thickness and velocity of the outer lamella rim. It is noted that Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) considered the initial droplet radius $R_0$ as the characteristic length scale for normalising relevant parameters, whereas this study considers the initial diameter $D_0 (= 2R_0)$ . This consideration leads to their dimensionless time being expressed as $t^* (= 2\tau )$ and the dimensionless spread radius as $r (\approx s(t^*)) = 2R^*(\tau )$ . For our impact cases, the impact substrate is hydrophilic, resulting in the geometric parameter $\phi = 1/2$ and the wettability-related parameter $\varepsilon = 0$ (see Gordillo et al. Reference Gordillo, Riboux and Quintero2019 for more details).

Equation (3.1) was numerically solved using the initial boundary conditions detailed in Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) using MATLAB ODE45 solver, while determining $u(r,t^*)$ and $h(r,t^*)$ using the analytical expressions by Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) as follows:

(3.2) \begin{align} u(r,t^*) = \frac {r}{t^*} - \frac {Re^{-1/2}}{t^*} \left [ \frac {\sqrt {3} \chi x}{2 h_a (x)} + \frac {2\sqrt {3} \lambda }{7 h_a (x) x^{5/2}} \big (t^{*7/2} - x^{7/2} \big ) \right ] , \\[-28pt] \nonumber \end{align}

(3.3) \begin{align} h(r,t^*) &= 9 \frac {t^{*2}}{r^4} h_a \big [ 3 (t^* / r)^2 \big ] + \frac {Re^{-1/2}}{r t^*} \left [ \frac {\sqrt {3}}{2} x^2 + \frac {\sqrt {3} (105\chi - 60\lambda )}{42} x^3 \big (t^{*-1} - x^{-1} \big ) \right ] \nonumber \\&\quad + \frac {24\sqrt {3} \lambda }{105} x^{-1/2} \big (t^{*5/2} - x^{5/2} \big ), \\[0pt] \nonumber \end{align}

where $x = 3 ({t^*}/{r} )^2$ . The function $h_a (x)$ was approximated using a polynomial function reported in Gordillo et al. (Reference Gordillo, Riboux and Quintero2019).

The solution process requires approximating two constants $\lambda$ and $\chi$ in (3.2) and (3.3), where $\lambda$ is a free constant that absorbs the assumptions in the theory ( $\lambda \neq 0$ in the presence of a viscous boundary layer on the impact substrate) and $\chi$ is a constant used for boundary condition approximation, as explained in Gordillo et al. (Reference Gordillo, Riboux and Quintero2019). Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) fixed $\lambda = 1$ based on the fit between the theoretical prediction and their experimental data. Additionally, they found $\chi = 0.6$ worked well for the Ohnesorge number range $10^{-3} \lesssim Oh \lesssim 10^{-2}$ and proposed $\chi \lt 0.6$ for droplet impacts with higher $ \textit{Oh}$ (viscosity), i.e. for higher-viscosity boundary layer thickness.

Figure 7 shows a comparison of the proposed model of Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) with the experimental data for $\phi _g = 70$ % ( $ \textit{Oh} = 0.067$ ) and $\phi _g = 80$ % ( $ \textit{Oh} = 0.177$ ), for an impact Weber number $ \textit{We} = 81$ . For these high-viscosity cases, the theory cannot be solved by applying $\chi = 0.6$ ; instead, as proposed by Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) for similar conditions, adjustment of $\chi$ to a new value of $\chi \lt 0.6$ was required to obtain solutions. Accordingly, for each case, starting from $\chi = 0.60$ , the $\chi$ value was decreased in $0.01$ increments until the first value resulting in a solution to the theory was identified, yielding $\chi \approx 0.46$ and $\chi \approx 0.30$ for $\phi _g = 70$ % and $\phi _g = 80$ %, respectively. The other constant $\lambda$ is fixed at $\lambda = 1$ , as set by Gordillo et al. (Reference Gordillo, Riboux and Quintero2019).

Figure 7(a) shows that the proposed model of Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) significantly overestimates the spread radius $R^*(\tau )$ , and this overestimation increases with increasing liquid viscosity (i.e. glycerol fraction $\phi _g$ ). Similar overestimations of theoretical $R^*(\tau )$ from the experimental data are also found for these viscosities for impact $ \textit{We} = 130$ and $ \textit{We} = 104$ . Such overestimations indicate that the theoretical model of Gordillo et al. (Reference Gordillo, Riboux and Quintero2019), with its proposed constants for high viscosities (i.e. $\chi \lt 0.6$ , $\lambda = 1$ ), cannot reliably capture droplet impacts involving larger Ohnesorge numbers (i.e. higher viscosities). In contrast, the constants originally suggested by Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) for the range $10^{-3} \lesssim Oh \lesssim 10^{-2}$ , i.e. $\chi = 0.6$ and $\lambda = 1$ , work well up to $\phi _g = 60$ % ( $ \textit{Oh}=0.031$ ) for the impacts with all considered Weber numbers ( $ \textit{We} = 81, 104$ and $130$ ), as presented later in figure 8. Therefore, before presenting the results for all viscosity cases, an evaluation is conducted to identify more appropriate values of $\lambda$ and $\chi$ that could align the theoretical $R^*(\tau )$ by Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) with the corresponding experimental data for the high-viscosity impact cases with $\phi _g = 70$ % ( $ \textit{Oh} = 0.067$ ) and $\phi _g = 80$ % ( $ \textit{Oh} = 0.177$ ).

Figure 7. (a) Comparison of the experimental dimensionless spread radius ( $R^*(\tau ) = R(\tau )/D_0$ ) with the $R^*(\tau )$ obtained by the model of Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) for the glycerol volume fractions $\phi _g = 70$ % ( $ \textit{Oh} = 0.067$ ) and $\phi _g = 80$ % ( $ \textit{Oh} = 0.177$ ). The spread radius $R(\tau )$ is schematically defined in the inset image. The colour of a theoretical line is the same as the edge colour of the corresponding experimental symbols. (b–c) The dependency of the theoretical $R^*(\tau )$ on (b) $\chi$ for a given $\lambda (=0.5)$ and (c) $\lambda$ for a given $\chi (=0.30)$ , for droplet impacts with $ \textit{We} = 81$ and $\phi _g = 80$ %.

Figure 7(b–c) shows a typical optimisation of $ \chi$ and $ \lambda$ values to minimise the deviation of theoretical $ R^*(\tau )$ from measurements for $\phi _g=80$ %. First, values of $ \chi$ that can allow a solution of the theory of Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) are identified as $ \chi \lt 0.6$ ; e.g. $ \chi = 0.3$ for this impact case. Figure 7(b) shows that $ \chi \gt 0.30$ cannot solve the theory, i.e. no curve could be obtained for $ \chi \gt 0.30$ , and $ \chi \lt 0.30$ worsens the theoretical prediction for a given value of $ \lambda$ (e.g. $ \lambda = 0.5$ in figure 7 b). Figure 7(c) shows the dependency of the theoretical $ R^*(\tau )$ on $ \lambda$ when $ \chi$ is fixed; $ \lambda$ values less than 0.5 cannot solve the theory up to $ \tau _{\textit{max}}$ (corresponding to $ R_{\textit{max}}$ ) and $ \lambda$ values greater than 0.5 result in larger deviations from the experimental data for $ \chi = 0.30$ . Thus, the optimised values for this case are $ \chi = 0.30$ and $ \lambda = 0.5$ , which minimise the deviation of the theoretical curve from the experimental data. However, differences between theory and experiments still remain. A similar approach is also applied to optimise these theoretical constants for the other high-viscosity cases. Table 2 lists the optimised $ \lambda$ and $ \chi$ values used to obtain the theoretical $ R^*(\tau )$ for different viscosity cases, as presented in comparison with experimental data in figure 8(a–c). It is noted that our previous work experimentally demonstrated that the spreading of individual lamellae remains unaffected by the interaction of lamellae at the central contact region, as evidenced by the comparison of $R^{*}(\tau )$ for single-droplet and droplet-pair impacts for water droplets (Goswami & Hardalupas Reference Goswami and Hardalupas2023). A similar spreading behaviour is also observed for high-viscosity droplets, as shown in Appendix C, confirming that the discrepancy with theory in these cases arises solely due to increased viscosity, rather than enhanced inter-droplet interactions at higher viscosities.

Table 2. Constants $\chi$ and $\lambda$ used to minimise deviations of $R^*(\tau )$ estimated from the theory of Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) for different droplet impacts with different Weber numbers $(W e)$ for different viscosities (i.e. glycerol volume fractions $\phi _g$ corresponding to $ \textit{Oh}=0.002$ to 0.177) in comparison with current measurements.

Figure 8. (a–c) Effect of liquid viscosity (i.e. liquid glycerol volume fraction $\phi _g=0$ % to $80$ % corresponding to $ \textit{Oh}=0.002$ to $0.177$ ) on the time-dependent dimensionless spread radius $R^*(\tau )$ for different impact Weber numbers ( $ \textit{We}=130,104$ and $81$ ). For all cases, the dimensionless inter-droplet spacing $\Delta x^*$ is kept constant at $\Delta x^* \approx 1.80$ . The scatter plots represent experimental $R^*(\tau )$ and the solid lines represent the $R^*(\tau )$ estimated from the theory of Gordillo et al. (Reference Gordillo, Riboux and Quintero2019), with optimised constants $\chi$ and $\lambda$ for $\phi _g=70$ % ( $ \textit{Oh}=0.067$ )and $80$ % ( $ \textit{Oh}=0.127$ ). The values of constants $\chi$ and $\lambda$ used for different viscosities are listed in table 2. The colour of a theoretical line is the same as the edge colour of the corresponding experimental symbols. Error bars represent standard deviations from mean data and are only visible where error bars are larger than the symbol size. (d) Comparison of the experimental maximum spread radius $R_{{max}}^* (=R_{{max}}/D_0)$ with $R_{{max}}^*$ obtained from (1) the theoretical model by Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) with constants listed in table 2 and (2) the scaling model by Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014). Different symbol colours represent liquids with different glycerol volume fractions: $\phi _g=0$ % ( $ \textit{Oh}=0.002$ , cyan), $40$ % ( $ \textit{Oh}=0.010$ , grey), $60$ % ( $ \textit{Oh}=0.031$ , black), $70$ % ( $ \textit{Oh}=0.067$ , red) and $80$ % ( $ \textit{Oh}=0.177$ , blue).

Figure 8(a–c) compares the experimental $ R^* (\tau )$ with its theoretical prediction by Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) for the droplet-pair impacts across all liquid viscosities (i.e. $\phi _g=0$ % to $\phi _g=80$ %) for impact Weber numbers $ \textit{We}=130,104$ and $ 81$ . For the low-viscosity droplet-impact cases, the agreement between the experimental data and theoretical prediction is excellent for all impact Weber numbers. However, for high-viscosity droplet impacts (i.e. $\phi _g=70$ % ( $ Oh=0.067$ ) and $\phi _g=80$ % ( $ Oh=0.177$ )), the deviations between theoretical predictions and experimental data remain significant even with the optimised $ \chi$ and $ \lambda$ values (see table 2), as revealed in figure 8(a–c). Also, it is noteworthy that the set of optimised constant values for these two high-viscosity cases differ appreciably for a given impact Weber number (table 2). Thus, no single pair of $ \chi$ and $ \lambda$ values could be identified to reliably fit experimental data for these high-viscosity cases. In summary, although the theory of Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) reliably predicts the time-dependent spread radius $ R^* (\tau )$ for droplets with lower viscosities, it cannot reliably predict $ R^* (\tau )$ for droplets with higher viscosities, even when employing optimised $ \chi$ and $ \lambda$ values, as revealed in figure 8(a–c).

Figure 8 (a–c) also presents the liquid viscosity (Ohnesorge number) effect on the temporal evolution of the dimensionless spread radius $ R^*$ . For all impact $ \textit{We}$ cases, it is evident that the liquid viscosity does not appreciably influence the initial droplet spreading stage (figure 8 a–c), primarily attributed to the dominance of inertial forces during this phase. However, beyond this initial spreading stage, viscous dissipation, which increases with increasing liquid viscosity, significantly influences the spreading behaviour, leading to an appreciable decrease in the maximum spread of the lamellae (i.e. $ R_{\textit{max}}$ ). For instance, for the same impact Weber number $ \textit{We}=130$ , while $ R_{\textit{max}}$ is $ \approx 1.9D_0$ for $ \phi _g=0$ % (minimum $ {Oh}$ ), $ R_{\textit{max}}$ is $ \approx D_0$ for $ \phi _g=80$ % (maximum $ {Oh}$ ). Similar reduction in $ R_{\textit{max}}$ with increasing liquid viscosity ( $ {Oh}$ ) is observed for the other examined impact Weber number cases, as revealed in figure 8(a–c). Additionally, it is worth noting that, for a given liquid viscosity ( $ {Oh}$ ), the dimensionless maximum spread radius $ R_{\textit{max}}^*$ ( $ =R_{\textit{max}}/D_0$ ) exhibits strong dependency on the impact Weber number; $ R_{\textit{max}}^*$ decreases appreciably with decreasing impact Weber number due to reduced impact inertia.

During droplet impacts, higher droplet liquid viscosity increases viscous dissipation in different regions of a spreading droplet (Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016): (i) within the viscous boundary layer near the impact substrate, (ii) in the flow from the descending bulk liquid into the radially expanding lamella and (iii) in the liquid flow from the lamella into the outer rim. Part of the initial kinetic energy of an impacting droplet contributes to the surface energy gain of the spreading droplet and the rest to the viscous dissipation. Recently, Sanjay & Lohse (Reference Sanjay and Lohse2025) demonstrated that the total viscous dissipation during droplet impact can be split into two temporal phases, reflecting the distinct underlying fluid dynamics: (i) dissipation during the initial impacting phase, $0 \le t \le t_i$ , with $t_i \sim D_0/V_0$ as the inertial time scale, and (ii) dissipation during the droplet spreading phase, $t_i \le t \le t_{\textit{max}}$ , where $t_{\textit{max}}$ corresponds to the maximum spread radius $R_{\textit{max}}$ . Accordingly, the total energy dissipated per unit mass up to any instant $t \gt t_i$ can be expressed as (Sanjay & Lohse Reference Sanjay and Lohse2025)

(3.4) \begin{align} E_d(t) = \int _{0}^{t} \epsilon (t)\, {\rm d}t = \int _{0}^{t_i} \epsilon _i(t)\, {\rm d}t + \int _{t_i}^{t} \epsilon _s(t)\, {\rm d}t , \end{align}

where $\epsilon (t)$ is the viscous dissipation rate, with subscripts $i$ and $s$ denoting the impacting and spreading phases, respectively. These dissipation rates, integrated across the growing boundary layer thickness $\delta (t) \sim \sqrt {\mu t/\rho }$ (Josserand et al. Reference Josserand, Lemoyne, Troeger and Zaleski2005; Roisman et al. Reference Roisman, Horvat and Tropea2006; Xu et al. Reference Xu, Barcos and Nagel2007; Bird et al. Reference Bird, Tsai and Stone2009) within the liquid volume $\varOmega (t) \sim R(t)^2 \delta (t)$ , scale with liquid viscosity $\mu$ as (Sanjay & Lohse Reference Sanjay and Lohse2025)

(3.5) \begin{align} \epsilon _i(t) \sim \frac {\mu }{\rho D_0^3}\, \frac {V_0^2}{\delta _i(t)^2}\, \varOmega _i(t) \sim \frac {V_0^3}{D_0^2}\, \sqrt {\frac {\mu t}{\rho }}, \\[-28pt] \nonumber \end{align}

(3.6) \begin{align} \epsilon _s(t) \sim \frac {\mu }{\rho D_0^3}\, \frac {V_L(t)^2}{\delta _s(t)^2}\, \varOmega _s(t) \sim \frac {V_L(t)^2 R(t)^2}{D_0^3}\, \sqrt {\frac {\mu }{\rho }}\, t^{-1/2}, \\[0pt] \nonumber \end{align}

with $V_L(t) = {\rm d}R(t)/{\rm d}t$ as the lamella-front velocity during spreading.

For the droplet impacts shown in figure 8, it is evident from figure 8(a–c) that, for a given impact $ \textit{We}$ , the spread radius $R(t)$ (or $R^{*}(\tau )$ in dimensionless form) for all viscosities is nearly comparable during the early spreading stage ( $\tau \lesssim 0.2$ ), indicating nearly comparable lamella-front velocities as well. However, the viscosity $\mu$ varies by over $90$ times, from $\mu = 1$ mPa s for water droplets ( $\phi _g = 0$ %) to $\mu = 91.46$ mPa s for glycerol–water droplets with $\phi _g = 80$ %, while the liquid density $\rho$ remains nearly unchanged (detailed in table 1). From (3.5)–(3.6), the droplets with higher viscosities therefore dissipate more energy during this spreading stage, leaving less kinetic energy available for subsequent spreading on the impact substrate, and thus achieve lower $R_{\textit{max}}^{*}$ , as revealed in figure 8(a–c). Consequently, for a given dimensionless inter-droplet spacing $\Delta x^{*}$ , the maximum combined liquid spread – defined as $2 R_{\textit{max}}^{*} + \Delta x^{*}$ (Goswami & Hardalupas Reference Goswami and Hardalupas2023) – also decreases with increasing liquid viscosity during droplet-pair impacts on solid surfaces.

In the literature, depending on the relative dominance of capillary and viscous forces on the spreading of impacting droplets, two spreading asymptotic regimes are reported (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004; Laan et al. Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014; Josserand & Thoroddsen Reference Josserand and Thoroddsen2016): (i) the capillary regime, where spreading is limited by capillary forces and governed by an inertia–capillary balance, and (ii) the viscous regime, where spreading is limited by viscous forces and governed by an inertia–viscous balance. For the capillary regime (or inviscid regime), the dimensionless maximum spread radius $R_{\textit{max}}^*$ scales with the Weber number as $R_{\textit{max}}^* \sim \textit{We}^{1/2}$ (Bennett & Poulikakos Reference Bennett and Poulikakos1993; Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010), whereas for the viscous regime, $R_{\textit{max}}^*$ scales with the Reynolds number as $R_{\textit{max}}^* \sim Re^{1/5}$ (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004; Du et al. Reference Du, Chamakos, Papathanasiou and Min2021). By interpolating between these asymptotic scaling laws (i.e. ${\sim} \textit{We}^{1/2}$ and ${\sim} Re^{1/5}$ ), Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014) introduced a cross-over and universal scaling for $R_{\textit{max}}^*$ as

(3.7) \begin{align} R_{\textit{max}}^* \sim \textit{Re}^{\tfrac {1}{5}} f(P)=\frac {1}{2} \frac {\sqrt {P}}{\sqrt {P}+A} \textit{Re}^{1 / 5} , \end{align}

where $P (\equiv \textit{WeRe}^{-2/5})$ is a dimensionless impact parameter that varies between zero (capillary regime) and infinity (viscous regime), and $A (= 1.24)$ is a fitting constant. Figure 8(d) compares the experimental $R_{{max}}^*$ with the predictions by (3.7) (i.e. Laan et al. Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014) and the model of Gordillo et al. (Reference Gordillo, Riboux and Quintero2019), with optimised $\chi$ and $\lambda$ values applied for high-viscosity cases (from table 2). For low-viscosity cases, the prediction by Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) is closer to the experimental cases compared with that by Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014). However, for high-viscosity cases, the scaling law of Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014) predicts $R_{\textit{max}}^*$ more reasonably compared with the already optimised model of Gordillo et al. (Reference Gordillo, Riboux and Quintero2019).

Figure 9. Effect of liquid viscosity (i.e. glycerol volume fraction $ \phi _g$ ; $ \textit{Oh}=0.002\text{ to }0.177$ ) on the temporal evolution of dimensionless central sheet height $ H_S^*$ ( $ = {H_S}/{D_0}$ ) for different impact Weber numbers ( $ \textit{We} = 130,104$ and $ 81$ ). No plot is presented for $ \textit{We} = 81$ and $\phi _g=80$ % ( $ \textit{Oh}=0.177$ ) as no sheet is formed for this condition. The inter-droplet spacing is constant at $ \Delta x^* \approx 1.80$ . Error bars represent standard deviations from mean data and are only visible where error bars are larger than the symbol size.

3.3. Central sheet dynamics

Figure 9 shows the temporal variation of the dimensionless central sheet height $H_S^* (= H_S / D_0; H_S$ is schematically shown in the inset image) for the droplet-pair impacts with different viscosities ( $\phi _g = 0$ % to $80$ % corresponding to $ \textit{Oh}=0.002$ to 0.177) for impact Weber numbers $ \textit{We} = 130, 104$ and $81$ and a fixed dimensionless inter-droplet spacing $\Delta x^* \approx 1.80$ (see figures 3–5 for related morphology). Note that the variation of $H_S^*$ for $\phi _g = 80$ % ( $ \textit{Oh}=0.177$ ) is not shown for $ \textit{We} = 81$ , as no central sheet is formed for this impact case. For all the cases, $H_S^*$ is a strong function of the liquid viscosity, i.e. the maximum $H_S^*$ , denoted as $H_{S,max }^*$ , decreases appreciably with increasing $\phi _g$ (viscosity or $ \textit{Oh}$ ) for a given impact $ \textit{We}$ and $\Delta x^*$ . Also, for a given $\phi _g$ (viscosity or $ \textit{Oh}$ ), $H_S^*$ decreases appreciably with decreasing impact $ \textit{We}$ as less kinetic energy is available for central sheet formation for a lower droplet impact $ \textit{We}$ .

The formation and growth of the central sheet depend on the lamella–lamella impact Weber number $ \textit{We}_{L,\textit{imp}}$ (Goswami & Hardalupas Reference Goswami and Hardalupas2023, Reference Goswami and Hardalupas2024), which is the ratio of the lamella–lamella impact inertia and surface tension per unit arclength of the lamellae, i.e. $ \textit{We}_{L,\textit{imp}} = 2 \rho V_L^2 T_L / \sigma$ , where $V_L (= {\rm d}R/{\rm d}t)$ and $T_L$ are the average velocity and average thickness of the lamella edges, respectively, at the moment of lamella collision, i.e. at $\tau = \tau _0$ (see figure 10 a for related schematic). As discussed in the previous section (§ 3.2), the droplet kinetic energy dissipates more with increased liquid viscosity due to viscous dissipation within the boundary layer region and in the liquid flow from bulk to lamella and from lamella to rim. This increased viscous dissipation during lamella spreading leads to decreased available kinetic energy (or $V_L$ ), thereby reducing $ \textit{We}_{L,\textit{imp}}$ at the instant of lamella collision, i.e. before the central sheet formation ( $\tau = \tau _0$ ). Moreover, higher liquid viscosity induces increased shear stress in the formed ‘semilunar’ central sheet, resisting its vertical growth in the ambient quiescent air. Therefore, the decrease in the dimensionless maximum sheet height $H_{S,max }^*$ with $\phi _g$ , observed in figure 9, is a cumulative effect of viscous dissipation both before the sheet formation (i.e. during lamella spreading) and during the subsequent vertical growth of the central sheet.

Figure 10. (a) Collision of two lamellae at $\tau =\tau _0$ , with definitions of lamella velocity $ V_L$ , lamella edge thickness $ T_L$ and collision force (inertia) $ F_C$ per unit lamella arclength. (b) The dimensionless maximum sheet height $ H_{S,\textit{max}}^*$ $ ( = {H_{S,\textit{max}}^*}/{D_0} )$ vs. lamella impact Weber number ( $ \textit{We}_{L,imp} = {F_C}/{\sigma }$ ) for droplet-impact cases with different viscosities (i.e. glycerol volume fraction $ \phi _g$ from $ 0$ % ( $ \textit{Oh}=0.002$ ) to $ 80$ % ( $ \textit{Oh}=0.177$ )). The dashed lines represent best-fit curves for individual viscosity cases. For all fitted curves, the R-squared value is greater than 0.95. ( $c$ ) The best-fit power laws in a log–log plot, where the slopes of individual fit lines correspond to the individual exponents of the power laws. Error bars represent standard deviations from mean data and are only visible where error bars are larger than the symbol size.

As demonstrated in our previous work (Goswami & Hardalupas Reference Goswami and Hardalupas2023, Reference Goswami and Hardalupas2024), the dimensionless maximum sheet height $ H_{S,{max}}^*$ scales with the lamella impact number $ \textit{We}_{L,imp}$ . Therefore, to characterise the effect of viscosity on $ H_{S,{max}}^*$ , it is crucial to maintain a nearly constant $ \textit{We}_{L,imp}$ for different viscosities. In this way, the effect of the viscous dissipation before the sheet formation (i.e. during lamella spreading) can be eliminated for different viscosity cases. For any given viscosity (i.e. $ \phi _g$ ), achieving a desired $ \textit{We}_{L,imp}$ primarily depends on regulating the lamella impact velocity, i.e. $ V_L$ at $ \tau =\tau _0$ . This can be accomplished by adjusting the droplet-impact Weber number $ \textit{We}$ (i.e. droplet release height) and/or inter-droplet spacing as required. Therefore, additional experiments were carried out, by changing the droplet release height (i.e. $ \textit{We}$ ) and/or inter-droplet spacing to achieve central sheet formations with nearly constant $ \textit{We}_{L,imp}$ for different viscosities (i.e. $\phi _g=0$ % ( $ \textit{Oh}=0.002$ ) to 80 % ( $ \textit{Oh}=0.177$ )). This approach ensures that the observed effects on the central sheet expansion are attributed to variations in viscosity while eliminating the effect of viscous dissipation before sheet formation.

Figure 10(b) shows $ H_{S,{max}}^*$ as a function of $ \textit{We}_{L,\textit{imp}}$ plots obtained from a series of simultaneous droplet-impact experiments conducted for different droplet liquid viscosities ( $ \phi _g=0$ % to 80 %). The impact conditions (i.e. $ \textit{We}$ and $ \Delta x^*$ ) with corresponding $ \textit{We}_{L,\textit{imp}}$ for individual viscosity cases are listed in table 3. Figure 10(b) reveals that $ H_{S,{max}}^*$ is not a function of $ \textit{We}_{L,\textit{imp}}$ only as the plots for individual viscosities do not collapse into a single curve. Instead, these individual plots are distinctly separated within the graph, where individual power laws can be considered to fit different viscosity cases with R-squared values greater than 0.95 (figure 10 b). Figure 10(c) shows these individual power laws in a log–log plot, where the slopes of individual fit lines correspond to the individual exponents of the power laws.

Table 3. Impact conditions, i.e. impact Weber numbers ( $ \textit{We}$ ) and inter-droplet spacing ( $ \Delta x^*$ ), and corresponding lamella impact Weber number ( $ \textit{We}_{L,\textit{imp}}$ ) and lamella Ohnesorge number ( $ Oh_L$ ) for droplet impacts across different viscosities (i.e. glycerol volume fractions ( $ \phi _g$ ) corresponding to $ \textit{Oh}=0.002$ to $0.177$ ). The impact conditions corresponding to $ \textit{We}_{L,\textit{imp}} = 14.3 \pm 0.6$ and $ \textit{We}_{L,\textit{imp}} = 28.1 \pm 0.7$ are presented in bold font, and these impact processes are analysed in figure 11(a–c).

The exponent of $\textit{We}_{L,\textit{imp}}$ in the individual power laws does not vary appreciably (figure 10 b); the average exponent value is $ 0.48 \pm 0.04$ , which is in good quantitative agreement with the near square-root scaling (i.e. $ H_{S,{max}}^* \propto \textit{We}_{L,\textit{imp}}^{1/2}$ ) obtained for specific viscosity cases, such as water droplet impacts, as examined in our previous studies (Goswami & Hardalupas Reference Goswami and Hardalupas2023, Reference Goswami and Hardalupas2024). However, the prefactors in these individual power laws vary up to one order of magnitude as $ \phi _g$ changes from 0 % to 80 %, indicating that introducing an additional dimensionless number incorporating the liquid viscosity can lead to a more general scaling law that might collapse all the data into a single curve. Considering a dimensionless lamella Ohnesorge number ( $ {Oh}_L$ ) as $ {Oh}_L = {\mu }/{\sqrt {\rho \sigma T_L}}$ , the dependence of $ H_{S,{max}}^*$ on liquid viscosity and a more general scaling law for $ H_{S,{max}}^*$ is determined as follows.

Figure 11. (a–b) Droplet-pair impacts with nearly constant lamella–lamella impact Weber numbers ( $ \textit{We}_{L,\textit{imp}}$ ) for different viscosities (i.e. glycerol volume fraction $ \phi _g$ from 0 % ( $ \textit{Oh}=0.002$ ) to 80 % ( $ \textit{Oh}=0.177$ )). For (a), $ \textit{We}_{L,\textit{imp}} = 14.3 \pm 0.6$ and for (b), $ \textit{We}_{L,\textit{imp}} = 28.1 \pm 0.7$ . Here, $ \tau _0$ is the dimensionless time corresponding to the lamella impact and $ \tau _{{max}}$ corresponds to the maximum dimensionless sheet height $ H_{S,{max}}^*$ . Scale bars represent 3 mm. The impact conditions (i.e. impact $ \textit{We}$ and dimensionless inter-droplet spacing $ \Delta x^*$ ) for individual $ \phi _g$ cases are presented in bold font in table 3. Panel (c) shows $ H_{S,{max}}^*$ vs. the lamella Ohnesorge number $ {Oh}_L = \mu /\sqrt {\rho \sigma T_L}$ for the impact cases shown in (a) and (b). Error bars represent standard deviation from mean data. The dashed lines (coloured with corresponding symbols’ colour) represent the best fits $ H_{S,{max}}^* \approx 0.32 {Oh}_L^{-0.20}$ and $ H_{S,{max}}^* \approx 0.42 {Oh}_L^{-0.21}$ for $ \textit{We}_{L,\textit{imp}} \approx 14.3$ and $ \textit{We}_{L,\textit{imp}} \approx 28.1$ , respectively, with R-squared value greater than 0.88. The inset image shows the dependence of $ {Oh}_L$ on the droplet liquid viscosity $ \mu$ . Panel (d) shows $ H_{S,{max}}^*$ as a function of $ {Oh}_L^{-1/5} {We}^{1/2}$ across different liquid viscosities ( $ \phi _g = 0$ % ( $ \textit{Oh}=0.002$ ) to 80 % ( $ \textit{Oh}=0.177$ )) for the droplet impacts listed in table 3. The dashed line represents the scaling law $ H_{S,{max}}^* \approx 0.088 {Oh}_L^{-1/5} \textit{We}_{L,\textit{imp}}^{1/2}$ .

From the conducted experiments listed in table 3, two nearly constant lamella–lamella impact Weber numbers can be identified across different liquid viscosities ( $ \phi _g$ from $ 0$ % to $ 80$ %, i.e. $ \textit{Oh}$ from 0.002 to 0.177) as $\textit{We}_{L,\textit{imp}} = 14.3 \pm 0.6$ and $\textit{We}_{L,\textit{imp}} = 28.1 \pm 0.7$ . Figures 11(a) and 11(b) show the morphologies of lamella collision (at $ \tau = \tau _0$ ) and the corresponding central sheets possessing their maximum heights ( $ H_{S,{max}}$ at $ \tau = \tau _{{max}}$ ) for these two lamella impact Weber number cases, i.e. $\textit{We}_{L,\textit{imp}} \approx 14.3$ and $\textit{We}_{L,\textit{imp}} \approx 28.1$ , respectively. The impact conditions (i.e. impact $ \textit{We}$ and dimensionless inter-droplet spacing $ \Delta x^*$ ) of these specific impact processes are presented in bold font in table 3, for clarity. In effect, figure 11(a–c) shows the sole viscosity effect on $ H_{S,{max}}$ as the lamella collision inertia (or $\textit{We}_{L,\textit{imp}}$ ) is nearly constant while varying the liquid viscosity or the lamella Ohnesorge number $ {Oh}_L$ (listed in table 3). The inset image of figure 11(c) shows that $ {Oh}_L$ is mainly a function of liquid viscosity $ \mu$ and an appropriate dimensionless number to represent the viscosity effect. The dimensionless maximum sheet height $ H_{S,{max}}^*$ decreases with increasing $ {Oh}_L$ (viscosity), as revealed in figure 11(c). This decrease in $ H_{S,{max}}^*$ is primarily attributed to the increased shear stress within the uprising liquid sheet, resisting its expansion. It is noteworthy that the viscosity effect on central sheet expansion includes the dissipation in the liquid flow from the lamellae to the sheet and from the inner portion of the sheet to its outer rim.

Figure 11(c) also reveals the dependence of $ H_{S,{max}}^*$ on $ {Oh}_L$ by two power-law fits, with $ H_{S,{max}}^* \approx 0.32 {Oh}_L^{-0.20}$ for $\textit{We}_{L,\textit{imp}} \approx 14.3$ and $ H_{S,{max}}^* \approx 0.42 {Oh}_L^{-0.21}$ for $\textit{We}_{L,\textit{imp}} \approx 28.1$ , demonstrating a nearly inverse fifth-root dependency, i.e. $ H_{S,{max}}^* \propto {Oh}_L^{-1/5}$ . Therefore, combining the dependence of $ H_{S,{max}}^*$ on $\textit{We}_{L,\textit{imp}}$ and $ {Oh}_L$ , a unified scaling law can be expressed as

(3.8) \begin{align} H_{S, max }^* \sim O h_L^{-1 / 5} W e_{L, i m p}^{1 / 2} . \end{align}

Figure 11(d) shows that the scaling law (3.8) is in good agreement with the experimental data for different viscosity cases; the dashed line, representing (3.8) with a numerical coefficient ${\approx} 0.088$ , fits the data with an R-squared value of 0.92. It is noteworthy that an independent fitting of a power-law relation, $ H_{S,{max}}^* = a {Oh}_L^b {We}_{L,\textit{imp}}^c$ (where $ a, b$ and $ c$ are constants) to the experimental data using the curve-fitting toolbox of MATLAB led to $ H_{S,{max}}^* \approx 0.09 {Oh}_L^{-0.19} {We}_{L,\textit{imp}}^{0.49}$ , which is nearly identical to (3.8) and thus further substantiates the developed scaling law. In summary, the scaling law $ H_{S,{max}}^* \sim {We}_{L,\textit{imp}}^{1/2}$ is inadequate to scale $ H_{S,{max}}^*$ for different viscosity cases, rather $ H_{S,{max}}^* \sim {Oh}_L^{-1/5} {We}_{L,\textit{imp}}^{1/2}$ can reliably scale the maximum dimensionless sheet height for different liquid viscosities. It is noted that, as shown in Appendix D, the lamella edge thickness $T_L$ at the onset of lamella–lamella interaction (i.e. at $\tau = \tau _0$ ), which appears in both $ \textit{We}_{L,imp}$ and $ \textit{Oh}_L$ , increases with viscosity and contributes to the scaling law as $H_{S,\textit{max}}^{*} \sim T_L^{0.6}$ , and its measurement uncertainty ( $6$ %– $13$ %) propagates to only $3.6$ %– $7.8$ % in $H_{S,\textit{max}}^{*}$ .

3.4. Non-simultaneous impacts

In the previous sections, the effect of viscosity on droplet-pair impacts has been characterised by investigating simultaneous droplet impacts. In addition, our recent study (Goswami & Hardalupas Reference Goswami and Hardalupas2024) comprehensively investigated non-simultaneous water droplet impacts, offering direct comparisons with their simultaneous counterparts. Based on the insights obtained from these earlier investigations, this section briefly explores the viscosity effect on the lamella spreading and central sheet evolution during non-simultaneous droplet impacts both qualitatively and quantitatively.

Figure 12 shows the temporal evolution of non-simultaneous droplet impacts, where a fixed time lag between impacts, i.e. $\Delta \tau ={\Delta t V_0}/{D_0}\approx 0.22$ , was maintained for different viscosities ( $\phi _g$ from $0$ % to $80$ %; $ \textit{Oh}=0.002$ to 0.177). For all cases, the impact Weber number ( $ \textit{We}$ ) and the dimensionless inter-droplet spacing ( $\Delta x^*$ ) are constant at $ \textit{We}=131\pm 2$ and $\Delta x^* = 1.79\pm 0.04$ , respectively. Due to the time lag $\Delta \tau$ between droplet impacts, the evolved central sheets for these cases become inclined, as observed in non-simultaneous water droplet impacts detailed in (Goswami & Hardalupas Reference Goswami and Hardalupas2024). This sheet inclination becomes less pronounced with increasing viscosity (i.e. $\phi _g$ ). Also, as expected, the length (or height) of the central sheet appears to decrease with increasing $\phi _g$ . Note that the variation of these central sheet characteristics is a combined effect of viscous dissipation before central sheet formation and during the subsequent central sheet expansion. Although simultaneous droplet impacts with the same lamella–lamella impact Weber number ( $ \textit{We}_{L,imp}$ ) for different viscosity cases could be obtained through experiments, as discussed in the previous section, it is very hard to achieve experimentally the same $ \textit{We}_{L,imp}$ for non-simultaneous impacts for different viscosity cases. Nevertheless, the viscosity effect on the dimensionless maximum sheet length $L_{S,\textit{max}}^*$ $(={L_{S,\textit{max}}}/{D_0}; L_S$ is defined in the inset of figure 13) for non-simultaneous impacts can still be assessed by quantitatively analysing the sheets observed in figure 12.

Figure 12. Impact morphologies of non-simultaneous droplet impacts with a fixed dimensionless time delay between impacts $\Delta \tau \approx 0.22$ for different viscosities (i.e. glycerol volume fraction $\phi _g$ from $0$ % ( $ \textit{Oh}=0.002$ ) to $80$ % ( $ \textit{Oh}=0.177$ )). The scale bar is $2 \;\text{mm}$ in length. The impact Weber number and the dimensionless inter-droplet spacing are fixed at $ \textit{We}=131\pm 2$ and $\Delta x^* = 1.79\pm 0.04$ , respectively.

Figure 13. The dimensionless maximum sheet length $L_{S,{max}}^{*}$ as a function of $ \textit{Oh}_L^{-1/5} We^{1/2}$ for different liquid viscosities ( $\phi _g=0$ % to $80$ %; $ \textit{Oh}=0.002 \text{ to } 0.177$ ) for the non-simultaneous impacts presented in figure 12. The sheet length $L_S$ is schematically defined in the inset image. The dashed line represents the scaling law $L_{S,{max}}^{*}\approx 0.088 Oh_L^{-1/5} We_{L,\textit{imp}}^{1/2}$ .

The experimental investigations of Goswami & Hardalupas (Reference Goswami and Hardalupas2024) revealed that the dimensionless maximum sheet length, $L_{S,{max}}^*$ , exhibits a near square-root dependence on the lamella–lamella impact Weber number for non-simultaneous water droplet impacts, like their simultaneous counterparts. In other words, $L_{S,{max}}^*$ approximately scales as $L_{S,{max}}^* \sim We_{L,\textit{imp}}^{1/2}$ , where $ \textit{We}_{L,\textit{imp}} = {(\rho V_{L1}^2 T_{L1} + \rho V_{L2}^2 T_{L2})}/{\sigma }$ where $V_{L1}$ and $V_{L2}$ are the velocities of lamella edges and $T_{L1}$ and $T_{L2}$ are the respective thicknesses of the lamella edges (see Goswami & Hardalupas Reference Goswami and Hardalupas2024 for more details). This summation form of $ \textit{We}_{L,\textit{imp}}$ captures the total inertial input from both lamellae at the onset of their interaction, consistent with its definition as the ratio of driving inertia to resisting surface tension in central sheet formation. In addition, similar to the simultaneous impacts, the viscosity effect can be characterised by the lamella Ohnesorge number $ \textit{Oh}_L$ as $L_{S,{max}}^* \propto \textit{Oh}_L^{-1/5}$ , where $ \textit{Oh}_L = (\textit{Oh}_{L1} + \textit{Oh}_{L2})/2$ with $ \textit{Oh}_{L1} = \mu / \sqrt {\rho \sigma T_{L1}}$ and $ \textit{Oh}_{L2} = \mu / \sqrt {\rho \sigma T_{L2}}$ as the Ohnesorge numbers of the respective lamellae. Incorporating the numerical coefficient derived for simultaneous impacts, the scaling law for the non-simultaneous impacts is approximated as $L_{S,{max}}^* \approx 0.088 Oh_L^{-1/5} We_{L,\textit{imp}}^{1/2}$ . Figure 13 compares this scaling law with the experimental $L_{S,{max}}^*$ for the central sheets of figure 12. The dashed line in figure 13 represents the scaling law, and its excellent agreement with the experimental data suggests the applicability of this scaling for both simultaneous and non-simultaneous droplet impacts for different viscosities. In sum, the scaling $L_{S,{max}}^* \sim Oh_L^{-1/5} We_{L,\textit{imp}}^{1/2}$ emerges as a general scaling law applicable to both simultaneous and non-simultaneous impacts involving droplets of identical or varying sizes, impacting with equal or differing velocities, across a range of liquid viscosities.

The spreading dynamics of the unaffected lamella portions on the impact substrate, as depicted in figure 12, exhibits a qualitative resemblance to those observed for non-simultaneous impacts of (Goswami & Hardalupas Reference Goswami and Hardalupas2024). Figure 14 shows the temporal evolution of dimensionless spread radii $R_1^* = {R_1}/{D_0}$ and $R_2^* = {R_2}/{D_0}$ (where $R_1$ and $R_2$ are defined in figure 14 a) for the impact cases presented in figure 12. For the different viscosity cases ( $\phi _g = 0$ % to $80$ %), $R_1^*$ and $R_2^*$ initially differ appreciably due to the time lag $\Delta \tau$ , but subsequently, they become equal as both lamellae reach their maximum spreads (figure 14). Therefore, any spreading theory capable of predicting $R^*(\tau )$ including the high-viscosity cases, would enable prediction of the temporal evolution of $R_1^*$ , $R_2^*$ and the combined liquid spread as $R^*(\tau )$ , $R^*(\tau - \Delta \tau )$ and $R^*(\tau ) + \Delta x^* + R^*(\tau - \Delta \tau )$ , respectively, as discussed in (Goswami & Hardalupas Reference Goswami and Hardalupas2024), for all the viscosity cases.

Figure 14. (a) Definition of spread radii of first and second droplets, i.e. $R_1$ and $R_2$ . (b–f) Temporal variation of the dimensionless spread radii ( $R_1^*$ and $R_2^*$ ) of individual droplets for different viscosities (glycerol volume fraction $\phi _g$ from $0$ % to $80$ %, corresponding to $ \textit{Oh}=0.002$ to 0.177). Error bars represent standard deviations from mean data and are only visible where error bars are larger than the symbol size.