Rheology of dilute bubble suspensions in unsteady shear flows

The viscoelasticity of a dilute bubble suspension is theoretically derived from the constitutive equation originally for a dilute emulsion proposed by Frankel & Acrivos (J. Fluid Mech., vol. 44, issue 1, 1970, pp. 65-78). Non-dimensionalization of the original tensor equation indicates that the viscoelasticity is systematized for a given void fraction by the capillary number Ca and dynamic capillary number Cd, representing the bubble deformability and unsteadiness of bubble deformation. Comprehensive evaluation of the viscoelasticity according to the volume fraction, Ca and Cd reveals that whether the viscosity increases or decreases depends on whether Ca or Cd exceeds a common critical value. In addition, it is indicated that the bubble suspension has the most prominent viscoelasticity when the time scale of the shear deformation is the same as the relaxation time of the suspended bubble and when the bubbles keep a spherical shape, that is, Ca << 1 and Cd = 1. The applicability of this theory in flow prediction was examined in a Taylor-Couette system, and experimentally good agreement was confirmed.