2.1 Theoretical framework & hypotheses

We consider throughout this study an idealized drop impact where a perfectly spherical liquid drop collides with a flat rigid surface. Though classic, this model situation relies on a number of physical hypotheses detailed in the following. Starting with the initial perfect spherical shape assumption, we may identify several typical causes for deviations from sphericity such as capillary drop oscillations during free fall (Engel Reference Engel1955; Thoroddsen et al. Reference Thoroddsen, Etoh, Takehara, Ootsuka and Hatsuki2005) or flowing air shaping (Pruppacher & Beard Reference Pruppacher and Beard1970). Such effects will be disregarded in the following even though they merely result in a local curvature radius modification in the contact region and hence could easily be incorporated in the discussion. The impact flow evolution will be considered incompressible. However, the question of the role of compressible effects within the liquid should not be eluded when considering impact phenomena. A large body of literature has been devoted to acoustic effects within (typically fast) impacting drops. Typically, such effects are considered to play a significant role over length scales of order $UR/c$ during time scales of order $UR/c^{2}$ , where $U$ stands for the impact velocity, $R$ for the drop radius and $c$ for the celerity of compressive sound waves in the liquid ( $(U/c)^{2}\ll (Ut/R)\ll 1$ ). Considering a millimetric drop impacting at a velocity of the order of $10~\text{m}~\text{s}^{-1}$ representative of e.g. a raindrop falling at terminal velocity, we estimate that acoustic effects matter only in a micron-sized region over a few nanoseconds (see Weiss & Yarin (Reference Weiss and Yarin1999), for a discussion and references). The following discussion will therefore be limited to those cases where the impact velocity is much lower than the speed of sound, as the falling raindrop, where acoustic effects can harmlessly be neglected and an incompressible description remains accurate. The high pressure and stresses generated upon impact can result in marked erosion or yielding (Rein Reference Rein1993). Furthermore substrate deformation has recently been shown to significantly alter drop impact in the limit of very soft (Mangili et al. Reference Mangili, Antonini, Marengo and Amirfazli2012) or very flexible substrates (Antkowiak et al. Reference Antkowiak, Audoly, Josserand, Neukirch and Rivetti2011). None of these effects will be considered in the following, yet an estimation of the net force exerted by the impacting drop on the underlying substrate will be provided in § 3. Another phenomenon potentially responsible for the cushioning of the impact is the thin air layer between the drop and the substrate just before impact. Due to lubrication effects, this air layer pressurizes and dimples the drop, eventually yielding a tiny entrapped gas bubble in the drop (Thoroddsen et al. Reference Thoroddsen, Etoh, Takehara, Ootsuka and Hatsuki2005). This phenomenon along with other roles of surrounding gas in impact dynamics will be disregarded in the forthcoming analysis (except where explicitly specified, the starting state for each of the simulations is a drop already touching the ground – see appendix A for details) and discussed in the last section. Finally the core hypothesis of the present study is the inertia-dominated character of impact. In particular, we assume that gravity, capillary and viscous effects are small with respect to inertial ones, i.e. Froude $Fr=U^{2}/gR$ , Weber $We={\it\rho}U^{2}R/{\it\sigma}$ and Reynolds $Re={\it\rho}UR/{\it\mu}$ numbers are all large with respect to unity. Here $g$ denotes the gravity, ${\it\sigma}$ the liquid–gas surface tension, ${\it\rho}$ the liquid density and ${\it\mu}$ its viscosity. These assumptions underpin the choice a purely inertial description free of these effects in the following. However, locally, these phenomena might become more important or even dominant, examples being viscosity near the boundaries or capillarity in high-curvature regions. In § 4 we will address viscous effects and develop a boundary layer correction to the inviscid solution, and capillary effects will eventually be discussed in § 5.

2.2 Governing equations and analogy with the water entry problem

2.3 Leading-order description for the drop impact problem

Interested in the early time behaviour of the impact-induced flow, we set out by examining time-dependent solutions of system (2.8)–(2.12) in the vicinity of the contact zone. To this end, we introduce ${\it\varepsilon}$ as a measure of the wet region: $d(t)/R=O({\it\varepsilon})$ (see figure 7). This ${\it\varepsilon}$ is the fundamental small parameter of our problem.

As typical in two-phase phenomena, the length scales for the dynamical fields and for the geometry of the free surface differ in this problem. Starting by considering the space variables $\bar{r}$ and $\bar{z}$ on which depend the dynamical fields (such as the velocity potential ${\it\phi}$ or the pressure $p$ ), we introduce the following rescaling: $\bar{r}={\it\varepsilon}_{r}\tilde{r}$ and $\bar{z}={\it\varepsilon}_{z}\tilde{z}$ , where $\tilde{r}$ and $\tilde{z}$ are $O(1)$ quantities and ${\it\varepsilon}_{r}$ and ${\it\varepsilon}_{z}$ are gauge functions. From the structure of the Laplace operator, we expect the dynamical fields to display identical length scales in each direction, so that ${\it\varepsilon}_{r}={\it\varepsilon}_{z}={\it\varepsilon}$ .

Insights into the relevant length scales for the description of the free surface geometry can be gained by decomposing the position of the surface into that of a translating sphere $\bar{z}_{S}(\bar{r},\bar{t})$ plus a surface disturbance $\bar{h}(\bar{r},\bar{t})$ (see figure 3 b). Assuming the drop falls with constant velocity, the shape of the unperturbed translating sphere obeys $\bar{r}^{2}+(\bar{z}_{S}-(1-\bar{t}))^{2}=1$ . Sufficiently close to the contact area, we introduce gauge functions for the vertical position of the moving sphere $\bar{z}_{S}$ and the time $\bar{t}$ : ${\it\varepsilon}_{z_{s}}\tilde{z}_{s}={\it\varepsilon}^{2}\tilde{r}^{2}/2-{\it\varepsilon}_{t}\tilde{t}$ . The equation for the sphere surface can be approximated by ${\it\varepsilon}_{z_{S}}\tilde{z}_{S}=({\it\varepsilon}^{2}\tilde{r}^{2}-{\it\varepsilon}_{t}\tilde{t})/2$ . As previously, the determination of these scaling functions is obtained by dominant balance arguments: ${\it\varepsilon}_{z_{S}}={\it\varepsilon}_{t}={\it\varepsilon}^{2}$ . Note that at short times the intersection radius between the sphere and the impacting plane is given by $\tilde{r}_{intersect}=\sqrt{2\tilde{t}}$ .

We remark that as in the original study of Wagner, a scale separation between $\bar{z}_{S}$ and $\bar{z}$ exists (small deadrise angle hypothesis, see e.g. Oliver Reference Oliver2002). This scale separation arises because the drop typical radius of curvature ( $O(1)$ ) is very large in comparison with the other length scales of the problem, see figure 7.

We now turn to the surface perturbation $\bar{h}(\bar{r},\bar{t})$ , that embodies the impact-induced flow. Recalling that $\bar{h}$ represents a perturbation around a falling sphere, we can express the position of the free surface with the following implicit equation: $\bar{{\mathcal{S}}}(\bar{r},\bar{z},\bar{t})=\bar{z}-(\bar{z}_{S}-\bar{h}(\bar{r},\bar{t}))$ . Introducing an appropriate gauge function ${\it\varepsilon}_{h}$ such that $\bar{h}={\it\varepsilon}_{h}\tilde{h}$ we obtain by dominant balance analysis that ${\it\varepsilon}_{h}={\it\varepsilon}^{2}$ . It follows that:

The kinematic boundary condition derives from the previous equation. At the free surface, we have:

where $\bar{{\it\phi}}={\it\varepsilon}_{{\it\phi}}\tilde{{\it\phi}}$ . It is impossible here to keep all terms at the same order; the dominant balance between the vertical velocities $\partial \tilde{h}/\partial \tilde{t}$ and $\partial \tilde{{\it\phi}}/\partial \tilde{z}$ implies that ${\it\varepsilon}_{{\it\phi}}={\it\varepsilon}$ . At leading order, the kinematic boundary condition is therefore reduced to:

It proves convenient to introduce a translation of the velocity potential such that $\tilde{{\it\phi}}=-\tilde{z}+\check{{\it\phi}}$ . This translation merely allows us to analyse the problem in the falling-drop reference frame. The kinematic boundary condition is then simply rewritten as:

Inserting these different scaled variables into Bernoulli’s equation, we obtain:

where $\bar{p}={\it\varepsilon}_{p}\tilde{p}$ . The scale of the pressure, ${\it\varepsilon}_{p}=1/{\it\varepsilon}$ , is here seen to be as large as the contact zone is small – as expected in an impact problem. At leading order, Bernoulli’s equation is therefore reduced to:

It follows from this equation that the constant pressure Dirichlet boundary condition at the free surface $p=0$ can be recast as a condition for the potential at the free surface: ${\it\phi}=\text{const.}$ , where the constant is arbitrary. Without loss of generality, we set from now on this constant to zero.

Table 1.

Résumé of the most important asymptotic scales of the problem.

Finally, as is classic in water wave theory, we exploit the shallowness of the gap between the free surface and the plane to transfer the boundary condition at the free surface onto the plane (see e.g. Van Dyke Reference Van Dyke1975, § 3.8).

Summarizing, using table 1 scalings, the near-field model problem takes the following expression:

the locus $\tilde{d}(\tilde{t})$ of the contact line is determined with the Wagner condition:

so that the boundary conditions at $\tilde{z}=0$ read:

and the far-field behaviour is given by: Finally, the corresponding model geometry is sketched figure 8. We remark that the previous set of equations resembles that of the classic water entry problem, and can be solved using the methodology described in e.g. Oliver (Reference Oliver2002). In the next section however, we will present an alternative method based on self-similar solutions.

Figure 8.

Leading-order outer problem for times of order ${\it\varepsilon}^{2}$ .

3.1 A self-similar problem

To reveal the self-similar nature of our problem, we classically seek scale invariance in the following (Darrozès & François Reference Darrozès and François1982). We start by expressing the fact that any variable $\tilde{q}$ in $(\tilde{r},\tilde{z},\tilde{t},\check{{\it\phi}},\tilde{h},\tilde{d},\tilde{p})$ can be rewritten as $\tilde{q}={\it\lambda}_{q}\hat{q}$ , where $\hat{q}$ is a rescaled variable and ${\it\lambda}_{q}$ a numerical stretching coefficient embodying the change of scale. Inserting these variables into the governing equations, it is straightforward to see that invariance of Laplace equation through this stretching requires ${\it\lambda}_{r}={\it\lambda}_{z}$ . Similarly, expressing the invariance of Wagner condition yields ${\it\lambda}_{h}={\it\lambda}_{t}$ , ${\it\lambda}_{r}=\sqrt{{\it\lambda}_{t}}$ and ${\it\lambda}_{d}=\sqrt{{\it\lambda}_{t}}$ . The same operation performed on the additional boundary conditions finally imposes ${\it\lambda}_{{\it\phi}}=\sqrt{{\it\lambda}_{t}}$ and ${\it\lambda}_{p}=1/\sqrt{{\it\lambda}_{t}}$ . Note that ${\it\lambda}_{t}$ remains here as the sole stretching parameter.

The pressure field can be written as an implicit function of time and space as follows: ${\mathcal{F}}(\tilde{p},\tilde{r},\tilde{z},\tilde{t})=0$ . Upon using the previous scale invariance arguments, this relation may be rewritten as ${\mathcal{F}}(\hat{p}/\sqrt{{\it\lambda}_{t}},\sqrt{{\it\lambda}_{t}}\hat{r},\sqrt{{\it\lambda}_{t}}\hat{z},{\it\lambda}_{t}\hat{t})=0$ . A simple algebraic manipulation allows us to remove the ${\it\lambda}_{t}$ dependence for all but one variable, so that finally ${\mathcal{G}}(\sqrt{\hat{t}}\hat{p},\hat{r}/\sqrt{\hat{t}},\hat{z}/\sqrt{\hat{t}},{\it\lambda}_{t}\hat{t})=0$ , for any ${\it\lambda}_{t}$ . Remarking that for a given $\hat{t}$ , this function has to cancel whatever the choice of the scale ${\it\lambda}_{t}$ , it readily appears that the last variable is superfluous. In other words, a relation linking $\sqrt{\hat{t}}\hat{p}$ to $\hat{r}/\sqrt{\hat{t}}$ and $\hat{z}/\sqrt{\hat{t}}$ only must exist.

The pressure field may therefore be rewritten explicitly as:

With a similar reasoning, and upon introducing the self-similar variables ${\it\xi}=\tilde{r}/\sqrt{\tilde{t}}$ and ${\it\eta}=\tilde{z}/\sqrt{\tilde{t}}$ , we readily obtain:

where ${\it\Phi}$ and ${\mathcal{H}}$ are unknown functions of the self-similar variables and ${\it\delta}$ a constant representing the (fixed) position of the contact line in self-similar space. This allows us to formulate the self-similar version of the drop impact problem:

the boundary conditions at ${\it\eta}=0$ take the following form: the far-field behaviour is: and the self-similar version of Wagner condition is finally given by:

This problem can now be solved in several steps.

3.2 Self-similar potential

In this geometry, the Laplace equation can be solved with variable separation, leading to a family of elementary cylindrical harmonic solutions with an exponential behaviour in ${\it\eta}$ and an oscillatory one in ${\it\xi}$ . We recompose by summation and obtain:

The weight function ${\mathcal{C}}(k)$ is determined with boundary conditions (3.6) and (3.7), leading to the following pair of dual integral equations:

Solving these dual integral equations using the technique described in Sneddon (Reference Sneddon1960), we obtain a closed-form expression for the weight function:

Anticipating the description of the contact-line dynamics, we now derive $\partial {\it\Phi}/\partial {\it\eta}$ at the substrate level ${\it\eta}=0$ :

where we recognize the sum of two Hankel transforms (see e.g. Sneddon Reference Sneddon1995, table IV, p. 528). This allows us to obtain the following explicit expression for $\partial {\it\Phi}/\partial {\it\eta}$ for ${\it\eta}=0$ :

This result was originally obtained by Schmieden (Reference Schmieden1953) in the water entry framework.

3.3 Wagner condition and contact-line dynamics

With the help of the vertical velocity expression in the near-wall region just derived, we can rewrite the kinematic boundary condition (3.5) as:

This inhomogeneous differential equation can be solved using variation of parameters, i.e. looking for a solution of the form ${\mathcal{H}}({\it\xi})={\it\xi}^{2}f({\it\xi})$ . This gives:

Upon using the far-field decaying behaviour of ${\mathcal{H}}$ (see (3.9)), this last equation reduces to $f({\it\delta})={\it\delta}^{-2}/2$ so that at the contact line the drop deformation is:

In the self-similar space, the Wagner condition therefore takes the following remarkably simple form:

from which we finally derive the position of the contact line:

We note that this result was found in a non-self-similar setting in the recent paper of Riboux & Gordillo (Reference Riboux and Gordillo2014), which also presents experimental data in excellent agreement with this theoretical prediction. It is interesting to remark that the contact-line motion $\tilde{d}(\tilde{t})=\sqrt{3\tilde{t}}$ just predicted within the framework of Wagner theory is quite close to the rough truncated sphere approximation $\tilde{r}_{intersect}=\sqrt{2\tilde{t}}$ (Rioboo et al. Reference Rioboo, Marengo and Tropea2002). Indeed figure 9 reports early post-impact successive positions of the contact line extracted from numerical simulations performed with Gerris  along with our theoretical prediction. Noticeably the superposition between theory and numerical results is excellent, at least until the moment of formation of a liquid corolla (here indicated with a red dot).

Figure 9.

Comparison between the theoretical position of the contact line as a function of time deduced from Wagner theory $\tilde{d}(\tilde{t})=\sqrt{3\tilde{t}}$ (dashed line) and the position of the contact line extracted from Gerris  computations of an impacting drop at $Re=5000$ and $We=250$ (grey dots). The red dot marks the birth of the corolla.

3.4 Analogy with the normal motion of an expanding disk in an infinite mass of liquid

In § 2.2.3 we proposed to visualize the flow in an impacting drop as analogous to the one induced by a flat rising disk expanding radially as the wet area (see also figure 6). We are now in a position to formally justify this water entry analogy. The axisymmetric flow induced by ‘the motion of a thin circular disk with velocity $U$ normal to its plane, in a infinite mass of liquid’ is for example analysed in Lamb’s classic textbook § 101 (Lamb Reference Lamb1932). After deriving some elementary axisymmetric solutions of the Laplace equation of the form $\exp (\pm kz)J_{0}(kr)$ in § 100, Lamb examined a variety of axisymmetric potential flows. Among these was the one (later connected to the flow around a flat circular disk in normal incidence) where at the symmetry plane $z=0$ the potential takes the value ${\it\phi}=C\sqrt{a^{2}-r^{2}}$ for $r<a$ and ${\it\phi}=0$ for $r>a$ , with $a$ the disk radius. The solution for this problem was stated under the following integral representation:

And from ‘a known theorem in Electrostatics’, Lamb obtained the expression for the vertical velocity in the symmetry plane:

This corresponds precisely to the flow within the impacting drop, after posing $C=-2/{\rm\pi}$ and $a={\it\delta}$ , thereby justifying formally our initial analogy between the impact-induced flow with the one associated with a flat rising disk rapidly expanding with the wet area. Setting $C=2U/{\rm\pi}$ , Lamb remarked that the above potential indeed describes the flow winding around a flat disk moving at velocity $U$ . He further noted that a simple expression for the fluid half-space kinetic energy could be derived from the previous relation:

This expression can immediately be transposed into the (non-dimensional) kinetic energy of the impact-induced flow within the drop:

or, equivalently, into its dimensioned counterpart:

We emphasize that this expression is derived within the frame of the falling drop and, as such, represents the kinetic energy of the defect flow associated with impact. Although a direct physical interpretation of this quantity is not straightforward, we will see in § 3.7 that the knowledge of this defect kinetic energy will allow for a direct determination of the impacting force.

3.5 Structure of the velocity field

We now investigate the structure of the velocity field in the contact region and search for exact closed-form expressions and convenient approximations for this field.

3.6 Self-similar pressure

From the knowledge of the velocity potential we are now in a position to derive the pressure field as the time derivative of the potential. In the self-similar space, the pressure field is given by (3.4). Figure 14 proposes a comparison between the structure of the self-similar pressure extracted from numerical computations performed with Gerris  and the theoretical prediction. There it can be seen that the overall structure of the pressure field developing in the impacting drop, and in particular the pressure peak in the vicinity of the contact line already pinpointed out in figure 1, nicely matches with the theory. Interestingly, the structure just described is at variance with the pressure distribution around a flat disk rising steadily (Lamb’s original problem). Indeed, in such a configuration the pressure is expected to be maximal in the stagnation point area, whereas in our model problem the pressure peaks near the contact line/disk edge. This is a consequence of motion unsteadiness: the pressure is here dominated by the $\partial \check{{\it\phi}}/\partial \tilde{t}$ contribution rather than the steady $(\tilde{\boldsymbol{{\rm\nabla}}}\check{{\it\phi}}^{2})/2$ term.

Figure 14.

Comparison between the pressure field developing inside an impacting drop (left) and around a rapidly expanding disk (Lamb analogy, right). The pressure field represented on the left is extracted from Gerris  computations and represented in the self-similar space ( $\bar{t}=10^{-3}$ , $Re=5000$ , $We=250$ ; The isovalues are: 0.12, 0.24, 0.36, 0.48, 0.6, 0.72, 0.84). The self-similar theoretical pressure field represented on the right is given by (3.4) (isovalues: 0.13, 0.28, 0.445, 0.57, 0.73, 0.9, 1.2). Though isovalues have been slightly changed between the two panels, theoretical and numerical results are in a good overall agreement.

As in § 3.5.2, closed-form expressions for the pressure can be obtained along the axis and the substrate plane. The radial structure of the self-similar pressure across the wet area reads ${\mathcal{P}}({\it\xi},{\it\eta}=0)=3/({\rm\pi}\sqrt{3-{\it\xi}^{2}})$ for $0\leqslant {\it\xi}<\sqrt{3}$ . This analytical prediction is confronted with Gerris  numerical results in figure 15. After a transient numerical initialization phase (corresponding to the red curves), the pressure profiles collapse on the self-similar analytical solutions (blue curves). In accordance with the overall pressure field structure depicted earlier, the pressure radial profile presents a local minimum at ${\it\xi}=0$ and a maximum in the vicinity of the contact line, that is for ${\it\xi}=\sqrt{3}$ – where the analytical solution exhibits an inverse square-root singularity, as in water entry problems (e.g. Cointe Reference Cointe1989; Howison et al. Reference Howison, Ockendon and Wilson1991). We note that for later times the pressure peak is smoothed out in the numerical simulations (grey curves). As this regularization coincides with the birth of the ejecta sheet, we conjecture that this fall-off can appropriately be described with a second-order Wagner theory (Korobkin Reference Korobkin2007; Oliver Reference Oliver2007).

Figure 15.

(a) Pressure trace on the substrate $\bar{z}=0$ obtained from the numerical simulations between $\bar{t}=5\times 10^{-4}$ and $10^{-1}$ for $Re=5000$ and $We=250$ . The colour code for each decade is the same as in figure 17. Note that the curves are equally distributed within each decade. (b) Same as in the left but in the self-similar space. The black dashed curve represent the analytical solution for ${\mathcal{P}}({\it\xi},{\it\eta}=0)$ .

Similarly the expression for the self-similar pressure along the symmetry axis can also be obtained analytically: ${\mathcal{P}}({\it\xi}=0,{\it\eta})=(3\sqrt{3})/({\rm\pi}(3+{\it\eta}^{2}))$ for ${\it\eta}\geqslant 0$ . Figure 16 compares this last result with rescaled axial pressure profiles extracted from numerical simulations. There again the agreement between the computations and the theory is seen to hold for a large time span.

Figure 16.

Pressure along the axis $\bar{r}=0$ obtained with Gerris  for $\bar{t}=5\times 10^{-3},10^{-2},5\times 10^{-2}$ and $10^{-1}$ (same as in figure 10 b) represented in the self-similar space ( $Re=5000$ and $We=250$ ). The red dashed line is the analytical solution for ${\mathcal{P}}({\it\xi}=0,{\it\eta})$ . Fluctuations of the pressure around the theoretical prediction is to be related with numerical projections errors.

The structure of the pressure field in the vicinity of the origin can be inferred from these last results. Re-expressing the pressure cuts determined in terms of $\tilde{r}$ , $\tilde{z}$ and $\tilde{t}$ , we obtain:

From these two relations, we deduce the following expansion for the pressure near the origin:

This expression provides with a local approximation for $\partial \check{{\it\phi}}/\partial \tilde{t}$ from which, after time integration and space differentiation, we readily recover the stagnation point flow structure found earlier: $(\tilde{u} _{\tilde{r}},\tilde{u} _{\tilde{z}})=2/({\rm\pi}\sqrt{3})(\tilde{r}/\sqrt{\tilde{t}},-2\tilde{z}/\sqrt{\tilde{t}}).$ This near-axis behaviour emphasizes again that simple intuitive dimensional analysis suggestion $r/t$ and $-z/t$ is here not relevant.

The leading-order term for the pressure at the origin follows:

This result extends the $t^{-1/2}$ scaling law proposed by Josserand & Zaleski (Reference Josserand and Zaleski2003) on the basis of scaling arguments. A comparison between this theoretical prediction and Gerris  numerical simulations is proposed figure 17, using the colour code of figure 15. After a numerical transient phase, the pressure rapidly reaches the self-similar regime. Remarkably, this short-time similarity regime is nicely captured with Wagner theory up to times as large as $\bar{t}=0.5$ . At this point, a sharp departure from similarity is observed and the pressure promptly drops to zero. The physical mechanisms involved in this transition regime are still to be elucidated.

Figure 17.

Time evolution of the pressure $\bar{p}(0,0,\bar{t})$ measured at the origin in the numerical simulations with Gerris  ( $Re=5000$ and $We=250$ ). Note that each decade is represented with a different colour. The theoretical prediction $\bar{p}(0,0,\bar{t})=(\sqrt{3}/{\rm\pi})\bar{t}^{-1/2}$ is superimposed with a black dashed line.

3.7 Normal force induced by drop impact

Building on the last set of results, we deduce the total net normal force imparted by an impacting drop on the underlying substrate at early times. Integrating the pressure on the wet surface, we have:

The dimensional counterpart of this net total force induced by the drop on the substrate therefore reads:

where the force is seen to increase as $t^{1/2}$ for short times. Interestingly, $F(t)$ could have been inferred directly from energy arguments, with no knowledge of the pressure distribution. Indeed, writing the global kinetic energy conservation for the upper semi-infinite space, we have:

In the context of a flat rising disk, the kinetic energy reduces to $T_{disk}=(4/3){\it\rho}a^{3}U^{2}$ (Lamb § 102). This expression can immediately be transposed to the impacting drop problem so that $T=4\sqrt{3}{\it\rho}U^{7/2}R^{3/2}t^{3/2}$ (see (3.25) in § 3.4). The power of pressure forces then follows as $(\text{d}/\text{d}t)T=6\sqrt{3}{\it\rho}U^{3}R^{2}(Ut/R)^{1/2}$ . Dividing this power by $U$ , we recover exactly the previously obtained result for the net normal total force. This alternate derivation of the normal force provides us with yet another illustration of the relevance of Lamb’s analogy for the drop impact problem.

4.1 A simple boundary layer problem?

Typically, the (inviscid) slip velocity $\tilde{u} _{e}(\tilde{r},\tilde{t})=(2/{\rm\pi})\tilde{r}/\sqrt{3\tilde{t}-\tilde{r}^{2}}$ , here first introduced (3.30), and the no-slip condition at the substrate, trademark of real fluids, are reconciled through the introduction of a viscous boundary layer. According to the classic boundary layer theory (e.g. Schlichting Reference Schlichting1968), the transverse scale of this layer is $Re^{-1/2}$ , so that an appropriate inner coordinate $\tilde{Z}$ can be defined via $\tilde{z}=Re^{-1/2}\tilde{Z}$ . The most simple idea at this point is to think that the outer variable scales defined in § 2.3 imply that the nonlinear terms of the boundary layer equation are negligible when compared to unsteady and viscous terms, so that this equation would simply read:

where capitalized variables refer to boundary layer quantities. Considering that the Euler equation in the inviscid outer domain reduces to $\partial \tilde{u} _{e}/\partial \tilde{t}=-(\partial \tilde{p}/\partial \tilde{r})$ , the boundary layer equation can be recast as the following diffusion equation for the defect velocity:

The corresponding solution can then be expressed as a convolution between the forcing term and the Green function of the heat equation:

Unfortunately, this solution leads to a paradoxical cancelling of shear stresses at the wall. We conjecture that this unreasonable result stems from the fact that the sharp longitudinal variations associated with the contact-line motion have here been disregarded. Specifically nonlinear terms do balance unsteady terms, at least near the contact-line location $\tilde{r}=\sqrt{3\tilde{t}}$ . As a result, the boundary layer actually grows from this moving point both in space and time. While a comprehensive analysis of this problem demands a careful balance of each term, likely resulting in a nonlinear boundary layer problem, beyond the scope of the present study, we nonetheless propose in the following an approximation based on an analogy with boundary layers developing behind shockwaves.

Figure 18.

(a) Sketch of the contact line during its motion and of the growing boundary layer in its trail, analogous to that developing behind a shockwave. (b) Shockwave-induced boundary layer, reproduced from the German edition of Schlichting textbook (Schlichting Reference Schlichting1968). Notations are from Schlichting, with a correspondence between $x$ and $\tilde{r}$ . Note that in the shockwave case, $U_{\infty }$ and $U_{s}$ are both constant.

4.2 Approximation of the drop impact boundary layer via an analogy with shock-induced boundary layers

We now depict qualitatively the inner viscous structure of the velocity field by using a simple analogy. First remembering the tank-treading movement in the vicinity of the contact line observed and discussed in § 2.2.2, we point out the violent change in radial velocity when passing through the contact line. In other words, the contact line embodies a neat discontinuity where the slip velocity sees its value suddenly change from 0 to $\tilde{u} _{e}$ . Building on this observation, we consider in the following the contact line as a kind of shockwave sweeping the substrate, and seeding a boundary layer in its trail (see figure 18). This problem is classic in compressible flows and was solved by Mirels (Reference Mirels1955) in the context of a shock tube (see Schlichting Reference Schlichting1968, for more details). In this study, a fluid initially at rest is swept by a shockwave travelling at celerity $U_{s}$ in the direction $x$ and instantly acquires an impulse of velocity $U_{\infty }$ in the process. Behind the normal shockwave is left a growing viscous boundary layer.

The ansatz for Mirel’s solution is to introduce ${\it\eta}_{m}=z/\sqrt{t-x/U_{s}}$ as the self-similar variable. This variable not only takes into account time variations but also longitudinal effects from the shock backwards in $x$ . Disregarding any pressure gradient but considering both unsteady and nonlinear effects, the momentum equation may be rewritten in terms of ${\it\eta}_{m}$ and of the velocity $U_{\infty }\,f^{\prime }({\it\eta}_{m})$ :

Note that compressible effects have here been absorbed via an appropriate Lees-Dorodnitsyn’s transformation (see Stewartson Reference Stewartson1964). Two limiting cases clearly emerge from the picture. For large $U_{\infty }/U_{s}$ (and after a rescaling and a change of sign due to the choice of origin), the velocity profile tends to a Blasius profile. Conversely, for small values of the velocity ratio, the velocity rather adopts an error function profile. Note that profiles corresponding to intermediate values of this ratio can be found in Schlichting’s textbook.

From this sound result we may by analogy transpose this approach to the drop impact problem (see figure 18). Obviously the outer solution for the drop impact problem is more complex as neither $U_{s}$ nor $U_{\infty }$ are constant. The core idea consists in drawing a parallel between the shock (at position $U_{s}t$ ) and the contact line (at position $\sqrt{3\tilde{t}}$ ) on the one hand, and between the steady slip velocity $U_{\infty }$ and $\tilde{u} _{e}(\tilde{r},\tilde{t})$ on the other hand. Following this simple analogy the longitudinal velocity is approximated with:

where $f^{\prime }$ is solution of an equation which is analogous to (4.4). The so-called composite solution (Van Dyke Reference Van Dyke1975), which is an expansion valid in the ideal fluid and in the boundary layer, then follows:

In practice we approximated $f^{\prime }$ with the erf function. Figure 19 proposes a comparison between the numerical velocity profiles extracted from Gerris  computations and this approximation, which proves to provide a fairly good description for the flow. We also tested this prediction for different values of the Reynolds number in the range 250 to 5000. The actual velocity profiles extracted from the simulations are confronted figure 20 with the theoretical prediction. Interestingly, we note that this asymptotic result remains accurate even for the lowest values of the Reynolds number. As a side note, we remark that replacing the error function with Blasius profile yields a slightly more marked deviation between theory and numerical results. That said, we chose not to tune the velocity ratio appearing in (4.4) as (i) this is too speculative and (ii) such adjustment is certainly beyond the limits of our analogy.

Figure 19.

Inner boundary layer radial velocity profiles at different locations ${\it\xi}:0.125$ , 0.25, 0.5, 1 and 1.5. Blue solid lines correspond to numerical solutions obtained with Gerris  at $\bar{t}=5\times 10^{-3},10^{-2},5\times 10^{-2}$ and $10^{-1}$ for $Re=5000$ and $We=250$ and represented in the self-similar space. Note that velocities are rescaled by their maximum value. The red dashed lines stand for the theoretical composite solution (4.6) blending the self-similar viscous boundary layer solution with the self-similar Wagner inviscid solution for impact. The composite solution is also rescaled by the edge velocity $\tilde{u} _{e}({\it\xi})$ given by (3.30).

It is interesting to note that for ${\it\xi}$ smaller than $\sqrt{3}$ , Mirel’s self-similar variable ${\it\eta}_{m}$ tends to $Re^{1/2}{\it\eta}=\tilde{Z}/\sqrt{\tilde{t}}$ , so that the vertical structure for solution (4.5) now simply involves $\text{erf}(Re^{1/2}{\it\eta}/2)$ . Actually, this solution is merely the purely diffusive solution of (4.2) for constant forcing ( $\tilde{u} _{e}$ constant in time).

From the previous results we may extract several quantities, such as the displacement thickness or the locus of isovelocities. The displacement thickness ${\it\delta}_{1}$ can readily be estimated with $f^{\prime }=\text{erf}$ as:

Similarly, isolines for the velocity can be extracted both for Gerris  computations and for boundary layer theory. Figure 21 provides with a qualitative comparison between theory and numerical results, and it can be remarked that the overall prediction is more than just qualitative.

Though the velocity ratio $(2r/{\rm\pi}/\sqrt{3\tilde{t}-\tilde{r}^{2}})/(\sqrt{3\tilde{t}}/2/\tilde{t})$ (counterpart of $U_{\infty }/U_{s}$ in (4.4)) is infinite near the shock, we note that the agreement between numerical and theoretical solutions is actually surprisingly good. We conjecture that whenever this ratio decreases to a value lower than one, i.e. near the centre and for large times where this ratio behaves as $(4\tilde{r})/(3{\rm\pi})$ so tends to zero, the error function approximation emerges as the solution of (4.4). We remark that an in-depth analysis of these phenomena demands a more involved description for the boundary layer. The Interactive Boundary Layer theory (see e.g. Lagrée Reference Lagrée2010), may be a candidate. Another promising approach might be the careful boundary layer analysis of Elliott & Smith (Reference Elliott and Smith2015) with intricate multiple structures. Nonetheless, a deeper analysis of the Wagner region (Cointe Reference Cointe1989; Oliver Reference Oliver2002; Korobkin Reference Korobkin2007; Oliver Reference Oliver2007) has to be performed.

Figure 20.

Same as figure 19, but for different Reynolds number $Re:250$ , 500, 1250, 2500 and 5000 and at the fixed location in self-similar space ${\it\xi}=1.5$ .

Figure 21.

Left: isolines for the radial velocity $\tilde{u} _{r}$ extracted from the numerical simulations. Right: theoretical isolines for the radial velocity given by the composite expansion $\tilde{u} _{r}^{comp}$ . Note that the transverse scale has here been stretched to visualize the boundary layer.

Figure 22.

(a) Numerical and theoretical shear stress distribution underneath the drop, represented in the self-similar space (the numerical data are taken at times $\bar{t}=5\times 10^{-3}$ , $10^{-2}$ , $5\times 10^{-2}$ and $10^{-1}$ ). (b) Same data represented as a function of the distance from the contact line (log plot). This representation reveals a cutoff distance ${\it\Delta}$ from which the $1/x$ singularity is screened. Importantly, the numerical mesh size has been chosen to be small enough ( ${\rm\Delta}x=5\times 10^{-4}$ ) to ensure the resolution of the fine-scale motion in the vicinity of the contact line.

4.3 Estimation of the shear stress and the total drag

With this boundary layer solution, we are now in a position to provide an estimation of the wall shear stress $\tilde{{\it\tau}}=(1/Re)\partial \tilde{U} _{r}/\partial \tilde{Z}|_{\tilde{Z}=0}$ , i.e. the viscous component of the stress which has been disregarded so far. And indeed this quantity is of paramount importance as far as raindrop-induced erosion of erodible beds is concerned (Ellison Reference Ellison1945; Rein Reference Rein1993; Lagrée Reference Lagrée2003; Leguédois et al. Reference Leguédois, Planchon, Legout and Le Bissonnais2005). Upon using (4.5) (with the same erf approximation for function $f^{\prime }$ as before), we readily obtain:

This theoretical prediction is confronted figure 22 with numerical profiles for the shear stress extracted from Gerris  computations, and is shown to nicely agree with observations. From this local distribution for the stress we may infer the total drag induced by a drop impact, by integration over the wet area:

Unfortunately this integral diverges due to the $1/x$ singularity developing in the near-contact-line region, and is visible in figure 22 on the left. Such singularities are usually a signature of additional physics in the diverging region which have not taken into account in the model. And indeed, figure 22 on the right reveals that the calculated shear stress significantly deviates from the theoretical prediction at some small distance ${\it\Delta}$ from the contact-line position to reach a maximum value. Now integrating the local shear stress up to $\tilde{r}=\sqrt{3\tilde{t}}-{\it\Delta}$ , where ${\it\Delta}$ is this small cutoff length, we can provide an estimation for the drag at leading order in $\log ({\it\Delta})$ :

Upon noting that this quantity can be dimensionalised with ${\it\rho}U^{2}R^{2}$ , the expression for the total drag in dimensional variables follows:

Noticeably, the departure from the theoretical prediction pinpointed out in figure 22 on the right seems to occur at a precise location in self-similar variables, therefore suggesting a $\sqrt{\tilde{t}}$ time dependence for ${\it\Delta}$ . From the numerical computations the value of ${\it\Delta}/\sqrt{\tilde{t}}$ can be estimated to be around $0.03$ . Note that this is obviously a crude estimation, which nonetheless allows us to propose the following estimate for the impact-induced drag:

To further refine this prediction, the true nature of the cutoff length ${\it\Delta}$ needs to be clearly identified. Several candidates for governing this quantity naturally emerge, with for example the viscous $1/Re$ regularisation length in the vicinity of the contact-line region or the inertial matching with the Wagner inner layer of typical size $(d(t)/R)^{2}$ . To probe the nature of the cutoff length ${\it\Delta}$ we performed extra simulations for Reynolds number varying over more than a decade. Indeed, should ${\it\Delta}$ arise from viscous effects it should vary appreciably with the Reynolds number. However an analysis of these various simulations revealed that ${\it\Delta}$ hardly changed with varying viscosities, as shown in table 2. These preliminary results advocate for an inviscid nature of the cutoff length which could be revealed with the next order in the ideal fluid expansion. This requires further investigation.

Table 2.

Cutoff length ${\it\Delta}$ (illustrated figure 22 b) estimated for different Reynolds numbers $Re:250$ , 500, 1250, 2500 and 5000. Although $Re$ varies over more than a decade, the value of ${\it\Delta}/\sqrt{\tilde{t}}$ is quite insensitive to this variation. This suggests that the physical nature of this cutoff length does not rely on viscosity.

5.1 Influence of capillary phenomena

The Weber number provides us with a global measure of the ratio of available kinetic energy to surface energy. For low values of this number (with respect to unity), drops typically bounce (Richard & Quéré Reference Richard and Quéré2000) while preserving their shape or gently spread (Pasandideh-Fard et al. Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996) according to the wetting properties of the underlying substrate. Note that even in this regime of droplet deposition, fast phenomena associated with the imbalance of surface stresses can set in (Stebnovskii Reference Stebnovskii1979). When the initial kinetic energy of the drop is comparable with the surface energy (We  ${\sim}$  1), a surface wave field starts to develop on the drop, shaping it into a characteristic liquid pyramid or torus (Renardy et al. Reference Renardy, Popinet, Duchemin, Renardy, Zaleski, Josserand, Drumright-Clarke, Richard and Quéré2003). For even higher values of the Weber number, such as the inertial limit We $\gg$ 1 investigated in the present paper, we do not expect capillary phenomena to have a significant influence on a global scale, but locally surface tension can still play a dominant role. For example, the high-curvature turnaround region at the lamella root is typically a place where capillarity presumably plays an important role. But due to scale separation, this region is invisible at our level of description. Indeed, in classic impact analyses see e.g. Oliver (Reference Oliver2002), the typical extent of this intermediate Wagner region associated with highly curved interfaces is found to be $O({\it\varepsilon}^{2})$ , to be compared both with the $O({\it\varepsilon})$ size of the main impact region considered throughout this paper (see § 2.3) and with the $O({\it\varepsilon}^{3})$ thickness of the lamella. In the framework of our first-order theory, we therefore do not anticipate appreciable deviations stemming from this zone. Conversely, for a correct description of the ejected liquid sheet feeding conditions and of the pressure fall-off near the lamella root reported figure 15, an accurate representation of this matching region appears mandatory.

The contact line is another region where marked effects from capillarity are to be expected. Drop impact is characterized with fast motions near the contact line. This violent dynamic wetting phenomenon can arguably bring about issues in our description of impact. Actually Blake, Bracke & Shikhmurzaev (Reference Blake, Bracke and Shikhmurzaev1999) demonstrated that non-local hydrodynamics could play a significant role in the dynamic contact angle selection. Based on experimental data, Blake et al. further put forward the possible ‘mutual interdependence’ between the phenomena in the near contact-line region and the far-field hydrodynamics. This complex interplay was further confirmed in the context of drop impact by Šikalo et al. (Reference Šikalo, Wilhelm, Roisman, Jakirlić and Tropea2005), especially for the late receding phase. Interestingly, these authors demonstrated that the early evolution of the dynamic contact angle was quite insensitive to the experimental conditions and fairly well captured by the contact angle of a truncated sphere. This nice agreement certainly advocates for a predominance of inertial effects over capillary corrections emanating from the dynamic contact line, at least in the early stage of impact. And indeed, remembering that shortly after impact the fluid motion in the contact area is essentially vertical, it appears likely that the point of contact can be determined with mere inertial arguments. In our simulations, dynamic effects have been disregarded in the description of the contact angle, which has been set to the constant value ${\rm\pi}/2$ . The agreement between our simulations and the purely inertial theory is again an indication of the unimportance of dynamic wetting. It might further be interesting to note that the surface energy gained by wetting the solid is of the order of $1/We$ when rescaled by the initial kinetic energy. Again this heuristically rules out any leading effect from wetting in the short-term dynamics. This ratio evolves with time though, and ultimately wetting phenomena become dominant, as evidenced by the late $t^{1/10}$ spreading behaviour consistent with Tanner’s law in the experiments of Rioboo et al. (Reference Rioboo, Marengo and Tropea2002) performed with purely wetting liquids.

5.2 Influence of ambient air

For approximately a decade or so, there has been an increasing realization of the role played by surrounding air in liquid impact in general, and drop impact in particular (Josserand & Thoroddsen Reference Josserand and Thoroddsen2016). Following key experiments performed by Xu, Zhang & Nagel (Reference Xu, Zhang and Nagel2005) on air-induced splash triggering, a number of studies have focused on the events preluding liquid sheet ejection. The first significant effect of surrounding air is to impart a dimple-like deformation in the bottom most region of the drop (see experimental observations of the dimple obtained by Thoroddsen et al. Reference Thoroddsen, Etoh, Takehara, Ootsuka and Hatsuki2005 and X-ray ultra-fast imagery of the complex dynamics of this air pocket by Lee, Weon, Je & Fezzaa Reference Lee, Weon, Je and Fezzaa2012). Smith, Li & Wu (Reference Smith, Li and Wu2003) first depicted theoretically this process by coupling lubrication in the squeezed air film and potential flow inside the drop. These authors notably evidenced the presence of off-axis pressure peaks. While more recent studies raised doubt about the link between this dimple formation and splash triggering per se – that might merely be a secondary independent consequence of the presence of surrounding gas (Duchemin & Josserand Reference Duchemin and Josserand2011), this gas pocket is nonetheless formed over time scales and length scales overlapping that of the phenomenon reported in the present paper (Mani, Mandre & Brenner Reference Mani, Mandre and Brenner2010). It is therefore legitimate to question the impact of this air entrapment phenomenon on our results.

Figure 23.

(a) Time evolution of the pressure as measured under an impacting drop with air-induced dimple formation (i.e. bubble entrapment) taken into account. The red trace monitors the pressure at the origin. The physical parameters of this Gerris  simulation are $Re=5000$ and $We=250$ . The superimposed black dashed line corresponds to the theoretical solution $\bar{p}(0,0,\bar{t})=(\sqrt{3}/{\rm\pi})(\bar{t}-\bar{t}_{impact})^{-1/2}$ delayed from $\bar{t}_{impact}$ , where $\bar{t}_{impact}$ is the real impact time (in the numerical simulations, $\bar{t}_{impact}$ corresponds to the time at which liquid and solid are just one grid cell away). (b) Close-up of the bottommost point of the drop in the numerical simulation (at $\bar{t}=1.54\times 10^{-1}$ ). The position of the interface is materialised with a blue line. The colour map illustrates the distribution of the pressure field within the drop and in the gas layer. Noteworthy enough the isopressure lines seamlessly cross the interface, revealing the transparency of the dimple to pressure. Note that for the sake of clarity, the vertical scale has here been magnified by a factor 22.

In order to investigate these effects, we performed a simulation of a liquid drop approaching a solid substrate, deforming as a result of lubrication pressure rise in the film, and finally impacting the substrate. Figure 23 represents the time evolution of the pressure exerted on the support at the axis. Considering that air delays the moment of impact (Mani et al. Reference Mani, Mandre and Brenner2010), we introduce a time shift $t_{impact}$ corresponding to the moment where the drop and the solid are only a grid cell apart. Interestingly, our results reveal that the pressure at the origin (measuring now the entrapped bubble pressure) is fairly well captured by relation (3.37) after replacing $\tilde{t}$ with the true time from impact $\tilde{t}-\tilde{t}_{impact}$ , that is:

This agreement between our prediction and a simulation incorporating air entrapment effects not only validates and extends our results beyond the initial scope of Wagner impact theory (disregarding air effects), but also suggests that the results of the present manuscript correspond to the far-field behaviour of an impacting drop in presence of surrounding gas. This observation outlines the appealing prospect of describing both the dimple geometry and associated dynamical fields by analytical means.

5.3 Main results

In this paper, the short-term dynamics of a drop impacting a rigid substrate has been elucidated. A self-similar solution for the impact-induced flow has in particular been unravelled and matched to a self-similar viscous boundary layer. This solution has been intensively validated with numerical Gerris  computations, and this constant cross-testing between asymptotic theory and multiphase adaptive flow simulations is one of the key features of the present approach. In the course of this investigation, several important results have been substantiated. These results allow both for a simple yet accurate qualitative depiction of drop impact along with an in-depth quantitative understanding of this phenomenon. These key results are summarised in the following:

1. (i) A fundamental analogy between the water entry of a solid object (Wagner’s original problem) and drop impact exists.

2. (ii) During the earliest moments post impact, the contact line follows a tank-treading motion. There is in particular no contact-line sweeping motion.

3. (iii) The impact-induced flow is concentrated in the contact zone, and the far field merely corresponds to an undisturbed rigid-body motion reducing to a global free flight at velocity $U$ . There is no global or large-scale drop deformation during impact.

4. (iv) The position of the contact line is given by the simple relation $d(t)=\sqrt{3RUt}$ . Though simple, this locus does not correspond to the cut radius of a truncated sphere.

5. (v) The wet footprint extent of the drop dictates the size of the impact-induced perturbed flow.

6. (vi) There is a consistent analogy between the impact-induced flow within the drop and the flow induced by a flat rising expanding disk (Lamb’s analogy).

7. (vii) The impact pressure is to be associated with the unsteady Bernoulli contribution $-\partial _{t}{\it\phi}$ . It cannot be inferred from usual inertial steady contribution $-{\it\rho}U^{2}$ .

8. (viii) As a corollary to the previous point, the impact pressure is extremal at the contact line. It is not maximal at the stagnation point.

9. (ix) A full three-dimensional self-similar solution for the impact-induced flow of an inviscid drop exists and matches quantitatively realistic numerical data on drop impact.

10. (x) Analytical solutions for this flow have been presented in integral forms (with some explicit closed-form expressions along some particular locations), see table 3 (in dimensional form).

11. (xi) An original inviscid stagnation point structure with an unexpected $r/\sqrt{t}$ slip velocity develops in the vicinity of the origin. The velocity field structure markedly differs from the classic $r/t$ prediction occurring for later times.

12. (xii) An approximate self-similar solution for the viscous boundary layer seamlessly matches with the inviscid impact flow (analogy with Mirels shockwave problem).

13. (xiii) Self-similar variables have the same structure $z/\sqrt{t}$ both in the outer region and in the boundary layer.

14. (xiv) From the knowledge of the distribution of the dynamical fields across the wet area, the expressions for the normal and tangential total force on the substrate are provided.

15. (xv) The asymptotic solution is found to be numerically valid over several decades in time up to $t=0.5R/U$ . This solution was found to be insensitive to air-induced dimple formation.

16. (xvi) For times of order one, the present results remain at least qualitative.

Table 3.

Summary of the main results of the paper in dimensioned form. The top part of the table refers to ideal fluid results (left: closed-form results along the axis of symmetry, right: along the substrate). The left middle part sums up inviscid stagnation point results, and the right middle part summarises the viscous boundary layer results. Observables such as the net normal force $F(t)$ and tangential force $D(t)$ are also given in the bottom part of the table.

5.4 Perspectives

The results obtained in the present manuscript offer several appealing prospects. On the role of surrounding air first, the last results of § 5.2 support the idea that the dimple geometry and characteristics could be derived analytically, with a far-field corresponding to the here presented flow. The question of the role of air in suppressing the pressure divergence (Josserand, Ray & Zaleski Reference Josserand, Ray and Zaleski2015), or in altering the shear stress distribution, is also a central point for a correct description of a single drop impact action. Similarly, the question of the scope of the present results for impact on a liquid film or on a soft/erodible substrate is also of interest. Indeed, the analytical and numerical toolboxes developed here might well be transposed to other rheologies (such as Bingham, see Staron et al. (Reference Staron, Lagrée, Ray and Popinet2013) or granular media, (Lagrée, Staron & Popinet Reference Lagrée, Staron and Popinet2011)). While the inverse-square-root singularity of the pressure is integrable and yields a finite normal force, the sharper singularity observed in the viscous shear stress distribution leads to a divergence for the total drag. From the simulations, it appears that the singularity is screened over a length scale ${\it\Delta}$ . As discussed in § 4 this cutoff length proves to be quite insensitive to variations of the Reynolds number. This result suggests that the nature of ${\it\Delta}$ would be purely inertial. A second-order correction of the ideal fluid solution is therefore probably necessary to explain this regularisation.

While the rudimentary description for the viscous boundary layer proposed in the manuscript certainly necessitates a refined analysis, the question of inertial effects at the Wagner region scale is also worth studying. Not only an in-depth investigation of such effects might provide an explanation for the regularisation of the shear stress, but it shall shed light over the onset of lamella formation, which still conceals mysteries. Finally the process responsible for the loss of self-similarity observed at intermediate times is still uncertain: confinement effects arising when the impact-flow length scale overlaps with the drop extent, eventual deceleration in the far-field region or contact-line geometrical departures from the square-root law are all a priori legitimate to explain the final pressure fall-off, and certainly needs further investigation.

5.5 To conclude

Within the numerous limits carefully drawn along this paper, a consistent asymptotic description of the dynamics and geometry of a drop impacting a solid surface has been proposed. The results may simply be summarised through three analogies: Wagner water entry (drop impact being the dual of this problem), Lamb’s disk winding flow (that accurately represents the flow induced with the impact) and Mirels shockwave-induced boundary layer (remarkably capturing the boundary layer developing in the contact-line’s trail). The original strategy developed throughout this paper has been to validate these three analogies through a constant confrontation between numerical simulations and asymptotic analysis. Our study revealed that very powerful state-of-the-art adaptive codes now allow us to probe all the dynamic features of realistic violent events such as drop impact, but in the meantime, it also emphasised how powerful and useful asymptotic analysis is in providing an in-depth understanding of such phenomena and in uncloaking the raw data delivered by the code. Finally, our study brought to light some interesting features and observables (such as the particular stagnation point structure, pressure distribution, contact-line motion, viscous total drag force) never observed to date in simulations or experiments. This certainly arouses the exciting prospect of their unveiling in future experimental studies.