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In order to model polymer fluid flows within the framework of continuum mechanics, it is necessary to write a theological state equation that establishes a relationship between the stress tensor for a polymer system and the velocity-gradient tensor. This can be done either by a phenomenological approach [1], generalizing the available experimental data, or by using some model concepts of the structure of polymer materials [2][3][4][5][6][7][8][9][10][11][12][13]. However, both approaches will probably not provide us .with a simple enough rheological constitutive relation suitable for a description of various flows of linear polymer solutions and melts. Therefore, the problem of construction of a succession of rheologieal constitutive relations taking new and more subtle effects into account at each step is of great importance. The success of such a procedure is determined by the selection of an initial approximation and by the rules of transition to subsequent approximations.At various times the well-known BKZ [1] or Doi-Edwards [2, 4] theological models have been proposed as initial approximations.In the present work, zeroth approximations of the molecular theory of viscoelasticity [5][6][7][8][9] for given small parameters are considered, and the possibility of using these relations as a first approximation in the construction of a sequence of rheological constitutive relations is demonstrated.Rheological Constitutive Relation. The model concepts coming from the simulation of polymerchain motion serve as a basis for different microstructural approaches to the description of the dynamics of polymer systems. Here, statement of the equations of dynamics of a macromolecule is not possible without some additional assumptions. Two essential assumptions are used most often: 1) a monomolecular approximation, in which a single selected macromolecule moving in an effective relaxing medium formed by a solvent and by the other macromolecules is considered instead of the entire set of macromolecules in the volume; 2) the ability to consider the motion of a selected macromolecule as the motion of N (N >> 1) centers of friction (beads) linked together one after another by elastic entropy forces (springs). These assumptions. which are only hypothetical, bring us to the equations of dynamics of a macromolecule [5][6][7][8]: D r -~ T 8 + Tg = -(r -coStp~{).Here p7 and ~b~ are the generalized coordinate and velocity; m is the mass of a bead; F 7 is the force of hydrodynamic entrainment; T/~ is the force of internal viscosity; q~7 is a random force; 2T#Aa is the coefficient of elasticity, 7-is the relaxation time of the environment; ( is the coefficient of friction of a bead in a monomeric fluid; B~ and E~ are the tensor coefficients of friction of a bead; l.,ij is the velocity-gradient tensor; ~ij is the antisymmetrized velocity-gradient tensor; D/Dt is the Jauman's tensor derivative; the Latin indices i,

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532.135The effect of the molecular mass of a polymer sample on the dependence of the stationary viscosity on the velocity gradient upon simple shear and uniaxial tension is studied. The model of the dynamics of a suspension of noninteracting dumbbells in the anisotropic medium is used. The theoretical results show that the asymptotic behavior of the shear viscosity does not depend on the molecular mass and corresponds to experimental data.The study of the technological processes of polymer processing is an important practical problem whose solution requires a mathematical fornmtation of the behavioral laws of polymer fluids. In mathematical modeling of the flows of linear-polymer solutions and melts, the greatest difficulties are connected with nonlinear effects. For their description it is necessary to obtain a rheological constitutive relation and check its correspondence to the real flows of polymer fluids. The rheological constitutive relation has already been formulated by Altukhov and Pyshnograi [1,2], and Kulicke and Wallbam calculated the simple stationary shear flow and their results agree with experimental data [3]. The indicated relation was derived as a zero approximation of the more general rheological constitutive relation with respect to small parameters related to the reaction of the macromolecular chain and the internal viscosity [4,5]; therefore, this relation needs to be additionally substantiated. AVe now construct a rheological zero-approximation model by a different method and study the effect of the molecular mass and concentration of the polymer on its parameters.Rheological Constitutive Relation. We use the microstructural approach that permits us to establish a relationship between the macro and microeharacteristics of a polymer system [6,7]. In the theory of polymer viscoelasticity, a monolnolecular approximation, in which one inacromolecule moving in an effective medium formed by a solvent and other macromolecules is considered instead of a set of macromolecules in the solvent volume, is the most efficient. To study relatively slow motions, one can use the Kargin-SlonimskiiRouse model. In this model, the macromolecular dynamics is simulated by the motion of N + 1 centers of friction (beads) connected sequentially by elastic forces (springs), and the equations of the dynamics of a macromolecule have the form dwhere m is the mass of a bead, u~ is the velocity of a particle with number a, P/~ is the force of hydrodynamic entrainment, Q~ is the elastic force, and (I)~ is a random (Brownian) force. It is noteworthy that the application of this approach leads to rheological constitutive relations of different complexity (see the review of publications in [6,7]). The simplest rheological model [1, 2] allows one to simulate the stationary viscosimetric flows of linear-polymer solutions and melts qualitatively and quantitatively. We now consider it in more detail. We simulate the dynamics of a macromolecule by an elastic dumbbell that corresponds to the slowest relaxational process of a polymer...