This thesis documents mathematical and computational studies of the hydrodynamic motion and stability of plasma-liquid interactions.

The thesis begins with a thorough review of the rapidly-emerging technological importance of plasma-liquid interactions in industry and nuclear fusion research. Deficiencies in current theories of plasma-liquid interactions are highlighted and used to identify those aspects which require significant improvements. The previously-unexplored effect of charge separation and sheath formation at a plasma-facing liquid surface is examined in detail. Two particular findings, which are supported by experimental observations, are the enhancement in droplet ejection rates from bursting bubbles at the surface of electrically-biased liquids and the pulsed emission of droplets from plasma-facing liquid surfaces which are deformed into Taylor cones by the electric field of the plasma sheath.

The behaviour of plasma-immersed droplets is also examined. Some commonly-used models of macroparticle charging are extended to include plasma flows, nonspherical particles or magnetic fields. The stability of droplets in plasmas is assessed and the rapid spinning of droplets in magnetised plasmas is shown to lead to their rotational breakup. This disruption mechanism is incorporated into simulations of droplet transport in tokamaks and results in forked trajectories which are strikingly-similar to those observed with fast cameras in tokamaks. The beneficial consequences of this rotational breakup process on the operation of next-generation fusion devices are emphasised.