Fractional diffusant model of a multicomponent inhomogeneous gas mixture as a basis for the diffusant approximation

It can be shown that for a correct description of the motion of an inhomogeneous multicomponent mixture of gases, such concepts as a multi-velocity continuum and interpenetrating motion are not enough. This is the basis for the introduction of the concept of “poly-velocity motion”. The reason for the need to use this concept is that to describe the motion of each component of an inhomogeneous mixture, it is necessary to use not one, but two velocity fields. The use of two velocity fields to describe the motion of a mixture component is the basis for constructing a fractional model of the mixture component, in which each component is divided into two fractions. This model serves as the basis for constructing the proposed fractional diffusant model of a multicomponent inhomogeneous gas mixture.


Introduction
The study of the behavior of multicomponent inhomogeneous gas-liquid mixtures is one of the main areas of research in hydrodynamics. The theories of these mixtures are based on the following concepts: multi-velocity continuum and interpenetrating motions [1], [2]. In addition, these theories postulate that for each component of the mixture, the equations of conservation of mass, momentum, and energy can be written.
Based on these theories, a hypothetical simplified model of a multicomponent inhomogeneous mixture, known as the diffusion approximation, is constructed [2].
The key concepts of the diffusion approximation are: barycentric velocity and diffusion velocity, and the main hypothesis of this approximation is the hypothesis that the density of the substance carried by diffusion flows can be assumed to be equal to the density of the considered component of the mixture.
In this paper, the inconsistency of this hypothesis is shown by a concrete example, from which the inconsistency of the diffusion approximation itself follows. The inadequacy of the diffusion approximation makes it relevant to search for alternative approximations. One of such alternative approximations is the diffusant approximation proposed in this paper.
The diffusant approximation is based on the fractional diffusant model of a component of an inhomogeneous multicomponent mixture proposed in [3]. In this fractional diffusant model, each i-th component Ni of an inhomogeneous multicomponent mixture N0 is divided into two fractions: the convective fraction Nvi and the molecular relay fraction or diffusant Nwi. The main classification distinguishing feature of the diffusant Nwi is: poly-velocity type of motion, which includes two 2 different mono-velocity types of motion. One of these types of motion is the mono-velocity convective type of motion, which is motion together with the environment. The second type of motion is the mono-velocity molecular relay race type of motion, which is a motion relative to the environment. The main difference between the proposed diffusant approximation and the traditional diffusion approximation is that the diffusant approximation is based on the concept of "poly-velocity type of motion" and on the postulate that the magnitude of the modulus of the diffusant velocity is one of the physical characteristics of an inhomogeneous multicomponent mixture.

Basic concepts and hypotheses of the diffusion approximation
The basic concepts of the mechanics of multi-velocity continuums in general and the diffusion approximation in particular are the concepts of multi-velocity continuum and interpenetrating motions [2]. While the two main hypotheses of the diffusion approximation are: -the hypothesis that the motion of each i-th {i=1,...,n} component Ni of a multicomponent mixture N0 can be described by a single velocity field vi; -the hypothesis that the value of the density i of the mass transferred by the diffusion fluxes of the mass Jρi can be assumed to be equal to the density of the component of the mixture Ni.
When using the termdiffusion flow, it should be taken into account that in physics there are two types of diffusion flows: the diffusion flow of the mass Jρi and the concentration diffusion flow Jni. The relation that establishes the relationship between these two types of diffusion flows has the form where µ is the mass of a single atom or molecule. In addition, it should be taken into account that there are two main forms of diffusion fluxes of mass Jρi: the physical form of recording, which has the form where Di is the diffusion coefficient, and the hydrodynamic form of the record, which has the form One of the main postulates of the diffusion approximation is the postulate that the value of the density Ji of the diffusion flow of mass Ji can be assumed to be equal to the value of the density i of the component Ni, that is, it is postulated that One of the key concepts of the diffusion approximation is the concept of the average -mass velocity or barycentric velocity v0. The ratio that allows you to determine the value of this speed has the form where 0 = n i.
Another basic concept of this approximation is the concept -diffusional velocity wi, defined as It is easy to check that with such a determination of the diffusional velocity wi, the expression of the form

The basic system of equations of the diffusion approximation
When using the diffusion approximation, the continuity equation for the density i of the component Ni is written as [2] ij where Jij are the source terms that characterize the intensity of the transition of the mass from the component of the mixture Ni to the considered component of the mixture Ni. Given the expressions (7) and (8), we obtain that the continuity equation for the mixture as a whole can be written as It is shown in [2] that when using the diffusion approximation, the equation of motion describing the motion of the ith component of the mixture has the form where σi = σ kl are the components of the surface force tensor, and g is the acceleration of gravity. Summing up the expressions (11) with respect to i, we can obtain the equation of motion for the mixture as a whole, which has the form [2] Neglecting the terms containing second-order quantities with respect to the diffusion velocities w k i, we obtain The energy equation for the i-th component of the mixture can be written as [2] where ui is the internal energy of the i-th component of the mixture, and q k is the components of the The energy equation written for the mixture as a whole has the form [2] The complete system of equations of the diffusion approximation includes the equations (9), (10), (13) and (15) [2]. where ii is the guiding ort of the concentration diffusion flow Jni, then it becomes obvious that to set the vector field Jni(ni,|wi|,ii), it is necessary to set two scalar functions ni and |wi|. In addition, it is necessary, using Fick's law (16), to set the unit vector field ii.

Diffusion and diffusant approaches to the construction of models of multicomponent inhomogeneous mixtures
With the diffusion approach to the construction of models of inhomogeneous multicomponent mixtures, it is assumed that the density value nJi of the same concentration diffusion flow Jni is equal to the concentration value nNi of the component of the mixture Ni.
With the diffusant approach to constructing models of inhomogeneous multicomponent mixtures, it is assumed that the magnitude of the modulus |wJi| velocity wJi of the concentration diffusion flow Jni is equal to the value of the average velocity of the thermal motion of the molecules vTi of the component of the mixture Ni.
From a physical point of view, the postulate that the values of the modulus |wJi| velocity wJi are equal to the value of the average velocity of thermal motion of molecules vTi is equivalent to the postulate that the magnitude of the velocity modulus of diffusion flows Jni belongs to the main thermodynamic characteristics of a multicomponent inhomogeneous mixture.
It is obvious that in order to establish the fact that the magnitude of the modulus |wJi| velocity wJi of the concentration diffusion flow Jni belongs to the main thermodynamic characteristics of multicomponent inhomogeneous mixtures, it is sufficient to establish this fact for any particular mixture.
One of such particular mixtures is considered in [3] and is a stationary inhomogeneous mixture of two isotopes Сis,1 and Сis,2 of an ideal gas C, located at constant temperature and pressure. In this work, it is shown that the values of the velocity modulus function of the diffusion fluxes contained in this stationary mixture should be considered equal to the values of the function of the average velocity of the thermal motion of the molecules of the isotopes Сis,1 and Сis,2.

Proof of the failure of the main hypothesis of the diffusion approximation
In this case, the main hypothesis of the diffusion approximation is understood to be the postulate that the concentration value nJi of molecules forming concentration diffusion flows Jni, can be assumed to be equal to the concentration value nNi of molecules forming the component Ni of a multicomponent inhomogeneous mixture N0.
It follows from the results of [3] that the use of this hypothesis in analyzing the behavior of a stationary inhomogeneous mixture of two isotopes Сis,1 and Сis,2 of an ideal gas C, located at constant temperature and pressure, leads to a violation of the law of conservation of energy.
Recall that in the thought experiment discussed in [3], an inhomogeneous mixture of isotopes Сis,1 and Сis,2 of an ideal gas placed in a rectangular parallelepiped, at the boundaries of which stationary boundary conditions are set, such that along the axis of this parallelepiped there should be a linear distribution of concentrations of nC1 and nC2 molecules of isotopes Сis,1 and Сis2, As a result of the existence of this stationary linear distribution of the concentrations nC1 and nC2 in the parallelepiped, there should be two equal in modulus and opposite in direction stationary concentration diffusion flows JxC1 and JxC2.
From the stationarity of the thought experiment considered in [3], it follows that the amount of substance and the amounts of momentum and the amount of kinetic energy of each isotope that enter and leave this parallelepiped through its end side surfaces must be equal to each other. In other words, in this experiment, a mixture of isotopes Сis,1 and Сis,2 placed in a parallelepiped acts no more than as an intermediary providing the exchange of matter and momentum and energy between external objects located to the left and right of the end surfaces of the parallelepiped.
It turns out that the absence of an exchange of matter and momentum and energy between external objects and a mixture of isotopes Сis,1 and Сis,2 located in a parallelepiped is possible only if the magnitude of the modulus |w| velocity w of the oncoming diffusion fluxes JxC1 and JxC2 existing in the parallelepiped is constant and equal to the value of the average velocity of thermal motion of molecules of isotopes Сis,1 and Сis,2.
From the above-mentioned basic hypothesis of the diffusion approximation, it follows that in the case of a linear decrease in the concentrations of nC1 and nC2 isotopes of Сis,1 and Сis,2 along the axis of the parallelepiped, the velocity of the oncoming diffusion fluxes JxC1 and JxC2 should simultaneously increase. In addition, with an increase in the velocity of diffusion flows, the amount of kinetic energy transferred by these flows should simultaneously increase. As a result, the amount of kinetic energy leaving the parallelepiped must exceed the amount of kinetic energy entering this parallelepiped. This means that this parallelepiped is an infinitely capacious source of kinetic energy.
This circumstance indicates the inadmissibility of using the basic hypothesis of the diffusion approximation in the study of the behavior of multicomponent inhomogeneous mixtures. Moreover, the failure of the main hypothesis of the diffusion approximation implies the failure of the diffusion approximation itself.

Fractional diffusant model of a component of a multicomponent inhomogeneous mixture
In [3], a fractional diffusant model of a component of a multicomponent inhomogeneous mixture was proposed. This model is based on the postulate that each component Ni of an inhomogeneous multicomponent mixture N0 consists of two fractions: the convective fraction Nvi and the diffusant fraction or simply diffusant Nwi.
In this case, the basis for dividing the component of the mixture Ni into the convective fraction Nvi and the diffusant Nwi is the presence of three classification distinguishing features in the diffusant Nwi, -poly-velocity type of diffusant motion, including portable motion described by the field of portable diffusant velocities w0, coinciding with the field of convective velocities v0 of the mixture as a whole, and relative motion described by the field of relative diffusant velocities wJi; -belonging of the modules |wJi| of the field of relative diffusion velocities wJi to the number of thermodynamic characteristics of a multicomponent inhomogeneous mixture While one distinctive mathematical feature of a diffusant is that the field of relative diffusant velocities wJi belongs to the number of layered vector fields [5].
The presence of any of these three distinctive features is sufficient for a reasonable introduction to the consideration of the concept of "diffusant Nwi". It makes sense to consider the concept of "diffusant" introduced in this way as a generalized concept that includes various particular types of this concept. These particular types of the generalized concept of "diffusant Nwi" include: -"particle quantity diffusant" or " concentration diffusant Nwni", consisting of the number of molecules of the type under consideration associated with particles of concentration diffusion flows Jni; -" mass diffusant Nwmi " associated with the amount of substance carried by particles forming concentration diffusion flows Jni; -"pulse diffusant Nwpi" associated with the amount of motion carried by particles forming concentration diffusion flows Jni; -"energy diffusant NwEi", associated with the amount of kinetic energy transferred by particles forming concentric diffusion flows Jni. It is essential that all the above-mentioned particular types of diffusants have the property of conservation.

Mono-velocity and poly-velocity types of multi-velocity continuum
It was mentioned above that in hydrodynamics, the construction of models of multicomponent inhomogeneous mixtures is based on the concepts of "multi-velocity continuum" and "interpenetrating motions". In addition, it was mentioned above that one of the main hypotheses currently used in the mechanics of inhomogeneous mixtures is the hypothesis that the motion of each component Ni of a multicomponent mixture N0 can be described by a mono-velocity field vi.
Considering that, by definition, a mono-velocity type of motion is a motion that can be described by a mono-velocity field, it is possible to introduce the concept of a "mono-velocity type of a multivelocity continuum" into consideration. By definition, a mono-velocity type of poly-velocity continuum is a multi-velocity continuum, which is a superposition of mono-velocity continuums. It follows from this definition that continuums that are currently used in hydrodynamics in constructing models of multicomponent inhomogeneous mixtures belong to this type of multi-velocity continuums.
When constructing fractional models of multicomponent inhomogeneous mixtures N0 based on fractional models of the component Ni of the mixture N0, it is necessary to take into account that the motion of the component Ni of the mixture N0 belongs to a poly-velocity type of motion, for the description of which it is necessary to use two velocity fields: the portable diffusant velocity field w0 = v0 and the relative diffusant velocity field wJi.
It is assumed that, by definition, a poly-velocity type of a multi-velocity continuum is called a continuum, which is a superposition of continuums, to describe the movement of some parts of which it is necessary to use several velocity fields. From this definition it follows that it is to this type of multi-velocity continuums that continuums belong, which must be used in hydrodynamics when constructing fractional models of multicomponent inhomogeneous mixtures.