Flowing Liquid He II: Critical Velocities and Dissipation Functions

Abstract

In Chapter 1, we remarked that the two-fluid equations of motion

$${\rho _s}D{v_s}/{D_t} = - ({\rho _s}/\rho )\nabla P + {\rho _s}s\nabla T $$

((8.1))

and

$${\rho _n}D{v_n}/Dt = - ({\rho _n}/\rho )\nabla P - {\rho _s}s\nabla T - {\eta _n}\nabla \times \nabla \times \nabla {v_n}$$

((8.2))

where

$$Dv/Dt = \partial v/\partial t + (v \cdot \nabla )v$$

((8.3))

are generally valid only in the limit of vanishing velocities. However, if, for example, v _n becomes large, we expect, in analogy with ordinary hydrodynamics, that the laminar flow described by equation (8.2) will be modified by the onset of turbulence. Similarly, with increasing values of v _s , it is physically reasonable that the frictionless character of the superfluid flow should be violated by the appearance of some sort of dissipation process. And, finally, at sufficiently large relative velocities, v _r = v _n - v _s , the assumption that the two velocity fields are independent ought to fail, and we should see some evidence for the mutual interaction between these fields. In other words, associated with each of the velocities v _n , v _s , and v _r , will be a sort of critical velocity, which we call v _n,c , v _s,c , and v _r,c , respectively, each signaling the onset of a flow regime different from that described by the steady-state linear equations of motion and therefore indicating the necessity for including one or more additional terms into these equations.