الفهرس 
General Introduction
Temporal, Spatial, and Spatio-temporally InstabilityOnly 14 pages are availabe for public view
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Abstract
The objective of this thesis is to study the linear instability of a non-Newtonian
(viscoelastic) liquid sheets or liquid jets issued into a streaming inviscid or viscous
gaseous environment for different models, like viscoelastic Walter B’ and RivlinEricsen models, Oldroyd 8-constants model in the presence or absence of a tangential
electric field, and porous medium using the normal modes analysis. The dispersion
relations for the system were derived and solved numerically using a new technique
via Mathematica software by taking the imaginary part of the angular frequency
K
Ωi = − where K is the non-dimensional wave number, and using the (FindRoot)
command taken into account that the growth rate equal to initial value as an initial
guess value for the root, we obtain by iteration, the solution as an ordered pairs
(K,Ωr ) for different values of the other physical parameters included in the analysis,
and then by using the (ListPlot) command we get the required figures. The effects of
various parameters on the instability of the considered system have been discussed,
and the thesis consists of the following six chapters:
Chapter (I):
In this chapter, we introduced a general introduction of the topic of this study
included the equations of motion, the associated boundary conditions, the concepts of
liquid sheets and liquid jets, temporal and spatial instability, linear hydrodynamic
stability, viscoelastic fluids, electrohydrodynamics, absolute and convective
instability, and the phenomena of liquid jet breakup.
Chapter (II)
In this chapter, we have investigated the hydrodynamic instabilities of two models of
viscoelastic liquid sheets (of Walters B′ and Rivlin-Ericksen types) issued in an
inviscid gas medium. The dispersion relations between the growth rate and wave
number of both symmetric and antisymmetric disturbances are derived for both
models, and solved numerically using a new technique. The effects of Weber number,
Ohnesorge number, gas-to-liquid density ratio, and viscoelasticity parameter on the
growth rates of two- and three-dimensional disturbances in both models are studied.
A comparison between the obtained results for both viscoelastic models has been
achieved. To the best of our knowledge, this study for liquid sheet instabilities of any
of these two types of viscoelastic fluids has not been investigated yet. The obtained
new results are listed in the concluding remarks section at the end of the chapter, and
this chapter accepted for publication in ”Atomization and Sprays, vol. 25, pp. 123-
151, 2015”.
Chapter (III): In this chapter, we have investigated the linear electrohydrodynamic instability of a
non-Newtonian dielectric liquid sheet issued in an inviscid dielectric gas medium of
different velocity by using the Oldroyd eight-constant model. the dispersion relations
between the growth rate and wave number of both symmetric and antisymmetric
disturbances are derived, and some limiting cases are recovered. Then, the effects of
various parameters included in the analysis, namely the electric field parameter, gas
to liquid velocity ratio, gas and liquid dielectric constants, time constant ratio, liquid
elasticity, liquid viscosity, gas to liquid density ratio, surface tension, and liquid sheet
velocity on the growth rates of symmetric and antisymmetric disturbances are studied
in detail using a simple numerical technique, and a number of quantitative
conclusions on the stability behavior of the considered system are drawn, in addition
to two appendices contain an explanation to the boundary conditions for both
symmetric and antisymmetric modes in details, the solutions, and the dispersion
relation in case of symmetric mode, and this chapter accepted for publication in
”Interfacial Phenomena and Heat Transfer, in press, 2015”.
Chapter (IV):
In this chapter, we have investigated the mechanisms of the temporal
electrohydrodynamic instability of non-Newtonian dielectric liquid jets which
ambient in gas medium, more specifically viscoelastic liquid jets, with a surface
tension gradient. The constitutive behavior of the viscoelastic jets is represented by
Oldroyd eight-constant model. The present work can provide a good foundation for
further investigations of the instability and breakup of viscoelastic liquid jets issued
in an ambient dielectric gas medium under the situation where the surface tension
gradient exists. Our analysis is restricted to an axisymmetric disturbance case. The
dispersion relation between the growth rate and the wave number for a viscoelastic
liquid jet has been derived. The effects of various parameters on the instability
behavior are studied, and a number of quantitative conclusions and sensitivities on
the instability behavior of non-Newtonian viscoelastic liquid jet are investigated , in
addition to an appendix contains an explanation to the using software program, and
this chapter accepted for publication in ”Atomization and Sprays, in press, 2015”.
Chapter (V):
In this chapter, we have investigated the mechanisms of temporal instability of nonNewtonian liquid jet moving in a streaming inviscid gas through porous media, and
to explore the differences between the instabilities of axisymmetric and asymmetric
disturbances. The present work is a good foundation for further investigations of the
instability and breakup of viscoelastic liquid jets through a porous medium. Hence,
the dispersion relation between the growth rate and the wave number for a
viscoelastic liquid jet with three-dimensional disturbances are derived in detail in
presence of porous medium. Then, the effects of various parameters on the instability
behavior are studied. Finally, a number of quantitative conclusions on the instability
behavior of viscoelastic jets with both axisymmetric and asymmetric disturbances are
summarized, and this chapter is submitted to publication in Journal of Porous Media.
Chapter (VI):
In this chapter, we have investigated the mechanisms of temporal instabilities of
viscoelastic liquid jets (of Rivlin-Ericksen type) issued in a viscous gas medium with
both axisymmetric and asymmetric disturbances. It is hoped that the present work can
provide a good foundation for further investigations of the instability and breakup of
viscoelastic liquid jets issued into a viscous gas. Hence, the dispersion relation
between the growth rate and wave number of the disturbance for a viscoelastic liquid
jet surrounded by a viscous (or inviscid) gas with three-dimensional disturbances are
derived in detail. The effects of various parameters, e.g. Weber number, Ohnesorge
number, gas to liquid density ratio, gas to liquid velocity ratio, gas-to-liquid viscosity
ratio, and viscoelasticity parameter, on the instability behavior are studied. To the
best of our knowledge, this study for liquid jet instability of this type of viscoelastic
fluids streaming into a viscous gas has not been investigated yet. A number of
quantitative conclusions on the instability behavior of viscoelastic liquid jets with
both axisymmetric and asymmetric disturbances are summarize. We end this chapter
by an appendix to write the constants of the solutions, and this chapter to submitted to
publication in Journal of Non-Newtonian Fluid Mechanics.
Finally, these studies which we presented enable us to interpret how the liquid sheet
and liquid jet breakup occurs, which has lots of applications in fluid mechanics like;
metal coating, milk powdered process, agriculture sprays, inkjet printing, and toxic
material removal.