TY - JOUR

T1 - Nonlinear long-wave stability of superposed fluids in an inclined channel

AU - Tilley, B. S.

AU - Davis, Stephen H

AU - Bankoff, S. G.

PY - 1994/1/1

Y1 - 1994/1/1

N2 - We consider the two-layer flow of immiscible, viscous, incompressible fluids in an inclined channel. We use long-wave theory to obtain a strongly nonlinear evolution equation which describes the motion of the interface. This equation includes the physical effects of viscosity stratification, density stratification, and shear. A weakly nonlinear analysis of this equation yields a Kuramoto-Sivashinsky equation, which possesses a quadratic nonlinearity. However, certain physical situations exist in two-layer flow for which modifications of the Kuramoto-Sivashinsky equation are physically pertinent. In particular, the presence of the second layer can mediate the wave-steepening instability found in single-phase falling films, requiring the inclusion of a cubic nonlinearity in the weakly nonlinear analysis. The introduction of the cubic nonlinearity destroys the symmetry-breaking bifurcations of the Kuramoto-Sivashinsky equation, and new isolated solution branches emerge as the strength of the cubic nonlinearity increases. Bistability between these new solutions and those associated with the Kuramoto-Sivashinsky equation is found, as well as the formation of a hysteresis loop from smaller-amplitude travelling waves to larger-amplitude travelling waves. The physical implications of these dynamics to the phenomenon of laminar flooding in a channel are discussed.

AB - We consider the two-layer flow of immiscible, viscous, incompressible fluids in an inclined channel. We use long-wave theory to obtain a strongly nonlinear evolution equation which describes the motion of the interface. This equation includes the physical effects of viscosity stratification, density stratification, and shear. A weakly nonlinear analysis of this equation yields a Kuramoto-Sivashinsky equation, which possesses a quadratic nonlinearity. However, certain physical situations exist in two-layer flow for which modifications of the Kuramoto-Sivashinsky equation are physically pertinent. In particular, the presence of the second layer can mediate the wave-steepening instability found in single-phase falling films, requiring the inclusion of a cubic nonlinearity in the weakly nonlinear analysis. The introduction of the cubic nonlinearity destroys the symmetry-breaking bifurcations of the Kuramoto-Sivashinsky equation, and new isolated solution branches emerge as the strength of the cubic nonlinearity increases. Bistability between these new solutions and those associated with the Kuramoto-Sivashinsky equation is found, as well as the formation of a hysteresis loop from smaller-amplitude travelling waves to larger-amplitude travelling waves. The physical implications of these dynamics to the phenomenon of laminar flooding in a channel are discussed.

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U2 - 10.1017/S0022112094002685

DO - 10.1017/S0022112094002685

M3 - Article

AN - SCOPUS:0028519599

VL - 277

SP - 55

EP - 83

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -