On the Transition from Potential Flow to Turbulence Around a Microsphere Oscillating in Superfluid ^4He

The flow of superfluid $^4$He around a translationally oscillating sphere, levitating without mechanical support, can either be laminar or turbulent, depending on the velocity amplitude. Below a critical velocity $v_c$ that scales as $\omega ^{1/2}$, and is temperature independent below 1 K, the flow is laminar (potential flow). Below 0.5 K the linear drag force is caused by ballistic phonon scattering that vanishes as T$^4$ until background damping, measured in the empty cell, becomes dominant for T $<$ 0.1 K. Increasing the velocity amplitude above $v_c$ leads to a transition from potential flow to turbulence, where the large turbulent drag force varies as $(v^2 - v_c^2)$. In a small velocity interval $\Delta v / v_c \le 3 \%$ above $v_c$, the flow is unstable below 0.5 K, switching intermittently between both patterns. From time series recorded at constant temperature and driving force, the lifetimes of both phases are analyzed statistically. We observe metastable states of potential flow which, after a mean lifetime of 25 minutes, ultimately break down due to vorticity created by natural background radioactivity. The lifetimes of the turbulent phases have an exponential distribution, and the mean increases exponentially with $\Delta v^2$. We investigate the frequency at which the vortex rings are shed from the sphere. Our results are compared with recent data of other authors on vortex shedding by moving a laser beam through a Bose-Einstein condensate. Finally, we show that our observed transition to turbulence belongs to the class of "supertransient chaos" where lifetimes of the turbulent states increase faster than exponentially. Peculiar results obtained in dilute $^3$He - $^4$He mixtures are presented in the Appendix.