^{1,a)}, Giuseppe Rocco

^{2,b)}and Luigi de Luca

^{2,c)}

### Abstract

Numerical simulations of early and intermediate instants of a plane two-dimensional dropimpact on a preexisting thin film of the same liquid are performed. The evolution of the phenomenon is analyzed by solving the free-surface Navier–Stokes equations by means of a volume of fluid (VOF) method. Viscous, inertial and surface tension forces are taken into account; gravity is neglected. The so-called splashing regime is emphasized, where the emergence of an initial horizontal ejecta sheet is followed by the formation of an almost vertical lamella sheet, which is the planar counterpart of the well known splashing-crown of spherical geometry. Overall velocity and pressure fields as well as detailed interface shapes are presented, and several insights on the relevant scaling laws are furnished. In the ejecta sheet (jet) regime a major result is the finding of a deviation from the standard square root behavior for the dependence on time of the contact length of sheet first emergence, which is proved to be crucial in the subsequent original application of the potential theory of Howison *et al.* [J. Fluid Mech.542, 1 (2005)]. In the lamella sheet regime, the outwards expansion of its base is discussed in connection with the theory of the formation of a kinematic discontinuity within the underneath film of Yarin and Weiss [J. Fluid Mech.283, 141 (1995)]. Analogies between planar and axysymmetric configurations are discussed.

The authors greatly thank Philip Yecko for useful discussions on the VOF technique.

I. INTRODUCTION

II. NUMERICAL PROCEDURE AND PROBLEM FORMULATION

A. VOF numerical method

B. Definition of the problem and numerical set-up

III. RESULTS

A. Formation of the ejecta sheet (jet)

B. Sheet evolution

C. Lamella sheet formation and onset of kinematic discontinuity

IV. CONCLUSIONS

### Key Topics

- Lamellae
- 40.0
- Reynolds stress modeling
- 30.0
- Fluid drops
- 29.0
- Ejecta
- 26.0
- Viscosity
- 20.0

## Figures

Sketch of problem definition.

Sketch of problem definition.

Definition sketch of the key points monitored during the evolution of the sheet. , . , , .

Definition sketch of the key points monitored during the evolution of the sheet. , . , , .

Contact length of emergence of the ejecta sheet as a function of Reynolds number.

Contact length of emergence of the ejecta sheet as a function of Reynolds number.

Time of emergence of the ejecta sheet as a function of Reynolds number. Continuous line reproduces the scaling law . In the inset the same data are reported in log-log scale.

Time of emergence of the ejecta sheet as a function of Reynolds number. Continuous line reproduces the scaling law . In the inset the same data are reported in log-log scale.

Contact length of emergence of the ejecta sheet as a function of the relative time instant for different Reynolds numbers. Continuous line reproduces the best fit power law .

Contact length of emergence of the ejecta sheet as a function of the relative time instant for different Reynolds numbers. Continuous line reproduces the best fit power law .

Dimensionless initial velocity of ejecta sheet as a function of Reynolds number. Continuous line reproduces the scaling law .

Dimensionless initial velocity of ejecta sheet as a function of Reynolds number. Continuous line reproduces the scaling law .

Jet evolution. (a) , (b) . For both cases and curves are relative to time instants ranging (from left to right) from to , with time step of . Vertical line indicates the location of first emergence of the sheet.

Jet evolution. (a) , (b) . For both cases and curves are relative to time instants ranging (from left to right) from to , with time step of . Vertical line indicates the location of first emergence of the sheet.

Rescaled jet base location (dots) as a function of rescaled time for , , and . Open circles are the simulated positions of the neck before the jet formation. The continuous line reproduces the square root law of Howison *et al.* (Ref. 7) and dashed line is the square root law of Josserand and Zaleski (Ref. 3) with the rescaled (i.e., using in place of as length scale) prefactor .

Rescaled jet base location (dots) as a function of rescaled time for , , and . Open circles are the simulated positions of the neck before the jet formation. The continuous line reproduces the square root law of Howison *et al.* (Ref. 7) and dashed line is the square root law of Josserand and Zaleski (Ref. 3) with the rescaled (i.e., using in place of as length scale) prefactor .

Characteristic curves of Eq. (6) for the functions represented by red lines. Dashed curves are relative to the square root law of Howison *et al.* (Ref. 7) and continuous curves correspond to the best fitting power law.

Characteristic curves of Eq. (6) for the functions represented by red lines. Dashed curves are relative to the square root law of Howison *et al.* (Ref. 7) and continuous curves correspond to the best fitting power law.

Values of the functions (lower lines) and (upper lines) on the curve . Circles refer to the square root law of Howison *et al.* (Ref. 7), while dots refer to the best fitting power law.

Values of the functions (lower lines) and (upper lines) on the curve . Circles refer to the square root law of Howison *et al.* (Ref. 7), while dots refer to the best fitting power law.

Jet thickness as a function of time for different Reynolds numbers. Red symbols refer to jet thickness measured as the distance between the points and of Fig. 2 and black symbols refer to the length of the horizontal projection of the segment . The continuous line reproduces the prediction of Howison *et al.* (Ref. 7), the dashed line refers to the scaling law of Josserand and Zaleski (Ref. 3) .

Jet thickness as a function of time for different Reynolds numbers. Red symbols refer to jet thickness measured as the distance between the points and of Fig. 2 and black symbols refer to the length of the horizontal projection of the segment . The continuous line reproduces the prediction of Howison *et al.* (Ref. 7), the dashed line refers to the scaling law of Josserand and Zaleski (Ref. 3) .

Velocity components at the jet base as a function of time for different Reynolds numbers. Upper symbols refer to and lower symbols to . Continuous line reproduces the prediction of Howison *et al.* (Ref. 7).

Velocity components at the jet base as a function of time for different Reynolds numbers. Upper symbols refer to and lower symbols to . Continuous line reproduces the prediction of Howison *et al.* (Ref. 7).

Time evolution of a typical lamella splashing at and . (a) , (b) , (c) , and (d) .

Time evolution of a typical lamella splashing at and . (a) , (b) , (c) , and (d) .

Velocity vector field of lamella splashing at , , and .

Velocity vector field of lamella splashing at , , and .

Normalized pressure fields of lamella splashing at and . (a) , (b) , (c) , and (d) .

Normalized pressure fields of lamella splashing at and . (a) , (b) , (c) , and (d) .

Velocity fields of lamella splashing at and . (a) , (b) , (c) , and (d) .

Velocity fields of lamella splashing at and . (a) , (b) , (c) , and (d) .

Traveling fronts of pressure (a) and velocity (b) along the wall liquid layer at and . Time instants are from to , with time step of 0.2 (a), and from to , with time step of 1 (b).

Traveling fronts of pressure (a) and velocity (b) along the wall liquid layer at and . Time instants are from to , with time step of 0.2 (a), and from to , with time step of 1 (b).

Functions (black symbols) and (red symbols) defined by Cossali *et al.* (Ref. 12), calculated with respect to four specifications of the position of the discontinuity. The continuous line and the dashed line correspond to the theoretical predictions and , respectively.

Functions (black symbols) and (red symbols) defined by Cossali *et al.* (Ref. 12), calculated with respect to four specifications of the position of the discontinuity. The continuous line and the dashed line correspond to the theoretical predictions and , respectively.

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