Nonliner Electrohydrodynamic Marangonl Stability :

Abdel Raouf Farid El-hefnawy. 

This thesis deals with the nonlinear electruhydrodynamicMaran~oni stability of both Rayleigh-Taylor and Kelvin-Helmholtzmodels in the presence of different electric field distributions.This thesis consists of six chapters:In chapter I, we explain the main aspects, the previous studiesof electrohydrodynamics and their various applications. We discussthe concepts of electrohydrodynamics and stability. The equations.governing the motion, the electric field and the associated boundaryconditions aPe introduced. We explain the surface ten:::ion and adsorptionand the related differential equations. The concept of Marangoniinstability is explained. We present a review of both Rayleigh-Taylorand Kelvin-Helmholtz models in the last section.In chapter II, we study the problem of electrohydrodynamicMarangoni stability in Kelvin-Helmholtz flow in the presence of atangential electric field of an interface between two semi-infinite,dielectric, inviscid and incompressible fluids. The two fluids areassumed to be streaming in the x-direction. Also, the motion in eitherfluids is assumed to be irrotational. We use the method of multiplescales to expand the various perturbation quantities to yield the linearand slcce!’lsive nonlinear partial differential equat.ions of the variousorders. The solutions of these equations are obtained.Chapter III is connected with the second chapter presented inthis thesis, where the linear electrohydrodynamic Marangoni stabilityis discussed in both Rayleigh-Taylor and Kelvin-Helmholtz models. Weobtained a third-order dispersion relation with real and complexcoefficients for the Raylei~h-Taylor and Kelvin-Helmholtz instabilitiesrespectively. The necessary and sufficient conditions of stabilityare discussed theoretically and numerically in both models. For RayleighTaylorinstability, we find that the critical value of the fieldincreases with the idbrease of adsorption which means that theadsorption is destabilizing while the surface tension is stabilizing.For Kelvin-Helmholtz instability, the behavior is similar to that ofRayleigh-Taylor instability. Only the Kelvin-Helmholtz instabilityrequires larger values of the field than those in the Rayleigh-Taylorinstability.In chapter IV, we study the nonlinear electrohydrodynamicMarangoni stability in both Rayleigh-Taylor and Kelvin-Helmholtzmodels. We get the nonlinear SchrBdinger equation with complexcoefficients. The surface elevation and the cutoff wavenumber in themarginal state are obtained. We discuss the stability conditions ofnonlinear Schrodinger equation with complex coefficients in differentcases. The numerical analysis in the marginal state is discussed forboth models. For nonlinear Rayleigh-Taylor instability, we observethat the nonlinearity plays a dual role in the stability criterionregarding the effect of the electric field, the adsorption and thesurface tension. For nonlinear Kelvin-Helmholtz instability,we ob~~rv~that the surface tension and the adsorption are playing a dual role,but in a manner different from that in the Rayleigh-Taylor instability.In chapter V, we study the stability of the system at thecritical point. We get the nonlinear Klein-Gordon equation with complexcoefficients, while the nonlinear Klein-Gordon equation with realcoefficients is obtained in the absence of interfacial adsorption •The stability conditions of the latter nonlinear Klein-Gordon equationare obtained and the stability analysis of that equation is discussed.We find that the necessary condition of stability is that either ofthe following conditions should be satisfied:(i) 0.283 < p < 1or(ii) p < 0.283wherep2p = isplthe ratio density (p1 and p2 are the densities ofthe lower and upper fluids,respectively) and £1 and £2 are thelower and the upper dielectric constants respectively.Also, we observe that the increase of the surface tensionresults in an increase of the critical value of the stabilizing fieldswhile the increase of the critical wavenumber results in a decreaseof the critical value of the stabilizing field.In chapter VI, we study· the problem of linear and nonlinearelectrohydrodynamic Marangoni stability in both Rayleigh-Taylor andKelvin-Helmholtz models in the case of a normal field in the absenceof surface charges on an interface of two dielectric, semi-infinite,inviscid, incompressible and immiscible fluids moving with uniformvelocities in the x-direction. We use the method of multiple scalesto expand the .variou:;. pertlrbation quant·it.iea t.o . yieldthe characteristic equation for the first-order and the solvabilityconditions for the second-and the third-orders. For the first-order,we get the dispersion relations in both Rayleigh-Taylor and KelvinHelmholtzmodels. In Rayleigh-Taylor instability, we observe that tnestabilizing field increases with the increase of the wavenumber,contrary to the case of the tangential field. Also, we observe thatthe decrease of adsorption and the increase of surface tension arestabilizing. The previous results are the same in the Kelvin-Helmholtzinstability, only the latter requires lesser values of the field thanthose in the Rayleigh-Taylor instability. Also this result is incontrast with the case of the tangential field.For the higher-orders, we get nonlinear Schrodinger equationwith complex coefficients and the conditions of stability for bothRayleigh-Taylor and Kelvin-Helmholtz models. For the nonlinearRayleigh-Taylor instability, we observe that the new unstable regionsin the stability diagram are decreased with the increase of adsorption,while the same regions are decreased with the decrease of surfacetension. For the nonlinear Kelvin-Helmholtz instability, we observethat the newly formed stable regions allow stability for smallwavenumbers while for large wavenumbers stability is possible fora small band of values of the electric field.Finally, we discuss the stability at the critical point toobtain the nonlinear Klein-Gordon equation with complex and realcoefficients. The stability conditions and stability analysis arediscussed. We observe that the necessary condition of stability tobe satisfied is that either of the following conditions should besatisfied:( 1} p < 0. 283or(2} 0.283 < p < 1