Flow onset for a single bubble in yield-stress fluids

We use computational methods to determine the minimal yield-stress required in order to hold static a buoyant bubble in a yield-stress liquid. The static limit is governed by the bubble shape, the dimensionless surface tension ($\gamma$) and the ratio of the yield-stress to the buoyancy stress ($Y$). For a given geometry, bubbles are static for $Y>Y_c$, which we determine for a range of shapes. Given that surface tension is negligible, long prolate bubbles require larger yield-stress to hold static compared to oblate bubbles. Non-zero $\gamma$ increases $Y_c$ and for large $\gamma$ the yield-capillary number ($Y\mathrm{/}\gamma$) determines the static boundary. In this limit, although bubble shape is important, bubble orientation is not. 2D planar and axisymmetric bubbles are studied.