Viscosity-ratio-based scaling for the rise velocity of a Taylor drop in a vertical tube.

Simulations of a silicon oil Taylor drop rising in a tube filled with a glycerol-water mixture are performed to investigate the viscosity ratio effects on the rise velocity of the Taylor drop. By varying the viscosity ratio λ between the drop and the suspending liquid from O(0.1) to O(10), a simple relationship of the nondimensional terminal velocity, the Froude number (Fr), is revealed as Fr ∝ λ(-0.27). This scaling is further confirmed by recently published experimental data [Hayashi, Kurimoto, and Tomiyama, Int. J. Multiphase Flow 37, 241 (2011)]. The simulated drop shapes also compare well with the experiments. Increasing the viscosity ratio elongates the drop and tends to make the tail bulge out. The correlation applies to small Reynolds numbers and finite viscosity ratios.

Numerical study of a Taylor bubble rising in stagnant liquids.

The dynamics of a Taylor bubble rising in stagnant liquids is numerically investigated using a front tracking coupled with finite difference method. Parametric studies on the dynamics of the rising Taylor bubble including the final shape, the Reynolds number (Re(T)), the Weber number (We(T)), the Froude number (Fr), the thin liquid film thickness (w/D), and the wake length (l(w)/D) are carried out. The effects of density ratio (η), viscosity ratio (λ), Eötvös number (Eo), and Archimedes number (Ar) are examined. The simulations demonstrate that the density ratio and the viscosity ratio under consideration have minimal effect on the dynamics of the Taylor bubble. Eötvös number and Archimedes number influence the elongation of the tail and the wake structures, where higher Eo and Ar result in longer wake. To explain the sudden extension of the tail, a Weber number (We(l)) based on local curvature and velocity is evaluated and a critical We(l) is detected around unity. The onset of flow separation at the wake occurs in between Ar=2×10(3) and Ar=1×10(4), which corresponds to Re(T) between 13.39 and 32.55. Archimedes number also drastically affects the final shape of Taylor bubble, the terminal velocity, the thickness of thin liquid film, as well as the wall shear stress. It is found that w/D=0.32 Ar(-0.1).

Numerical studies of bubble necking in viscous liquids.

The pinch-off of a gas bubble from a tiny nozzle immersed vertically in another quiescent viscous fluid due to buoyancy is numerically investigated. The dynamics of bubble growth and pinch-off are described by the full Navier-Stokes equations for both gas and liquid phases. The equations are solved with a finite-volume method based on the SIMPLE scheme, coupled with a front tracking method to locate the interface between the two phases. The effects of liquid viscosity, surface tension, and gas density on the bubble pinch-off dynamics, which are always coupled in experiments, are investigated separately through simulations. The numerical results are compared with experimental observations on the bubble pinch-off for validation purposes. The simulation results show that the radius of the necking region decreases in a power law mode with time as r approximately tau;{alpha} , where tau is the time to pinch-off and the exponent alpha varies in the range 0.5-1.0 depending strongly upon the liquid properties such as viscosity and surface tension. In addition, the surface tension can significantly affect the bubble pinch-off exponent alpha when the surface tension coefficient is smaller than 0.030 N/m with a Bond number higher than 0.72. It is also found that both higher viscosity of the liquid phase and higher surface tension may result in a delayed pinch-off process and a larger bubble. The effect of gas phase density on the pinch-off is also investigated. As reported in the literature, the gas density variation has minimal effect on the necking process because the density ratio of the gas phase to the liquid phase is small.

Numerical study of wall effects on buoyant gas-bubble rise in a liquid-filled finite cylinder.

The wall effects on the axisymmetric rise and deformation of an initially spherical gas bubble released from rest in a liquid-filled, finite circular cylinder are numerically investigated. The bulk and gas phases are considered incompressible and immiscible. The bubble motion and deformation are characterized by the Morton number (Mo), Eötvös number (Eo), Reynolds number (Re), Weber number (We), density ratio, viscosity ratio, the ratios of the cylinder height and the cylinder radius to the diameter of the initially spherical bubble ( H*=H/d0, R*=R/d0). Bubble rise in liquids described by Eo and Mo combinations ranging from (1,0.01) to (277.5,0.092), as appropriate to various terminal state Reynolds numbers (ReT) and shapes have been studied. The range of terminal state Reynolds numbers includes 0.02 or =3 , is noted to correspond to the rise in an infinite medium, both in terms of Reynolds number and shape at terminal state. In a thin cylindrical vessel (small R*), the motion of the bubble is retarded due to increased total drag and the bubble achieves terminal conditions within a short distance from release. The wake effects on bubble rise are reduced, and elongated bubbles may occur at appropriate conditions. For a fixed volume of the bubble, increasing the cylinder radius may result in the formation of well-defined rear recirculatory wakes that are associated with lateral bulging and skirt formation. The paper includes figures of bubble shape regimes for various values of R*, Eo, Mo, and ReT. Our predictions agree with existing results reported in the literature.