Michael Loewenberg

Rheology of bubble-bearing magmas

The rheology of bubble-bearing suspensions is investigated through a series of three-dimensional boundary integral calculations in which the effects of bubble deformation, volume fraction, and shear rate are considered. The behaviour of bubbles in viscous flows is characterized by the capillary number, Ca, the ratio of viscous shear stresses that promote deformation to surface tension stresses that resist bubble deformation. Estimates of Ca in natural lava flows are highly variable, reflecting variations in shear rate and melt viscosity. In the low capillary number limit (e.g., in carbonatite flows) bubbles remain spherical and may contribute greater shear stress to the suspension than in high capillary number flows, in which bubble deformation is significant. At higher Ca, deformed bubbles become aligned in the direction of flow, and as a result, contribute less shear stress to the suspension. Calculations indicate that the effective shear viscosity of bubbly suspensions, at least for Ca<0.5, is a weakly increasing function of volume fraction and that suspensions of bubbles are shear thinning. Field observations and qualitative arguments, however, suggest that for sufficiently large Ca (Ca greater than about 1) the effective shear viscosity may be less than that of the suspending liquid. Bubbles reach their quasi-steady deformed shapes after strains of order one; for shorter times, the continuous deformation of the bubbles results in continual changes of rheological properties. In particular, for small strains, the effective shear viscosity of the suspension may be less than that of the liquid phase, even for small Ca. Results of this study may help explain previous experimental, theoretical, and field based observations regarding the effects of bubbles on flow rheology.

Drop breakup in three-dimensional viscous flows

A new three-dimensional boundary integral algorithm is presented that is capable of simulating the process of drop breakup in viscous flows. The surface discretization is fully adaptive, thus providing accurate resolution of the highly deformed drop shapes that are characteristic of breakup events. Our algorithm is used to study drop breakup in shear flow and in buoyancy; the predictions are compared with experimental observations.

Collision of two deformable drops in shear flow

A boundary integral formulation is used to investigate the interaction between a pair of deformable drops in a simple shear flow. The interactions do not promote appreciably the breakup of the drops. For certain ratios of the viscosities of the drops and the suspending fluid, the lubrication gap that separates the two drops can diminish rapidly in the extensional quadrant of the flow. Slight deformation endows the drops with an apparent short-range repulsive interaction: drop coalescence requires van der Waals attraction which was not included in this study. From the trajectories of different collisions, the self-diffusion coefficients that describe the cross-flow migration of the non-Brownian drops in a dilute sheared emulsion are obtained. The self-diffusivities are very anisotropic, depend strongly on the viscosity ratio, and depend modestly on the shear rate.

Drop breakup and fragment size distribution in shear flow

We report a study on the deformation and breakup of drops in an impulsively started shear flow under Stokes flow conditions using boundary-integral simulations and video-microscopy experiments. Two independent techniques are used for determining the physical parameters of the system from the combined use of numerical simulations and experiments. Accurate breakup criteria (critical capillary numbers) are presented for a range of viscosity ratios. The time required for breakup events has a broad minimum corresponding to moderate shear rates. The size distribution of droplets produced by breakup events is shown to scale with the critical size drop for breakup in shear. A simplified model, based on this finding, is developed for the size distribution in a sheared emulsion. According to the model, the drop size distribution in a given emulsion depends only on the average initial drop size and the shear rate.

Viscosity of magmas containing highly deformable bubbles

Abstract The shear viscosity of a suspension of deformable bubbles dispersed within a Newtonian fluid is calculated as a function of the shear rate and strain. The relative importance of bubble deformation in the suspension is characterized by the capillary number (Ca), which represents the ratio of viscous and surface tension stresses. For small Ca, bubbles remain nearly spherical, and for sufficiently large strains the viscosity of suspension is greater than that of the suspending fluid, i.e. the relative viscosity is greater than 1. If Ca>O(1) the relative viscosity is less than one. In the limit that Ca→∞ (surface tension is dynamically negligible), numerical calculations for a suspension of spherical bubbles agree well with the experimental measurements of Lejeune et al. (1999, Rheology of bubble-bearing magmas. Earth Planet. Sci. Lett., vol. 166, pp. 71–84). In general, bubbles have a modest effect on the relative viscosity, with viscosity changing by less than a factor of about 3 for volume fractions up to 50%.

Deformation of a surfactant-covered drop in a linear flow

We study the effect of adsorbed surfactant on drop deformation in linear flows by means of analytical solutions for small perturbations of the drop shape and surfactant distribution, and by numerical simulations for large distortions. We consider a drop with the same viscosity as the suspending fluid. Under these conditions, the problem simplifies because the disturbance flow field results solely from the interfacial stresses that oppose the distortion of shape and surfactant distribution induced by the incident flow. A general form of perturbation analysis valid for any flow is presented. The analysis can be carried out to arbitrary order given its recursive structure; a third-order perturbation solution is explicitly presented. The expansions are compared to results from boundary integral simulations for drops in axisymmetric extensional and simple shear flows. Our results indicate that under weak-flow conditions, deformation is enhanced by the presence of surfactant, but the leading-order perturbation of the drop shape is independent of the (nonzero) surfactant elasticity. In strong flows, drop deformation depends nonmonotonically on surfactant elasticity. The non-Newtonian rheology in a dilute emulsion that results from drop deformation and surfactant redistribution is predicted. Shear thinning is most pronounced for low values of the surfactant elasticity. In the weak-flow limit with finite surfactant elasticity, the emulsion behaves as a suspension of rigid spheres. In strong flows, the stresses can approach the behavior for surfactant-free drops.We study the effect of adsorbed surfactant on drop deformation in linear flows by means of analytical solutions for small perturbations of the drop shape and surfactant distribution, and by numerical simulations for large distortions. We consider a drop with the same viscosity as the suspending fluid. Under these conditions, the problem simplifies because the disturbance flow field results solely from the interfacial stresses that oppose the distortion of shape and surfactant distribution induced by the incident flow. A general form of perturbation analysis valid for any flow is presented. The analysis can be carried out to arbitrary order given its recursive structure; a third-order perturbation solution is explicitly presented. The expansions are compared to results from boundary integral simulations for drops in axisymmetric extensional and simple shear flows. Our results indicate that under weak-flow conditions, deformation is enhanced by the presence of surfactant, but the leading-order perturbation ...

Near-contact motion of surfactant-covered spherical drops

A lubrication analysis is presented for the near-contact axisymmetric motion of spherical drops covered with an insoluble non-diffusing surfactant. Detailed results are presented for the surfactant distribution, the interfacial velocity, and the gap width between the drop surfaces. The effect of surfactant is characterized by a dimensionless force parameter: the external force normalized by Marangoni stresses. Critical values of the force parameter have been established for drop coalescence and separation. Surfactant-covered drops are stable to rapid coalescence for external forces less than 4π kTac 0 , where c 0 is the surfactant concentration at the edge of the near-contact region and a is the reduced drop radius. For subcritical forces, the behaviour of surfactant-covered drops is described by two time scales: a fast time scale characteristic of near-contact motion between drops with clean interfaces and a slow time scale associated with rigid particles. The surfactant distribution evolves on the short time scale until Marangoni stresses approximately balance the external force. Supercritical values of the external force cannot be balanced; coalescence and separation occur on the fast time scale. The coalescence time normalized by the result for drops with clean interfaces is independent of the viscosity ratio and initial gap width. Under subcritical force conditions, a universal long-time behaviour is attained on the slow time scale. At long times, the surfactant distribution scales with the near-contact region and the surface velocity is directed inward which impedes the drop approach and accelerates their separation compared to rigid particles. For drops pressed together with a sufficiently large subcritical force, a shrinking surfactant-free clean spot forms. Surfactant-covered drops exhibit an elastic response to unsteady external forces because of energy stored in the surfactant distribution.

Hydrodynamic interactions and collision efficiencies of spherical drops covered with an incompressible surfactant film

A theory is developed for the hydrodynamic interactions of surfactant-covered spherical drops in creeping flows. The surfactant is insoluble, and flow-induced changes of surfactant concentration are small, i.e. the film of adsorbed surfactant is incompressible. For a single surfactant-covered drop in an arbitrary incident flow, the Stokes equations are solved using a decomposition of the flow into surface-solenoidal and surface-irrotational components on concentric spherical surfaces. The surface-solenoidal component is unaffected by surfactant; the surface-irrotational component satisfies a slip-stick boundary condition with slip proportional to the surfactant diffusivity. Pair hydrodynamic interactions of surfactant-covered bubbles are computed from the one-particle solution using a multiple-scattering expansion. Two terms in a lubrication expansion are derived for axisymmetric near-contact motion. The pair mobility functions are used to compute collision efficiencies for equal-size surfactant-covered bubbles in linear flows and in Brownian motion. An asymptotic analysis is presented for weak surfactant diffusion and weak van der Waals attraction

Rheology of a dilute emulsion of surfactant-covered spherical drops

The rheology of a diluted emulsion of surfactant-covered spherical drops has been investigated. A diluted film of insoluble surfactant is assumed. A matrix formulation of the problem is derived and analyzed by perturbation expansions for low- and high-shear rates, and for high-viscosity drops; the high-viscosity expansion converges rapidly for a wide range of parameters. Our theory provides a quantitative description of shear thinning and normal stress differences that occur as a result of surfactant redistribution.

Small-deformation theory for a surfactant-covered drop in linear flows

A small-deformation perturbation analysis is developed to study the effect of surfactant on drop dynamics in viscous flows. The surfactant is assumed to be insoluble in the bulk-phase fluids; the viscosity ratio and surfactant elasticity parameters are arbitrary. Under small-deformation conditions, the drop dynamics are described by a system of ordinary differential equations; the governing equations are given explicitly for the case of axisymmetric and two-dimensional imposed flows. Analytical results accurate to third order in the flow-strength parameter (capillary number) are derived (i) for the stationary drop shape and surfactant distribution in simple shear and axisymmetric straining flows, and (ii) for the rheology of a dilute emulsion in shear flow which include a shear-thinning viscosity and non-zero normal stresses. For drops with clean interfaces, the small-deformation theory presented here improves the results of Barthes-Biesel & Acrivos ( J. Fluid Mech ., vol. 61, 1973, p. 1). Boundary integral simulations are used to test our theory and explore large-deformation conditions.