الفهرس 
CHAPTER 1 INTRODUCTIONOnly 14 pages are availabe for public view
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Abstract
Aly Maher Abourabia
Rabab Abd-Allah Shahein
This thesis consists of four chapters, which are organized as follows, the
introduction, three chapter and a list of references. It is concerned of the
theoretical study of the convection phenomenon in incompressible fluids.
The study is based on analytical solutions of the Camassa Holm Degasperis
Procesi equation (CH-DP), the Perturbed KdV equation (PKdV) and the
Navier stokes system of fundamental equations.
In chapter one: We present an introductory survey of the following points
1. Definition of the convection phenomenon that includes a historical
background.
2. The fundamental equations of conservation of mass, momentum and
energy.
3. The non-dimensional numbers that describes the properties of the
system.
4. The methods of solution used through the following chapters.
In chapter two: This chapter is devoted to the study of the CH-DP
equation that describes the relationship between convection and stretching
in a unideirectional motion in shallow water. We solved the equation by
three different methods which are the Factorization technique, the ColeHopf transformation and the Schwarzian derivatives method, plus studying
the stability of the system through the Phase Portrait method that shows a
saddle and a center at the two critical points. llustrations of the solution are
presented using symbolic software which shows different pattern
formations. The patterns obtained are compared with previous work in
literature, showing agreement for some of the obtained patterns while we
also have different exchange of stability between the patterns.
In chapter three: Various techniques are applied to solve the perturbed KdV
equation (PKdV), which describes the evolution of surface waves velocities
in convecting fluids. Under certain conditions use is made of the
characteristic Galileo and Prandtl numbers of water to plot the resulting
solutions, by which a variety of pattern formations for the wave velocities
(in mm/s) at different temperatures are illustrated. Some solutions resulted
by applying factorization technique represent bright solitons, the others give
a combination of bright and dark solitons. A comparison is made with the
solution of the same problem tackled in one of the references. The
Hamiltonian method of solution gives solitary wave behaviours. Kink
solutions are emerged through the application of Painleve analysis. The
resulting nonlinear second order differential equation is dealt with in the
phase portrait, which reveilles the stability of the system by demonstrating
that the corresponding eigenvalues indicate saddles and centers.
In chapter four: We studied and solved the system of the fundamental
equations of Navier Stokes that describes convection enhanced by the
thermal effect of ultrasonic waves using the space-time Minkowski space.
We suggested, the use of a phase variable method to solve the system and
studied the different pattern formations. We found that the Rayleigh number
is very large which shows a turbulent flow while Nusselt number is >> 1
which shows that the dominant way of heat transfer is through convection
instead of conduction. We studied two different cases; the first is when the
ultrasonic transducer is placed below the container while the second case is
when the transducer is placed at the left side of the container. We illustrated
the solutions in the two cases. We noticed that the effect of the ultrasonic
waves in the first case is leading to acoustic streaming only while in the
second case we had the acoustic cavitaion together with the acoustic
streaming. After that, we studied the entropy generation by calculating the
two different components of the entropy generation r ate at the top and the bottom layers of the fluid. Furthermore, we calculated the ratio of the Bejan
number at each layer to find that the dominant contribution to entropy
generation comes from heat transfer irreversibility in case of placing the
transducer below the container. On the other hand, the dominant contribution
to entropy generation comes from fluid friction irreversibility when the
transducer is placed at the side. Finally we studied the stability of the
temperature by the phase plane method algebraically, through eigenvalues,
and graphically which shows a stable spiral in both cases.