SOLITARY WAVES IN A LAYER OF VISCOUS LIQUID Vhh" +@BULLET ] h'+.h-E (original) (raw)

Solitary waves in a thin layer of viscous liquid which is running down a vertical surface under the action of gravity are investigated. The existence of such waves was demonstrated in the experiments of [i, 2]. The difficulties that must be faced in a theoretical computation were also noted in these studies. Below a solution of the problem of stationary waves is obtained by the method of expansion in the small parameter in two regions with subsequent matching and also by a numerical integration method. It is shown that in each case a solution of solitary wave type exists along with the single-parameter family of periodic solutions (parameter the wave number ~). On decreasing the wave number, the periodic waves go over into a succession of solitary waves. As the basis of the investigation we take the equation for the thickness of the layer h(~:), which is obtained by integrating the basic equations of motion of a viscous liquid transverse to the layer. In the integration it is assumed that the boundary-layer approximation can be used and a parabolic profile of the longitudinal velocity is taken. In the coordinate system attached to the wave this equation has the following form: Here c is the wave velocity; Uo and ao are the characteristic values of the velocity and the thickness of the layer. In the case of a solitary wave Uo and ao denote the mean values of the velocity and thickness of the unperturbed layer. The nonlinear periodic solutions of Eq. (i) were investigated in [3]. The method of Fourier series expansion was used and explicit expressions were obtained for the waveform, the phase velocity, and the layer thickness. For a fixed number of terms considered in this solution, the accuracy decreases with the decreasing wave number ~ due to the fact that for small values of ~ the wave profile is very different from a harmonic wave. As an example, some results of direct numerical integration of Eq. (I) in the case of periodic waves in a layer of water are shown in Fig. i for Re = 3aoUov-I = 24.41 and ~ = 0.107 (curve I) and = 0.051 (curve 3). The integration was done over a wavelength ~io ~ ~ ~ ~Io + 2~-~ The initial point ~1owas chosen at the crest of the wave h'(~1o) = 0; for given values of Re and ~, the values of c, ao, and the initial data h(~1o), h"(~o) were chosen in such a way that for ~ = ~o + 2~-~ the periodicity condition is satisfied. For the values of a lying close to the neutral stability curve in the Re, ~ plane the wave profiles are almost sinusoidal. The effect of the nonlinear terms in Eq. (i) increases with the decrease of a, and the profiles become noticeably deformed, acquiring the form of solitary waves. The computations were carried out with a small step along parameter a. For obtaining the wave for ~: = ~ + As the characteristics of the wave solution corresponding to ~ were Moscow.