Pressure generated at the instant of impact between a liquid droplet and solid surface

The prime objective of this study is to answer the question: How large is the pressure developed at the instant of a spherical liquid droplet impact on a solid surface? Engel first proposed that the maximum pressure rise generated by a spherical liquid droplet impact on a solid surface is different from the one-dimensional water-hammer pressure by a spherical shape factor (Engel 1955 J. Res. Natl Bur. Stand. 55(5), 281–298). Many researchers have since proposed various factors to accurately predict the maximum pressure rise. We numerically found that the maximum pressure rise can be predicted by the combination of water-hammer theory and the shock relation; then, we analytically extended Engel’s elastic impact model, by realizing that the progression speed of the contact between the gas–liquid interface and the solid surface is much faster than the compression wavefront propagation speed at the instant of the impact. We successfully correct Engel’s theory so that it can accurately provide the maximum pressure rise at the instant of impact between a spherical liquid droplet and solid surface, that is, no shape factor appears in the theory.


Introduction
An understanding of liquid droplet impact onto a rigid solid surface is needed in a number of technological situations such as cleaning of surfaces, spray coating, spray cooling and ink-jet printing. Important practical situations are the erosion of blades in a steam turbine, in the wet steam erosion of forward-facing components on an aircraft, and erosion of hydraulic machinery including ship propellers [1][2][3][4][5]. When water drops impact a & 2018 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
surface above a threshold velocity, they can eventually lead to erosion [6]. Liquid droplet impact is also one of the mechanisms leading to wall thinning, which causes rupture of the piping and unplanned outages of pressurized water reactors and boiled water reactors [7]. In order to determine the threshold velocity for the liquid droplet impact damage, a detailed knowledge of the liquid droplet impact process is required [4,8]. Obreschkow et al. [9] summarized the most important mechanisms of erosion that are known to occur during the impact of a liquid drop. A rigorous evaluation of the impact pressure generated on the solid surface at the instant of impact is essential for studying the very early stage of liquid droplet impact.
For the accurate prediction of the maximum pressure generated at the instant of impact, a large number of models have been proposed. Engel [6] estimated the maximum pressure rise DP E from the atmosphere developed after a spherical liquid drop impact on a solid surface as where V is the impact velocity, and r 0 and c 0 are the density and speed of sound, respectively, in the undisturbed liquid before the impact. The factor of 1 2 is due to the 'so-called' spherical shape of the drop. It was found that the coefficient a E is 0.4 using the Schlieren patterns produced by the collision of a water droplet with a glass plate. Later, Engel [10] proposed a general form of a E . The factor proposed by Engel became a controversial subject in droplet impact thereafter.
Jenkins & Booker [1] observed the impingement of water droplets, nominally of 2 mm diameter and roughly of spherical shape on a surface, moving at speeds ranging from 91 to 114 m s 21 . They found that their result agrees with the so-called classical water-hammer pressure rise in water DP w , which is based on the relation DP w ¼ r 0 c 0 V: (1:2) Note that these studies [1,6,10] did not directly measure the pressure; instead, they evaluated the pressure using the experimentally obtained velocity. Rochester [11] and Rochester & Brunton [12] observed the impact pressure of a 5.0 mm diameter cylindrical water droplet when it was struck by a bullet at 100 m s 21 . The pressure rise at instant of the impact is approximately 0.7r 0 c 0 V. Further, Rochester determined that the pressure is reduced at the centre of impact because the surface or the bullet deforms slightly during impact. The implication of this is that the pressure rises in the very early stage of impact against a solid surface are approximately 1.4 times those measured in the experiments. Therefore, he concluded that in an impact against a solid surface, these pressures become r 0 c 0 V. Bourne [13] evaluated the impact response of polymethylmethacrylate (PMMA) by measuring pressure in the region beneath the impacting liquid droplet with a radius of 2 mm by using polyvinylidene fluoride gauges at an impact velocity of 600 m s 21 . He found that the initial impact pressure was approximately 0.8 GPa, which was just below the Hugoniot elastic limit for PMMA. Heymann [14] introduced a two-dimensional model that represents a thin parallel slice taken vertically through the impacting drop. He inferred that the pressure rises are virtually uniform and equal to DP s ¼ r 0 sV, ( 1 :3) at the first instant of contact for the impact of a liquid droplet, where s in equation (1.3) is the compression wavefront speed [15] with a pressure that increases by DP s with respect to the one in the undisturbed liquid pressure. His inference that the pressure rise at the instant of the impact for a liquid drop can be predicted by equation (1.3) is far from obvious. It is this inference that we primarily discuss in detail throughout this paper. Lesser [16] also claimed that the impact pressure at the instant of the impact on the centre can be estimated from the shock relation. Later, Field [17] explained that for the impact of a liquid droplet on a solid surface, the pressure rise on the central axis at the instant of impact is estimated by equation (1.3). Korobkin [18,19] generalized Lesser's result to the whole region behind the shock front in his analysis on the impact by a flat plate on a circular liquid drop. The numerical analysis conducted by Hwang & Hammitt [20] explored a new aspect of the study of liquid droplet impact. They investigated the impact process of cylindrical, spherical and conical water droplets on a solid plane surface using numerical methods. They found that the impact pressure rise reaches a peak of 0.7r 0 c 0 V. Haller [21] and Haller et al. [22] also numerically investigated the fluid dynamics of high speed (500 m s 21 ) small size (200 mm in diameter) droplet impact on a solid surface using a high-resolution axisymmetric solver for the Euler equations. Their numerical results support Lesser's assumption.
These studies show that extensive efforts have been made toward the understanding of the pressure generated at droplet impact; however, the maximum pressure rise at the impact on the central axis has yet to be indisputably determined. Recent various numerical analyses indeed provide a variety of results. Sanada et al. [23] carried out a numerical simulation of an axisymmetric spherical liquid droplet impact on a solid surface, using the ghost fluid method [24]. They found that the maximum pressure rise of the point closest to the solid surface can be well predicted by equation (1.2). Xiong et al. [7] found that the maximum average impact pressure given by their simulation agrees well with the Heymann correlation pressures [14] using the moving particle semi-implicit method. Sanada et al. [25] and Kondo & Ando [26] numerically studied axisymmetric and cylindrical droplet impact on a solid surface, respectively, using the shock-interface capturing scheme [27]. Their results support Engel [6], i.e. equation (1.1). Hsu et al. [28] also supported equation (1.1) as an empirical formulation that can predict the maximum pressure rise for the impingement of a spherical liquid drop.
Note that pressure impulse theory, which Cooker & Peregrine [29] and Philippi et al. [30] used, and most of classical work on the fluid-solid impact, including the seminal paper by Wagner [31], Howison et al. [32] and Scolan & Korobkin [33], were carried out under the assumption of an ideal and incompressible liquid. However, the compressibility of the liquid plays the most significant role in the prediction of the maximum pressure rise. The most general methods to investigate the pressure generated at the instant of impact between a liquid droplet and solid surface are those that appropriately take the liquid compressibility into consideration.
The prime objective of this study is to answer the question: How large is the pressure developed at the instant of a liquid droplet impact on a solid surface? by shedding light on the shape factor that is still controversial. To answer this question, we consider the impact of a spherical liquid droplet on a solid surface under the assumption that liquid droplet keeps its spherical shape at the instant of the impact, although it is well known that a liquid droplet impacting on a solid surface deforms before the liquid droplet makes contact with the surface [34 -36] or entraps a small air bubble under its centre under atmospheric conditions [37,38]. The reason is that we primarily discuss the role of the spherical shape factor introduced by Engel [6], who also used the assumption that the liquid droplet keeps its spherical shape at the instant of the impact, in the central impact pressure rise. We numerically estimate the central impact pressure (defined here as the pressure developed at the instant of the impact on the contact point on the axis of symmetry) by solving two-dimensional axisymmetric compressible Euler equations. We then extend Engel's elastic impact model [6] taking the limit of the time from impact for the pressure to reach its maximum to 0, by realizing that the progression speed of the contact between the gas -liquid interface and the solid surface is much faster than the compression wavefront propagation speed at the instant of the impact. We correct Engel's elastic impact model to find that the factor a E /2 appearing in equation (1.1) converges to 1 at the instant of the impact.

Previous models for one-dimensional analyses of impact pressure rise
As stated in the previous section, the water-hammer theory has been often used in the analysis of liquid droplet impact pressure rise. We also contrast the liquid droplet impact pressure with the water-hammer pressure rise in later sections; hence, in this section, we briefly review a couple of previous one-dimensional models used in the estimation of impact pressure.
A sudden halt of flow in a pipe is considered to be equivalent to a liquid droplet impact on a solid surface. Cook [39] was one of the first to realize that the pressure generated on an element of a surface at its first encounter with water moving at a finite velocity would be given by the water-hammer pressure rise. Gardner [40] and de Haller [41] extended this analysis by accounting for the compressibility of the solid.
Suppose a uniform flow of a liquid moving with a speed of V in a pipe to the right in figure 1a. When the pipe is suddenly closed at the valve, a high pressure builds and propagates upstream (to the left in figure 1a) with a speed of (s 2 V ), where s is the compression wavefront speed in a quiescent liquid. Figure 1a can be redrawn as figure 1b using the coordinate system moving with a speed of (s 2 V ).
Heymann [14], on the other hand, used the so-called shock relation to estimate the pressure rise developed at a liquid droplet on a solid surface. He considered that a moving solid plate with a velocity V collides with a stationary liquid droplet. Then, a shock front is generated with a speed of s moving into a quiescent medium, as shown in figure 2a. The liquid velocity induced behind the shock front is the same as the plate velocity V. Figure 2a can be redrawn as figure 2b using the coordinate system moving with a velocity of s. Note that both figures 1b and 2b are identical; hence, the conservation equations of mass, momentum and energy for both cases should be identical and are, respectively, written as (2:1) and Equations (2.1) and (2.2) lead to the following expression of pressure rise DP s : The classical water-hammer pressure rise (1.2) can be obtained by simplifying equation (1.3) using the commonly used assumption that the compression wavefront speed can be replaced with c 0 , the speed of sound in an undisturbed liquid: Further review on this topic can be found in Brunton & Rochester [42] and Lesser & Field [43]. We now consider s as a variable instead of a constant to more accurately predict the maximum pressure rise. We need an additional equation to close the equation set. We examine two different equations for properly evaluating s. The first equation was empirically developed by Heymann [15] and is the approximate shock velocity -particle velocity relationship: where is the 'impact Mach number' and k is a dimensionless constant. Heymann [14] obtained the impact pressure rise using the approximate shock velocity -particle velocity relationship DP H : r 0 , p 0 r 0 , p 0 r 1 , p 1 r 1 , p 1 Figure 1. Water-hammer pressure rise in (a) a stationary coordinate system, and (b) a moving coordinate system with a speed of (s 2 V ), where s is the compression wavefront speed in a quiescent liquid and r 0 , p 0 , and V are density, pressure and speed of the uniform flow of the undisturbed liquid, respectively. A high pressure builds and propagates upstream (to the left in (a)) with a speed of (s 2 V ). r 1 and p 1 are the density and pressure of the liquid that the pressure wave has reached, respectively. Heymann [15] found that the value k ¼ 2 fits the data for water up to about M 0 ¼ 1.2. Substituting the values c 0 ¼ 1497 m s 21 and V ¼ 304.8 m s 21 used by Engel [44] into equation (2.7), an impact pressure of 1.41r 0 c 0 V can be obtained with the assumption of a rigid solid impact plate. The second equation is the stiffened-gas equation of state proposed by Harlow & Amsden [45]: where e is the internal energy per unit mass, g and P are constants (g ¼ 1.4 and P ¼ 0 for air; g ¼ 4.4 and P ¼ 6.0 Â 10 8 for water). Equation (2.8) is widely used for computational analyses to investigate compressible gas -liquid two-phase flow [46][47][48]. Saurel & Abgrall [46] reported that it is possible to describe compressible liquid under high pressure with reasonable accuracy using the stiffened-gas equation of state, and that each phase of the flows involving various materials may be described by an equation of state of this type. We also implemented equation (2.8) in our numerical analysis, as explained in the next section. Equations (2.1), (2.2) and (2.3) with the stiffened-gas equation of state (2.8) give the speed of sound for the stiffened-gas [49]: (2:9) Using equation (2.9), we obtain the impact pressure rise with the use of stiffened-gas equation of state DP sg : (2:10) We found in this section that the one-dimensional liquid impact pressure rise DP s on a solid surface can be formally written regardless of the choice of the model. All these models were derived from the same conservation equations; however, different equations for the compression wavefront speeds were used. We refer to these pressure rises, DP w : equation (1.2), DP H : equation (2.7) and DP sg : equation (2.10), as the classical water-hammer impact pressure rise, Heymann impact pressure rise and stiffened-gas impact pressure rise, respectively, in the following discussion.
3. Numerical evaluation of the pressure developed immediately after the liquid droplet impact

Numerical method
We numerically investigated the liquid droplet impact pressure rise on a solid surface. We analysed the pressure generation at droplet impact by solving the two-dimensional axisymmetric compressible Euler equations [22,50]. To close the Euler equations, we implemented the stiffened-gas equation of state (2.8) in our numerical analysis; hence, we compare the pressure rise due to the liquid droplet impact obtained by numerical analysis with the stiffened-gas pressure rise DP sg (2.10). We solved the flow field using the level-set method [51,52] combined with the ghost fluid method [24,53,54]. We used a finite difference method for discretization of the system. The third-order TVD Runge -Kutta scheme was employed to march the equations forward in time, and the third-order ENO-LLF scheme was employed to calculate the convection term [55,56]. The third-order TVD Runge -Kutta scheme and fifth order WENO scheme [57] was employed to advance the level-set function c, where sets of c ¼ 0, c , 0 and c . 0 represent the gas -liquid interface, one fluid and the other fluid, respectively. We calculated the physical properties only in the liquid phase with constant gas pressure. Haller et al. [22] carried out a numerical simulation of droplet impact on a solid surface by solving flow fields of both liquid and gas phases. Their results succeeded to reproduce the theoretical results by Lesser [16] who did not consider the dynamics of the gas phase. Hence, we can neglect the contribution of gas that was also neglected by the theoretical models for the purpose of this study. We introduced the assumption that the portion of the droplet that has not collided remains spherical. This should hold in the computation partly because we mainly investigate the pressure development rsos.royalsocietypublishing.org R. Soc. open sci. 5: 181101 immediately after the droplet impact, where the deformation may be insignificant, and partly because we discuss the legitimacy of the theoretical models by comparing them with the numerical results. Therefore, the contour line of c ¼ 0 is enforced to maintain a circle whose centre moves with impact velocity V in z . 0.
Impact velocity V was set to 100 m s 21 because this order of magnitude of impact velocity can be typically observed in cleaning technologies for semiconductor device processes [23,50,58,59], in single water droplet impact experiments [11,60,61] and in numerical simulations of liquid droplet impact [9,26]. Note that the difference between c 0 and s is hardly noticeable if the value of V is much smaller, and the stiffened-gas equation of state (2.8) is not applicable if the value of V is much larger.
We calculated the pressure of the fluid in the cell contacting the solid surface. We then identified this pressure using the pressure on the solid surface. We refer to these cells adjacent to the solid surface as the surface cells. We define Cell-1 to be the left-most surface cell contacting with the z-axis, as shown in figure 3c. We consider the gas -liquid interface to impact on the solid surface when the fluid in the surface cell that is adjacent to the point turns from gas to liquid, i.e. when the interface crosses the centre of the surface cell. We also consider the droplet to impact on the solid wall when the gas -liquid interface impacts on the solid surface in Cell-1. The central pressure is defined as the pressure in Cell-1. We discuss the temporal change of the central pressure after the droplet impacts on the solid surface. We also investigated the pressures in other surface cells, particularly Cell-2, Cell-3, Cell-4 and Cell-5, as shown in figure 3c.

Estimation of the central pressure
We now examine the maximum central pressure rise DP max at the instant of impact. Figure 4 shows the temporal changes of the pressure in Cell-1 calculated with various cell sizes, where the nondimensionalized time t Ã was defined by t Ã ¼ Vt=R. We use the following expression for the maximum central pressure rise: where a is defined as the ratio of the calculated maximum central pressure rise to the stiffened-gas impact pressure rise, and s is defined by equation (2.9). Throughout this paper, we use the constant speed of compression wavefront s ¼ 1760 m s 21 , which is obtained by substituting V ¼ 100 m s 21 and c 0 ¼ 1625 m s 21 in equation (2.9). It is a remarkable finding that a depends on Dr Ã ; a reaches 0.994 in the case of Dr Ã ¼ 2 À15 , while it is 0.455 in the case of Dr Ã ¼ 2 À4 , as shown in figure 4. Surprisingly, the variation in the value of a is more than double. We evaluate a for each Dr Ã . Figure 5a shows that a monotonically converges to 1 when Dr Ã decreases. Therefore, we conclude that the maximum central pressure rise at the instant of impact DP max is given by Note that the results of the numerical simulation provide the maximum central pressure rise that can be expressed by equation (3.2) under very restricted conditions, where Dr Ã is sufficiently small. For instance, we should choose a Dr Ã that is less than 2 214 to obtain an a greater than 0.99. We found that significantly small calculation cells are required to resolve the magnitude of the pressure at the instant of impact. We also found in figure 4 that t e , the elapsed time for the pressure to reach its maximum from the instant of impact, decreases as Dr Ã decreases. Non-dimensionalized elapsed time t Ã e ( ¼ t e V=R) is plotted in figure 5b. Note that the impact pressure attains the maximum pressure because of the elastic compression of the fluid in the cell; hence, t Ã e decreases as Dr Ã decreases. Therefore, in the limit taken as Dr Ã approaches 0, the elapsed time to reach the maximum central pressure converges to 0.  Figure 4. Convergence of the maximum central pressure obtained by numerical analysis as the non-dimensional calculation cell size Dr Ã decreases: (a) 2 À9 Dr Ã 2 À4 and (b) 2 À15 Dr Ã 2 À10 . The calculated pressure rise was normalized by the stiffened-gas impact pressure rise DP sg defined by equation (2.10). Note that the scale of the abscissa of (b) is 25 times greater than that of (a). We found that a can converge to 1 as long as Dr Ã is sufficiently small in §3.2; however, a can reach at most 0.455 when Dr Ã ¼ 2 À4 : We now discuss the dependency of a on Dr Ã . For a large cell size (Dr Ã ¼ 2 À4 in figure 6a), the pressure in Cell-2 begins to rise only after the pressure in Cell-1 reaches the maximum; that is, Cell-2, Cell-3, Cell-4 and Cell-5 are filled with gas while the pressure in Cell-1 is increasing; then, a is 0.455. For a medium cell size (Dr Ã ¼ 2 À7 in figure 6b), the pressure in Cell-2 begins to rise when the pressure in Cell-1 is still increasing; that is, the fluid in Cell-2 turns from gas to liquid while the pressure in Cell-1 increases; then, a is much greater than 0.455 but much less than 1.0. For a small cell size (Dr Ã ¼ 2 À10 in figure 6c), the pressure in Cell-2 begins to rise immediately after the pressure in Cell-1 has begun to rise; that is, Cell-2 is almost filled with liquid while the pressure in Cell-1 is increasing; then, a is very close to 1. These results suggest that the status of Cell-2, i.e. either liquid or gas, should greatly affect the central maximum pressure. We now introduce the ratio of time f: where t Ã 1 is the non-dimensional time which it takes for the pressure in Cell-1 to rise from p 0 to P max , and t Ã 2 is the non-dimensional duration over which the fluid in Cell-2 is gas for t Ã 1 . Figure 7 shows that a monotonically increases as f decreases. We can see that the time duration for which the fluid in Cell-2 is liquid while the pressure in Cell-1 is increasing totally dominates the value of a. When the fluid in Cell-2 turns to liquid immediately after the pressure rises in Cell-1, the liquid in Cell-1 is confined to the space bounded by the liquid in Cell-2, for which the pressure is about the same as that of Cell-1; hence, the liquid in Cell-1 can be compressed primarily in the direction normal to the solid surface. This compression can be thought of as the one-dimensional liquid compression by which the pressure generated can be predicted well by equation (2.10). Therefore, a is essentially equal to 1 when f is sufficiently close to 0.
When the fluid in Cell-2 remains gas after the pressure rises in Cell-1, the liquid in Cell-1 is confined to the space bounded by the gas in Cell-2; hence, the liquid in Cell-1 is able not only to be compressed in the normal direction but also to be expanded in the direction parallel to the solid surface. In other words, the liquid in Cell-1 can flow in the direction parallel to the solid surface. As Bowden & Field [2] explained, a cannot reach 1 when the liquid in Cell-1 can flow. According to them, a can reach 1 when the liquid in Cell-1 cannot flow, that is, the flow can be assumed to be one-dimensional. Therefore, a is significantly smaller than 1 when f is close to 1.

Impact of an interface with a finite curvature
According to the numerical analyses conducted by Sanada et al. [25] and Kondo & Ando [26], the pressure rise in liquid impact on a solid wall can be well predicted by using the Heymann impact pressure rise (equation (2.7)) when the impact can be considered as a one-dimensional impact, where the radius of curvature of the impacting interface is infinite, i.e. a plane interface. However, they also found that liquid droplet impact pressure rise on the same solid wall can be significantly reduced when the radius of curvature of the impacting interface is finite: they numerically reproduced a cylindrical droplet impact on a solid wall. The latter pressure was approximately 0.2 of the former pressure. Their results apparently supported Engel [6], i.e. equation (1.1). We now examine whether the liquid impact pressure rise can be reduced by the stiffened-gas pressure rise (equation (2.10)) when the interface that impacts on a solid surface has a finite radius of curvature using the numerical analysis.
We found that the impact pressure rise can reach the one-dimensional liquid impact pressure rise when Dr Ã is sufficiently small in our numerical simulation. We must carefully examine the possibility that the interface of the liquid droplet might be plane with an infinite radius of curvature when the liquid droplet impacts on a solid surface for sufficiently small Dr Ã because the plane interface impact might be equivalent to a one-dimensional impact, which leads to the stiffened-gas pressure rise (equation (2.10)). Therefore, we need to confirm whether the liquid droplet impacts a solid surface with a finite radius of curvature.
When a gas -liquid interface impacting on a solid surface is a plane, the impacts of the interface simultaneously occur in adjacent multiple cells; hence, the impacts of the interface should not simultaneously occur in adjacent cells in figure 8 when the interface has a finite radius of curvature. Figure 9a shows the pressure development at the five surface cells in figure 8, although the pressures observed at different surface cells are identical. Figure 9b shows an enlarged view of the very early stage of droplet impact. Therefore, we conclude that we successfully reproduced the impact of a droplet which has a finite radius curvature with a solid surface.
We now consider the maximum central pressure rise. Figure 9a shows that the maximum pressure rises observed at the surface cells Cell-1 to Cell-5 shown in figure 8 are identical. We find that the factor a defined by equation (3.1) is 0.994 in figure 9. Therefore, we conclude that the maximum central pressure rise that develops when a droplet impacts a solid surface can be accurately predicted by equation (2.10), even with a finite radius of curvature interface.
To close this section, we add the following as the final remarks. It requires extremely high resolutions in both the spatial and temporal domains to obtain the numerical results that confirm that the maximum central pressure rise can be predicted by equation (2.10). We are convinced that insufficient resolutions in either the numerical or experimental studies have caused the confusion in determining the factor.

Equation for the central pressure resulting from a liquid droplet impact
We now return to the Engel pressure rise DP E : Engel stated with regard to equation (1.1) that this is the well-known water-hammer equation multiplied by factor a E /2, which comes from the spherical shape of the liquid droplet; indeed, it has long been understood that the factor of 1 2 is considered to be a consequence of the spherical shape of the drop [28]. However, our numerical analysis showed that the impact pressure rise due to the spherical liquid droplet can be predicted by equation (2.10), where no shape factor is required. In this section, we reconsider the origin of the factor of 1 2 and then discuss the legitimacy of using equation (1.1) for predicting the maximum central pressure rise.

Maximum pressure rise proposed by Engel
Engel [6] derived equation (1.1) for the maximum pressure rise DP E that develops at the time Dt after a liquid droplet impacts on a solid surface. We now briefly introduce the analysis by Engel. She considered that the liquid droplet impact consists of two sequential (not simultaneous) processes: a compression which is an instantaneous process and a subsequent displacement which takes a finite time Dt, as shown in figure 10 drawn based on fig. 14 in [6]. Engel introduced a coefficient a E , which is an index of the fraction of velocity V that is imparted to the liquid molecules on average. She assumed that the coefficient a E must be governed mainly by the extent of divergence of the compression wave as it spreads through the spherical droplet. She also assumed that the amount of divergence of the wave decreases and the value of the coefficient a E approaches unity as the impact velocity increases.
Note that the ordinates in (b -d) are greatly magnified with respect to (a To obtain the maximum pressure rise DP at time t ¼ Dt, Engel considered the total mass of liquid m that gains momentum during a finite time interval Dt, and consequently all the force that acts to produce the momentum over time interval Dt. Engel considered a separated pair of regions. First, she considered the mass m 0 of the spherical cap, which gains a momentum that causes a radial flow. She defined the l-plane to be the upper boundary of the radial flow and considered it to be an effective striking surface with which compression wavelets are initiated at the points of contact of the remaining liquid droplet. The distance between the d-and l-planes is (v 2 1)d where v ! 1. She noted that in the limit, the volume of a spherical cap is one-half the volume of a cylinder that has the same height and base area.  [6]. First, the impact of a droplet of radius R (dashed curve) with a solid plane surface forces the radius of curvature of the droplet in the vicinity of the impact to change instantaneously to R 0 (solid curve), and a compression wavefront to propagate through the liquid droplet, which induces a velocity in the z-direction in the liquid (the compression process). Second, the moving solid plane surface of velocity V travels through the liquid whose average velocity may be written as a E V in the time interval Dt to reach the d-plane. The relative velocity of the moving solid plane surface to the liquid droplet therefore becomes (1 2 a E )V; hence, the compressed cap, which is shown as the light grey region, with height d, d ¼ (1 2 a E )VDt, is displaced into a very thin cylinder of liquid that is in radial flow (the displacement process). The liquid cylinder with a base radius of r vd formed between the d-plane and the l-plane has the same volume as that of the spherical cap whose base is on the l-plane with a height of vd.
rsos.royalsocietypublishing.org R. Soc. open sci. 5: 181101 Engel then assumed that the first liquid of the droplet that encounters the moving solid plane surface during the time interval Dt is displaced to form a thin cylinder the height of which is determined so that the volume of the spherical cap below the l-plane before the displacement (the union of medium grey and light grey regions in figure 10) is equal to that of the cylinder (the union of medium grey and dark grey regions in figure 10); hence, v is equal to 2. During time Dt, which is the time needed for the moving solid plane surface with the relative velocity (1 2 a E )V to move a distance d through the droplet, the l-plane moves a distance vd. Velocity V l of the l-plane is then given by The mass of the spherical cap (the union of medium grey and light grey regions in figure 10) whose base radius is r vd , is given by where v m 0 is the volume of the spherical cap. Engel mentioned that although the mass of liquid m 0 moves horizontally after time Dt, it was given as a velocity in the collision direction z during this time interval because it was traversed by the compression wave. Engel next considered the region (with mass m 00 ) of the liquid droplet that has been traversed by the compression wave during the time interval Dt, which is shown as a hatched region in figure 10; then, she assumed that the average velocity of the liquid in this region may be written as a E V. Suppose that the radius of the circular intersection between the l-plane and the surface of the spherical cap is r. The compression wavelet can be generated on this circular intersection with 0 z vd when the l-plane hits the surface of the spherical cap as shown in figure 10. The l-plane is relatively stationary with respect to the droplet in the compression process because it keeps contact with a point at the first instant of impact, while it moves a distance vd in the displacement process over time Dt. The compression wave consists of each compression wavelet, which travels with the velocity s. Engel assumed that the compression wavelet travels only in the z-direction. Recall that z ¼ vd at r ¼ r vd ; hence, Let z 0 be the distance that the compression wavelet that originated from the circular intersection with radius r travels above the l-plane at z ¼ vd. The distance Dz between the circular intersection and the l-plane can be written as Suppose that the l-plane moves by Dz to reach z ¼ vd within time Dt after the compression wavelet was generated: then, for Dz, 0 Dz 2d, where b is defined as b ¼ s=V l À 1. Note that the locus of points z 0 gives the envelope of the compression wavelets. Next, Engel evaluated the volume of the region v m 00 that has been traversed by the compression wave during the time interval Dt, which is indicated by the hatched region in figure 10. Volume v m 00 is that of the body of revolution bounded by the envelope and l-plane at z ¼ vd. Then, v m 00 ¼ p Engel estimated that the average rate of change of momentum of liquid in volume (v m 0 þ v m 00 ) is (m 0 þ m 00 )a E V/Dt. She also assumed that all of the force that has acted to produce the liquid momentum is rsos.royalsocietypublishing.org R. Soc. open sci. 5: 181101 written as pr 2 vd DP E . She finally obtained the following: r 0 sV: (4:8)

Instantaneous maximum pressure at the instant of liquid droplet impact
The most significant aspect of the analysis by Engel is that she assumed that Dt is finite; however, the results obtained by numerical analysis in §3 show that the time elapsed for the pressure on the central axis to reach the maximum pressure converges to 0. Therefore, we should examine whether equation (4.8) is valid even for the instantaneous maximum pressure rise. In other words, we should answer the question: Is equation (4.8) valid in the limit of Dt to 0, although Dt does not explicitly appear in equation (4.8)?
First, we discuss the compression process. Engel assumed that the radius of curvature of the droplet in the vicinity of the impact changes instantaneously with the point impact remaining in the compression process, as shown in figure 10. We should understand that it takes a finite compression time Dt 0 for the liquid in the vicinity of the impact to be compressed, where Dt 0 is the time that it takes for the compression wavefront to travel through the compressed region. We now reexamine how the droplet can be compressed in Dt 0 .
Our numerical results in §3.4 showed that while the intersection advances a distance of 5Dr Ã ( ¼ 5 Â 2 À15 ) in the r-direction during this compression time of 6Dt Ã ( ¼ 2:28 Â 10 À9 ), the solid surface moves by a distance 7:32 Â 10 À5 Dr Ã and the compression wavefront travels in the liquid droplet by a distance 1:30 Â 10 À3 Dr Ã in the z-direction. We are hence considering an extremely small compression time. This consecutive impact of the interface on the solid surface in the surface cells corresponds to the expansion of the circular intersection between the solid surface and the surface of the liquid droplet.
The expansion speed of the circular intersection is much faster than the compression wavefront speed, as Lesser [16] also pointed out. When the circular intersection expansion is caused purely by compression, the circular intersection expansion speed at Dt 0 after the impact V R can be written as (4:9) Note that s/V R is a good measure to evaluate which process dominates in the compression process, the expansion of the circular intersection or the change in the curvature due to the compression wavefront propagation. We evaluate s/V R , taking the limit of Dt 0 to 0, obtaining We now understand that the compression deformation in the z-direction, as shown in figure 10, never occurs because the compression wavefront should pass through a portion in the liquid droplet for that portion to deform. We conclude that the circular intersection between the solid surface and the surface of the liquid droplet expands at an extremely high speed, much greater than the compression wavefront speed, compressing the spherical cap over which the circular intersection has moved while keeping the radius of curvature of the droplet unchanged in the compression process.
Our new finding leads to the following. At the end of the droplet compression process, where neither displacement of liquid nor propagation of compression wavelets has occurred, we assume that the equation of the l-plane is z ¼ 0, with no change in the radius of curvature of the droplet in the vicinity of the impact, but a compression of the spherical cap, as shown in figure 11. We set the z-coordinate of the first impact point between the solid plane surface and the liquid droplet to be z ¼ 2j. We assume that no point contact can be sustained, even at the end of the compression process; hence, the contact area between the liquid droplet and solid surface should be finite at the end of the compression process.
We next discuss the displacement process. After the compression process, the solid plane surface, i.e. the d-plane, moves a distance d and the l-plane moves a distance of (h 2 j) over Dt, i.e. the l-plane reaches z ¼ 1h at Dt, as shown in figure 11, where 1 is defined by (4:11) As we discussed above, the circular intersection expansion speed is much faster than both the compression wavefront speed and the impact speed, especially at the very early stage of impact; rsos.royalsocietypublishing.org R. Soc. open sci. 5: 181101 13 hence, the volume of the compressed spherical cap can also be much larger than the volume of the displaced portion of the liquid droplet, which is indicated by the light grey region in figure 11. Note that V/V R is a good measure for evaluating which process dominates the deformation of the spherical cap, the compression process or the displacement process. We evaluate V/V R , taking the limit of Dt to 0 using equation (4.10), obtaining lim Dt!0 V/V R ¼ 0. We conclude that 1 converges to 0 in the limit of Dt to 0, i.e. at the instant of the droplet impact, because the compression process completely dominates the deformation of the spherical cap.
Using the notation used in §4.1, we obtain the relation vd ¼ eh; then, h is written as (4:12) We now investigate the pressure rise in the limit of Dt to 0. First, we consider the volume v 0 m 0 of m 0 . Recall that v m 0 is the volume of a spherical cap with a height of vd and the radius of the base of r vd . v 0 m 0 is the volume of a spherical segment with a radius of the base of r h and a height of h(1 2 n). Simple algebra gives We estimate the average rate of change of momentum of the liquid in volume v 0 , which is to be (m 0 þ m 00 )a E V/Dt. Noting that instead of equation (4.1), the velocity V l of l-plane is given by (4: 16) We obtain the equation for the instantaneous pressure rise DP i at the instant of impact: Figure 11. Cross section of a liquid droplet. The original shape of the droplet before compression is indicated by a dotted line. The liquid cylinder with base radius r vd formed between the dand l-planes has the same volume as that of the spherical cap whose base is on the l-plane with height vd. We consider the limit of h to 0 as we take the limit of Dt to 0. Note that the z-direction distance travelled by the compression wavefront generated at the compression process is uniform (the straight horizontal line on top of the hatched region) because the circular intersection expansion speed is much faster than the speed of sound. Note that the magnitude of j is greatly exaggerated here, i.e. j ( R.
rsos.royalsocietypublishing.org R. Soc. open sci. 5: 181101 To obtain these results, we used the fact that 1 converges to 0 in the limit of Dt to 0. If the compression process completely dominates, as in the case we examined, 1 becomes 0. On the contrary, if the displacement process completely dominates, 1 could become 1. When 1 is 1, the coefficient of r 0 sV in equation (4.17) becomes a E /2, which gives equation (1.1). We found that the coefficient 1 2 appears when the displacement process completely dominates or the radial flow becomes prominent. Recall that we found in §3.2 that a is 0.455 when the liquid in a significantly large Cell-1 can flow. We hence conclude that the coefficient 1 2 that appears in equation (1.1) originates from the spherical shape of the liquid droplet that can allow the radial flow. We emphasize that a can be 1 2 only when 1 ¼ 1; however, a cannot be 1 2 in the limit of Dt to 0, rather a is 1, i.e. 1 is 0. Indeed, 1 can be nothing but 0 at the instant of the droplet impact.
Note that the traversal of the compression wavefront in the droplet may be severely impulsive; hence, the divergence of the compression wave may be negligibly small. Then, we may regard a E as 1, which also enforces the point impact because h in equation (4.12) converges to 0 in the limit of a E to 1. Then, we finally obtain DP i ¼ r 0 sV: (4:18) We have successfully extended the equation for the maximum pressure rise DP E developed at droplet impact given by equation (4.8) in the limit of Dt to 0, to obtain the equation for the instantaneous maximum pressure rise DP i at the instant of impact given by equation (4.18). Equation (4.18) is exactly the same as equation (2.11), which predicts the impact pressure rise under the assumption of a one-dimensional flow. Equation (4.10) shows that the central pressure develops much more slowly than the neighbouring contact between the gas -liquid interface at the instant of liquid droplet impact, i.e. in the limit of Dt 0 to 0, as we have observed in the numerical results in §3.4; hence, the pressure development on the z-axis is forced to be confined to one dimension, as we have discussed in §3.3. Therefore, we conclude that the instantaneous pressure rise due to the spherical liquid impact can be well predicted by equation (4.18).

Conclusion
The prime objective of this study was to answer how high is the pressure that develops at the instant of a liquid droplet impact on a solid plate. To achieve this objective, we investigated the following issue: At the instant of impact, is the assumption that the generated pressure coincides with the water-hammer pressure rise relevant? The water-hammer theory and the shock relation yield the one-dimensional liquid impact pressure rise DP s : where r 0 is the density of the undisturbed liquid, s is the compression wavefront propagation speed in the disturbed liquid and V is the impact velocity. We showed both numerically and analytically that the liquid droplet impact pressure, which is referred to as the pressure developed at the instant of the liquid droplet impact on the contact point, can be predicted by equation (5.1). We examined the pressure rise at the impact of the liquid droplet by numerically solving twodimensional axisymmetric compressible Euler equations using the level set ghost fluid method and the stiffened-gas equation of state. We investigated the pressure of the fluid in the surface cells that are adjacent to the solid surface. We defined Cell-1, which is the surface cell that contains the z-axis, and the central pressure, which is the pressure in Cell-1. We found that the maximum central pressure rise DP max depends on the calculation cell size, and it converges to the one-dimensional liquid impact pressure rise DP s (equation (5.1)) as the cell size decreases. When the non-dimensional calculation cell size Dr Ã , which is the dimensional cell size divided by the droplet radius, is 2 215 , DP max reaches 0.994DP s . We also found that the elapsed time for the pressure to reach the maximum from the instant of impact t Ã e decreases as Dr Ã decreases and converges to 0. We found that the maximum central pressure DP max is greatly affected by the status of Cell-2, which is the surface cell adjacent to Cell-1, while the central pressure increases. When the cell size is sufficiently small, the pressure in Cell-2 begins to rise immediately after the pressure in Cell-1 begins to rise. They rise practically simultaneously. When the fluid in Cell-2 remains a gas after the pressure rises in Cell-1, the liquid in Cell-1 is confined to the space bounded by the gas in Cell-2; hence, the liquid in Cell-1 can be compressed in directions both normal and parallel to the solid surface. In other words, the liquid in Cell-1 can flow in a direction parallel to the solid surface. When the liquid in Cell-1 cannot flow, the flow can be assumed to be one-dimensional. Therefore, DP max can reach DP s rsos.royalsocietypublishing.org R. Soc. open sci. 5: 181101 We then discussed whether the impact pressure rise can be reduced from stiffened-gas pressure rise DP sg when the interface that impacts on a solid surface has a finite radius of curvature by using the numerical analysis. Numerical calculations were carried out under the conditions Dr Ã ¼ 2 À15 and Dt Ã ¼ 3:73 Â 10 À10 . Under these conditions, we successfully reproduced the droplet impacts with a solid surface with a finite radius curvature because no simultaneous interface impacts occurred in any adjacent cells. The maximum central pressure rise DP max was 0.994DP s . We hence concluded that the maximum central pressure rise that develops when a droplet impacts on a solid surface can be accurately predicted by equation (5.1), even with a finite radius of curvature interface. We emphasize that it requires extremely high resolution in both the spatial and temporal domains to obtain the numerical result that the maximum central pressure rise can be predicted by equation (2.10). We are convinced that insufficient resolutions in either the numerical or experimental studies caused the previous confusion in determining the factor.
We shed light on the prediction of the maximum pressure rise DP E by [6], where, for comparison, the compression wavefront propagation speed s is used instead of c 0 , the sound velocity in the undisturbed liquid that was used in the original equation. It is widely recognized that a E /2 comes from the spherical shape of the liquid droplet; however, our numerical analysis showed that the impact pressure rise due to the spherical liquid droplet can be predicted by equation (5.1), where no shape factor is required. We discussed the origin of the mysterious factor a E /2; then, revealed the paradox.