The structural model of the multiple-transient networks and the mechanism of the multiple-reptation entangled chains due to the dynamic reorganization in the entangled sites were extensively applied on the die swell of polymeric liquids in the steady simple shear flow. The total (recoverable and unrecoverable) viscoelastic free energy of deformation and flow, the constitutive equation and the expression of the simple shear (tanψ)were deduced from the conformational probability distribution function of the entangled polymer chains. It found that: (1) the magnitudes of simple shear (tanψ) depended not only on the free recoil (or recoverable strain) but also on the viscous heating (or unrecoverable strain); (2) the total recoil may be resolved into the instantaneous and delayed recoil. Based on these facts, the functions of the partition and two experiential fractions of the recoverable (1(-W)γ) and the unrecoverable ((-W)γ) conformations for the recoil and viscous heating of polymeric liquids were defined correspondingly. Then the correlation of the instantaneously and ultimately (or total) recoverable strains to the (N1/τ12)w and the fraction of trans-form conformation was obtained. After introducing the condition of uniform two-dimensional extension (αx=αy=α, αz=α-2) and the swell ratio (B=α), two sets of equations on the instantaneous and ultimate swelling ratios (BE, BEVT) were obtained, and a method to determine the fraction of the recoverable transform conformation were proposed.The equations of BE and BE5 were verified by the experimental data of HDPE (high denisity polyethylene)at two different high temperatures. It shows that the molecular theory of die swell can be used to predict the correlation of the swelling to the (N1/τ12)w and the fraction of trans-form conformation.