Stability analysis of cusped bubbles in viscoelastic flows

<jats:p>Experiments have established that an axisymmetric sharp trailing edge of a gas bubble in a viscoelastic fluid can develop into a three-dimensional ‘knifelike’ shape under certain conditions (high capillary number, large bubble size). A numerical study is conducted to discover the physics of this phenomenon. The axisymmetric deformation of a bubble rising buoyantly in a viscoelastic fluid is simulated by solving the axisymmetric flow equations coupled with the constitutive equations of the finitely extensible nonlinear elastic Chilcott–Rallison (FENE-CR) model. The three-dimensional temporal linear stability analysis of this axisymmetric base state is carried out. The dominant eigenvalue which is indicative of the growth rate of the perturbations is computed. The only unstable eigenmode has azimuthal wavenumber <jats:italic>m</jats:italic> equal to 2. The corresponding eigenfunction shows that indeed a sharp axisymmetric tail develops a knife-edge form. A further investigation of the energy budget of the disturbances for <jats:italic>m</jats:italic> = 2 is performed to determine the production and dissipation terms affecting the growth of this instability. It is shown that the normal gradient of the base-state pressure along the free surface plays an important role in the evolution of the instability.</jats:p>