Dissipative superfluid mass flux through solid $^4$He

The thermo-mechanical effect in superfluid helium is used to create a chemical potential difference, $\Delta \mu$, across a liquid or solid $^4$He sample and induce a mass flux. With an improved technique, measurements of the mass flux, $F$, through a solid-filled sample cell at several fixed helium sample temperatures, $T$, have been done as a function of $\Delta \mu$. And, measurements of $F$ (in the range $100 < T < 550$~mK) have been done as a function of temperature for several fixed values of $\Delta \mu$. The temperature dependence of the flow through solid helium above $100$~mK is confirmed to show a reduction of the flux with increasing temperature, while for liquid helium there is no marked temperature dependence in the temperature range studied. The dependence of $F$ on $\Delta \mu$ documents in some detail the dissipative nature of the flow for the case of a solid helium- filled sample cell. In the case of solid helium we observe $F\sim \Delta \mu^b$ with $b \approx 0.3$, which is consistent with expectations for 1D superfluidity. The relationship between this work and the various torsional oscillator NCRI results is not clear. We may be exploring different phenomena.