Rheology of a dilute bubble suspension in unsteady shear flows

Bubbly flows are ubiquitous in both industry and nature. A small amount of bubbles changes the original rheological properties and modifies, for example, the flow behavior of liquid products and the eruption pattern of volcanoes. In this study, the viscoelasticity of a dilute bubble suspension is elucidated based on the constitutive equation theoretically derived by Frankel and Acrivos (The constitutive equation for a dilute emulsion, J. Fluid Mech. 44(1), 1970). Nondimensionalization of the constitutive equation reveals that the viscoelasticity is systematized by three parameters, volume fraction, capillary number Ca, and dynamic capillary number Cd, where Ca and Cd are non-dimensional parameters indicating deformability of the suspended bubble in a Newtonian medium and unsteadiness of the bubble deformation. The viscoelasticity was comprehensively investigated according to the volume fraction, Ca, and Cd, and it was revealed that the viscosity increase or decrease depends on whether Ca or Cd exceeds a common critical value. In addition, it was turned out that the most prominent viscoelasticity emerges when the time scale of the shear deformation is as same as the relaxation time of the bubble and when the bubbles keep in a spherical shape, that is, Cd = 1 and Ca << 1. To further show that this theoretical work is applicable to flow prediction beyond just the rheological evaluation, a typical case of unsteady laminar flows in a Taylor-Couette geometry is demonstrated and compared with experimental results.