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Coalescence of Tollmien-Schlichting and interfacial modes of instability in two-fluid flows

An extensive parametric study is performed to bring out the details of the mode coalescence phenomenon in a two-fluid boundary-layer stability problem. The solution consists of numerically solving the Orr-Sommerfeld equation. The mode coalescence occurs between the Tollmien-Schlichting (TS) mode also found in the classical hydrodynamic stability theory and the interfacial mode that exists in the presence of a viscosity stratification in two-fluid flows. It was previously shown that the coalescence occurs in the unstable part of the domain when the sheared fluid thickness is sufficiently large. It was also shown that if the sheared fluid thickness is further increased, a composite mode dominates the stability problem, which carries the characteristics of the TS mode at moderate wave numbers and the interfacial mode at low and high wave numbers. In this study, the influence of sheared fluid thickness, viscosity ratio, surface tension, density stratification, and gravity on the composite mode is further elaborated. It is demonstrated that for certain parameter combinations, the mode coalescence occurs not only in the unstable part of the domain but also in the stable part. This feature had been previously reported by other researchers by using a different solution method and is now demonstrated in the Orr-Sommerfeld context as well. Moreover, mode coalescence occurs in the presence of density stratification and gravity also.(C) 2008 American Institute of Physics.