Calculation of He II flow in tubes

Abstract

The flow of superfluid helium through a tube with different temperatures at the ends differs considerably from that of a Newtonian fluid. The strong dependence of the thermodynamic properties on temperature, the internal convection mechanism and the structure of superfluid turbulence causes unusual flows. The equations for the flow of He II are integrated using a new one-dimensional, steady state model to study the flow in a tube. A wide range of driving conditions is studied. The temperature and pressure profiles along the tube fall into four classes. A dimensionless parameter called σ is defined which determines the progression through the four classes of behavior. The deviation of the flow from Newtonian is measured by σ. Significant maxima of the temperature and pressure can occur between the ends of the tube for large values of σ. The shapes of the profiles and the mass flux depend primarily on σ, the geometry and the boundary conditions. Formulas are presented which relate the variables of interest to the boundary conditions. These formulas result from averaging the equations of motion along the tube. A general and unified approach, based on σ, is presented for analyzing experimental data and designing new experiments. It is shown that the common practice of neglecting the pressure term in the energy equation results in poor prediction for many situations. The occurrence of large maxima of pressure and temperature imply that the interpretation of some of the experimental data of the literature should be reconsidered.