Mathematical Models For Meso- And Nano-Domain Heat, Mass, Pulse Transfer Processes

The results for development of new physical and mathematical processes of heat, mass, pulse transfer processes based on the local non-equilibrium thermodynamics principles


Introduction
The traditional theory of transfer processes is based on the local thermodynamic equilibrium principle and continuous medium theory, according to which a local equilibrium state is observed in every minor medium component irrespective of presence of gradients of temperatures, concentrations etc.in the system in general, which is possible only in the case, when the changing rate of the system microparameters is considerably lower than the local equilibrium setting rate. However, any transfer process is non-local since energy transfer from one point to another is not immediate, but it takes quite a finite amount of time instead. Consequently, when assuming the local equilibrium principle, finite transfer process velocity is neglected. Otherwise, transfer process will be non-local and local non-equilibrium transfer model has to be used to describe it [1 -4].

Mathematical models of local non-equilibrium transfer processes
In this paper, local non-equilibrium transfer process equations are derived by using the summands in the formulas of phenomenological Fourier's law, Newton's law and Hooke's law, which consider time acceleration of the driving forces (reasons -temperature, velocity, displacement gradients) and consequences caused by them (heat flow, shear stress, displacement). Their general formula has the form of where i τrelaxation factors; ηheat flow q shear stress τ and normal stress σ respectively in the formulas of the Fourier's law, Newton's law and Hooke's law; Rtemperature T , velocity  and displacementU in the formulas of the above laws; ttime; xcoordinate (in the Newton's law formula where where 1 z , 2 z , 3 z -the roots of characteristic equation k D 1 , k D 2 , k D 3 -integration constants determined based on the initial conditions of the boundary problem; Calculations made by the formula (6) allowed determining the fact of time delay in taking the firstclass boundary condition, which evidences that the immediate heating of the body at the boundary is impossible under any conditions of heat exchange with the ambient medium.
; Θ , Fo , ξ -timeless pressure, time, coordinate 1 Fo , r Fo -non-dimensional relaxation and resistance factor; p -pressure; 0 pinitial pressure ; 1 p -pipe inlet pressure; x -coordinate; l -pipe length; c -sound velocity in fluid; An exact analytical solution of the task (7) -(9) is in the form where k c 1 , k c 2 , k c 3 are integration constants determined by the initial conditions (8); k z 1 , k z 2 , k z 3roots of the characteristic equation; . The calculation results of by the formula (10) are given in fig. 2 (for 0 ξ  ). Their analysis allows concluding that solution of the task (7) -(9) in the quasi-static setting (at 0 Fo 1  ) is slightly different from that one obtained in [5] with the employment of the Bernoulli-Fourier method. However, the results of the two theoretical methods are considerably different from the experimental data [6]. Consideration of non-stationary of the velocity gradient and shear stress results in a considerable approximation of the calculation data to the experiment results. Based on the experimental data, by using the relation (10) the relaxation factor с 0108 , 0 τ 1  was found in solving the reverse task.

Heat exchange in the moving fluid considering its relaxation properties
In the above model (7) -(9) pressure change is studied under the hydraulic impact conditions. Let's consider the equation derivation applicable to the non-stationary heat exchange at laminar flow in the two-dimensional duct (energy equation). Formulas for heat flow by axes x and y have the form [7] i t x where x q , y qheat flows; x ω , y ω − velocities; i − heat content; x , y − lateral and transverse coordinates; 1 τrelaxation factor.
By substituting (11), (12), in the heat balance equation with neglecting the lateral heat conductivity direction, we find where To solve the task (13), (14), the finite difference method was employed. It follows from the calculation results analysis (for

Oscillations of elastic solids
By using the modified formula of the Hooke's law in the form (4) and equilibrium equation , the following elastic solid oscillation equation was obtained velocity. The boundary conditions for the rod deformed in the initial time moment, whose one end is rigidly fixed with applying load to another end, which is changing according to the harmonic law, have the form The analysis of results obtained by the finite difference method allows concluding that in the coincidence of own oscillation frequencies of ( c ω ) the road and external load ( 4 F ) resonance oscillations are observed, whose amplitude, while increasing in the initial time period, then becomes stabilized at some constant value. At frequencies, which are close to resonance, ones, bifurcational flutter fluctuations (beats) occur, at which the rod ocsillation frequency increases periodically from zero to some maximum value in the undamped oscillation process ( fig. 3). At frequencies, which are far from resonance ones, each road point participates in two oscillating processes with high-frequency and low-amplitude oscillations taking place in the first process and low-frequency and high-amplitude oscillations in the second one.   Fo -non-dimensional relaxation factors; 3 Fo -resistance factor ; α -linear expansion factor; V -volume unit; 1 τ , 2 τ -relaxation factors; β , Α -constants. An exact analytical solution of the task (17)