Linear and nonlinear free surface flows in electrohydrodynamics

摘要

This thesis examines free surface flows in electrohydrodynamics under forcing in the form of a moving pressure distribution or topography. The ideas from examining free surface flows with forcing and those ideas andmethods coming from examining solitary waves within electrohydrodynamics are combined to study free surface flows under forcing in electrohydrodynamics. Chapter 1 gives a brief introduction to the ideas and work that have gone into investigating free surface flows and solitary waves in general and gives an idea of what will happen in the thesis. Chapter 2 formulates the general problem for the full nonlinear case and then examines the linear solution for both a moving pressure distribution and topography and presents profiles of the free surfaces and then shows that the solutions are nonuniform by examining the deep water case. Chapter 3 introduces the scaling for the weakly nonlinear problem and produces an equation which there is no nonuniformity and the amplitude of the free surface is finite. The case when the Bond number is around a 1/3 is also examined. Stokes analysis is performed to look for Wilton ripples. Chapter 4 examines conducting fluids adhering to an upper surface, the basic equations are set up and then the dispersion relation is derived to examine the existence of linear waves for certain values of the wavenumber k. A set of weakly nonlinear equations are examined and then solved numerically with examples of periodic profiles presented. A Stokes analysis is carried out for small amplitudes to look for Wilton ripples. An analysis is carried out for the approximation of long wavelength but finite depth, where the wave amplitude is the depth of the fluid. Chapter 5 considers surface flows and generalised the results from chapters 2 and 3 from two dimensions to three, linear free surface profiles are calculated and plotted and the weakly nonlinear equation is derived for the cases where the Bond number is close to 1/3 and not close to 1/3 giving a 5th order (Kadomstev-Petviashvili) (KP) equation. Chapter 6 is the set of conclusions and avenues of future research.