Full Text
HYDROMECHANICS
Yu. P. Krasovskii
ON THE EXISTENCE OF APERIODIC FLOWS WITH A FREE BOUNDARY
(Presented by Academician M. A. Lavrent’ev on February 8, 1960)
Many problems in the theory of wave motions of a liquid lead to the study of aperiodic flows whose free boundary has horizontal asymptotes. These include problems on the motion of a body beneath a free surface, on the flow of a liquid over an uneven bottom, and the problem of a solitary wave. Here the existence of a solution to the problem of the motion of a flow over an uneven bottom will be proved in the case of large Froude numbers. A special case of this problem, when the bottom of the flow descends in the direction of the current, was considered by R. Gerber (¹). In addition, we shall show that in a liquid of infinite depth a solitary wave cannot exist. The existence of solitary waves in a liquid of finite depth, as is known, was first proved by M. A. Lavrent’ev.
The motion is assumed to be plane, steady, and irrotational; the liquid is ideal and incompressible.
Fig. 1 Fig. 2
- Let a flow of liquid with free boundary \(\Gamma\) flow over an uneven bottom \(D\) (see Fig. 1). The bottom is horizontal everywhere except for a segment \(AB\) of finite length \(L\). We shall specify the curve \(D\) by the parametric equation
\[ \alpha = \alpha [l], \]
where \(\alpha\) is the angle formed by the tangent to \(D\) with the horizontal; \(l\) is the arc length measured from the point \(C\), the midpoint of the arc \(AB\). We have
\[ \alpha [l] = 0 \quad \text{for } |l| > L/2. \]
The function \(\alpha [l]\) is assumed piecewise smooth.
Problem 1. Determine the liquid flow and the curve \(\Gamma\), if the following are given: the function \(\alpha [l]\); the discharge \(Q\); \(L\), the length of the arc \(AB\); and the mean velocity
\[ c = \frac{1}{L} \int_{-L/2}^{+L/2} |\bar v|\, dl . \]
To derive the basic equations, map the flow region onto the infinite strip \(0 \le \psi \le h\) of the plane \(w = \varphi + i\psi\). Under this mapping the free boundary \(\Gamma\) becomes the straight line \(\psi = h\), the bottom \(D\) becomes the straight line \(\psi = 0\), and the arc \(AB\) becomes the segment \(0 \le \varphi \le 1\). We express the complex velocity \(\bar v\) through the function
\[ \omega(w) = c\Phi(\varphi;\psi) + i\Phi(\varphi;\psi), \]
putting
\[ \bar v = c e^{-\omega(w)} , \]
Using the conditions of constancy of the pressure along the free boundary and of flow past the bottom, we arrive at a system of equations, in analogy with how this is done in paper \((^{1})\):
\[ \Phi(\varphi)=-\frac{1}{2h}\int_{-\infty}^{\infty} \frac{c\Phi(\varphi)-c\Phi(u)}{\operatorname{sh}\frac{\pi}{2h}(u-\varphi)}\,du +\frac{1}{2h}\int_{-\infty}^{+\infty} \frac{\alpha[l(u)]}{\operatorname{ch}\frac{\pi}{2h}(u-\varphi)}\,du, \]
\[ \frac{dc\Phi}{d\varphi}=\frac{gL}{c^{2}}e^{3c\Phi(\varphi)}\sin\Phi(\varphi), \qquad \frac{dl}{d\varphi}=Le^{c\Phi^{*}(\varphi)}. \]
Here the functions \(\Phi(\varphi)=\overline{\Phi}(\varphi;h)\), \(c\Phi(\varphi)=c\overline{\Phi}(\varphi;h)\), \(l(\varphi)\) are unknown, while the function \(c\Phi^{*}(\varphi)=c\overline{\Phi}(\varphi;0)\) is determined through \(\Phi(\varphi)\) and \(\alpha[l(\varphi)]\) as the limiting value of the conjugate \(\overline{\Phi}(\varphi;\psi)\). We reduce this system to the operator equation
\[ x=Ax, \tag{1} \]
where \(A\) is a completely continuous operator in a certain Banach space. If the Froude number is sufficiently large, then an a priori estimate for the solution of equation (1) can be obtained, after which the existence of a solution is a consequence of Schauder’s principle.
Theorem 1. If \(\max|\alpha|<\pi/2\), then one can indicate a positive number \(\varepsilon\) such that, for \(gL/c^{2}\leqslant \varepsilon_h\), a solution of problem 1 exists.
This solution has the following properties:
1) \(|\Phi[l]|\leqslant M_1e^{-k|l|}\), \(k>0\), \(M_1\) does not depend on \(l\);
2) \(\max_l|\Phi[l]|\leqslant M_2h/L\), where \(h=Q/c\) is the mean depth and \(M_2\) does not depend on the ratio \(h/L\). Here \(*\ \Phi=\Phi[l]\) is the parametric equation of the free boundary.
The following problem concerns flows symmetric with respect to the vertical axis.
Problem 2. Determine the fluid flow and the curve \(\Gamma\) so that the conditions
\[ \Phi[l]=-\Phi[-l];\qquad \Phi[l]\geqslant 0\quad \text{for } l\leqslant 0, \]
are satisfied, if the following are given: the function \(\alpha[l]\); the discharge \(Q\); \(L\), the length of the arc \(AB\); and \(c_0\), the velocity at the point \(C'\), the vertex of the free boundary.
For problem 2 a more precise result is valid, obtained by the same methods as for problem 1.
Theorem 2. Suppose the following conditions are fulfilled:
1) \(gQ/c_0^3<1\);
2) \(gQ/c_0^3+\max|\alpha|<\pi/2\);
3) \(\alpha[-l]=-\alpha[l]\), \(\alpha[l]\geqslant 0\) for \(l\leqslant 0\).
Then a solution of problem 2 exists.
- Suppose that on the surface of a fluid a solitary wave propagates with constant velocity \(c\) without change of form. At infinity the fluid is at rest. We shall assume that the solitary wave (see Fig. 2) has the following properties: 1) the free boundary has one vertex; 2) the solitary wave is symmetric with respect to the vertical axis passing through the vertex; 3) the height of the solitary wave is finite.
Theorem 3. Solitary waves in a fluid of infinite depth do not exist.
Let us note that in a fluid of infinite depth there also cannot exist periodic waves of arbitrarily great length, if their propagation velocity is bounded \((^{3})\).
* \(\Phi\) is the angle formed by the tangent to the profile of the free boundary with the horizontal.
For a liquid of finite depth \(h\) (\(h\) is the depth of the liquid at infinity) the following result is valid:
Theorem 4. If the Froude number is less than one, i.e. \(gh/c^2 > 1\), then solitary waves do not exist.
A consequence of the last theorem is the fact that a solitary wave possessing properties 1), 2), 3) must necessarily be a wave of elevation.
The proof of Theorems 3 and 4 is based on a method for estimating the bounds of the positive spectrum, widely used in the theory of positive operators \((^2)\).
The work was carried out in the seminar on nonlinear problems of mechanics at Rostov State University under the direction of I. I. Vorovich.
Rostov-on-Don State
University
Received
6 II 1960
REFERENCES
\(^1\) R. Gerber, in: On exact solutions of the equations of motion of a heavy fluid with a free surface. Theory of surface waves, IL, 1959, pp. 218–308. \(^2\) M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, 1956. \(^3\) Yu. P. Krasovskii, DAN, 130, No. 6 (1960).