Abstract Generated abstract
This paper examines the velocity distribution function of gas molecules near a solid wall when a temperature gradient is applied tangentially to the wall, a problem relevant to the calculation of thermal slip. The authors solve the stationary Boltzmann kinetic equation more directly than in their earlier treatment based on the Bhatnagar, Gross, and Krook approximation, representing the correction to the Chapman, Enskog distribution as several terms that separately satisfy linearized kinetic equations. By applying boundary conditions with a diffuse reflection coefficient, they obtain explicit distance-dependent exponential contributions to the distribution function for molecules moving toward and away from the wall. The resulting function differs substantially from the previous approximation, particularly because the incident-molecule distribution depends explicitly on distance from the wall, suggesting a revised thermal slip velocity to be determined subsequently.
Full Text
UDC 533.72
PHYSICS
Yu. I. Yalamov, I. N. Ivchenko,
Corresponding Member of the Academy of Sciences of the USSR B. V. Deryagin
THE VELOCITY DISTRIBUTION FUNCTION OF GAS MOLECULES NEAR A SOLID WALL IN A NONUNIFORMLY HEATED GAS
In our work \((^1)\) it was shown that, for the most rigorous determination of the thermal slip velocity of a gas along a solid wall in the presence of a temperature gradient tangential to the latter, it is necessary to compute the velocity distribution function of the gas molecules by solving the Boltzmann kinetic equation.
In \((^1)\) an attempt was made to carry out this task by a rough approximation of the collision integral in the Boltzmann kinetic equation by the Bhatnagar–Gross–Krook method \((^2)\). In this case the distribution function, as is to be expected, depends on the distance to the solid wall, but the slip velocity turns out to be equal to
\[ u = {^3/_4}\, v\, d \ln T / dy, \tag{1} \]
which exactly coincides with the result obtained earlier by Maxwell \((^3)\).
In the present work the distribution function is determined considerably more correctly than in \((^1)\). The distribution function will be determined as the sum of three terms, each of which satisfies the Boltzmann equation. This approach is more convenient (in the sense of overcoming mathematical difficulties) than the Gross–Ziering method \((^{4,5})\).
Let us consider a gas situated in the field of a temperature gradient tangential to an infinite wall. We choose the origin of coordinates on the surface of the wall. The \(x\)-axis is directed perpendicular to the wall, and the \(y\)-axis along the surface. Along the \(y\)-axis there is a temperature gradient in the gas. Under these conditions the velocity distribution function of the gas molecules will, generally speaking, depend on the coordinates \(x\) and \(y\), and can be found from the Boltzmann kinetic equation, which in the stationary case has the form
\[ (\mathbf{c}\nabla) f = \partial f / \partial t_{\text{coll}}, \tag{2} \]
where \(\mathbf{c} = (m/2kT)^{1/2}\mathbf{v}\) is the dimensionless velocity; \(f = f(x,y,\mathbf{c})\) is the distribution function; \((\partial f/\partial t)_{\text{coll}}\) is the collision integral. The collision integral in Boltzmann form is given by the expression
\[ \left(\frac{\partial f}{\partial t}\right)_{\text{coll}} = \int d\mathbf{v}_1 \int gb\,db\,d\varepsilon\, (f'f_1' - ff_1), \tag{3} \]
where \(g = |\mathbf{c}_1 - \mathbf{c}|\) is the relative velocity of two colliding molecules; \(b\) is the impact parameter of the collision; \(\varepsilon\) is the azimuthal scattering angle.
At large distances from the wall the gas is described by the Chapman–Enskog distribution \((^6)\), which may be written in the form
\[ f = f^{(0)} [1 + 2c_y u + \tau c_y S_{3/2}^{(1)}(c^2)\,\partial \ln T/\partial y], \tag{4} \]
where
\[ f^{(0)} = n (m/2\pi kT)^{3/2} \exp(-mv^2/2kT); \]
\(u\) is the dimensionless mass velocity; \(\tau = {^{15}/_{16}}\lambda \pi^{1/2}\) (\(\lambda\) is the mean free path); \(S_{3/2}^{(1)}(c^2) = {^5/_2} - c^2\).
It is assumed that at infinity from the wall there is a concentration gradient \(\partial n/\partial y\), which in relation (4) is expressed through \(\partial \ln T/\partial y\), as well as a mass transport velocity \(u\).
We shall seek the distribution function in the form:
\[ f^{\pm}=f^{(0)}[1+\Psi(\infty,y,\mathbf{c})+\Phi^{\pm}(x,y,\mathbf{c})], \tag{5} \]
where the signs \(\pm\) denote the distribution functions for molecules flying away from the wall and toward the wall, respectively; \(f^{(0)}[1+\Psi(\infty,y,\mathbf{c})]\) is the Chapman–Enskog distribution (4).
From the estimates given in work (1), it follows that
\[ |\partial\Phi/\partial y| \ll |\partial\Phi/\partial x|, \]
which makes it possible to regard \(\Phi\) as a function only of \(x\) and \(\mathbf{c}\).
Taking into account the form of the distribution function at large distances from the wall (4), and the fact that at arbitrary distances the distribution function contains terms proportional to \(c_xc_y\partial u/\partial x\), we shall seek the correction to the distribution function in the form
\[ \Phi^{\pm}(x,\mathbf{c}) = \Phi_0^{\pm}(x,\mathbf{c}) + \Phi_1^{\pm}(x,\mathbf{c}) + \Phi_2^{\pm}(x,\mathbf{c}) + \Phi_3^{\pm}(x,\mathbf{c}), \tag{6} \]
where
\[ \Phi_0^{\pm}(x,\mathbf{c})=a_0^{\pm}(x)c_y,\quad \Phi_1^{\pm}(x,\mathbf{c})=a_1^{\pm}(x)c_xc_y,\quad \Phi_2^{\pm}(x,\mathbf{c})=a_2^{\pm}(x)c_xc_y, \]
\[ \Phi_3^{\pm}(x,\mathbf{c})=a_3^{\pm}(x)c_yS_{3/2}^{(1)}(c^2). \]
Substituting the distribution function (5) into equation (2), we obtain
\[ c_x\frac{\partial \Phi}{\partial x} + c_y\frac{1}{f^{(0)}}\frac{\partial f^{(0)}}{\partial y} = J(\Phi)+J(\Psi), \]
where \(J(\Phi)\) and \(J(\Psi)\) are the linearized collision integrals. The function \(\Psi(\infty,y,\mathbf{c})\) satisfies the equation
\[ c_y\frac{\partial f^{(0)}}{\partial y}=f^{(0)}J(\psi), \]
therefore \(\Phi^{\pm}(x,\mathbf{c})\) will be found from the solution of the equation
\[ c_x\partial\Phi/\partial x=J(\Phi). \tag{7} \]
We choose the functions \(\Phi_i^{\pm}(x,\mathbf{c})\) so that they satisfy the equations:
\[ c_x\frac{\partial}{\partial x}(\Phi_0+\Phi_1)=J(\Phi_0+\Phi_1), \tag{8} \]
\[ c_x\frac{\partial}{\partial x}\Phi_2=J(\Phi_2), \tag{9} \]
\[ c_x\frac{\partial}{\partial x}\Phi_3=J(\Phi_3). \tag{10} \]
The solution of equation (8) is contained in works \((5,7)\). Using the data of work \((7)\), it is easy to write the expressions for the functions \(a_0^{\pm}(x)\) and \(a_1^{\pm}(x)\):
\[ a_0^+=\alpha_0+c_1\exp(-7.56x/\lambda),\quad a_0^-=c_1\exp(-7.56x/\lambda), \tag{11} \]
\[ a_1^+=\alpha_1+c_1\exp(-7.56x/\lambda),\quad a_1^-=\alpha_1-c_1\exp(-7.56x/\lambda), \]
where \(\alpha_0^+=-4.487;\ \alpha_1^+=4.643;\ \alpha_1^-=1.551;\ c_1\) is a constant determined from the boundary conditions at the wall.
Let us find the solution of equation (9). It is convenient to introduce the auxiliary function \(\operatorname{sign}c_x=+1\) for \(c_x>0\) and \(\operatorname{sign}c_x=-1\) for \(c_x<0\). With its aid \(\Phi_2\) is expressed through \(\Phi_2^+\) and \(\Phi_2^-\) as follows:
\[ \Phi_2 = \Phi_2^+\frac{1+\operatorname{sign}c_x}{2} + \Phi_2^-\frac{1-\operatorname{sign}c_x}{2} = \frac{a_2^+ + a_2^-}{2}c_xc_y + \frac{a_2^+ - a_2^-}{2}c_xc_y\operatorname{sign}c_x. \tag{12} \]
Since \(J(\Phi_2)\) is a linear operator, we have
\[ J(\Phi_2)=\frac{a_2^+ + a_2^-}{2}J(c_xc_y)+ \frac{a_2^+ - a_2^-}{2}J(c_xc_y\operatorname{sign}c_x). \tag{13} \]
Substituting (12) and (13) into equation (9), multiplying both sides of the resulting equation by
\(c_xc_y(1\pm \operatorname{sign}c_x)\exp(-c^2)\,dc\), and integrating over the entire velocity space, we obtain
\[ \frac{d}{dx}a_2^+ = a_2^+\frac{I_1+I_2}{\pi} + a_2^-\frac{I_1-I_2}{\pi}, \qquad \frac{d}{dx}a_2^- = -\,a_2^+\frac{I_1-I_2}{\pi} - a_2^-\frac{I_1+I_2}{\pi}, \tag{14} \]
where
\[ I_1=[c_xc_y,c_xc_y] = \int c_xc_y\exp(-c^2)J(c_xc_y)\,dc = -0.400\,\frac{\pi}{\lambda}, \]
\[ I_2=[c_xc_y\operatorname{sign}c_x,\ c_xc_y\operatorname{sign}c_x] = -1.698\,\frac{\pi}{\lambda}. \]
The solution of system (14) has the form
\[ a_2^+=2.897\,c_2\exp(-1.65x/\lambda), \qquad a_2^-=c_2\exp(-1.65x/\lambda), \tag{15} \]
where \(c_2\) is an arbitrary constant determined from the boundary conditions at the wall.
Equation (10) for the function \(\Phi_3\) is solved in the same way. The function \(\Phi_3\) is given by the expression
\[ \Phi_3(x,c)= \frac{a_3^+ + a_3^-}{2}\,c_yS_{3/2}^{(1)}(c^2) + \frac{a_3^+ - a_3^-}{2}\,c_yS_{3/2}^{(1)}(c^2)\operatorname{sign}c_x, \tag{16} \]
and the collision integral of \(\Phi_3\) has the form
\[ J(\Phi_3)= \frac{a_3^+ + a_3^-}{2} J\!\left(c_yS_{3/2}^{(1)}(c^2)\right) + \frac{a_3^+ - a_3^-}{2} J\!\left[c_yS_{3/2}^{(1)}(c^2)\operatorname{sign}c_x\right]. \tag{17} \]
Substituting (16) and (17) into equation (10), multiplying both sides of the resulting equation by
\[ c_yS_{3/2}^{(1)}(c^2)(1\pm \operatorname{sign}c_x)\exp(-c^2)\,dc, \]
and integrating over the entire velocity space, we obtain
\[ \frac{d}{dx}a_3^+ = a_3^+\frac{4(I_3+I_4)}{13\pi} + a_3^-\frac{4(I_3-I_4)}{13\pi}, \]
\[ \frac{d}{dx}a_3^- = -\,a_3^+\frac{4(I_3-I_4)}{13\pi} - a_3^-\frac{4(I_3+I_4)}{13\pi}, \tag{18} \]
where
\[ I_3= \left[ c_yS_{3/2}^{(1)}(c^2),\, c_yS_{3/2}^{(1)}(c^2) \right] = -1.333\pi/\lambda, \]
\[ I_4= \left[ c_yS_{3/2}^{(1)}(c^2)\operatorname{sign}c_x,\, c_yS_{3/2}^{(1)}(c^2)\operatorname{sign}c_x \right] = -1.262\pi/\lambda. \]
The solution of system (18) has the form:
\[ a_3^+=-73.09c_3\exp(-0.798\,x/\lambda), \qquad a_3^-=c_3\exp(-0.798\,x/\lambda); \tag{19} \]
\(c_3\) is an arbitrary constant of the system.
The arbitrary constants of the differential equations are determined from the boundary conditions at the wall, which for the complete distribution function have the form (1)
\[ f^+(0,c)=qf^{(0)}+(1-q)f^-(0,-c_x,c_y,c_z), \tag{20} \]
where \(q\) is the coefficient of diffuse reflection.
Substituting the distribution function (5) into (20), it is easy to obtain the boundary condition for the function \(\Phi(x,c)\)
\[ \Phi^+(0,c)=-q\Psi(\infty,c)+(1-q)\Phi^-(0,-c_x,c_y,c_z). \tag{21} \]
Equating the coefficients of \(c_y\), \(c_xc_y\), \(c_yS_{3/2}^{(1)}(c^2)\) in relation (21), we obtain the boundary conditions for the functions \(a_i^\pm\)
\[ \begin{gathered} a_0^+(0)=-q\cdot 2u+(1-q)a_0^-(0),\qquad a_1^+(0)+a_2^+(0)=\\ =q-1[a_1^-(0)+a_2^-(0)],\qquad a_3^+(0)=-q\tau\,\partial\ln T/\partial y+(1-q)a_3^-(0). \end{gathered} \tag{22} \]
Substituting expressions (11), (15), (19) into the boundary conditions (22), we obtain
\[ \begin{gathered} c_1=-q\,\frac{2u}{q-5.487},\qquad c_2=q\,\frac{2u(6.194-1.551q)}{-q^2+9.384q-21.38},\\ c_3=q\,\frac{\tau}{74.09-q}\,\frac{\partial\ln T}{\partial y}. \end{gathered} \tag{23} \]
The correction to the distribution function has the form
\[ \begin{aligned} \Phi(x,c)=&\frac{a_0^+ + a_0^-}{2}c_y+\frac{a_0^+ - a_0^-}{2}c_y\operatorname{sign}c_x +\frac{a_1^+ + a_1^-}{2}c_xc_y+\frac{a_1^+ - a_1^-}{2}c_xc_y\operatorname{sign}c_x+\\ &+\frac{a_2^+ + a_2^-}{2}c_xc_y+\frac{a_2^+ - a_2^-}{2}c_xc_y\operatorname{sign}c_x +\frac{a_3^+ + a_3^-}{2}c_yS_{3/2}^{(1)}(c^2)+\\ &+\frac{a_3^+ - a_3^-}{2}c_yS_{3/2}^{(1)}(c^2)\operatorname{sign}c_x . \end{aligned} \tag{24} \]
From (24), (11), (15), (19), after carrying out the calculations we shall have
\[ \begin{aligned} \Phi(x,c)=&-3.487\,\frac{c_1}{2}c_y\exp(-7.56x/\lambda)-\\ &-5.487\,\frac{c_1}{2}c_y\operatorname{sign}c_x\exp(-7.56x/\lambda) +6.194\,\frac{c_1}{2}c_xc_y\exp(-7.56x/\lambda)+\\ &+3.092\,\frac{c_1}{2}c_xc_y\operatorname{sign}c_x\exp(-7.56x/\lambda) +3.897\,\frac{c_2}{2}c_xc_y\exp(-1.65x/\lambda)+\\ &+1.897\,\frac{c_2}{2}c_xc_y\operatorname{sign}c_x\exp(-1.65x/\lambda) -72.09\,\frac{c_3}{2}c_yS_{3/2}^{(1)}(c^2)\exp(-0.798x/\lambda)-\\ &-74.09\,\frac{c_3}{2}c_yS_{3/2}^{(1)}(c^2)\operatorname{sign}c_x\exp(-0.798x/\lambda). \end{aligned} \]
The distribution function obtained differs substantially from the one derived by us earlier \((^1)\) in that the distribution function for the incident molecules depends explicitly on the distance to the wall; consequently, one may expect that on the basis of the distribution function obtained there should result a slip velocity different from the expression obtained in \((^1)\). This question will be the subject of our next paper.
Institute of Physical Chemistry
Academy of Sciences of the USSR
Received
27 III 1967
REFERENCES
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