V. A. SUCHKOV

PLANE POTENTIAL FLOWS WITH STATIONARY STREAMLINES

(Presented by Academician M. A. Lavrent’ev on February 20, 1965)

Hydromechanics

§ 1. Consider the equations of plane potential motion of a polytropic gas

\[ \frac{\partial u_i}{\partial t}+\frac{\partial \Delta}{\partial x_i}=0, \qquad \frac{\partial u_1}{\partial x_2}=\frac{\partial u_2}{\partial x_1}; \tag{1,1} \]

\[ \frac{\partial \Delta}{\partial t} +2u_k\frac{\partial \Delta}{\partial x_k} -u_i u_k\frac{\partial u_i}{\partial x_k} +c^2\frac{\partial u_k}{\partial x_k}=0; \tag{1,2} \]

\[ c^2=\chi\left(\Delta-\frac{u_k^2}{2}\right), \qquad i,k=1,2. \tag{1,3} \]

Here \(x_i\) are Cartesian coordinates, \(t\) is time, \(u_i\) are the components of the velocity vector, \(c\) is the speed of sound, and \(\Delta\) is the total energy per unit mass. For an adiabatic flow \(\chi=\gamma-1\); for an isothermal flow \(\chi=0\), \(\chi\Delta=1\).

Plane flows with stationary streamlines are characterized by the differential relation

\[ u_2\frac{\partial u_1}{\partial t} - u_1\frac{\partial u_2}{\partial t}=0. \tag{1,4} \]

It is necessary to investigate the compatibility of the overdetermined system (1,1)—(1,4) (see \((^1)\)). This class of flows includes stationary flows and nonstationary flows with plane or axial symmetry, where the stationary streamlines are rectilinear \((^2,^3)\).

§ 2. Take as new unknown functions the velocity modulus \(v\) and \(\theta\), the angle of inclination of the velocity vector to the \(x_1\)-axis; then

\[ u_1=v\cos\theta,\qquad u_2=v\sin\theta. \tag{2,1} \]

Substituting (2,1) into (1,4), we obtain

\[ \frac{\partial \theta}{\partial t}=0. \tag{2,2} \]

Let us write the equations of potential flows with stationary streamlines in curvilinear orthogonal coordinates \(q_1,q_2\) (\(q_2=\mathrm{const}\) are the streamlines, \(q_1=\mathrm{const}\) are the lines orthogonal to them):

\[ \frac{\partial \psi}{\partial t} +\frac{\partial \Delta}{\partial q_1}=0, \qquad \frac{\partial \psi}{\partial q_2}=0, \qquad \frac{\partial \Delta}{\partial q_2}=0; \tag{2,3} \]

\[ \frac{\partial \Delta}{\partial t} +2\psi\frac{\partial \Delta}{\partial q_1}l -\psi^2\frac{\partial \psi}{\partial q_1}l^2 -\psi^3 m +c^2\left(l\frac{\partial \psi}{\partial q_1}+\psi n\right)=0; \tag{2,4} \]

\[ \frac{\partial}{\partial q_1} \left(\frac{1}{H_1}\frac{\partial H_2}{\partial q_1}\right) + \frac{\partial}{\partial q_2} \left(\frac{1}{H_2}\frac{\partial H_1}{\partial q_2}\right)=0, \tag{2,5} \]

where

\[ \psi=vH_1,\qquad l=\frac{1}{H_1^2},\qquad m=\frac{l}{2}\frac{\partial l}{\partial q_1}, \tag{2,6} \]

\[ n=\frac{1}{2}\frac{\partial l}{\partial q_1} +\frac{l}{H_2}\frac{\partial H_2}{\partial q_1}, \qquad c^2=\chi\left(\Delta-\frac{\psi^2}{2}\,l\right). \]

§ 3. Suppose that the Lamé coefficients are known and satisfy equation (2,5) \((^4)\). Consider the compatibility of the system (2,3), (2,4). The func-

\(\chi\), \(\Delta\), and \(\psi\) depend only on \(q_1, t\), while the functions \(l, m, n\) depend on \(q_1, q_2\). The variable \(q_2\) enters only into equation (2.4) as a parameter. Differentiating this equation with respect to \(q_2\) and setting the differential consequences equal to zero, we obtain compatibility conditions for the functions \(H_1(q_1,q_2)\), \(H_2(q_1,q_2)\).

Theorem 1. A plane unsteady potential flow with rectilinear stationary streamlines is a motion with plane or axial symmetry.

Theorem 2. A plane flow with curvilinear stationary streamlines, possessing a functional Jacobian, is either stationary or belongs to the class of solutions:

1. \(\chi=0,\ \psi=1,\ \Delta=\alpha t+\beta,\ \alpha,\beta\) are constants;
2. \(\chi\ne0,\ \psi=t^{-1},\ \Delta=q_1t^{-2}\).

The functions \(H_1, H_2\) are determined from a system of two differential equations (2.5) and, respectively, (3.1) or (3.2):

\[ \alpha+n=m; \tag{3.1} \]

\[ \chi n(q_1-l/2)-m-2(q_1-l)=0. \tag{3.2} \]

Theorem 3. The functions \(\psi=t^{-1}\), \(\Delta=q_1t^{-2}+s\), where for \(\chi\ne2\), \(s=s_0t^{-4\chi/(\chi+2)}\); for \(\chi=2\), \(s=-t^{-2}\ln t\); \(s_0\) is constant, describe a plane flow with curvilinear streamlines having a constant Jacobian. The Lamé coefficients \(H_1,H_2\) are known functions of \(q_1,q_2\), for example, for \(\chi=2\)

\[ H_1^2+H_2^2=1,\qquad H_1^{-2}+\ln(H_1^{-2}-1)=2(q_1+q_2). \tag{3.3} \]

§ 4. From the condition of stationarity of the streamlines we have obtained an explicit dependence of the solutions on time in the variables \(q_1,q_2,t\). This dependence is unchanged also in the variables \(x_1,x_2,t\), so that the solution must have the form

\[ u_i=\psi V_i,\qquad \Delta=-\psi'V+s, \tag{4.1} \]

where \(\psi,s\) are known functions of \(t\); \(V\) and \(V_i=\partial V/\partial x_i\) are functions only of \(x_1,x_2\).

For \(\chi=0,\ \psi=1,\ s=\alpha t+\beta\), for the function \(V(x_1,x_2)\) from (1.2) we obtain the equation

\[ (V_iV_k-\delta_{ik})\cdot V_{ik}=\alpha. \tag{4.2} \]

The function \(V(x_1,x_2)\) for \(\chi\ne0,\ \psi=t^{-1},\ s=0\) must satisfy the equation

\[ \bigl[V_iV_k-\chi(V-V_i^2/2)\delta_{ik}\bigr]V_{ik}-2V_i^2+2V=0. \tag{4.3} \]

The Jacobian of the solution of equations (4.2) and (4.3) is two functions of one argument.

For \(\chi=2,\ \psi=t^{-1},\ s=-t^{-2}\ln t\), integrating the system

\[ \partial x_1/\partial q_1=H_1\cos\theta,\qquad \partial x_1/\partial q_2=-H_2\sin\theta, \]

\[ \partial x_2/\partial q_1=H_1\sin\theta,\qquad \partial x_2/\partial q_2=H_2\cos\theta, \tag{4.4} \]

we obtain the solution

\[ q_1=x_1+\frac{1}{2}(x_2^2+1)=V(x_1,x_2). \tag{4.5} \]

Similarly, for \(\chi\ne2,\ \psi=t^{-1},\ s=s_0t^{-4\chi/(\chi+2)}\) we find

\[ q_1=\frac{1}{2}\left(x_2^2+\frac{2-\chi}{2+\chi}x_1^2\right)=V(x_1,x_2). \tag{4.6} \]

I take this opportunity to express my deep gratitude to N. N. Yanenko for his constant attention and interest in the work.

REFERENCES

1. N. N. Yanenko. Proceedings of the IV All-Union Mathematical Congress, Leningrad, July 1961, vol. 2, p. 247.
2. B. Bernstein, T. G. Thomas, J. Rat. Mech. and Anal., 1, 703 (1955).
3. L. D. Landau, E. M. Lifshitz, Mechanics of Continuous Media, Moscow, 1954.
4. S. P. Finikov, A Course of Differential Geometry, Moscow, 1952.