Abstract
Full Text
Physics
I. I. Olkhovsky
Generalized Hydrodynamic Theory of the Ultrasonic Interferometer
(Presented by Academician N. N. Bogoliubov on 22 IX 1959)
-
In work \((^{1a})\) we proposed a method for solving the equations of generalized hydrodynamics; in \((^{1b})\) we considered the question of boundary conditions of the Maxwell type. On the basis of these works, in \((^{1b})\) the problem of the ultrasonic interferometer was solved in the 4-moment approximation, i.e., without taking thermal conductivity into account. In the present work we give a generalized hydrodynamic theory of the ultrasonic interferometer in the 5-moment approximation, i.e., in an approximation that takes into account the thermal conductivity of the gas and of the boundary.
-
Let us consider a one-dimensional model of the interferometer \((^{1b})\) with linearized Maxwell boundary conditions. In this case the behavior of a monatomic gas is governed by the following equations \((^{1r})\):
\[ \frac{\partial n'}{\partial t} + (C_e)_0 \frac{\partial u'}{\partial x} = 0,\qquad \frac{\partial u'}{\partial t} + (C_e)_0 \frac{\partial P'_{11}}{\partial x} = 0, \]
\[ \frac{\partial P'}{\partial t} + \frac{5}{3}(C_e)_0 \frac{\partial u'}{\partial x} + \frac{1}{3}(C_e)_0 \frac{\partial S'_1}{\partial x} = 0, \]
\[ \frac{\partial P'_{11}}{\partial t} + 3(C_e)_0 \frac{\partial u'}{\partial x} + \frac{3}{5}(C_e)_0 \frac{\partial S'_1}{\partial x} = \frac{6}{\tau}(P' - P'_{11}), \tag{1} \]
\[ \frac{\partial S'_1}{\partial t} + 2(C_e)_0 \frac{\partial P'_{11}}{\partial x} + 3(C_e)_0 \frac{\partial P'}{\partial x} - 5(C_e)_0 \frac{\partial n'}{\partial x} = -\frac{4}{\tau}S'_1 . \]
As boundary conditions for system (1) we take the conditions on \(u'\) and \(S'_1\) \((^{1b})\):
\[ u'\big|_{x=0} = \frac{1}{(C_e)_0}\operatorname{Re}\,\dot{\xi}_0 e^{i\omega t}, \qquad u'\big|_{x=a}=0; \tag{2} \]
\[ \pm S'_1 + \left.\frac{1}{(2\pi)^{1/2}}\frac{2s}{2-s} (3P' + P'_{11} - 4n')\right|_{x=0,a} =0, \tag{3} \]
where \(\omega\) is a real quantity; \(s\) is the fraction of gas atoms reflected diffusely from the surface, and \((1-s)\) the fraction reflected specularly. (3) is a generalized Smoluchowski boundary condition,* which we have given for the case of a boundary that is an ideal heat conductor and \(\sigma=1\) (\(\sigma\) is the accommodation coefficient). The case of an ideal thermal insulator is obtained from (3) by putting \(s=0\).
- We shall seek the solution of (1) in the form
\[ n'=\operatorname{Re}\,[e^{i\omega t}n'(x)],\qquad u'=\operatorname{Re}\,[e^{i\omega t}u'(x)],\ldots \tag{4} \]
* If one passes to the classical limit \(\left(S_1\to -2\lambda \frac{\partial T}{\partial x}\right)\) and sets \(P_{11}=P\), then from an expression analogous to (3) one obtains the known expression for the temperature jump at the boundary.
Then the solutions of system (1) will have the form
\[ x'=\beta_0 x;\quad \beta_0=\frac{\omega}{v_{\mathrm{ad}}};\quad v_{\mathrm{ad}}=\sqrt{\frac{5}{3}(C_v)_0};\quad r=\frac{1}{2\pi\varepsilon};\quad \varepsilon=\nu\frac{\mu}{p}; \tag{5} \]
\[ \mu\text{ is the coefficient of viscosity of the gas;}\quad \nu=\frac{\omega}{2\pi}; \]
\[ y_j=\sum_{j'=1}^{4} C_{j'} A_{1j}(K_{j'}) e^{iK_{j'}x'}, \tag{6} \]
where \(K_{j'}\) are the roots of the characteristic determinant (5) \((1^{\mathrm{r}})\); \(A_{1j}(K_{j'})\) are the cofactors of this determinant, taken at \(K=K_{j'}\); \(C_{j'}\) are constants determined from (2), (3).
Substituting (6) into (2) and (3), we obtain a system of equations linear with respect to \(C_j\), from which we find \(C_j=\dfrac{\xi_0}{iv_{\mathrm{ad}}}\dfrac{\alpha_{1j}}{d}\), where \(d\) is the determinant of the system; \(\alpha_{1j}\) are the cofactors of \(d\). Thus, the solution of (1) together with (2) and (3) is determined. For example,
\[ \Delta P_{11}(x')=-i\lambda \xi_0 \sum_{j=1}^{4}\frac{-\alpha_{1j}K_j e^{iK_jx'}}{d}; \tag{7} \]
(7) makes it possible to determine the reaction on the radiator or reflector.
- Let us now consider the necessary conditions for the formation of a standing wave, which follow from the obvious requirement
\[ L_\alpha \gg a \gtrsim L_\beta, \tag{8} \]
where \(L_\alpha=\dfrac{1}{\alpha}\); \(\alpha\) is the attenuation coefficient; \(L_\beta\) is the wavelength with wave number \(\beta\). Let us note that the solution of the characteristic equation of system (5) satisfies (8) only for \(r\gg1\) and \(r\ll1\); indeed \((1^{\mathrm{r}})\),
\[ \begin{aligned} &\text{for } r\gg1 \qquad L_{\alpha_1}\gg L_{\beta_1}>L_{\beta_0}\gg L_{\alpha_2}\sim L_\beta \gtrsim l,\\ &\text{for } r\ll1 \qquad L_{\alpha_1}>L_{\alpha_2}>l\gg L_{\beta_1}>L_\beta>L_{\beta_2}, \end{aligned} \tag{9} \]
where the quantities with indices 1, 2 refer to the sound and thermal waves, respectively; \(l\) is the mean free path length. Thus, it follows from (8), (9) that the regions of formation of a standing wave are
\[ \text{I. } r\gg1,\quad a\gg l. \qquad \text{II. } r\ll1,\quad a\lesssim l, \tag{10} \]
or (taking into account the numerical solution) \((1^{\mathrm{r}})\)
\[ \text{I. } r>5,\quad \frac{a}{l}>10. \qquad \text{II. } r<0.5,\quad \frac{a}{l}\lesssim1.5. \tag{10'} \]
Let us give an approximate solution of the dispersion equation in these regions \((1^{\mathrm{r}})\):
\[ \text{I. }\quad \frac{\beta_1}{\beta_0}=1-1.08\frac{1}{r^2},\qquad \frac{\alpha_1}{\beta_0}=0.7\frac{1}{r},\qquad \frac{\beta_2}{\beta_0}=\frac{\sqrt{5}}{3}r^{1/2}\left(1+\frac{1.05}{r}\right), \]
\[ \frac{\alpha_2}{\beta_0}=\frac{\sqrt{5}}{3}r^{1/2}\left(1-\frac{1.05}{r}\right). \tag{11} \]
\[
\text{II.}\quad \frac{\beta_1}{\beta_0}=0.61,\qquad
\frac{\alpha_1}{\beta_0}=(\alpha_l)_{1\infty}\frac{5}{16}\sqrt{\frac{10\pi}{3}}\,r,\qquad
\frac{\beta_2}{\beta_0}=1.59,
\]
\[
\frac{\alpha_2}{\beta_0}=(\alpha_l)_{2\infty}\frac{5}{16}\sqrt{\frac{10\pi}{3}}\,r,\qquad
(\alpha_l)_{1\infty}=0.20,\qquad
(\alpha_l)_{2\infty}=0.35.
\tag{11'}
\]
Let us give very simple formulas for the attenuation in region II:
\[ \alpha_1=\frac{(\alpha_l)_{1\infty}}{l},\qquad \alpha_2=\frac{(\alpha_l)_{2\infty}}{l}. \tag{11''} \]
- We now analyze the reaction on the reflector in regions I and II. Setting \(x=a\) in (7) and using (11), with accuracy up to \(1/r\) in region I and up to \(r\) in region II, we obtain, respectively:
\[ \frac{\Delta P_{11}(a)}{\rho_0 v_{\mathrm{ad}}\dot{\xi}_0} = \frac{s_1^2+A_1/r^{1/2}+B_1/r} {\left(s_1^2+A_1/r^{1/2}+B_2/r\right)\operatorname{sh}1+ \left(s_1^2\Delta_1+B_3/r\right)\operatorname{ch}1}; \tag{12\(_{\mathrm{I}}\)} \]
\[
\frac{\Delta P_{11}(a)}{\rho_0 v_{\mathrm{ad}}\dot{\xi}_0}
=
\frac{1}{d}\left\{(a_1-ib_1 r)\operatorname{sh}1+(a_2-ib_2 r)\operatorname{ch}1+\right.
\]
\[
\left.
+(a_3-ib_3 r)\operatorname{sh}2+(a_4-ib_4 r)\operatorname{ch}2\right\},
\tag{12\(_{\mathrm{II}}\)}
\]
where
\[ s_1=\frac{s}{2-s},\qquad \operatorname{sh}1=\operatorname{sh}(ik_1a) \quad\text{(similarly below)};\qquad k_i=K_i\beta_0, \]
\[ A_1=s_1\frac{5\sqrt{6\pi}}{8}(1+i),\qquad B_1=i\left(\frac{75\pi}{64}+\frac{9s_1^2}{5}\right),\qquad B_2=i\left(\frac{75\pi}{64}+\frac{9s_1^2}{10}\right), \]
\[ \Delta_1=\frac{2}{\sqrt{5}}(1+i)\frac{1}{r^{1/2}},\qquad B_3=s_1 i\sqrt{\frac{15\pi}{8}}, \]
\[ a_1=0.37+0.58s_1^2,\qquad a_2=0.94s_1,\qquad a_3=1.45+0.41s_1^2,\qquad a_4=1.59s_1, \]
\[ b_1=0.45+1.05s_1^2,\qquad b_2=1.42s_1,\qquad b_3=2.35+0.97s_1^2,\qquad b_4=3.21s_1, \]
\[
d=(a_5-ib_5 r)\operatorname{sh}1\cdot\operatorname{sh}2
+(a_6-ib_6 r)\operatorname{sh}2\cdot\operatorname{ch}1+
\]
\[
+(a_7-ib_7 r)\operatorname{sh}1\cdot\operatorname{ch}2
+(a_8-ib_8 r)(\operatorname{ch}1\cdot\operatorname{ch}2-1),
\]
\[ a_5=1.45+0.46s_1^2,\qquad a_6=1.39s_1,\qquad a_7=1.01s_1,\qquad a_8=0.44s_1^2, \]
\[ b_5=2.54+0.99s_1^2,\qquad b_6=2.44s_1,\qquad b_7=2.34s_1,\qquad b_8=1.02s_1^2. \]
From (12) one can determine the maxima of the reaction on the reflector. In case I, setting \(\beta_1a=\pi n_{\max}\) \((n_{\max}=1,2,\ldots)\) and neglecting the quantities \(1/r\), \((\alpha_1a)^3\), we obtain from (12) the known result \({}^{(3-6)}\)
\[ \Delta P_{11}(a)_{\max} = \rho_0 v_{\mathrm{ad}}\dot{\xi}_0\, \frac{1}{\alpha_1a+\Delta_1}. \tag{13\(_{\mathrm{I}}\)} \]
In region II, for an insulator \((s_1=0)\), we obtain a system of maxima determined by the factors \(\frac{1}{\alpha_1a}\), \(\frac{1}{\alpha_2a}\); the positions of the maxima are found from the conditions \(\beta_1a=\pi n_{1\max}\), \(\beta_2a=\pi n_{2\max}\) \((n_{1\max}, n_{2\max}\) are integers).
In region II, for \(s\ne0\), a maximum can occur only due to the coincidence (with some accuracy) of positive and negative maxima of \(\cos\beta_1a\) and \(\cos\beta_2a\). This is approximately reduced to the requirement
\[ \cos\delta_1=\cos\delta_2=1-\delta,\qquad \sin\delta_1=-\sin\delta_2=\gamma,\qquad |\delta|,\ |\gamma|\ll1, \tag{14} \]
where \(\delta_1=1.38\pi x\), \(\delta_2=0.62\pi x\); \(\beta_2a=\pi x\), \(\beta_1a=0.38\pi x\).
From (14) we determine the positions of the reaction maxima: \(x_{\max}=3, 6, 10, 13, 16*\), etc. At the points \(x_{\max}\), neglecting second-order terms in \(x_1a, x_2a, r, \delta, \gamma\), from \((12_{11})\) we obtain:
\[ \frac{\Delta P_{11}(a)_{\max}}{\rho_0 v_{\mathrm{ad}}\xi_0} = \frac{s_1+[-c_1\delta+\alpha'_{12}a]+i[c_2\delta^{1/2}-c_3r]} {s_1\alpha_{12}a+[-\delta s_1\alpha_{12}a+c_4\gamma r-c_5\delta]+i[\gamma\alpha''_{12}a+\delta c_6r+c_7\gamma]}, \tag{15} \]
\[ c_1=0.63s_1,\quad c_2=0.80+0.23s_1^2,\quad c_3=1.83s_1,\quad c_4=0.92s_1,\quad c_5=0.17s_1^2, \]
\[ c_6=0.40s_1^2,\quad c_7=0.40s_1, \]
\[ \alpha'_{12}=0.55\alpha_2+0.40\alpha_1, \]
\[ \alpha_{12}=(0.15+0.23s_1^2)\alpha_1+(0.57+0.16s_1^2)\alpha_2, \]
\[ \alpha''_{12}=(0.57+0.18s_1^2)\alpha_2+s_1^2 0.17\alpha_1. \]
Fig. 1. Regions of existence of a standing wave. The hatched region is the measurement region in (4)
From (15), requiring \(r\ll a/l\), we obtain:
\[ \frac{\Delta P_{11}(a)_{\max}}{\rho_0 v_{\mathrm{ad}}\xi_0} = \frac{s_1+[-c_1\delta+\alpha'_{12}a]+i\delta^{1/2}c_2} {\alpha a+\Delta_2}, \tag{16} \]
\[ \alpha=s_1(1-\delta)\alpha_{12}+i\gamma\alpha''_{12}; \]
\[ \Delta_2=-c_5\delta+ic_7\gamma. \]
For not too small \(a/l\), the principal term in (16) is
\[ \Delta P_{11}(a)_{\max}=\rho_0 v_{\mathrm{ad}}\xi_0/\alpha_{12}a. \tag{17} \]
From (14) and (17) it follows that in the region \(r<0.5,\ r\ll a/l\), the reaction maximum occurs at the points \(\beta_1a=0.38\pi x_{\max}\), and the value of the maximum is determined by the attenuation of both the sound wave and the thermal wave.
Thus, for recalculating the value \(\beta_1/\beta_0\), calculated by the usual formula
\[ \beta_1/\beta_0=\frac{L_{\mathrm{ad}}}{2a}n_{1\max}, \]
one should take into account that
\[ 0.38x_{\max}/n_{1\max}\simeq 1.2 \tag{18} \]
From (16) one can obtain the criterion for the disappearance of the standing wave
\[ a/l\lesssim 0.1s_1. \tag{19} \]
Fig. 2. Dispersion in argon. \(a\)—6-moment approximation, \(b\)—4-moment approximation, points—experiment\({}^{(4)}\) (up to \(r\le 0.5\) recalculated according to (19))
The comparison of \((10')\), (19), and (18) with experiment is given, respectively, in Figs. 1 and 2.
In conclusion I express my deep gratitude to Academician N. N. Bogolyubov for valuable comments.
Moscow State University
named after M. V. Lomonosov
Received
8 IX 1959
REFERENCES
- I. I. Olkhovskii, a) DAN, 126, No. 4, 748 (1959); b) DAN, 123, No. 2, 262 (1958); c) DAN, 123, No. 5, 821 (1958); d) Scientific Reports of Higher Education Institutions, ser. phys.-math. sciences, No. 4, 143, 181 (1958).
- J. C. Hubbard, Phys. Rev., 38, 5, 1011 (1931).
- K. F. Herzfeld, Phys. Rev., 1, 53 (1938).
- B. P. Konstantinov, ZhTF, 9, issue 3, 226 (1939).
- R. S. Alleman, Phys. Rev., 1, 55 (1939).
- E. Meyer, G. Sessler, Zs. f. Phys., 149, 15 (1957).
* We have omitted \(x_{\max}=7\), which gives a maximum equal to \((\Delta P_{11})_{x_{\max}}=6\).