Non-equilibrium process of transfer of momentum in medium with relaxation microstructure

The non-equilibrium process of transfer of the momentum of polymer media with an evolving nonequilibrium relaxation microstructure is considered. For the basic conditions of deformation, the regularities for the structural and dynamic components of the stress tensor with conjugate forces are obtained and analyzed, which are consistent with experiment.


Introduction
Concentrated solutions of polymers, copolymers, viscoelastic oils, colloy systems, emulsions, latexes, paints, polymerized mixed fuels, biophysical cultures and various modifications of such media are endowed with a complex internal microstructure and, as a result, have special physicochemical properties. Depending on the conditions of deformation, even near equilibrium, they are able to exhibit nonlinear -viscous and nonlinear -relaxation properties, partially store energy supplied from the outside in the form of the first and second difference of normal stresses (tensor deviator), and relax the stressed state of the medium.
The consequence of these properties represents non-instantaneous and non-local (delayed) responses of the medium to external disturbances, which determine the processes of transfer of momentum to the state of equilibrium.
Despite the great variety of such media and their flows in the processes and apparatuses of chemical technology, they are characterized by one fundamental property: all transport processes by a nonequilibrium relaxation mechanism are irreversibly aimed at neutralizing external disturbances that deflect the system from thermodynamic equilibrium.
At the present time, the phenomenological approach is widespread in the methods of describing the momentum of ordinary structureless media, endowed with the properties of viscosity, thermal conductivity, and diffusion near their equilibrium [1,2]. The phenomenological theory, based on the principles of linear nonequilibrium thermodynamics, is used to model transport processes in continuous, structureless media endowed with the properties of viscosity, thermal conductivity, and diffusion near their equilibrium. It is assumed that the mechanical reaction of the medium at a given time is determined by the instantaneous values of the strain rate and remains in a deformed state after the removal of disturbances. The linear rheological relation for momentum transfer (Newton model) has the character of an instantaneous (without relaxation time lag in the development of the process) and local response of the medium to external influences, that is, regardless of the state of a representative point of a continuous medium from the relaxation and thermal state of points of its immediate environment. In the study of irreversible (due to the production of entropy) transfer processes in the whole of a nonequilibrium system, it is assumed that the relationships between the For a boundary value problem, the state variables of an open nonequilibrium system, as a result of energy and matter exchange between subsystems, become dependent on spatial coordinates and time.
The media considered in this work have a relaxation nonequilibrium microstructure, which manifests itself in the field of shear, entropy, and diffusion forces [3].
If the time of a non-stationary technological process is less than the relaxation time of the medium to a new equilibrium state, then such a state of the medium is no longer equilibrium. In this case, a non-instantaneous (delayed in time) response of the medium to external disturbances is manifested.
The process of transfer of momentum, together with the equation of motion, determines the thermodynamic characteristics of the motion of the medium and underlies numerous processes of convective transfer of heat, mass, generalized charge, etc. Research in this direction is important not only from an applied, but also from a scientific point of view. Non-instantaneous and non-local phenomena in transport processes in media with a nonequilibrium microstructure lead to the need to formulate and solve a wide range of transport phenomena problems taking into account the coordinated influence of the medium relaxation time and the technological process time.
In this regard, the work has both applied and scientific value; it is aimed at closing the fundamental law of conservation of momentum relative to a nonequilibrium momentum flow with forces coupled to them, which allows the analyze of the influence of microstructural and relaxation (delayed noninstantaneous and non-local) properties on transfer processes.

Investigation of the processes of transferring the momentum
In the isothermal approximation, the system of equations for studying the process of transfer of momentum in media with a nonequilibrium microstructure includes [4]: a set of continuity equations = 0; momentum conservation equations: To clarify the clear influence of non-instantaneous and non-local effects on the dynamic characteristics of the nonequilibrium process of transfer of momentum, we will perform an analytical solution to the problem.
Let us consider a quasi-stationary disturbance when the system is instantly set in motion ⁄ = 0 and is maintained by a constant velocity gradient ̇2 1 = . For this case, from by the method of variation of arbitrary constants, we find the corresponding moments 〈 〉. In this case, the initial condition t=0 is used: 〈 1 2 〉 = 〈 2 2 〉 = 〈 3 2 〉 = 1; 〈 3 1 〉 = 〈 1 2 〉 = 〈 2 3 〉 = 0. A consistent solution leads to the following expressions for non-zero moments: 〈 2 2 〉 = 〈 3 2 〉 = 1; 〈 1 2 〉 = ae 2 (1 − (−2ae −1 )); (2) Equations (2) reflect the structural changes in the polymer system during deformation. The velocity gradient deforms the diagonal component 〈 1 2 〉 of the moment tensor and elastically stretches the component with respect to the initial values 〈 1 2 〉 − 1 It is seen that the values for the moments in the transient conditions of deformation are characterized by the Deborah number = ae⁄ and the Weissenberg number = ae. In accordance with equation (2), at t → , we have: Here 〈̃〉 is the tensor macroparameter characterizing the change in the components of the structure of the deformed system as a result of relaxation phenomena, figure 1. In the region of 0<De<1, a nonlinear correlation of the transition from one conformational state to another occurs, as well as changes in the microcomposition of the system that affect the transfer processes. With an increasing De number, changes in the microcomposition of the system stabilize. In the studied range of We, the correlation is linear.   Figure 3 shows the nature of the change in the first difference in the normal components of ̃≡ 2( 11 − 22 )/ of the deviator of the stress tensor of the deviation of the system from the equilibrium state caused by the gradient of the shear rate G.