Formation of liquid drops at an orifice and dynamics of pinch-off in liquid jets

This paper presents a numerical investigation of the dynamics of pinch-off in liquid drops and jets during injection of a liquid through an orifice into another fluid. The current study is carried out by solving axisymmetric Navier-Stokes equations and the interface is captured using a coupled level-set and volume-of-fluid approach. The delicate interplay of inertia and viscous effects plays a crucial role in deciding the dynamics of the formation as well as breakup of liquid drops and jets. In the dripping regime, the growth and breakup rate of a drop are studied and quantified by corroborating with theoretical predictions. During the growth stage of the drops, a self-similar behavior of the drop profile is identified over a relatively short duration of time. The viscosity of the drop liquid shows substantial influence on the thinning behavior of a liquid neck and a transition is observed from an inertia dominated regime to an inertia-viscous regime beyond a critical minimum value of the neck radius. The phenomenon of interface overturning is fundamentally related to the magnitude of drop viscosity. The variation of overturning angle as a function of drop viscosity is computed and a critical value of Ohnesorge number is obtained beyond which overturning ceases. Increasing the inertia of drop liquid transforms the system from a periodically dripping regime to a quasiperiodic regime and finally it culminates into an elongated liquid jet. Another interesting transition from dripping to jetting regime is demonstrated by varying the viscosity of the ambient medium. The breakup of jets in Rayleigh mode is explored and the breakup length obtained from our computations shows excellent agreement with the theoretical predictions owing to Rayleigh's analysis. The ambient medium is entrained as the jet moves downstream with the creation of a vortical structure just outside the jet signifying increased participation of the ambient medium in the dynamics of jet breakup at higher inflow rates.

Figure 1

Schematic diagram (not to scale) showing the computational domain and boundary conditions in cylindrical coordinate system $(r,z,\theta )$. A fully developed flow is imposed at the orifice inlet of radius $R$ initialized with a hemispherical drop. Gravity acts along positive $z$ axis. The size of the domain is taken as $5R$ in radial $\left(r\right)$ direction whereas $30–60R$ in axial $\left(z\right)$ direction

Figure 2

Qualitative comparison of drop formation between (a) current computations and (b) experimental results of Subramani et al. [9] under conditions of $\mathrm{Oh}=0.13,\mathrm{Bo}=0.33,\mathrm{We}=0.119,\eta =0.001$, and $\lambda =0.0005$.

Figure 3

Numerical prediction of dimensionless drop diameter $D_d^{*}$ as a function of dimensionless time $t^{*}$ and comparison with analytical expression Eq. (22) for $\mathrm{Oh}=0.003,\mathrm{Bo}=0.18,\mathrm{We}=0.2,\eta =0.001$, and $\lambda =0.02$.

Figure 4

(a) Instantaneous interface profiles and (b) collapsed profiles during drop growth stage from $t=0.025\text{s}$ to $t=0.039\text{s}$ for $\mathrm{Oh}=0.002,\mathrm{Bo}=0.18,\mathrm{We}=0.2,\eta =0.001,\lambda =0.02$. (c) Instantaneous interface profiles and (d) collapsed profiles during drop growth stage from $t=0.025\text{s}$ to $t=0.039\text{s}$ for $\mathrm{Oh}=0.014,\mathrm{Bo}=0.18,\mathrm{We}=0.2,\eta =0.001,\lambda =0.004$. (e) Instantaneous interface profiles and (f) collapsed profiles during drop growth stage from $t=0.025\text{s}$ to $t=0.035\text{s}$ for $\mathrm{Oh}=0.1,\mathrm{Bo}=0.18,\mathrm{We}=0.2,\eta =0.001,\lambda =0.0006$. The axes has been rescaled following $z^{*}/t^{*\alpha }=a{r}^{*}{t}^{*\beta }$.

Figure 5

Relation between minimum neck radius ${r^{*}}_{\min }$ and time before pinch-off $\tau$ during collapse stage. The solid line represents ${r^{*}}_{\min }\sim {\tau }^{2/3}$. Parameters are $\mathrm{Oh}=0.003,\mathrm{Bo}=0.18,\mathrm{We}=0.2,\eta =0.001$, and $\lambda =0.02$.

Figure 6

Velocity field inside drop of (a) pure water (b) 91% glycerine. Only selected velocity vectors are plotted to maintain clarity. It is seen that interface overturning occurs for a pure water drop whereas it vanishes for highly viscous drops.

Figure 7

Effect of $\mathrm{Oh}$ on the overturning angles at pinch-off (a) large angle (b) small angle. The dashed line represents the limit when overturning vanishes $\left({\theta }_l={90}^{\circ }\right)$. The other relevant parameters are $\mathrm{Bo}=0.18,\mathrm{We}=0.2,\eta =0.001$, and $\lambda =0.02–0.0006$.

Figure 8

Effect of Weber number on the droplet formation process. (a)–(d) shows the drop formation for $\mathrm{We}=0.2$ at time $t^{*}10.90$, 20.83, 30.92, and 41.30, respectively. (e)–(h) shows the drop formation for $\mathrm{We}=1.0$ at time $t^{*}11.50$, 16.89, 22.22, and 27.50, respectively. (i)–(l) shows the drop formation for $\mathrm{We}=1.2$ at time $t^{*}11.05$, 17.15, 21.30, and 28.82, respectively. Other parameters remain fixed at $\mathrm{Oh}=0.003,\mathrm{Bo}=0.18,\eta =0.001$, and $\lambda =0.02$.

Figure 9

Transition from dripping to jetting regime with variation of viscosity ratio $\lambda$. (a) and (b) shows the drop formation in dripping mode for $\lambda =7.5$ and 10.0, respectively. (c) shows the jetting regime at $\lambda =15.0$. (d) shows the variation of limiting length at breakup $L_d^{*}$ with increasing $\lambda$. Other parameters remain fixed at $\mathrm{Oh}=0.02,\mathrm{We}=0.4,\mathrm{Bo}=0.18$, and $\eta =0.74$.

Figure 10

Variation of computed jet breakup length along with predictions from theory [Eq. (25)] as a function of inflow velocity. The breakup length show very good agreement with the analytical results of Rayleigh's theory.

Figure 11

(a) Instantaneous jet radius as a function of downstream distance showing varicose undulations on its surface. The radial direction has been magnified for clear visualization of the perturbations. (b) Closeup view of the jet between $z^{*}=31$ and $z^{*}=36$ showing pressure contours and also streamlines in the ambient air. A vortex is seen developing just outside the jet indicating air entrainment. The relevant parameters are $\mathrm{We}=3.0,\mathrm{Oh}=0.003$, and $\mathrm{Bo}=0.18$.