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Linear stability of an interface between two incompressible fluids
Fu, Yun

2006, Doctor of Philosophy, Ohio State University, Mathematics.
The linear stability of an interface between two incompressible fluids has been an interesting yet challenging topic for many years. Mathematically, the stability problem is governed by a particular velocity profile, the incompressibility conditions, the moment equations, as well as the interfacial conditions. This paper uses asymptotic and numerical approaches to study the two-dimensional stability problem with laminar base profile. Growth rate, the zones of amplification and zones of decay with respect to different wave lengths and wind speeds are found. A possible solution for the growth rate from continuous spectrum is identified. We first identify the physical equations, namely the incompressible conditions and the momentum equations, and the interfacial conditions for the air-water interface. Then, by assuming that the height of the interface is small, we perform a linear expansion on the velocity and surface pressure profile, substitute into the interfacial conditions, and obtain the linearized interfacial conditions. A set of PDEs is obtained as the physical equations, and the interfacial conditions. Finally, under the assumption that the solution has a particular form, we convert the PDE sets to non-linear ODE sets, and seek for analytical and numerical solutions. The asymptotic approach is used to construct approximate analytical solutions in case the wind speed is low. By linearly expanding and balancing the powers of wind speed, a series of linear ODE sets, including the base case and the linear correction are obtained. The linear correction part is solved through a tremendous amount of integration, plus the proper choice of integration constants to ensure far-field conditions. The solutions of velocity profile are then used in the interfacial conditions to determine the growth rate. While this analytical result is not applicable to a wide range of wind speeds, nevertheless it provides us an easy way to check the numerical velocity profile. The numerical velocity profile is solved by boundary value problem solver of MATLAB, and the growth rate is determined through either the quasi-Newton method or continuation method. All numerical codes are tested by asymptotic results with low wind speed and numerical simulation approaches by other researchers.

# APA Citation

Fu, Y. (2006). Linear stability of an interface between two incompressible fluids. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

# MLA Citation

Fu, Yun. "Linear stability of an interface between two incompressible fluids." Electronic Thesis or Dissertation. Ohio State University, 2006. OhioLINK Electronic Theses and Dissertations Center. 25 Feb 2018.

# Chicago Citation

Fu, Yun "Linear stability of an interface between two incompressible fluids." Electronic Thesis or Dissertation. Ohio State University, 2006. https://etd.ohiolink.edu/