# Stress distribution and floor shock wave of drop influence

*by*Phil Tadros

### Excessive-speed stress microscopy

We embed low-concentration (0.23% v/v) fluorescent polystyrene particles of diameter 30 μm in a cross-linked polydimethylsiloxane (PDMS) gel as tracers to trace the deformation of the gel beneath influence. The PDMS gel floor is hydrophobic with the water contact angle ~90^{∘}. Younger’s modulus of the gel is fastened at *E* = 100 kPa in our experiments, though gels with *E* as much as 420 kPa and with hydrophilic surfaces have additionally been examined (Strategies). A skinny laser sheet of 30-μm thickness illuminates the gel from a facet and excites the fluorescent tracers throughout the sheet (Fig. 2a). The sheet is finely adjusted to be regular to the impacted floor and to go by means of the middle of impacting drops. A high-speed digicam specializing in the sheet photos the movement of tracers at 40,000 frames per second.

We monitor the displacements of the tracers from the high-speed video utilizing digital picture correlation (DIC). An interrogation window of 384 μm by 384 μm with 70% overlap is adopted in DIC, which supplies a spatial decision of 115 μm. The temporal decision is 0.025 ms, set by the body charge of high-speed images. To scale back measurement errors, we first common the cross-correlation fields of DIC from 5 repeated impacts on the identical gel on the identical influence location. The floor is absolutely dried between two consecutive experimental runs. The stress measurements are the end result of an additional common over three totally different averaged displacement fields for impacts on totally different gels or impacts on the identical gels at totally different influence places. Thus, one information level represents the common results of whole 15 totally different experimental runs.

Stress fields depend upon pressure fields, that are the by-product of the displacement fields obtained from DIC. A smoothing process is important with a view to cut back the noise of differentiation. We implement the shifting least squares (MLS) interpolation technique to acquire a repeatedly differentiable displacement discipline *u*(*r*, *z*) = [*u*_{r}(*r*, *z*), *u*_{z}(*r*, *z*)] from the discrete displacement discipline of DIC^{30}. A 3rd-order polynomial foundation is adopted within the interpolation (Supplementary Data (SI) Part 1).

When the deformation is small, the pressure elements in cylindrical coordinates comply with:

$${varepsilon }_{rr}=frac{partial {u}_{r}}{partial r},{varepsilon }_{zz}=frac{partial {u}_{z}}{partial z},{varepsilon }_{theta theta }=frac{{u}_{r}}{r},{varepsilon }_{rz}=frac{1}{2}left[frac{partial {u}_{r}}{partial z}+frac{partial {u}_{z}}{partial r}right],$$

(1)

the place we take the benefit of the cylindrical symmetry of the drop-impact geometry. By assuming the PDMS gels are isotropic and linear following the generalized Hooke’s legislation at small strains, we calculate the stress fields utilizing the linear stress–pressure relation:

$${sigma }_{ij}=lambda {varepsilon }_{b}{delta }_{ij}+2G{varepsilon }_{ij},$$

(2)

the place *λ* = *E**ν*/[(1 + *ν*)(1 − 2*ν*)] is the Lamé coefficient, *G* = *E*/[2(1 + *ν*)] is the shear modulus, *δ*_{ij} is the Kronecker delta and *ε*_{b} ≡ *ε*_{zz} + *ε*_{rr} + *ε*_{θθ} is the majority pressure. *σ*_{rz} offers the shear stress *τ*, whereas *σ*_{zz} offers the strain *p*. PDMS gels are almost incompressible with Poisson’s ratio *ν* near 0.5, which lead to a big *λ*. However the bulk pressure *ε*_{b} is near 0 on this restrict. Due to this fact, the influence strain can’t be precisely decided from the product of *λ**ε*_{b} in Eq. (2). As an alternative, we undertake a quasi-steady state assumption to calculate the strain^{30}, a process detailed additional in SI Part 2. We’ve got verified the idea by evaluating the inertial drive and the elastic drive within the influence course of (SI Part 2) and by evaluating experimental and numerical outcomes on the influence strain of solid-sphere influence (see under). The shear stress, then again, just isn’t affected by the almost incompressible situation. The floor stresses and displacements are lastly obtained at a location barely under the unique impacted floor (Strategies).

As a calibration and the idea of comparability, we first measure the strain and shear stress induced by the influence of a strong metal sphere of diameter *D* = 3.16 mm at influence velocity *U* = 0.49 m/s and evaluate the outcomes with these from finite ingredient simulations (Strategies). Experimental measurements agree effectively with the numerical outcomes, validating the accuracy of high-speed stress microscopy (Fig. 2b).

For drop influence, our drops are manufactured from an aqueous resolution of sodium iodide (60% w/w), which has a density *ρ* = 2.2 g/ml and a viscosity *η* = 1.12 mPa s^{31}. The floor rigidity of the answer *σ* ≈ 81.3 mN/m from pendant-drop tensiometry^{32}, which is barely bigger than that of water. We repair the diameter *D* and influence velocity *U* of drops in our experiments. Drops of *D* = 3.49 mm influence usually on the floor of PDMS gels at *U* = 2.97 m/s, yielding a Reynolds quantity *R**e* = *ρ**U**D*/*η* = 20,360 and a Weber quantity *W**e* = *ρ**D**U*^{2}/*σ* = 833. Thus, the drop influence is dominated by fluid inertia at early occasions. We give attention to drop influence at early occasions under, when the shear stress and strain of impacting drops are excessive for robust erosion. Positions and occasions are reported in dimensionless kinds utilizing *D* and *D*/*U* because the corresponding size and time scale, respectively.

### Influence shear stress

Floor erosion is the direct consequence of the shear stress of drop influence. Determine 3a, b compares the temporal evolution of the shear stress of solid-sphere influence and liquid-drop influence. Upon the influence, spatially non-uniform shear stresses shortly develop in each instances. Nonetheless, whereas the place of the utmost shear stress of solid-sphere influence is stationary close to the influence axis at *r* = 0.095, the utmost shear stress of drop influence propagates radially with the spreading drop. The kymographs of the floor shear stress, *τ*(*r*, *z* = 0, *t*), of the 2 influence processes are proven in Fig. 3c, d, highlighting additional the quick propagation of the utmost shear stress of drop influence.

To know the origin of the utmost shear stress of drop influence, we correlate the place of the utmost shear stress *r*_{s} with the form of impacting drops (Fig. 4a). Two kinematic options are analyzed: the tip of the increasing lamella *r*_{lm} and the turning level *r*_{t}, the place the drop physique connects to the basis of the lamella (Fig. 4a inset). It ought to be emphasised that *r*_{t} just isn’t the contact line of the drop. The ejection of the lamella happens round *t* ≈ *W**e*^{−2/3} = 0.0104^{33}, which is shorter than the temporal decision of our experiments. Whereas *r*_{lm} strikes quickest, *r*_{s} follows intently behind *r*_{t}. Thus, the utmost shear stress arises from the robust velocity gradient close to the turning level, the place the movement adjustments quickly from the downward vertical path (the −*z* path) throughout the drop physique to the horizontal radial path (the *r* path) contained in the slim lamella^{34}. Quantitatively, *r*_{t}(*t*) follows the well-known square-root scaling ({r}_{t}(t)=sqrt{6t}/2approx 1.22sqrt{t}) established by many earlier experiments^{23,33,35,36,37,38,39}. As compared, *r*_{s}(*t*) additionally exhibits a square-root scaling with a barely smaller prefactor ({r}_{s}(t)approx sqrt{t}). The positions of the utmost shear stress and the turning level are unbiased of the wettability or Younger’s modulus of PDMS gels (Fig. 4b).

Philippi et al.^{40} proposed that the shear stress of incompressible drops on infinitely inflexible substrates possesses a self-similar dynamic construction when *t* → 0^{+},

$$tau (r,z=0,t)=2sqrt{frac{6}{{pi }^{3}Re}}frac{1}{sqrt{t}}fleft(frac{r}{sqrt{t}}proper)quad {{{{{{{rm{for}}}}}}}}quad rle {r}_{t}(t),$$

(3)

the place the scaling operate *f*(*x*) = *x*/(3 − 2*x*^{2}) dictates a finite-time singularity on the turning level *r*_{t}(*t*). Right here, *τ*(*r*, *t*) is non-dimensionalized by the inertial strain *ρ**U*^{2}. Word that the a lot stronger water-hammer strain *ρ**U**c* related to the compression wave happens on the time scale of some nanoseconds, which is just too brief to be related in our present experiments^{7,20,23}. Right here, *c* is the pace of sound in liquid. Impressed by the self-similar speculation, we plot (tau sqrt{t}) versus (r/sqrt{t}) of our experimental outcomes (Fig. 4c), which exhibits a very good collapse at a small *r* away from the singular area. With a modified scaling operate *f*(*x*) = *x*/(1 − *x*^{2}) to depend the totally different temporal scalings of *r*_{t} and *r*_{s}, the collapsed information quantitatively agrees with Eq. (3) (the dashed line in Fig. 4c). Thus, our examine supplies experimental proof on the propagation of shear stress of drop influence and demonstrates the self-similar construction of shear stress at early occasions.

Regardless of the overall settlement with Eq. (3) at *r* < *r*_{t}, our experiments additionally reveal the distinctive options of drop influence on elastic deformable substrates, absent within the theoretical consideration of drop influence on infinitely inflexible substrates. The shear stress on the floor of an elastic substrate is given by *τ* = *G*(∂*u*_{r}/∂*z* + ∂*u*_{z}/∂*r*) (Eqs. (1) and (2)), the place *G* is the shear modulus of the substrate and *u*_{r} and *u*_{z} are the radial and vertical displacement of the substrate floor. We discover ∣∂*u*_{z}/∂*r*∣ > ∣∂*u*_{r}/∂*z*∣ (Fig. 4d–f), suggesting the dominant function of the vertical velocity of the impacting drop on the contact floor *v*_{z}(*r*, *z* = 0) on the shear stress. Word that ({u}_{z}(r,z=0,t)=intnolimits_{0}^{t}{v}_{z}(r,z=0,t)dt). For drop influence on infinitely inflexible substrates, *v*_{z}(*r*, *z* = 0, *t*) = 0 due to the no-penetration boundary situation, which inevitably offers ∂*u*_{z}/∂*r* = 0. As an alternative, the shear stress of drop influence on infinitely inflexible substrates arises from the gradient of the radial velocity, ∂*v*_{r}/∂*z*, throughout the boundary layer close to the contact floor. As *u*_{z}(*r*, *z* = 0) is principally decided by the strain distribution on the contact floor at excessive *R**e*, this discovering illustrates the intrinsic coupling between the influence strain and shear stress of drop influence on elastic substrates.

The impact of the finite stiffness of the impacted substrate additionally manifests within the shear drive of impacting drops. By integrating the shear stress over the contact space, we get hold of the shear drive, ({F}_{d}(t)=2pi intnolimits_{0}^{{r}_{lm}}tau (r,z=0,t)rdr), which quantifies the overall erosion energy of drop influence. Though *F*_{d}(*t*) is unbiased of the wettability of the impacted floor, it will increase with Younger’s modulus following a scaling ({F}_{d} sim sqrt{E}) throughout the vary of our experiments (Fig. 5a, b). Due to the spreading of the utmost shear stress, drop influence and solid-sphere influence present comparable peak shear forces beneath related influence circumstances (Fig. 5a).

### Influence strain and floor shock wave

Though topic to bigger experimental errors as a result of almost incompressibility of PDMS, the strain (i.e. regular stress) distribution beneath impacting drops *p*(*r*) might be additionally measured by high-speed stress microscopy (SI Part 2). Just like the shear stress, we observe a non-central strain most propagating radially with the spreading drop (Fig. 6b, d). The dynamics are once more in sharp distinction to the strain of solid-sphere influence, the place the utmost influence strain is fastened on the influence axis *r* = 0 (Fig. 6a, c). The existence of the propagating non-central strain most has been predicted by a number of theories and simulations of drop influence^{38,40,41,42,43}. However, to the very best of our information, such a counter-intuitive prediction has not been instantly verified in experiments heretofore. Whereas our measurements qualitatively verify the prediction, we discover that the utmost strain falls behind the utmost shear stress (Fig. 4a), a characteristic sudden from drop influence on infinitely inflexible substrates^{40}.

Extra apparently, a destructive strain emerges in entrance of the turning level *r*_{t} at *t*_{c} ≈ 0.106 (Fig. 6b, d). Within the meantime, we additionally observe the propagation of floor disturbance on the gel floor away from the stress maxima above *t*_{c} (Fig. 7a). Each recommend the formation of a floor acoustic wave—the basic Rayleigh wave—within the gel. Because the pace of the turning level ({V}_{t}(t)=d{r}_{t}/dt=sqrt{6}/(4sqrt{t})) will increase with lowering *t*, the stress maxima related to the turning level unfold supersonically at early occasions (Fig. 7b). Thus, a shock entrance kinds close to *r*_{t} on the impacted floor when *t* < *t*_{c}. The Rayleigh wave lastly overtakes the turning level and is launched in entrance of the spreading drop in an explosion-like course of above *t*_{c}, giving rise to the destructive strain and the propagation of floor disturbance. Primarily based on the above image, the pace of the floor wave might be estimated as ({V}_{t}({t}_{c})=sqrt{6}/(4sqrt{{t}_{c}})=1.88), which quantitatively matches the pace of the Rayleigh wave^{44}

$${V}_{R}=frac{1}{M}left[sqrt{frac{1}{2(1+nu )}}frac{0.862+1.14nu }{1+nu }right]=1.89.$$

(4)

Right here, the Mach quantity (Mequiv Usqrt{{rho }_{s}/E}=0.292) with *ρ*_{s} = 0.965 g/cm^{3} and *ν* = 0.49 is the density and Poisson’s ratio of PDMS.

Inspired by the quantitative settlement between Eq. (3) and experiments, we couple the theoretical influence strain and shear stress of incompressible drops on infinitely inflexible surfaces with the Navier-Lamé equation of semi-infinite elastic media (SI Part 3). The dimensional evaluation of the governing equation and the boundary circumstances means that the shear drive ought to scale as ({F}_{d} sim {E}^{1/2}{(rho {U}^{2})}^{1/2}{D}^{2}), agreeing with our measurements at totally different *E* (Fig. 5b). Furthermore, the numerical resolution of the coupled equations qualitatively reproduces the formation of the shock-induced Rayleigh wave of drop influence, the place a pointy floor wave with a well-defined peak emerges at *t*_{c} ≈ 0.1 and propagates with *V*_{R} (Fig. 7c). The robust and sharp floor wave is produced by the mechanical resonance occurring when the pace of the stress maxima approaches the pace of the Rayleigh wave close to *t*_{c}. Such a resonant phenomenon doesn’t exist for solid-sphere influence with stationary stress maxima. In consequence, the floor Rayleigh wave of solid-sphere influence is extra diffusive (Supplementary Fig. 2).