The rheology of a bubbly liquid

A semiempirical constitutive model for the visco-elastic rheology of bubble suspensions with gas volume fractions phi < 0.5 and small deformations (Ca much less than 1) is developed. The model has its theoretical foundation in a physical analysis of dilute emulsions. The constitutive equation takes the form of a linear Jeffreys model involving observable material parameters: the viscosity of the continuous phase, gas volume fraction, the relaxation time, bubble size distribution and an empirically determined dimensionless constant. The model is validated against observations of the deformation of suspensions of nitrogen bubbles in a Newtonian liquid (golden syrup) subjected to forced oscillations. The effect of and frequency of oscillation f on the elastic and viscous components of the deformation are investigated. At low f, increasing phi leads to an increase in viscosity, whereas, at high f, viscosity decreases as phi increases. This behaviour can be understood in terms of bubble deformation rates and we propose a dimensionless quantity, the dynamic capillary number Cd, as the parameter which controls the behaviour of the system. Previously published constitutive equations and observations of the rheology of bubble suspensions are reviewed. Hitherto apparently contradictory findings can be explained as a result of Cd regime. A method for dealing with polydisperse bubble size distributions is also presented.