Three-dimensional boundary integral simulations of motion and deformation of bubbles with viscous effects

• We erect a 3d boundary element model of bubble with viscous effects.
• The normal second derivative of the velocity potential is got numerically.
• The loss of the mechanical energy transforms into the viscous dissipation energy.
• The jet velocity is slowed due to viscous dissipation.

Based on the velocity potential theory, a three-dimensional bubble model with the viscous effects is explored by using boundary element method. The velocity is irrotational, and the pressure is given by Bernoulli’s equation. The viscous component of the normal stress is evaluated and the viscous correction pressure is introduced to compensate for the non-zero shear stress. In this way, both normal stress and tangential stress boundary conditions are satisfied, and the weak viscous effects confined to the thin boundary layer around bubble surface are included. A spherical bubble moving in a viscous fluid is chosen as the case, and the 3d numerical results are compared with axi-symmetric ones and the analytical results. The good agreements between them validate the 3d model erected in this paper. On this basis, the non-spherical deformation of a single bubble under buoyancy force and the interaction of two bubbles are studied. The results of the bubbles with and without viscous effects are compared, and it is found that the jet velocity is slowed due to the viscous dissipation. The effects of Reynolds number and Froude number on the bubble dynamics are also investigated.