Chemical Engineering Science. Vol 41, No IO. pp. 2477-2486. 198b. OOK-2509/86 $3 oO+O.OO Printed an Great Bntain Pergamon Journals Ltd. NONLINEAR WAVES ON LIQUID FILM SURFACES-II. BIFURCATION ANALYSES OF THE LONG-WAVE EQUATION LIANG-HENG CHEN and HSUEH-CHIA CHANG+ Department of Chemical Engineering, University of Houston, University Park, Houston, TX 77004, U.S.A. (Received 31 August 1984) Abstract-The long-wave equation for film flow in vertical pipes with and without interfacial stress is studied numerically. Two sets of boundary conditions are imposed-periodic and infinite, corresponding to monochromatic and multi-frequency waves. For the periodic boundary conditions, waves with increasingly many subharmonic modes bifurcate from the flat-film base state as the system parameter 0 decreases below unity. Eventually, a strange attractor appears at c - 0.028. Thechaotic solution for 0 c 0.028 corresponds to the observed turbulent film surface near flooding conditions for systems with upward gas flow. The solutions of the infinite domain problem indicate that pulse-like initial profiles collapse into wave packets which propagate in both directions with speeds governed by the pulse heights. Interactions of these solitary waves are studied numerically. INTRODUCIION In Part I of this series (Chang, H.-C., 1986), we showed that weakly inertial waves of annular film flow are described by the equation [eq. (75) of Part I] ~++uu~+U<<+au~~,z<=O (1) with Zn-periodic boundary conditions if monochro- matic waves are considered. If one needs to investigate the interaction of several waves of different wave- lengths, an infinite domain problem results. The only parameter, 6, in eq. (1) contains the wavelength, Froude number, Weber number, Reynolds number and the counter-flowing gas rate [see eq. (2)]. Both free-flowing films and flows with interfacial stress are included. Consequently, the dynamics of these two cases can be conveniently studied together. The free-flowing case has been investigated by several workers who have reported seemingly con- tradictory results. Atherton (1972) and Tougou (1981) obtained finite-amplitude periodic waves whereas Sivashinsky and Michelson (1980) observed chaotic behaviour in their numerical studies. We will resolve the apparent controversy in the present paper by carrying out a detailed bifurcation analysis of the spatially periodic solutions. Our results show a strong dependence of the solution branches on the parameter n. A finite-amplitude wave with only one harmonic first bifurcates from the flat-film solution as 0 decreases below unity. Other waveforms with more unstable modes then bifurcate successively until a critical d value is reached below which no equilibrium states are found. Instead, time integration reveals a strange attractor in this region. The discovery of a strange attractor for small c values is especially gratifying when the results are applied to films with interfacial stress. The linear analysis of Part I reveals that the flat-film base states +To whom correspondence should be addressed. yield a sharp maximum in the growth term 4 (see Fig. 9 of Part I) for gas rate in the proximity of the flooding condition. The quantity q. which is the real part of the eigenvalue, is then carried into the weakly nonlinear analysis and the resulting equation, eq. (l), contains q in the denominator of (T: EFW a=3x. (2) Hence, a sharp maximum in q corresponds to a sharp minimum in 0 with respect to the gas rate, indicating that the gas rate at the minimum is where the most chaotic behaviour should be observed. This is exactly what Dukler et al. (1984) found in their experiments. We also carry out an analysis on the collapse of an initial pulse in an infinite domain. A wave packet containing several frequencies results and the waves propagate in both directions with equal speed relative to the moving coordinate. When two pulses collapse in the same neighbourhood, the resulting wavetrains seem to interact linearly with corresponding cancel- lation and reinforcement. Such behaviour of solitary waves have never been studied before. Insomuch as eq. (1) has appeared recently in a variety of contexts, our bifurcational analyses of its spatially periodic solutions and numerical investiga- tions of the corresponding initial value problem in an infinite domain are of particular interest. In fact, eq. (1) has come to be known as the Kuramoto-Sivashinsky equation in the physics literature (Nicolaenko et al., 1985). Other than falling films, the equation also models unstable flame fronts (Michelson and Sivashinsky, 1977; Sivashinsky, 1977, 1980), Belouzov-Zabotinskii reaction patterns (Kuramoto and Tsuzuki, 1975, 1976; Kuramoto, 1978), interfacial instabilities between two viscous fluids (Hooper and Grimshaw, 1985) and unstable drift waves in plasma (Cohen et al., 1976). In spite of such widespread application, the spatially periodic solutions of eq. (1) are still poorly understood. While the linear instability 2477