Momentum transfer between ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ quasiparticles and surfaces: The effective viscosity of dilute solutions of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in ${}_{}{}^4{}_{}{}^{}\mathrm{He}$

We present the first measurements at millikelvin temperatures of both the real and the imaginary parts of the transverse-acoustic impedance of an oscillating surface in contact with dilute solutions of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, as well as with pure ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ with and without a surface boundary layer of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. Except for pure ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, large departures from hydrodynamic behavior are found which are inconsistent with slip theories incorporating specular scattering at the boundary. The effects of the ${}_{}{}^4{}_{}{}^{}\mathrm{rich}$ surface boundary layer and of surface roughness are discussed.