2.1. Experimental set-up

A schematic of the experimental set-up used in this study is shown in figure 1(a). It consists of a syringe pump, a vertical stage, a stainless steel spherical target, and two perpendicularly mounted high-speed CMOS cameras, all mounted on a passive pneumatic vibration isolation table. An AL 300 Aladdin InfusionONE Syringe (World Precision Instruments Germany GmbH, Germany) is set to flow rate $0.3\ {\rm ml}\ {\rm min}^{-1}$, with a blunt bevel facet 21G needle with external diameter 0.8 mm allowing the generation of drops with consistent size, 2.2–3 mm diameter. The size of the drop is determined by the balance of surface tension and the weight of the drop. Impact velocities are determined by changing the vertical impact distance between the needle tip and the apex of the sphere. The height and position are adjusted using a millimetre-precise stepper motor vertical stage, followed by finer adjustment with an XYZ-microstage. In this way, velocities up to $4.5\ {\rm m}\ {\rm s}^{-1}$ are achieved. A protective acrylic tube helps to stabilize the trajectory of the drop.

Figure 1. Experimental set-up: (a) sketch of the main systems of the set-up, and (b) schematic view of the spherical target and impacting drop.

Mirror-polished stainless steel (AISI 304) spherical targets with diameters 6, 30, 40 and 50 mm are used. The roughness is $R_a = 0.15 \pm 0.05\ \mathrm {\mu } {\rm m}$ as reported by the manufacturer. The static contact angle for an equivalent flat AISI 304 surface is $85 \pm 5^\circ$. Contact angle hysteresis is known to be small for this type of material (AISI304), at ${\approx }\,7^\circ$ (Pou et al. Reference Pou, Del Val, Riveiro, Comesaña, Arias-González, Lusquiños, Bountinguiza, Quintero and Pou2019). The spheres are cleaned with acetone, isopropanol and distilled water, and dried before every experiment. Two CMOS high-speed cameras (Vision Research Inc., Germany) were utilized to image the impact from two perpendicular angles. A Phantom v12 with resolution $1024 \times 512$ pixels, combined with a macro lens to obtain a wider view (pixel size $25\ \mathrm {\mu }{\rm m}\ {\rm pixel}^{-1}$), captured the impact and spreading of the drop at 10 000 frames per second (fps). A Phantom v2012 with resolution $1280 \times 800$ pixels, combined with a $\times$5 zoom macro lens to obtain a higher spatial resolution (pixel size $4\ \mathrm {\mu }{\rm m}$), is used to measure the thickness of the lamella at 10 000 fps. Both cameras measured the offset of the drop from the centre of the target to ensure that the drop impact was exactly at the centre of the spherical target. Both cameras have an image-based auto-trigger that engages when the drop arrives in the proximity of the target. In order to achieve shadowgraphy high-speed imaging, backlighting Constellation 120E LED illumination (Imaging Solution GmbH, Germany) is placed behind the drop. A diffuser sheet with diffusion angle $80^\circ$ is placed between the LED illumination and the impact zone of the drop in order to achieve more uniform illumination of the falling drop. A Brookfield DV-III Ultra programmable rheometer (AMETEK GmbH, Germany) equipped with cone/plate is used to measure the viscosities of all the samples, using 0.5 ml sample size with adequate modes, plate type and frequency modulation based on the manufacturer's recommendations. Surface tension is measured using a Tensíío force tensiometer (KRÜSS GmbH, Germany) with the recommended settings by the manufacturer for the Wilhelmy plate method. The error for viscosity measurements in table 1 is below 1 % in all the samples. The repeatability of the surface tension measurement is below $0.2 \ {\rm mN}\ {\rm m}^{-1}$ for all samples.

Table 1. Properties of liquids at temperature $T=20\,^\circ {\rm C}$ used in this work.

The drops used in all experiments are made of ${\geqslant }99.5\,\%$ anhydrous glycerol (Carl Roth GmbH, Germany) and distilled water aqueous solutions with varying volume fractions. Prior to performing the experiments, the critical properties of the liquids are measured. Table 1 presents an overview of the liquids used in the experiments, and their physical properties. For brevity, the samples will be referred to as H2O for distilled water, and Gly40, Gly60, Gly80, Gly90, Gly95 and Gly100 for glycerol solutions of corresponding percentage.

In this study, the drop spreading factor and the evolution of the lamella thickness are measured for the spherical targets of diameters from the moment of impact until times well beyond the maximum spreading. The impact velocities are between $1$ and $4\ {\rm m} {\rm s}^{-1}$, while drop diameters were in the range 2.2–3 mm. The reason for the variation in drop size is that liquids also varied in surface tension.

2.2. Observations of axisymmetric and non-axisymmetric drop spreading

Exemplary images of the liquid film profile acquired at different instants after impact are shown in figures 2 and 3 for axisymmetric and non-axisymmetric impacts, respectively. The impact parameters of the experiments shown in these figures correspond to Reynolds number $Re=5200$ and Weber number $We = 142$. The off-axis distance for figure 3 is 1.12 mm.

Figure 2. Axisymmetric impact of (a) water, (b) Gly80 and (c) Gly90 drops. The impact velocity is ${U_0=3\ {\rm m}\ {\rm s}^{-1}}$, $10 < D_s/D_0 < 11.4$, and the target diameter is $D_s=30$ mm. The frames correspond to the initial drop deformation, end of the phase of inertial spreading, instant of minimum lamella thickness, and instant of maximum spreading diameter, respectively.

Figure 3. Non-axisymmetric impact of a water drop. The impact parameters are: initial drop diameter $D_0 = 3$ mm, impact velocity $U_0 = 2\ {\rm m}\ {\rm s}^{-1}$, off-axis distance $b=3.76$ mm, target diameter $D_s=15\ {\rm mm}$.

For larger off-axis distances as shown in figure 3, the droplet grazes on the edge of the target and leaves behind a small coated part on the target surface. The larger part of the liquid mass continues to move as a smaller droplet. The smaller droplets impacting onto a larger target spread, recede, and then remain deposited on the surface.

The main stages of drop collision with a spherical target and the geometrical parameters used in the further analysis are shown in figure 4, as observed from the high-speed video system I. The initial radius of the drop is given as $R_0 = D_0/2$, and that of the rigid spherical target as $R_s = D_s/2$. The impact velocity is denoted as $U_0$, and the off-axis distance is denoted as $b$ (see figure 4a). Let us indicate the intersection point of the impact axis with the target surface as the impact point. Its position at the spherical surface of the target can be defined by angle $\alpha _i = \arcsin [b/R_s]$ with the vertical axis. The typical velocity parallel to the target surface $U_\tau$ can be estimated from the geometrical considerations $U_\tau = U_0 b/R_s = U_0 \sin \alpha _i$. The corresponding normal velocity component is defined as $U_n = U_0 \cos \alpha _i$.

Figure 4. Sketch of non-axisymmetric drop impact and its initial deformation: (a) instant of impact when drop touches the sphere ($t = 0$); (b) typical drop deformation at $t=t_i$, at which the virtual centre of the drop arrives on the target's surface; (c) spreading of a drop lamella, $t\gg t_i$.

The instant $t=0$ is the first instant when the drop touches the target (figure 4a). The position of the point at the target's surface where the drop first touches the target is determined by the angle $\alpha _0 \equiv \arcsin [b/(R_s+R_0)]$. At $t>0$, a drop starts to deform and spread. The instant corresponding to the case shown in figure 4(b) can be estimated as $t_i = R_0 \cos \alpha _0/U_0$. The drop spreading in a thin lamella is observed at times $t\gg t_i$.

2.3. Image processing and post-processing

A Matlab code has been developed to perform the image processing and post-processing of the videos obtained from the cameras. First, the code converts the images obtained from each camera into binary images. Then, to detect the drop, the target and the background, it sorts all the detected objects in each image based on their roundness. Consequently, any disturbance aside from the drop and the target are filtered out, and the diameter and velocity of the drop right before impact is then measured. The equivalent diameter of a disc with an equal area method is used in all the calculations, i.e. ${\rm diameter} = 2\times {\rm area} / {\rm \pi}$. Both the vertical diameter and horizontal diameter are measured for every experiment. The difference was not significant in any of the experiments. Additionally, drop oscillations are dampened by the liquid viscosity, especially at higher velocities. To guarantee the alignment of the impact, the code identifies the centre of the drop and the apex of the target, and measures their corresponding offset. Finally, the code subtracts the spherical target from the images to measure the thickness of the lamella and the spreading diameter throughout the experiment using the videos from the Phantom v2012 and Phantom v12 cameras, respectively. Demonstration of the target subtraction is shown in figure 5.

Figure 5. Image of the film produced by drop impact: (a) the original image; (b) the image of a film profile obtained by subtracting the dry target.

The cases with an overall offset of less than 1 % with respect to the target are considered axisymmetric. For the offset experiments, the two cameras are used simultaneously to make sure that the offset is in only one direction, while the other is centred within 1 % with respect to the target. This guarantees that the drop will not fall behind or in front of the contact line in the plane of the view of the camera observing the offset.

3.1. Remote asymptotic solution

Consider a flow generated by a single drop impact. If the Reynolds and Weber numbers are high enough, then the flow in the lamella is governed mainly by the inertia, if the lamella is much thicker than the viscous boundary layer. Therefore, we start now with the solution for the inviscid flow in a thin liquid film on a spherical surface.

The flow dynamics in a thin drop lamella is described in a spherical coordinate system $\{r,\theta,\varphi \}$ with the origin at the sphere centre, where $r$ is the radial coordinate, $\theta$ is the polar angle, and $\varphi$ is the azimuthal angle. Denote as $u_\theta$ the average through the lamella cross-section velocity component of the liquid in the $\theta$-direction, and $u_\varphi$ in the $\varphi$-direction. A sketch of the spherical coordinate system and the corresponding spherical coordinates of an arbitrary point r on the sphere surface is presented in figure 6. In a thin film, the radial component of the velocity, normal to the substrate, is much smaller than $u_\theta$. We consider only the impact with high Weber, Froude and Reynolds numbers, for which the effects of surface tension, gravity and viscosity are minor in comparison with the inertia. The momentum balance equations for the flow in the thin lamella whose thickness $h(\theta,t)$ is much smaller than $R_s$ are reduced to

(3.1)$$\begin{gather} \frac{1}{\rho}\,\frac{\partial p}{\partial r} = \frac{u_\theta^2+u_\varphi^2}{R_s}, \end{gather}$$

(3.2)$$\begin{gather}\frac{1}{\rho}\,\frac{\partial p}{\partial \theta} = R_s\,\frac{\partial u_\theta}{\partial t} + u_\theta\,\frac{\partial u_\theta}{\partial \theta} + \frac{u_\varphi}{\sin\theta}\,\frac{\partial u_\theta}{\partial \varphi}-u_\varphi^2 \cot\theta, \end{gather}$$

(3.3)$$\begin{gather}\frac{1}{\rho \sin\theta}\,\frac{\partial p}{\partial \varphi}= R_s\,\frac{\partial u_\varphi}{\partial t} +u_\theta\, \frac{\partial u_\varphi}{\partial \theta}+\frac{u_\varphi}{\sin\theta}\,\frac{\partial u_\varphi}{\partial \varphi}+ u_\theta u_\varphi \cot\theta. \end{gather}$$

Figure 6. Sketch of the spherical coordinate system and the corresponding spherical coordinates of an arbitrary point $\boldsymbol {r}$ on the sphere surface.

The pressure in the lamella, estimated from (3.1), is $p\sim \rho h u^2/ R_s$, where $h$ is the lamella thickness, and $u \equiv u_\theta ^2+ u_\varphi ^2$. The pressure gradient along the trajectory of an arbitrary material point, moving with the flow, is

(3.4)\begin{equation} \frac{\partial p}{\partial s} \sim \rho u\,\frac{\partial u}{\partial s}\,\frac{h}{R_s} \ll \rho u\, \frac{\partial u}{\partial s}, \end{equation}

where $s$ is the path measured along the particle trajectory. The pressure gradients are thus negligibly small in comparison with the inertial effects. An arbitrary material point in the lamella flows along a geodesic line with a constant absolute velocity. It is known that the geodesic lines on a sphere are the great circles.

Consider an arbitrary material point, initially located at $\boldsymbol {r}_0=\{\theta _0,\varphi _0\}$. The corresponding components of the initial velocity $\boldsymbol {u}_0(\theta _0,\varphi _0)$ of this point are denoted $u_{\theta 0}$ and $u_{\varphi 0}$. The radius vector of this point at $t>0$ is denoted $\boldsymbol {r}(t)$. The unit vector $\boldsymbol {n}$ normal to the great circle along which the considered point moves is

(3.5a,b)\begin{equation} \boldsymbol{n} = \frac{\boldsymbol{r}_0\times \boldsymbol{u_0}}{R_s u_0}, \quad u_0 = \sqrt{u_{\theta 0}^2+ u_{\varphi 0}^2}. \end{equation}

Now, the trajectory of the material point, corresponding to the initial position $\{\theta _0, \varphi _0\}$, is determined through

(3.6a,b)\begin{equation} \boldsymbol{r}(t) = \boldsymbol{r}_0 \cos\omega t + \boldsymbol{n}\times\boldsymbol{r}_0\sin\omega t, \quad \omega = \frac{u_0}{R_s}, \end{equation}

and its velocity vector is

(3.7)\begin{equation} \boldsymbol{u} = u_0\,\frac{\boldsymbol{n}\times \boldsymbol{r}}{R_s}. \end{equation}

The expressions for the coordinates $\theta (t)$ and $\varphi (t)$ of the material point can be derived from (3.5a,b):

(3.8)$$\begin{gather} \theta = \arccos\left[\cos\theta_0 \cos\omega t - \frac{u_{\theta 0}\sin\theta_0}{u_0} \sin\omega t\right], \end{gather}$$

(3.9)$$\begin{gather}\varphi = \arctan\left[\frac{u_0 \sin\varphi_0\sin\theta_0 \cos\omega t + (u_{\varphi 0}\cos\varphi_0 + u_{\theta 0} \cos\theta_0 \sin\varphi_0)\sin\omega t}{u_0 \cos\varphi_0\sin\theta_0 \cos\omega t -(u_{\varphi 0}\sin\varphi_0 - u_{\theta 0} \cos\theta_0 \cos\varphi_0)\sin\omega t}\right]. \end{gather}$$

The evolution of the lamella thickness can be expressed using the mass balance of the flow. An infinitesimal volume $\Delta W$ of an element of the lamella is determined in the form

(3.10)\begin{equation} \Delta W = h \left|\frac{\partial \boldsymbol{r}}{\partial\theta_0} \times \frac{\partial \boldsymbol{r}}{\partial\varphi_0} \right| \Delta\theta_0\,\Delta\varphi_0 = h_0 \left|\frac{\partial \boldsymbol{r}_0}{\partial\theta_0} \times \frac{\partial \boldsymbol{r}_0}{\partial\varphi_0} \right|\Delta\theta_0\,\Delta\varphi_0 , \end{equation}

therefore

(3.11)\begin{equation} h = h_0(\theta_0,\varphi_0)\,\frac{R_s^2 \sin\theta_0 }{\left|\dfrac{\partial \boldsymbol{r}}{\partial\theta_0} \times \dfrac{\partial \boldsymbol{r}}{\partial\varphi_0} \right|}. \end{equation}

In spherical coordinates, this expression yields

(3.12)\begin{equation} h = h_0(\theta_0,\varphi_0) \left|\frac{\sin\theta_0}{\sin\theta}\right| \left|\frac{\partial\theta}{\partial\theta_0}\,\frac{\partial\varphi}{\partial\varphi_0}-\frac{\partial\theta}{\partial\varphi_0}\,\frac{\partial\varphi}{\partial\theta_0}\right|^{{-}1}. \end{equation}

3.2. Kinematics of the drop profile at the symmetry plane

Let us take the $z$-axis to connect the sphere centre and the point of the drop impact. The initial distribution of the flow at $t=0$ consists of the axisymmetric part, determined by the normal component of the impact velocity $U_n$, and the transverse flow, determined by the tangential component $U_\tau$. At a point $\{\theta _0,\varphi _0\}$ on the sphere surface in the vicinity of the impact point, $\theta _0 \ll 1$, the flow can be linearized, leading to

(3.13a,b)\begin{equation} u_{\theta 0} = U_{n} \kappa \theta_0 + U_\tau \cos\varphi_0, \quad u_{\varphi 0} ={-}U_\tau \sin\varphi_0, \end{equation}

where $\kappa$ is an unknown dimensionless constant. Consider the flow near the symmetry plane $\varphi _0\rightarrow 0$ and $\varphi _0\rightarrow {\rm \pi}$. The linearized solution yields

(3.14)$$\begin{gather} \theta(t) = \theta_0 + \varOmega t, \end{gather}$$

(3.15)$$\begin{gather}\varphi(t) =\left[ 1-\frac{ U_\tau \sin\varOmega t}{u_{\theta 0}\sin(\theta_0+\varOmega t)}\right]\varphi_0,\quad \varphi\rightarrow 0, \end{gather}$$

(3.16)$$\begin{gather}\varphi(t) =\left[ 1+\frac{ U_\tau \sin\varOmega t}{u_{\theta 0}\sin(\theta_0+\varOmega t)}\right](\varphi_0-{\rm \pi}),\quad \varphi\rightarrow {\rm \pi}, \end{gather}$$

(3.17)$$\begin{gather}\varOmega =\frac{U_{n} \kappa \theta_0 + U_\tau\cos\varphi_0}{R_s}. \end{gather}$$

Expressions (3.14) and (3.17) allow us to derive the expression for the velocity in the lamella in the Eulerian form

(3.18a–c)\begin{equation} u_\theta = \frac{R_s (\theta + \Delta\theta)}{t+\Delta t},\quad \Delta t = \frac{R_s}{\kappa U_n},\quad \Delta\theta = \frac{\Delta t\, U_\tau}{R_s}\cos\varphi_0, \end{equation}

where $\Delta t$ and $\Delta \theta$ are constants determined by the ratio $k=R_s/R_0$.

The expression for the lamella profile at the symmetry plane is then obtained in the form

(3.19)\begin{equation} h = h_0(\theta_0,\varphi_0) \left|\frac{\sin\theta_0}{\sin\theta} \right| \left|\left(1+\frac{\kappa t U_n}{R_s}\right)\left[ 1-\frac{ U_\tau \cos\varphi_0 \sin\varOmega t}{u_{\theta 0}\sin\theta}\right]\right|^{{-}1}. \end{equation}

The remote asymptotic solution for the evolution of the lamella profile on the sphere is obtained, assuming $\theta _0 \ll 1$. Linearization yields

(3.20)\begin{equation} h = h_0(\theta_0,\varphi_0)\,\frac{\theta_0}{|{\sin\theta}|}\,\frac{\Delta t}{t+\Delta t} \left| 1-\frac{\Delta \theta}{\Delta t}\,\frac{ R_s \sin\varOmega t}{u_{\theta 0} \sin\theta}\right|^{{-}1}. \end{equation}

This equation is written in the dimensionless form

(3.21)$$\begin{gather} h^* = h_0^*(\theta_0,\varphi_0)\,\frac{\theta_0}{|{\sin\theta}|}\,\frac{\tau}{t^*+\tau} \left| 1- \frac{\chi \sin(\varOmega^* t^*)}{\varOmega^*\sin\theta} \right|^{{-}1}, \end{gather}$$

(3.22)$$\begin{gather}\varOmega^* = \frac{\theta_0}{\tau} + \chi, \end{gather}$$

(3.23a,b)$$\begin{gather}\chi = \frac{2 \cos\varphi_0}{k}\tan\alpha_i, \quad \cos\varphi_0={\pm} 1, \end{gather}$$

(3.24a,b)$$\begin{gather}\theta(t^*) = \theta_0 + \varOmega^* t^*, \quad \alpha = \alpha_i + \theta \cos\varphi_0, \end{gather}$$

where $\tau = \Delta t\,U_n/D_0$ is a dimensionless constant. The shape $h^*(\alpha )$ is represented in the parametric form (3.21) with a small positive value $\theta _0$ serving as a parameter.

3.3. Experimental characterization of the drop profile on a spherical surface

In order to characterize the evolution of the drop height in time, we measure only the drop height $h_i$ at the position corresponding to $\alpha =\alpha _i$ at the impact axis; $h_i$ is measured normal to the sphere surface along the $z$-axis, as shown in figure 4. In figures 7(a) and 7(b), the measurements are shown in the dimensionless form, using the drop diameter as the length scale, and the characteristic normal component of the velocity $U_n = U_0 \cos \alpha _i$ as the velocity scale. Correspondingly, $D_0/U_n$ is used as the time scale. At short times ($t^* < 10$), it is obvious that the evolution of the drop height depends weakly on $\alpha _i$ if these angles are relatively small, $\alpha _i<20^\circ$. The dependence becomes more significant for higher values of $\alpha _i$, shown in figure 7(b). The height $h$ initially reduces in time, but at some instant reaches a plateau value, the residual thickness. The residual liquid film appears as a result of the flow-damping by viscosity when the thickness of the boundary layer is comparable with the lamella thickness. This problem has been solved for the case of normal (Roisman Reference Roisman2009) and oblique (Roisman Reference Roisman2010) impact onto a flat substrate. It is remarkable that the thickness of the residual film at $\alpha = \alpha _i$ almost does not depend on the impact inclination. This is an important practical result that can allow for improving the modelling of the particle encapsulation by drop impact.

Figure 7. (a) Water drop impact at various off-axis lengths at the longer times after impact. The dimensionless thickness $h^*_i = h_i/D_0$ of the liquid film at the impact point $\alpha =\alpha _i$ is a function of the dimensionless time $t^* = t U_0 \cos \alpha _i/D_0$. The impact parameters are $D_0 = 2.6$ mm, $U_0 = 2.0\ {\rm m}\ {\rm s}^{-1}$, $Re = 5200$, $We = 142$, and the target diameter is 6 mm. (b) Isopropanol drop impact at various off-axis lengths at the early stages of spreading. The dimensionless thickness $h^*_i = h_i/D_0$ of the liquid film at the impact point $\alpha =\alpha _i$ is a function of the dimensionless time $t^* = t U_0 \cos \alpha _i/D_0$. The impact parameters are $D_0 = 2.2$ mm, $U_0 = 1.37\ {\rm m}\ {\rm s}^{-1}$, $Re = 985$, $We = 154$, and the target diameter is $D_s = 6$ mm.

The effect of viscosity on the evolution of the film thickness is shown in figure 8. As expected, the value of the residual film thickness increases with the liquid viscosity.

Figure 8. Effect of the liquid viscosity on the evolution of the drop height for an axisymmetric drop impact. The dimensionless thickness $h^*_i = h_i/D_0$ of the liquid film at the impact point $\alpha =0$ is a function of the dimensionless time $t^* = t U_0/D_0$. The impact parameters are $D_0 = 2.66 \pm 0.06$ mm, $U_0 = 4\ {\rm m}\ {\rm s}^{-1}$, and the target diameter is $D_s = 40$ mm.

The influence of the impact angle $\alpha _i$ in the drop shape at the same time instant is shown in figure 9. The position of the peak of the curve, $h^*(\theta,t)$, propagates in time along the spherical surface. The average velocity of propagation of the position of the peak is compared with our estimation $U_\tau \approx U_0 \sin \alpha _i$ in figure 10. The agreement is rather good for angles $\alpha _i < 20^\circ$. This result can be used as a justification of the assumed initial velocity distribution (3.13a,b) in the impacting drop.

Figure 9. Distilled water drop impact. The dimensionless thickness of the liquid film is a function of the zenith angle $\theta$ for various impact angles at $t = 1.2$ ms ($t U_0/D_0 = 0.92$). The impact parameters are $D_0 = 2.6$ mm, $U_0 = 2.0\ {\rm m}\ {\rm s}^{-1}$, $Re = 5200$, $We = 142$, and the target diameter is $D_s = 6$ mm.

Figure 10. Tangential film velocity. Comparison of the average velocity of the film propagation with the estimated magnitude $U_0 \sin \theta _i$ at various impact angles. The impact parameters are $D_0= 2.6$ mm, ${U_0 = 2.0\ {\rm m}\ {\rm s}^{-1}}$, $Re = 5200$, $We = 142$, and the target diameter is $D_s = 6$ mm.

The shape of the drop has been captured using the camera settings that provide a spatial resolution $\Delta x =5\unicode{x2013}9\ \mathrm {\mu }{\rm m}$. The possible error $\delta _{peak}$ of the estimation of the peak position depends on its curvature. In our case, it is $\delta _{peak} \approx \sqrt {D_0\,\Delta x}= 1.6\times 10^{-4}$ m. The velocity is estimated on the time interval $\Delta t = 1$ ms. Therefore, the estimated error of the velocity estimation is $\Delta x/\Delta t=0.16\ {\rm m}\ {\rm s}^{-1}$.

Finally, the theoretical predictions (3.21) for the shape of the spreading drop are compared with the experimental data in figure 11 for impact angles $\alpha _i = 6.5^\circ$ and $22^\circ$. As an initial distribution of the drop height, the result from Roisman, Berberović & Tropea (Reference Roisman, Berberović and Tropea2009), developed for an impact onto a flat target from the mass balance equation, is reformulated in the spherical coordinate system:

(3.25)\begin{equation} h_0^*(\theta_0) = \frac{\eta}{\tau^2}\exp\left[-\frac{6\eta \theta_0^2 k^2}{4\tau^2}\right]. \end{equation}

Figure 11. Measured shape of the drop profile $h^*(\alpha )$ in comparison with the theoretical predictions (dashed lines). The impact parameters are $D_0 = 2.6$ mm, $U_0 = 2.0\ {\rm m}\ {\rm s}^{-1}$, $Re = 5200$, $We = 142$, and the target diameter is $D_s = 6$ mm. The predicted curves are computed using (3.21) with $\tau = 0.1$, $\eta = 0.38$. The impact angle is (a) $\alpha _i = 6.5^\circ$ and (b) $\alpha _i = 22^\circ$.

This approximate solution is valid if the thickness of the film is much smaller than the sphere radius. The theory correctly predicts the elevation of the film thickness with the angle $\alpha$. The agreement between the theoretical predictions and the experiments is rather good, which means that the theory accounts for the main physical effects.

The parameters $\tau = 0.1$, $\eta = 0.38$, used for the calculation of both cases, are obtained by fitting the theory with the experimental data for time $t^*\approx 1$. It is obvious that these values should depend on the diameter ratios of the drop and target. The values of $\tau$ and $\eta$ obtained by fitting the experimental data for various drop and target diameters are shown in figure 12.

Figure 12. Estimated values of the dimensionless parameters $\tau$ and $\eta$ obtained by fitting the measured evolution of the lamella height at $\alpha = \alpha _i$ using the theoretically predicted expression $h_i^* = \eta (t^* + \tau )^{-2}$ for various values of the ratio $D_0/D_s$.

It is important to note that this inviscid solution is valid only for cases when the viscous effects are negligibly small. At larger times, when the lamella thickness and the thickness of the viscous boundary layer are comparable, the viscous effects have to be taken into account even for high Reynolds numbers. The effect of viscosity on the drop spreading will be considered in the next section.

4.1. Evolution of the lamella thickness

In the axisymmetric cases, the inviscid solution (3.21) can be simplified, accounting for $\chi =0$:

(4.1)$$\begin{gather} h_{inv}^* = h_0^*\left(\frac{\theta \tau}{t^*+\tau}\right) \frac{\theta}{\sin\theta}\,\frac{\tau^2}{(t^*+\tau)^2},\end{gather}$$

(4.2)$$\begin{gather}\alpha ={\pm}\theta. \end{gather}$$

Here and later in the text, the term $h_{inv}$ is associated with the inviscid approximation of the flow.

Spreading of a drop on a solid substrate at $Re \gg 1$ is accompanied by an expansion of a viscous boundary layer of thickness $h_\nu \sim \sqrt {\nu t}$. It can be written in the dimensionless form $h^*_\nu \sim \sqrt { t^*}\,Re^{-1/2}$. The flow velocity slows down near the wall, leading to the generation of the normal component of the velocity. This flow has been analysed in the exact solution (Roisman Reference Roisman2009) for drop impact onto a dry, solid, planar substrate. This solution can be applied to the flow on a sphere if $h\ll R_s$:

(4.3)\begin{equation} h^* \approx h^*_{inv} + a \sqrt{t}\,Re^{{-}1/2}, \end{equation}

where $a\approx 0.48$ for a planar substrate. This value is applicable to the case of a spherical drop if the boundary layer thickness is much smaller than the sphere radius.

At some instant $t_\nu$, the thickness of the boundary layer is equal to the lamella thickness. Its expression can be obtained using (4.1) and (4.3), leading to

(4.4)\begin{equation} t^*_\nu \sim \left[\frac{\theta}{ \sin\theta}\right]^{2/5} Re^{1/5}. \end{equation}

At times $t>t_\nu$, the flow in the lamella is driven by the viscous stresses. The residual film thickness is obtained substituting the expression (4.4) for the time $t_\nu$ in (4.3), which leads to

(4.5a,b)\begin{equation} h^*_{res}(\theta) = A(k)\,Re^{{-}2/5} \left[\frac{\theta }{ \sin\theta}\right]^{1/5}, \quad D_0 = 2 R_0, \end{equation}

where $A(k)$ is a dimensionless parameter determined by the geometry of impact, namely by the ratio $k=R_s/R_0$.

The results of measurements of the residual film thickness at the sphere tip for the cases $k \gg 1$ are shown in figure 13. The best fit of the data is obtained as

(4.6)\begin{equation} h^*_{res}(0) \approx 0.72\,Re^{{-}0.409},\quad \mathrm{for}\ k \gg 1. \end{equation}

Figure 13. Non-dimensional residual thickness $h^*_{res}(0)$ at the sphere tip as a function of $Re$ for the experiments with $9.91< k <19.32$. All individual experiments are plotted. The error in each experiment is less than $50\ \mathrm {\mu }{\rm m}$.

This result is in very good agreement with the theory presented in (4.5a,b). It should be noted that for many practical reasons – for example, in the aim of viscosity estimation using the measurements of $h^*_{res}(0)$ – it could be recommended to use an empirical correlation (4.6), which can be considered as a result of the instrument calibration.

The experimental data on $h^*_{res}(0)$ were used for solving the inverse problem, namely for measurements of the viscosities of the liquids used in this study with the help of (4.6). For the viscosity estimations, only the cases have been used for which a clear plateau in the residual film thickness was observed. Certainly, the cases leading to the drop splash or cases of low Reynolds numbers have been considered invalid for viscosity measurements.

Table 2 shows a comparison between the viscosities measured using drop impact on a spherical target method as compared to traditional rheometry and corrected for the same temperatures based on the formulas given by Cheng (Reference Cheng2008) with the adjustments from Volk & Kähler (Reference Volk and Kähler2018). The results agree with each other for most viscosity values, as well as with the values from the literature, demonstrating the applicability of the drop impact technique. Traditional rheometry has error approximately 1 %. Significant deviation is obtained for low viscosity, such as distilled water. The reason is a very thin residual film whose thickness is comparable with the pixel size of the camera. Additionally, higher errors are associated with higher offset, while more axisymmetric impacts yield more accurate results. The criterion for the results in table 2 is that all impacts have less than 1 % offset. The target sphere diameter $D_s$ is 30 mm, which has the highest spreading, lowest residual thickness, and the best chance not to be affected by the rim formation that blocks the view. The calibration equation $h^* = 0.7379\,Re ^ {-0.422}$ is used for more accurate results for the range $10< k<11.4$.

Table 2. Drop impact viscometry results as compared to traditional rheometry results. The temperature is $T = 22 \pm 1\,^\circ {\rm C}$. The error is calculated as ${\rm error} = (\mu _{impact}-\mu _R)/\mu _R$.

It is important to note that the relevance of the drop impact viscometry (DIV) method is not in its accuracy but in its applicability to situations that are not measurable by existing methods. It excels in measuring the viscosity of non-Newtonian liquids at high shear rates, a regime inaccessible to traditional rheometry. While commercial rheometers typically reach 1000 Hz, DIV experiments can achieve shear rates exceeding $10^6$ Hz, depending on the liquid and impact velocity. This capability makes DIV valuable in analysing industrial processes such as pesticide application, paint droplet behaviour, and fuel injection. Additionally, DIV's rapid measurement, where the impact occurs in milliseconds, is ideal for characterizing liquids with time-dependent properties, such as volatile paints, suspensions, and emulsions prone to evaporation at low temperatures. Finally, DIV requires only microlitre-sized drops, making it suitable for analysing rare or expensive liquids such as biological fluids (blood, tears or insulin) or even mercury solutions.

4.2. Maximum drop spreading

Denote $S= k \theta _{rim}$ as the arc length of the wetted spot of the sphere. Here, $\theta _{rim}$ is the polar angle corresponding to the position of the drop rim. The rim initially spreads. Then the value of $\theta _{rim}$ at some instant reaches a maximum. Finally, if the surface is hydrophobic, then the rim can start to recede. The value of $\theta _{max}$ is an important parameter characterizing the level of the particle encapsulation after drop impact. It is determined mainly by the impact parameters, drop and sphere diameters, impact velocity, and liquid properties, including viscosity and surface tension.

In the case of drop impact onto a flat substrate, the expression for the maximum spreading diameter is obtained in Roisman (Reference Roisman2009) from the analysis of the drop spreading on a plane accounting for the dynamics of the rim (1.2). The same ideas as in Roisman (Reference Roisman2009) will be used in this study also for the description of the maximum spreading on a sphere. Let us assume first that the effects of surface tension can be neglected. Following (1.2), the volume of the rim can be neglected in comparison to the initial volume of the drop if $We^{1/2}/Re^{1/5}\gg 1$ and thus the second term on the right-hand side of (1.2) can be neglected in comparison with the first term.

The mass balance of the total volume of the residual lamella yields

(4.7)\begin{equation} \frac{{\rm \pi} k^2}{2} \int_0^{\theta_\nu} h_{res}^* \sin{\theta}\,\mathrm{d}\theta = \frac{\rm \pi}{6}, \end{equation}

which can be simplified with the help of (4.5a,b) as

(4.8)\begin{equation} \int_0^{\theta_\nu}\theta^{1/5} \sin^{4/5}\theta \,\mathrm{d}\theta = \frac{Re^{2/5}}{3 A k^2}, \end{equation}

where $\theta _\nu$ is the approximate value of the maximum spreading angle if the effects of surface tension are neglected.

This equation does not have an exact solution. An approximate expression for the root of this equation, $\theta _\nu$, valid in the range $\theta \in [0,{\rm \pi} ]$, is obtained after series expansions around $\theta =0$ of the integrand on the left-hand side of (4.8) with precision $\theta ^6$:

(4.9)\begin{equation} \theta_\nu \approx \sqrt{15} \sqrt{1-\left(1- \frac{2\,Re^{2/5}}{15 A k^2}\right)^{1/3}} ,\quad \mathrm{if} \ We^{1/2}/Re^{1/5}\gg 1. \end{equation}

The expression for $\theta _\nu$ can be represented as an upper bound for the value $\theta _{max}$ since the considerations leading to (4.9) do not take into account the dynamics of the rim. The known characteristic rim velocity $u_{rim}^*\sim We^{-1/2}\, (h_{res}^*)^{-1/2}$, the Taylor rim velocity (Taylor Reference Taylor1959; Roisman et al. Reference Roisman, Rioboo and Tropea2002), is determined by the balance of inertial and interfacial forces applied to the rim. The corrected increment of the dimensionless spreading factor due to the rim motion is $\Delta S^* \sim t_\nu ^* u_{rim}^*$. The final expression for the maximum spreading angle on a sphere $\theta _{max} = \theta _\nu -\Delta S^*/k$ is obtained with the help of (4.4), (4.5a,b) and (4.9) in the form

(4.10)\begin{equation} \theta_{max} = \sqrt{15} \sqrt{1-\left(1- \frac{2\,Re^{2/5}}{15 A k^2}\right)^{1/3}} -\frac{B\, Re^{2/5}}{k\,We^{1/2}\,A^{1/2}} \left[\frac{\theta_\nu }{ \sin\theta_\nu}\right]^{3/10}, \end{equation}

where $B$ is a fitting parameter.

It can be shown that the linearization of (4.10) for small values of $\theta _\nu \ll 1$, associated with an impact onto a flat substrate, leads to its reduction to the form (1.2). Moreover, noting that (1.2) is obtained in (Roisman Reference Roisman2009) for the value $A=0.79$, the parameter $B\approx 0.36$ can be estimated. The measured values for the maximum spreading angle $\theta _{max}$ are in good agreement with the theoretical predictions of (4.10), as shown in figure 14.

Figure 14. Comparison of the measured values for the maximum spreading angle $\theta _{max}$ with the theoretical predictions (4.10). The dashed line corresponds to perfect agreement. All individual experiments are plotted.

The model is applicable only to the cases where the sphere is not completely encapsulated, namely when $\theta _\nu <{\rm \pi}$. This condition, analysed with the help of (4.9), is satisfied if

(4.11)\begin{equation} Re <154.0\,A^{5/2} k^5 . \end{equation}

The Reynolds number determined on the right-hand side of (4.11) is the threshold value for the complete sphere encapsulation.

For large values of $k$, the value of the parameter $A$ approaches a constant corresponding to a flat target. If $\theta _\nu < 1$, then the term $(\theta _\nu /\sin \theta _\nu )^{3/10}$ is very close to unity. In this case, the expression (4.10) for $\theta _{max}$ can be further simplified to

(4.12)\begin{equation} \theta_{max} = \sqrt{15} \sqrt{1-\left(1- \frac{2\,Re^{2/5}}{15 A k^2}\right)^{1/3}} - \frac{0.40\, Re^{2/5}}{k\,We^{1/2}}. \end{equation}

It should be noted that some advanced theory for the lamella spreading and rim dynamics on a flat solid substrate has been developed recently by Gordillo, Riboux & Quintero (Reference Gordillo, Riboux and Quintero2019). It could be interesting to examine how this theory works in the case of not axisymmetric impact onto a curved surface. However, this topic is out of the framework of this study. Nevertheless, the current estimation of the maximum spreading on a spherical surface is sufficient to estimate the validity limits for the rheological measurements by drop impact, considered in this study.