Thermodynamics and hydrodynamics of \u3csup\u3e3\u3c/sup\u3eHe-\u3csup\u3e4\u3c/sup\u3eHe mixtures

\u3cp\u3eThe specific heat of liquid \u3csup\u3e3\u3c/sup\u3eHe–\u3csup\u3e4\u3c/sup\u3eHe mixtures is usually written in terms of the sum of the specific heat of a \u3csup\u3e3\u3c/sup\u3eHe-quasiparticle gas and the specific heat of the pure \u3csup\u3e4\u3c/sup\u3eHe component. The thermodynamics based on this starting point is derived. Relations of important quantities and their low- and high-temperature limits are given. These are used to derive expressions for the velocity of second sound. This latter quantity is a very important source of information for the Fermi gas properties. Finally, the Fermi gas parameters are summarized in the chapter. The experimental aspects of the \u3csup\u3e3\u3c/sup\u3eHe–\u3csup\u3e4\u3c/sup\u3eHe hydrodynamics are treated. The appearance of mutual friction that has long been neglected in this field is discussed, together with the properties of the critical velocities. The phenomenological equations of motion are given. The occurrence of mutual friction is a strong indication that \u3csup\u3e4\u3c/sup\u3eHe vortices play an important role in \u3csup\u3e3\u3c/sup\u3eHe–\u3csup\u3e4\u3c/sup\u3eHe hydrodynamics. From the equation of motion of quantized \u3csup\u3e4\u3c/sup\u3eHe vortices, the observed cubic velocity dependence of the \u3csup\u3e4\u3c/sup\u3eHe chemical potential difference is explained on purely dimensional grounds. A differential equation is given from which the temperature profile in a cylindrical tube in which \u3csup\u3e3\u3c/sup\u3eHe flows through superfluid \u3csup\u3e4\u3c/sup\u3eHe can be calculated.\u3c/p\u3