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J. Hüpfl:
"Bjerknes Forces: a layer potential approach";
Supervisor: J. Melenk; Institut für Analysis und Scientific Computing, 2021; final examination: 2020-12-22.

The movements of one and two interacting bubbles inside a weak acoustic incident pressure field have been extensively studied in the past. The results have focused on bubble systems with limited or no interaction, which greatly simplified the problem. It has been shown that considering low order couplings between the bubbles produce sign reversal effects of the forces. This can produce multi bubble systems with stable separation distances, which cannot be explained by a model with no coupling. This thesis covers theoretical derivations of the movement of two fully interacting bubbles inside an acoustic incident pressure field. A novel approach under the assumption of an inviscous fluid is introduced. The total forces are approximated in a new way and the governing equations are linearised, allowing the use of layer potential techniques. We further calculate the asymptotic forces on one and two bubbles and show that the results are consistent with past literature. Common approaches with only low order interactions produce growing errors for smaller bubble separations due to higher order coupling terms getting more significant. In contrast, our approach considers all coupling terms in the linearised model and should more accurately approximate bubbles with strong interactions. Partial results are verified by numerical computations and simulations of two bubble systems are made. Multiple sign reversal effects are observed which show the complex behaviour of the forces. Further applications and analysis of this approach could be a fruitful path of understanding the Bjerknes forces near resonance frequencies and for bubbles in close proximity with each other.

Bjerkne forces; layer potentials; bubbles in accoustic fieldasymptotic analysis