Viscosity of magmas containing highly deformable bubbles

The shear viscosity of a suspension of deformable bubbles dispersed within a Newtonian ¯uid 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 suf®ciently large strains the viscosity of suspension is greater than that of the suspending ¯uid, 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! 1 (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 t