METHODOLOGICAL NOTES

ON SOME HYDRODYNAMIC QUANTITIES FOR A MIXTURE

I. G. Shaposhnikov

This note proposes one possible mode of presentation when introducing certain basic hydrodynamic quantities for a mixture, namely: the mass density, the mass-flux density, and the hydrodynamic velocity. The note pursues only didactic aims; the occasion for this kind of discussion of the question of defining the indicated hydrodynamic quantities was the appearance in print of a number of works\(^{1—3}\) proceeding from incorrect notions of these quantities even in the simple case of a pure medium, to which attention had already been drawn\(^{4}\).

1. Let us consider a hydrodynamic medium (a liquid or a not too rarefied gas) consisting of molecules that differ in their properties, i.e. a mixture of several components. For the quantitative characterization of the macroscopic spatial distribution of mass, the macroscopic motion, and the macroscopic transfer of mass in this medium, one introduces, respectively, the mass density \(\rho\), the hydrodynamic velocity \(\mathbf{v}\), and the mass-flux density \(\mathbf{j}\) of the medium. How are these quantities to be defined from the molecular point of view?

2. The mass density \(\rho\) should, of course, be defined as the ratio of the mass of a physically infinitely small element of the medium to the volume of this element. If \(m_s\) is the mass of a molecule of the \(s\)-th component, and \(\nu_s\) and \(\rho_s=\nu_s m_s\) are, respectively, the number density of molecules and the mass density of this component, defined as the ratios of the number of such molecules in the physically infinitely small element of the medium under consideration and of their mass to the volume of this element, then we shall obviously have:

\[ \rho=\sum \rho_s=\sum \nu_s m_s . \tag{1} \]

1. The hydrodynamic velocity $\mathbf{v}$ must be defined as the velocity of the center of inertia of a physically infinitesimal element of the medium—only then can the equation of motion of the medium be obtained, as usual, by equating to each other the sum of the forces acting on a physically infinitesimal element of the medium and the product of the mass of this element by the substantial derivative of the hydrodynamic velocity with respect to time. Let us denote by $\mathbf{u}_s$ the mean velocity of the molecules of the $s$-th component over the aggregate of these molecules in the physically infinitesimal element of the medium under consideration; it is easy to verify that

\[ \mathbf{v}=\frac{1}{\rho}\sum \rho_s \mathbf{u}_s . \tag{2} \]

For the case of a pure medium this gives:

\[ \mathbf{v}=\mathbf{u}. \tag{3} \]

1. In order to introduce the mass flux density $\mathbf{j}$, let us first define the flux density $\mathbf{I}$ of any scalar quantity transported by the molecules of the medium (see, for example, $^{5}$).

Suppose we have an area element with vector $d\mathbf{S}$, moving translationally with velocity $\mathbf{c}$. During the time $dt$ some molecules pass through this area element in the positive direction, and some in the negative direction. Suppose that the motion of the molecules of each component is accompanied by the transport by these molecules of some additive scalar quantity $Q(\Gamma_s)$ (mass, kinetic energy, etc.), determined by the aggregate of quantities $\Gamma_s$ which fully characterize the state of a molecule of the component under consideration. Let $dQ_+$ and $dQ_-$ denote the values of the quantity $Q$ transported during the time $dt$ through the area element under consideration in the positive and negative directions, respectively, and let us call

\[ dQ=dQ_+-dQ_- \]

the total value of the quantity $Q$ transported through this area element in the positive direction. Dividing $dQ$ by $dS$ and by $dt$, we obtain a natural quantitative characteristic of the transport of the quantity $Q$ by the molecules of the medium through the area element under consideration. The flux density $\mathbf{I}$ of the quantity $Q$ should evidently be defined as such a vector that its component $I_n$, normal to the area element under consideration, is equal to the above quantitative characteristic of the transport of the quantity $Q$ through this area element.

Let us first consider the case of a pure medium. Denote by $n(\Gamma)\,d\Gamma\,dV$ the number of molecules of “type $\Gamma$” in the volume $dV$ near the point under consideration at the moment under consideration. Let us take, near this moment, a time interval $dt$ so small that during it the velocities of the overwhelming majority of molecules near the point under consideration may be regarded as not changing to any significant extent. For $dQ$ we shall evidently have,

have:

\[ dQ=dS\,dt\int Q(\Gamma)(u_n-c_n)n(\Gamma)\,d\Gamma =dS\,dt\,\nu\,\overline{Q(\Gamma)(u_n-c_n)}, \tag{4} \]

where \(u_n\) and \(c_n\) are the components, normal to the area under consideration, of the molecular velocity \(\mathbf u\) and of the velocity of the area \(\mathbf c\), \(\nu\) is the number of molecules per unit volume near the point under consideration at the instant in question, and the bar denotes averaging over the ensemble of molecules in a physically infinitely small element of the medium taken around this point at this instant. From (4) it now follows:

\[ \mathbf I=\nu\,\overline{Q(\Gamma)(\mathbf u-\mathbf c)}. \tag{5} \]

In the case of a mixture, the same consideration can be carried out for each of the components, and since, obviously, \(dQ=\sum dQ_s\), we have

\[ \mathbf I=\sum \nu_s\,\overline{Q(\Gamma_s)(\mathbf u_s-\mathbf c)}, \tag{6} \]

where the bar denotes averaging over the ensemble of molecules of the \(s\)-th component in the physically infinitely small element of the medium under consideration.

If we introduce the velocities \(\mathbf w_s=\mathbf u_s-\mathbf v\), characterizing the disordered part of the molecular motion, then the flux density of the quantity \(Q\) given by (6) may be divided into microscopic and macroscopic parts:

\[ \mathbf I=\sum \nu_s\,\overline{Q(\Gamma_s)\mathbf w_s} +(\mathbf v-\mathbf c)\sum \nu_s\,\overline{Q(\Gamma_s)} =\mathbf I_{\text{micro}}+\mathbf I_{\text{macro}}. \tag{7} \]

1. To obtain an expression for the mass-flux density \(\mathbf j\), one must put in (7) \(Q(\Gamma_s)=m_s\), which, taking (1) into account, gives:

\[ \mathbf j_{\text{micro}}=\sum \rho_s\,\overline{\mathbf w_s},\qquad \mathbf j_{\text{macro}}=\rho(\mathbf v-\mathbf c). \tag{8} \]

But by virtue of (1) and (2) \(\mathbf j_{\text{micro}}=0\), so that finally:

\[ \mathbf j=\rho(\mathbf v-\mathbf c); \tag{9} \]

in particular, if the area under consideration is at rest \((\mathbf c=0)\), then:

\[ \mathbf j=\rho\mathbf v. \tag{10} \]

It is seen from (9) that the mass flux through an area moving together with the medium \((\mathbf c=\mathbf v)\) is equal to zero; in particular, the mass flux through an area at rest in a liquid at rest \((\mathbf c=\mathbf v=0)\) is equal to zero.

Thus, both in the case of a pure medium and in the case of a mixture, for the mass-flux density, independently of the causes by which mass transfer is produced, expression (10) holds (for the case of an area at rest), so that the assertion of the authors mentioned at the beginning of this note is incorrect.

works¹–³ that, in the expression for the mass-flux density in the case of a pure medium, there must be, in addition to the term \(\rho \mathbf{v}\), also terms proportional to the density gradient and the temperature gradient, which, in the opinion of the authors of these works, should account for “self-diffusion phenomena” in a pure medium. We note that, in order to prove the incorrectness of this assertion, there is no need to appeal to the kinetic equation, as was done earlier.⁴

1. In the study of diffusion phenomena, which it makes sense to discuss, of course, only in the case of a mixture, one must find an expression not for the total mass-flux density \(\mathbf{j}\), but for the mass-flux density \(\mathbf{j}_k\) of some one, \(k\)-th component. For this purpose it is necessary to write expression (5) for the \(k\)-th component and put in it \(Q_k(\Gamma_k)=m_k\); for the case \(C=0\) this gives:

\[ \mathbf{j}_k = \nu_k m_k \mathbf{u}_k, \tag{11} \]

where the bar denotes averaging over the ensemble of molecules of the \(k\)-th component in a physically infinitesimal element of the medium. Introducing \(\mathbf{w}_k \equiv \mathbf{u}_k-\mathbf{v}\) and taking into account that \(\nu_k m_k=\rho_k\), from (11) we obtain:

\[ \mathbf{j}_k = \rho_k \mathbf{v}+\rho_k \mathbf{w}_k = \mathbf{j}_{k\,\mathrm{macro}}+\mathbf{j}_{k\,\mathrm{micro}}, \tag{12} \]

thus, the mass-flux density of the \(k\)-th component is composed of a macroscopic, convective part \(\mathbf{j}_{k\,\mathrm{macro}}=\rho_k\mathbf{v}\) and a microscopic, diffusive part \(\mathbf{j}_{k\,\mathrm{micro}}=\rho_k\mathbf{w}_k\), caused by the nonuniformity of the composition and state of the medium at different points.

References

1. N. A. Slezkin, DAN 77, 205, 1951; DAN 79, 33, 1951; Vestnik MGU, No. 10, 3, 1951.
2. S. V. Vallander, DAN 78, 25, 1951; PMM 15, 409, 1951.
3. S. V. Vallander and M. P. Elovskii, DAN 79, 37, 1951.
4. I. G. Shaposhnikov, ZhETF 21, 1309, 1951.
5. S. Chapman and T. Cowling, The Mathematical Theory of Non-uniform Gases. Cambr. Univ. Press, 1939.