Bubble Dynamics and Breakup in Straining Flows

Citation

Kang, In Seok (1988) Bubble Dynamics and Breakup in Straining Flows. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/y91q-ff03. https://resolver.caltech.edu/CaltechETD:etd-11072007-112829

Abstract

The dynamics and breakup of a bubble in axisymmetric flow fields has been investigated using numerical and analytical techniques. In particular, the transient bubble deformation, oscillation, and overshoot effects are considered in conjunction with the existence of steady-state solutions.

To explore the dynamics of a bubble with a high degree of deformation, a numerical technique suitable for solving axisymmetric, unsteady free boundary problems in fluid mechanics has been developed. The technique is based on a finite-difference solution of the equations of motion on a moving orthogonal curvilinear coordinate system, which is constructed numerically and adjusted to fit the boundary at any time. For example, the steady and unsteady deformations of a bubble in uniaxial and biaxial straining flows are examined for wide ranges of the Reynolds number and the Weber number. The computations reveal that a bubble in a uniaxial straining flow extends indefinitely if the Weber number is larger than a critical value (W > W_c). Furthermore, it is shown that a bubble may not achieve a stable steady state even at subcritical values of the Weber number if the initial state is sufficiently different from the steady state. Potential flow solutions for uniaxial straining flow show that an initially deformed bubble undergoes oscillatory motion if W < W_c, with a frequency of oscillation that decreases as the Weber number increases and equals zero at the critical Weber number.

In contrast to the uniaxial straining flow problem, a bubble at a finite Reynolds number in a biaxial straining flow has a stable steady state even though the deformation is extremely large. However, it is found that a bubble in a biaxial straining flow in the potential flow limit has exactly the same steady-state shape as in a uniaxial straining flow and a critical Weber number for breakup exists. Comparison of the results for the cases of high Reynolds numbers with the potential flow results suggests that the potential flow solution does not provide a uniformly valid approximation to the real flow at a high Reynolds number in the biaxial straining flow.

As a complementary analytical study to the numerical analysis, the method of domain perturbations is used to investigate the problem of a nearly spherical bubble in an inviscid, axisymmetric straining flow. The steady-state solutions suggest the existence of a limit point at a critical value of the Weber number. Furthermore, the asymptotic analysis for oscillation has provided a formula of oscillation frequency for the principal mode such as ω² = ω²₀(1 - 0.31W), where ω₀ is the oscillation frequency of a bubble in a quiescent fluid.

To include the weak viscous effect on the oscillation, a general formula for viscous pressure correction for a spherical bubble in an arbitrary axisymmetric flow has been derived in terms of the vorticity distribution. This formula has been applied to obtain the drag coefficient C_D = 48/R by directly integrating the normal stress over the surface for a spherical bubble in a uniform streaming flow at a high Reynolds number, which has so far been possible only via indirect macroscopic balances. The direct method also reveals that the drag coefficient up to O(R)⁻¹ depends only on the O(1) vorticity distribution right on the bubble surface, and is independent of the vorticity distribution inside the fluid.

Finally, a voidage bubble in a fluidized bed is considered in the low Reynolds number limit. The problem has been formulated as a generalized drop problem with one additional parameter. The analysis shows that the steady and unsteady deformations in the creeping flow limit are exactly the same as the conventional drop problem even though the flow fields are different. The effect of the additional parameter on deformation first appears when inertial effects are considered.

Thesis Files