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Aerodynamics
A. A. Nikolsky
THREE-DIMENSIONAL HOMOGENEOUS EXPANSION—COMPRESSION OF A RAREFIED GAS WITH POWER-LAW INTERACTION FUNCTIONS
(Presented by Academician A. A. Dorodnitsyn, 23 X 1962)
All quantities in the paper are assumed to be dimensionless.
In paper \((^{1})\) the motions of three-dimensional homogeneous expansion—compression in an unbounded space of a monatomic gas or of mixtures of monatomic gases were considered, when the functions \(f_i\) of the distribution of the peculiar velocities of the molecules are the same for all points of space and depend only on time. The Boltzmann equations for the functions \(f_i = f_i(t, c_i)\) of the distribution of the velocity vectors of molecules of the \(i\)-th species then have the form
\[ \frac{\partial f_i}{\partial t} -\frac{1}{t}\left( u_i\frac{\partial f_i}{\partial u_i} + v_i\frac{\partial f_i}{\partial v_i} + w_i\frac{\partial f_i}{\partial w_i} \right) = I_i(t,c_i), \tag{1} \]
where \(u_i, v_i, w_i\) are the components of the vectors \(c_i\); \(t\) is time. The collision integrals \(I_i\), as usual, have the form \((^{2})\)
\[ I_i(t,c_i)=\sum_j \iiint \bigl[ f_i(t,c'_i) f_j(t,c'_j) - f_i(t,c_i) f_j(t,c_j) \bigr] q_{ij} b\, db\, d\varepsilon\, dc_j . \tag{2} \]
Here \(g_{ij}=|c_i-c_j|\); \(b\) is the dimensionless impact distance; \(dc_j=du_jdv_jdw_j\); \(\varepsilon\) is the angle of rotation in the planes normal to the vector \(c_i-c_j\); \(c'_i, c'_j\) are the velocity vectors after collision of particles that before the collision had velocity vectors respectively \(c_i, c_j\).
Let us consider a model of point centers of potential repulsion or attraction. For it, on the basis, for example, of the book \((^{2})\) (Chap. III, § 4), we have the relations
\[ c'_i=\varphi_{ij}(c_i,c_j,\chi_{ij},e) =\frac{m_i c_i+m_j c_j}{m_0} +\frac{m_j}{m_0}\bigl[(c_i-c_j)\cos\chi_{ij}-e g_{ij}\sin\chi_{ij}\bigr]; \tag{3} \]
\[ c'_j=\psi_{ij}(c_i,c_j,\chi_{ij},e) =\frac{m_i c_i+m_j c_j}{m_0} -\frac{m_i}{m_0}\bigl[(c_i-c_j)\cos\chi_{ij}-e g_{ij}\sin\chi_{ij}\bigr]. \tag{4} \]
Here \(m_i\) and \(m_j\) are the masses of the colliding particles; \(m_0=m_i+m_j\); \(\chi_{ij}=\chi_{ij}(b,g_{ij})\) is the angle of rotation of the vector \(c_i-c_j\) as a result of the collision; \(e\) is a unit vector orthogonal to the vector \(c_i-c_j\). The magnitude of the increment of this vector is equal to the quantity \(d\varepsilon\) in relation (2) and has the same meaning.
Suppose now that between particles \(i\) and particles \(j\) there acts a force \(P_{ij}=\varkappa_{ij}/r^\nu\), where \(r\) is the distance between the particles, the quantity \(\nu\) is constant in all cases, and the constants \(\varkappa_{ij}\) either all have the same sign, or their sign depends on the combination of indices \(i\) and \(j\). In the case under consideration the angle \(\chi_{ij}\) is determined by the expression (see \((^{2})\), Chap. X, § 3):
\[ \chi_{ij}=\chi_{ij}(b,g_{ij}) =\pi-2\int_0^{v_{00}} \left\{ 1-v^2-\frac{2}{\nu-1} \left(\frac{v}{v_0}\right)^{\nu-1}\operatorname{sign}\varkappa_{ij} \right\}^{-1/2} dv, \tag{5} \]
where \(v_{00}\) is the root of the equation
\[ 1-v^2-\frac{2}{\nu-1} \left(\frac{v}{v_0}\right)^{\nu-1}\operatorname{sign}\varkappa_{ij}=0, \]
and the quantity \(v_0\), on which alone \(\chi_{ij}\) depends, is determined by the relation
\[ v_0=b\left(\frac{m_i m_j g_{ij}^2}{m_0|\varkappa_{ij}|}\right)^{1/(\nu-1)} . \tag{6} \]
By virtue of the linearity of relations (3), (4), we have the similarity property
\[ \begin{aligned} \varphi_{ij}(\lambda c_i,\lambda c_j,\chi,e)&=\lambda\varphi_{ij}(c_i,c_j,\chi,e),\\ \psi_{ij}(\lambda c_i,\lambda c_j,\chi,e)&=\lambda\psi_{ij}(c_i,c_j,\chi,e), \end{aligned} \tag{7} \]
where \(\lambda\) is an arbitrary positive quantity. Relations (5), (6) give one more similarity property
\[ \chi_{ij}(b,g_{ij}/\lambda)=\chi_{ij}(\lambda^{-2(\nu-1)}b,g_{ij}). \tag{8} \]
Now, in the same way as we did for the case of hard elastic spheres in [1], let us try to associate with each homogeneous state of a gas of the type under consideration (when the macroscopic velocities are equal to zero) some motion of expansion–compression of the same gas. In any homogeneous state the functions \(f_i=f_i^0(t,c_i)\) of the velocity distribution \(c_i\) satisfy the Boltzmann equations
\[ \frac{\partial f_i^0(t,c_i)}{dt} = \sum_j \iiint \left[ f_i^0(t,c_i')f_j^0(t,c_j') - f_i^0(t,c_i)f_j^0(t,c_j) \right]g_{ij}b\,db\,d\varepsilon\,dc_j = I_i^0(t,c_i). \tag{9} \]
We shall seek a solution of equations (1) in the form
\[ f_i=f_i^0(z,A_i),\qquad A_i=|t|c_i,\qquad z=z(t), \tag{10} \]
where \(z(t)\) is to be determined. Then equalities (2) take the form
\[ I_i(t,c_i) = \sum_j \iiint \{f_i^0(z,|t|c_i')f_i^0(z,|t|c_j') - \]
\[ -f_i^0(z,|t|c_i)f_i^0(z,|t|c_j)\}\,g_{ij}\,db\,d\varepsilon\,dc_j, \tag{11} \]
where \(c_i'\), \(c_j'\) are determined by relations (3), (4), in which for \(\chi_{ij}\) relations (5), (6) hold, and the similarity properties (7), (8) take place. Our considerations are valid only for \(\nu>2\), for at \(\nu\leqslant 2\) the integrals \(I_i\) diverge (see [2], Ch. 10, § 3, item 3). Setting \(\lambda=|t|\) in equalities (7), (8), we obtain:
\[ |t|\,c_i'=\varphi_{ij}[A_i,A_j,\chi_{ij}(b_1,G_{ij}),e], \tag{12} \]
\[ |t|\,c_j'=\psi_{ij}[A_i,A_j,\chi_{ij}(b_1,G_{ij}),e], \tag{13} \]
where \(G_{ij}=|t|g_{ij}=|A_i-A_j|\), \(b_1=|t|^{-2/(\nu-1)}b\).
Introducing in the right-hand side of (11) new variables of integration according to the formulas \(b_1=|t|^{-2/(\nu-1)}b\), \(dA_j=|t|^3dc_j\), we obtain
\[ I_i(t,c_i)=|t|^{4\frac{2-\nu}{\nu-1}}I_i^0[z(t),A_i]. \tag{14} \]
The left-hand side of equation (1) takes the form:
\[ \frac{\partial f_i}{\partial t} -\frac{1}{t} \left( u_i\frac{\partial f_i}{\partial u_i} + v_i\frac{\partial f_i}{\partial v_i} + w_i\frac{\partial f_i}{\partial w_i} \right) = \frac{\partial f_i^0(z,A_i)}{\partial z}\frac{dz(t)}{dt}. \tag{15} \]
Taking relations (14) and (15) into account, we reduce equation (1) to the form
\[ \frac{dz(t)}{dt}\, \frac{\partial f_i^0(z,A_i)}{\partial z} = |t|^{4\frac{2-\nu}{\nu-1}}I_i^0(z,A_i). \tag{16} \]
However, by virtue of equations (8), (3), (4), (5), (6), (9) we have \(\partial f_i(t,c_i)/\partial t=I_i^0(t,c_i)\), where, obviously, an arbitrary positive scalar quantity may be substituted for \(t\), and an arbitrary vector for \(c_i\). Therefore we have \(\partial f_i^0(z,A_i)/\partial z=I_i^0(z,A_i)\), and equations (16) will be satisfied if we set
\[ \frac{dz(t)}{dt}=|t|^{4\frac{2-\nu}{\nu-1}}. \tag{17} \]
For the case of expansion \(t>0\), we have:
\[ z(t)=\beta+\frac{\nu-1}{7-3\nu}t^{\frac{7-3\nu}{\nu-1}}, \qquad \beta=\mathrm{const}. \tag{18} \]
Thus, in the case of spreading we have
\[ f_i(t,c_i)=f_i^0\left[\left(\beta+\frac{\nu-1}{7-3\nu}t^{\frac{7-3\nu}{\nu-1}}\right),\, tc_i\right]. \tag{19} \]
The solution in the case of expansion, when at \(t=1\) (this is the general case, in view of the arbitrariness of the time scale) \(f_i=\Phi_i(c_i)\), is obtained in the form
\[ f_i=f_i(t,c_i)=f_i^0\left[\left(\frac{\nu-1}{3\nu-7}+\frac{\nu-1}{7-3\nu}t^{\frac{7-3\nu}{\nu-1}}\right),\, tc_i\right], \tag{20} \]
where \(f_i^0(t,c_i)\) is the solution for the homogeneous state with initial data \(f_i^0(0,c_i)=\Phi_i(c_i)\). The expressions given are not suitable for \(\nu=7/3\). Here we have \(z(t)=\ln(t/\beta)\), where \(\beta=\mathrm{const}\). The solution of the problem posed above has the form
\[ f_i=f_i(t,c_i)=f_i^0(\ln t,tc_i). \tag{21} \]
The quantity \(\dfrac{\nu-1}{3\nu-7}\left(1-t^{\frac{7-3\nu}{\nu-1}}\right)>0\) for \(t>1\) and for any \(\nu\) increases as \(t\) increases. For \(\nu>7/3\), as \(t\to\infty\) it tends to a definite limit, namely the value \((\nu-1)/(3\nu-7)\). Equality (20) gives
\[ f_i(\infty,c_i)=f_i^0\left(\frac{\nu-1}{3\nu-7},tc_i\right). \tag{22} \]
Thus, under expansion, in infinite time a distribution is attained that corresponds to the distribution attained for the homogeneous state already at \(t=(\nu-1)/(3\nu-7)\). For \(\nu\leqslant 7/3\) we have
\[ f_i(\infty,c_i)=f_i^0(\infty,tc_i). \tag{23} \]
The distribution \(f_i^0(t,c_i)\) as \(t\to\infty\) tends to the Maxwell distribution \((^2)\). Therefore, for \(\nu<7/3\), under expansion the velocity distribution also tends to the Maxwellian.
For the case of compression \(t<0\), \(dz(t)/dt=-dz(t)/d|t|\), integration of equation (17) gives
\[ z(t)=\beta-\frac{\nu-1}{7-3\nu}|t|^{\frac{7-3\nu}{\nu-1}}. \tag{24} \]
Analogously to the preceding, we obtain that the solution for the case of compression, coinciding at \(t=-1\) with the homogeneous state, has the form
\[ f_i(t,c_i)=f_i^0\left[\left(\frac{\nu-1}{7-3\nu}-\frac{\nu-1}{7-3\nu}|t|^{\frac{7-3\nu}{\nu-1}}\right),\, |t|c_i\right]. \tag{25} \]
For \(\nu\geqslant 7/3\), the velocity distribution tends to the Maxwellian as \(t\to-0\). For \(\nu<7/3\), as \(t\to-0\) it tends to the distribution
\[ f_i=f_i^0\left(\frac{\nu-1}{7-3\nu},\, |t|c_i\right) \]
and thus does not attain the Maxwell distribution, despite the fact that the particle density per unit volume increases as the quantity \(1/|t|^3\). Apparently this is explained by the fact that the growth of density in the latter case does not compensate for the boundedness of the time interval \(-1<t<0\). Formally putting \(\nu\to\infty\), we obtain the results of article \((^1)\), which pertain to spherical elastic molecules.
Institute of Mechanics
Academy of Sciences of the USSR
Received
23 X 1962
References Cited
- A. A. Nikol’skii, DAN, 151, No. 2 (1963).
- S. Chapman, T. Cowling, The Mathematical Theory of Non-Uniform Gases, IL, 1960.