The formation of gas bubbles and their subsequent rise due to buoyancy are very important fundamental phenomena that contribute significantly to the hydrodynamics in gas−liquid reactors. The rise o...

Bubble Formation and Bubble Rise Velocity in Gas−Liquid Systems: A Review

A constitutive model is developed to characterize a general class of polymer and polymer-like materials that displays hyperelastic orthotropic mechanical behavior. The strain energy function is derived from the entropy change associated with the deformation of constituent macromolecules and the strain energy change associated with the deformation of a representative orthotropic unit cell. The ability of this model to predict nonlinear, orthotropic elastic behavior is examined by comparing the theory to experimental results in the literature. Simulations of more complicated boundary value problems are performed using the finite element method. ©2002 ASME

A microstructurally based orthotropic hyperelastic constitutive law

A front tracking method for a deformable intravascular bubble in a tube with soluble surfactant transport

A finite element code based on the level-set method is used to perform direct numerical simulations (DNS) of the transient and steady-state motion of bubbles rising in a viscoelastic liquid modelled by the Oldroyd-B constitutive equation. The role of the governing dimensionless parameters, the capillary number (Ca), the Deborah number (De) and the polymer concentration parameter c, in both the rising speed and the deformation of the bubbles is studied. Simulations show that there exists a critical bubble volume at which there is a sharp increase in the terminal velocity with increasing bubble volume, similar to the behaviour observed in experiments, and that the shape of both the bubble and its wake structure changes fundamentally at that critical volume value. The bubbles with volumes smaller than the critical volume are prolate shaped while those with volumes larger than the critical volume have cusp-like trailing ends. In the latter situation, we show that there is a net force in the upward direction because the surface tension no longer integrates to zero. In addition, the structure of the wake of a bubble with a volume smaller than the critical volume is similar to that of a bubble rising in a Newtonian fluid, whereas the wake structure of a bubble with a volume larger than the critical value is strikingly different. Specifically, in addition to the vortex ring located at the equator of the bubble similar to the one present for a Newtonian fluid, a vortex ring is also present in the wake of a larger bubble, with a circulation of opposite sign, thus corresponding to the formation of a negative wake. This not only coincides with the appearance of a cusp-like trailing end of the rising bubble but also propels the bubble, the direction of the fluid velocity behind the bubble being in the opposite direction to that of the bubble. These DNS results are in agreement with experiments.

Transient and steady state of a rising bubble in a viscoelastic fluid

The flow field around air bubbles rising in aqueous polyacrylamide (PAAm) solutions was studied using a particle image velocimetry (PIV) system. This flow was analyzed in the vicinity of the critical bubble volume where the discontinuity of the terminal bubble velocity occurs. It was found that the flow configuration changes drastically below and above the critical bubble volume. The flow just below the critical bubble volume shows an upward flow at the front and back of the bubble with a symmetrical vortex around the bubble. The flow field obtained for a volume just above the critical one shows the appearance of the so-called negative wake behind the bubble. Additionally, it was found that the container walls affect significantly the magnitude of the terminal velocity as well as the velocity jump. However, the critical volume at which the velocity jump appears does not change for different container sizes.

The flow of non-Newtonian fluids around bubbles and its connection to the jump discontinuity

The motion of a gas bubble in a non-Newtonian fluid has been further examined in order to determine the conditions for the possible existence of a discontinuity in the bubble velocity-bubble volume log–log plot. It has been proposed in the past that this phenomenon was the result of a sudden change in the hydrodynamics of the moving bubble, resulting in a transition from a Stroke to a Hadamard regime. Furthermore, this abrupt transition was only qualitatively attributed to the elasticity of the fluid. Using our data as well as those of Leal et al., we demonstrate here that the discontinuity results as a balance between elastic and Marangoni instabilities, providing another major difference between Newtonian and non-Newtonian hydrodynamics.

Bubble velocities: further developments on the jump discontinuity

The drag force experienced by bubbles, rising freely in an inelastic shear-thinning fluid is calculated. In particular, the effect of contaminants is considered when the continuous phase can be represented by the power-law or the three parameter Carreau models. Two cases are examined in order to take into account the effect of surfactants: (a) interfacial mass transfer is taken into account via a perturbation around the thermodynamic equilibrium values using Gibbs elasticity, (b) a continuous distribution of impurities over the whole interface (uniform retardation) leading to Marangoni stresses. The relations obtained are based on perturbation methods around the Newtonian solution. The drag force can be evaluated via the rheological model parameters, the physical properties of the system and the concentration of impurities in the fluid. The presence of contamination was found to increase the drag resulting in a decrease of the rise velocity. Inversely, the effect of shear thinning is to decrease the drag, thus leading to a balance between both effects.

The slow motion of a single gas bubble in a non-Newtonian fluid containing surfactants

A stress jump, defined as the instantaneous gain or loss of stress on startup or cessation of a deformation, has been predicted by various models and has relatively recently been experimentally observed. In this paper, the internal viscosity idea is incorporated into the transient network model. Via appropriate approximations, we obtain a closed constitutive equation where the total stress equals the sum of an elastic contribution and a viscous contribution. As the latter is rate dependent, the model predicts a stress jump and we consider data on shear flow in this contribution. We successfully compare the model predictions with the stress jump measurements of Liang and Mackay [C.H. Liang, M.E. Mackay, J. Rheol. 37 (1993) 149]. The model yields good quantitative predictions of the steady, transient and dynamic material functions.

C.F. Chan Man Fong

Papers

Bubble velocities: further developments on the jump discontinuity

The slow motion of a single gas bubble in a non-Newtonian fluid containing surfactants

Network constitutive equation with internal viscosity: application to stress jump prediction