Shear viscosity of bubble suspensions under time-varying pressures

The rheology of bubble suspensions is critical for the prediction and control of bubbly flows in various industrial processes. It is well known that bubble suspensions exhibit shear-thinning behaviour due to bubble shape deformation under pure shear, but the shear rheological response to dilatation under time-varying pressure remains unexplored. Here, we propose a constitutive equation for dilute bubble suspensions that accounts for both shear and dilatational effects, demonstrating that bubble compressibility can significantly influence the shear viscosity under time-varying pressures. Under constant-rate pressure variations, compression leads to a progressive reduction in viscosity, whereas decompression induces an increase. This peculiar compression-thinning behaviour arises microscopically from the fact that a shrinking bubble surface effectively weakens the flow resistance of the surrounding liquid. Under oscillatory pressure, the amplitude of the dilatation-induced viscosity grows with frequency, and the phase shift of the shear stress response transitions from $-\pi /2$ to $-\pi$ . Notably, viscosity troughs emerge during oscillation cycles, leading to transient flow resistance lower than that in pure shear, indicating a potential route for drag reduction when combined with shear-thinning. These findings highlight bubble compressibility as a controllable factor for tuning bubbly flow rheology, with potential for enhanced flow efficiency in practical applications.