Increase of effective viscosity in bubbly liquids from transient bubble deformation

The effective viscosity of bubbly liquids is measured using the Stokes drag of a falling sphere, i.e. falling sphere viscometry. This method can evaluate the influence of a bubble's transient deformation. Viscosity relative to a single-phase fluid is directly obtained by the terminal falling velocities of the sphere. When bubbles are distributed around the sphere up to a void fraction of α, the following results are obtained. The relative viscosity for spherical bubble dispersion agrees with the Stokes-Einstein formula; 1+α. For large capillary numbers, relative viscosity converges to approximately 1-(5/3)α because bubble deformation is fully yielded. Between these two states, relative viscosity has a value higher than in simple shear flow. The critical capillary number is found to be 3.5, being five times as that of simple shear flow. The viscosity-increasing mechanism for trans-critical capillary numbers is deduced from the fact that bubbles have transient deformation along the streamline.