Recently we developed a theory for fast flows of entangled polymer melts which includes the processes of reptation, convective and reptation-driven constraint release, chain stretch and contour length fluctuations. The theory is derived from a stochastic microscopic equation of motion of the chain inside the tube and of the tube itself. As a result we obtain a partial differential equation for the tube tangent correlation function, the solution of which requires quite intensive calculations. At the same time the application of this theory to realistic flows (which is anything other than the laboratory rheometer) requires a simple and less computationally intensive set of equations for the stress tensor similar to the Giesekus, PTT, Larson or Pom-Pom equations. In particular, the last was derived from molecular theory for a generic type of branched polymer. In this paper we demonstrate that molecular tube theory can also provide a route to constructing a family of very simple differential constitutive equations for linear polymers. They capture the full model quite well and therefore can be used in flow solving software to model spatially inhomogeneous flows. We present a comparison of the proposed equations with our full model and with experimental data. (C) 2003 Elsevier B.V. All rights reserved.