HYDROMECHANICS

Corresponding Member of the Academy of Sciences of the USSR L. N. SRETENSKII

ON THE THEORY OF GAS JETS

1°. The S. A. Chaplygin equation for determining the stream function \(\psi(\theta,\tau)\) of a steady gas flow,

\[ \frac{\partial}{\partial \tau}\left\{\frac{2\tau}{(1-\tau)^\beta}\frac{\partial \psi}{\partial \tau}\right\} + \frac{1-(2\beta+1)\tau}{2\tau(1-\tau)^{\beta+1}} \frac{\partial^2\psi}{\partial \theta^2} =0 \tag{1} \]

possesses, in addition to those particular solutions which were indicated and used by S. A. Chaplygin, one more family of particular solutions, making it possible to solve a number of new problems on the jet motion of a gas.

This family consists of functions that are the product of a function \(\Theta(\theta)\) by a function \(T(\tau)\), where these functions are integrals of the differential equations

\[ \frac{d^2\Theta}{d\theta^2}-n^2\Theta=0, \]
\[ \frac{d}{d\tau}\left\{\frac{2\tau}{(1-\tau)^\beta}\frac{dT}{d\tau}\right\} + n^2\frac{1-(2\beta+1)\tau}{2\tau(1-\tau)^{\beta+1}}\,T=0, \tag{2} \]

in which \(n\) is an arbitrary constant number. Two linearly independent particular solutions of equation (2) can be represented in the following form:

\[ T'_n = M(\tau,n)\cos\left(\frac{1}{2}n\ln\frac{\tau_2}{\tau}\right) - N(\tau,n)\sin\left(\frac{1}{2}n\ln\frac{\tau_2}{\tau}\right), \]
\[ T''_n = M(\tau,n)\sin\left(\frac{1}{2}n\ln\frac{\tau_2}{\tau}\right) + N(\tau,n)\cos\left(\frac{1}{2}n\ln\frac{\tau_2}{\tau}\right), \tag{3} \]

where \(\tau_2\) is some number, and the functions \(M\) and \(N\) are defined by the series

\[ M(\tau,n) = 1+\frac{\beta}{2}\frac{n^2(1-\tfrac{1}{2}\beta n^2)}{n^2+4}\tau^2+\cdots, \]
\[ N(\tau,n) = \left[ -\frac{1}{2}\beta\tau - \frac{1}{2}\beta \frac{(1+\tfrac{1}{2}\beta)n^2-2(\beta+1)}{n^2+4}\tau^2 +\cdots \right]n. \tag{4} \]

Let us note that, in solving the simplest Sturm–Liouville problem for equation (3) with the conditions \(T=0\) for two values of \(\tau\) smaller than

\[ \frac{1}{2\beta+1}, \]

the corresponding fundamental number \(\lambda=n^2\) will always be positive.

We transform equation (3) to new variables \(z\) and \(u(z)\), putting

\[ z=\int_{\tau}^{\tau_2}\frac{d\tau}{2\tau} \sqrt{\frac{1-(2\beta+1)\tau}{1-\tau}}; \qquad u(z)=T(\tau)\sqrt[4]{\frac{1-(2\beta+1)\tau}{(1-\tau)^{2\beta+1}}}; \]

we obtain:

\[ \frac{d^2u}{dz^2} + \left[ n^2+ \frac{\beta(2\beta+1)\tau^2}{1-\tau} \frac{4-2(\beta+2)\tau-\beta(2\beta+1)\tau^2}{[1-(2\beta+1)\tau]^3} \right]u =0. \tag{5} \]

We shall construct for this differential equation the inversion formulas, starting from its particular solution \(\omega(z,\lambda)\), which becomes equal to,

zero at \(z=0\) and considered for all positive values of \(z\) from zero to \(\infty\). In view of the above-noted property of the number \(n^2\) and the convergence of the integral

\[ \int_0^\infty \frac{\beta(2\beta+1)\tau^2}{1-\tau}\, \frac{4-2(\beta+2)\tau-\beta(2\beta+1)\tau^2}{[1-(2\beta+1)\tau]^3} \,dz \]

equation (5) has a continuous spectrum located on the positive part of the \(\lambda\)-axis \((^1)\).

The indicated inversion formulas, which are constructed using the expansions (3) and (4), have the form

\[ F(\lambda)=\int_0^\infty f(x)\,\omega(x,\lambda)\,dx, \qquad f(x)=\int_{-\infty}^{\infty}\omega(x,\lambda)F(\lambda)\,d\rho(\lambda), \]

where

\[ \omega(x,\lambda)=\frac{4\tau_2^2}{\sqrt{\lambda}}\, \sqrt[4]{\frac{1-(2\beta+1)\tau}{[1-(2\beta+1)\tau_2]^3}\, \frac{(1-\tau_2)^{3-2\beta}}{(1-\tau)^{2\beta+1}}}\, T(\tau,n); \]

\[ \lambda=n^2;\qquad T(\tau,n)=T_n'(\tau_2)T_n''(\tau)-T_n''(\tau_2)T_n'(\tau); \]

\[ d\rho(\lambda)= \frac{1}{16\pi\tau_2^2} \sqrt[4]{\frac{[1-(2\beta+1)\tau_2]^3}{(1-\tau_2)^{3-2\beta}}}\, \frac{\sqrt{\lambda}\,d\lambda}{M(\tau_2,n)^2+N(\tau_2,n)^2}. \]

\(2^\circ\). Consider a vessel bounded by two parallel walls extending to infinity in one direction, and provided with a nozzle consisting of two small straight-line segments, equal to one another, inclined to the centerline of the vessel and departing from the free ends of the indicated parallel walls (Fig. 1). From such a vessel gas under pressure issues into free space in the form of a jet. Our problem consists in constructing the stream function of this gas flow.

Fig. 1

Let \(\tau_1\) and \(\tau_2\) denote the values of Chaplygin’s variables \(\tau\), respectively, in the remote parts of the vessel and on the jet. Further, let \(\theta_0\) denote the angle of inclination of the straight-line segments to the centerline of the vessel; let \(q\) be the value of the stream function along the centerline of the vessel, if the value of this function on the line \(ABCD\) is taken to be zero.

To construct the stream function of the gas flow under consideration, we take the solution of Chaplygin’s equation indicated in \(1^\circ\) in the form

\[ \psi=T_n(\tau)\frac{\operatorname{sh} n(\theta_0-\theta)}{\operatorname{sh} n\theta_0} \]

and apply the inversion formulas to it. We then obtain, after carrying out a number of operations, the following result:

\[ \psi=\frac{2q}{\pi} \int_0^\infty \left[ \frac{2\tau}{(1-\tau)^\beta}\frac{dT}{d\tau} \right]_{\tau_1}^{\tau_2} \frac{\operatorname{sh} n(\theta_0-\theta)}{n^2\operatorname{sh} n\theta_0}\, \frac{T(\tau,n)\,dn}{M(\tau_2,n)^2+N(\tau_2,n)^2}. \]

With the aid of similar calculations, a number of other problems in the theory of gas jets, considered for an incompressible fluid by N. E. Zhukovsky, can also be solved.

Received
29 I 1958

CITED LITERATURE

1. B. M. Levitan, Expansions in Eigenfunctions of Second-Order Differential Equations, chs. II, III, Moscow—Leningrad, 1950.