Modelling droplet collision outcomes for different substances and viscosities

Abstract

The main objective of the present study is the derivation of models describing the outcome of binary droplet collisions for a wide range of dynamic viscosities in the well-known collision maps (i.e. normalised lateral droplet displacement at collision, called impact parameter, versus collision Weber number). Previous studies by Kuschel and Sommerfeld (Exp Fluids 54:1440, 2013) for different solution droplets having a range of solids contents and hence dynamic viscosities (here between 1 and 60 mPa s) revealed that the locations of the triple point (i.e. coincidence of bouncing, stretching separation and coalescence) and the critical Weber number (i.e. condition for the transition from coalescence to separation for head-on collisions) show a clear dependence on dynamic viscosity. In order to extend these findings also to pure liquids and to provide a broader data basis for modelling the viscosity effect, additional binary collision experiments were conducted for different alcohols (viscosity range 1.2–15.9 mPa s) and the FVA1 reference oil at different temperatures (viscosity range 3.0–28.2 mPa s). The droplet size for the series of alcohols was around 365 and 385 µm for the FVA1 reference oil, in each case with fixed diameter ratio at Δ= 1. The relative velocity between the droplets was varied in the range 0.5–3.5 m/s, yielding maximum Weber numbers of around 180. Individual binary droplet collisions with defined conditions were generated by two droplet chains each produced by vibrating orifice droplet generators. For recording droplet motion and the binary collision process with good spatial and temporal resolution high-speed shadow imaging was employed. The results for varied relative velocity and impact angle were assembled in impact parameter–Weber number maps. With increasing dynamic viscosity a characteristic displacement of the regimes for the different collision scenarios was also observed for pure liquids similar to that observed for solutions. This displacement could be described on a physical basis using the similarity number and structure parameter K which was obtained through flow process evaluation and optimal proportioning of momentum and energy by Naue and Barwolff (Transportprozesse in Fluiden. Deutscher Verlag fur Grundstoffindustrie GmbH, Leipzig 1992). Two correlations including the structure parameter K could be derived which describe the location of the triple point and the critical We number. All fluids considered, pure liquids and solutions, are very well fitted by these physically based correlations. The boundary model of Jiang et al. (J Fluid Mech 234:171–190, 1992) for distinguishing between coalescence and stretching separation could be adapted to go through the triple point by the two involved model parameters C a and C b, which were correlated with the relaxation velocity $$ u_{\text{relax}} = {\sigma \mathord{\left/ {\vphantom {\sigma \mu }} \right. \kern-0pt} \mu } $$ . Based on the predicted critical Weber number, denoting the onset of reflexive separation, the model of Ashgriz and Poo (J Fluid Mech 221:183–204, 1990) was adapted accordingly. The proper performance of the new generalised models was validated based on the present and previous measurements for a wide range of dynamic viscosities (i.e. 1–60 mPa s) and liquid properties. Although the model for the lower boundary of bouncing (Estrade et al. in J Heat Fluid Flow 20:486–491, 1999) could be adapted through the shape factor, it was found not suitable for the entire range of Weber numbers and viscosities.