Hydrodynamic interaction between coaxially rising bubbles in elasto-visco-plastic materials: Bubbles with a wide range of relative sizes

We consider the buoyancy-driven rise and interaction between two initially stationary and gravity-aligned bubbles of wide radii ratio and constant volume in an elasto-visco-plastic material, extending our previous work regarding bubbles of equal radii Carbopol solution and model its rheology with the Saramito-Herschel-Bulkley constitutive model. Initially, we investigate the dynamics for a specific initial separation distance in a wide range of bubble radii, and we determine the conditions leading to three distinct patterns: bubble approach, bubble separation, and establishment of a constant distance between them. Specifically, when the leading bubble (LB) is smaller than the trailing bubble (TB), the bubbles approach each other due to the smaller buoyancy of the leading bubble. Strong attraction also occurs when the ratio of buoyant over viscous force of both bubbles is considerable. On the other hand, when the size of the TB is such that this ratio is moderate or small, the pattern is dictated by the size of the LB: A significantly larger LB compared to the trailing one causes separation of the pair. On the contrary, an only slightly larger LB may result in the bubbles rising with the same terminal velocity establishing a constant distance between them, the magnitude of which is mainly determined by the elastic response of the surrounding medium. The coupling of a negative wake behind the LB with a slight modification of the stresses exerted at its rear pole generates this dynamic equilibrium. The same equilibrium may be achieved by other specific pairs of bubble sizes for different initial distances of the pair if a critical initial distance is exceeded. Below this critical value, the bubbles approach each other. Finally, we construct maps of the three patterns with trailing bubble radius versus bubble radii ratio for different initial separation distances and material properties.