Models for supercritical motion in a superfluid Fermi liquid

We study the drag force on objects moving in a Fermi superfluid at velocities on the order of the Landau velocity $v_L$. The expectation has been that $v_L$ is the critical velocity beyond which the drag force starts to increase towards its normal-state value. This expectation is challenged by a recent experiment measuring the heat generated by a uniformly moving wire immersed in superfluid $^3$He. We introduce the basis for the calculation of the drag force on a macroscopic object using the Fermi-liquid theory of superfluidity. As a technical tool in the calculations we propose a boundary condition that describes diffuse reflection of quasiparticles from a surface on a scale that is larger than the superfluid coherence length. We calculate the drag force on steadily moving objects of different sizes. For an object that is small compared to the coherence length, we find a drag force that is in accordance with the expectation. For a macroscopic object we need to take into account the spatially varying flow field around the object. At low velocities this arises from ideal flow of the superfluid. At higher velocities the flow field is modified by excitations that are created when the flow velocity locally exceeds $v_L$. The flow field causes Andreev reflection of quasiparticles and thus leads to change in the drag force. We calculate multiple limiting cases for a cylinder-shaped object. In the absence of quasiparticle-quasiparticle collisions we find that the critical velocity is larger than $v_L$ and the drag force (per cross-sectional area) at $2v_L$ is reduced by an order of magnitude compared to the case of a small object. In a collision-dominated limit the flow shows signs of instability at a velocity below $v_L$.