Abstract:
The work in this thesis presents some new results concerning the flow behavior
of some non-Newtonian fluids through a cylinder and in annular regions. First of all
in preliminaries a brief discussion about Newtonian and non-Newtonian fluids, con-
stitutive equations, equation of motion, continuity equation and integral transforms
are given. In the following chapters some new exact solutions for fractional second
grade, fractional Maxwell, ordinary Oldroyd-B and fractional Oldroyd-B fluids are
established.
In chapters 2 and 3 we study the motion of second grade fluid with fractional
derivatives. In chapter 2 we obtained the exact solutions for the flow of a second
grade fluid with fractional derivatives in an annular region between two infinite coax-
ial circular cylinders and in chapter 3 obtained exact solutions for the same fluid
through a straight circular cylinder, by means of the Laplace and finite Hankel trans-
forms. These solutions are specialized to give the similar solutions for ordinary second
grade and Newtonian fluids performing the same motion. The required time to reach
the steady-state is obtained by graphical illustrations at the end of chapter 2. Also, in
the last part of the chapter 3, the influence of the material constants and of the frac-
tional parameter on the velocity and shear stress variations is underlined by graphical
illustrations.
In chapter 4, the axial Couette flow of a Maxwell fluid with fractional derivatives
is discussed. The velocity field and the shear stress corresponding to the flow in an
infinite circular cylinder are obtained by means of Laplace and Hankel transforms.
The motion is caused by the infinite cylinder which applies a longitudinal time depen-
dent shear stress to the fluid. Both solutions are written in terms of the generalized
Ra, b (·, t) and Ga, b, c (·, t) functions. The solutions for ordinary Maxwell and Newto-
nian fluids are obtained as limiting cases of general solutions. Finally, in this chapter
the influence of material and fractional parameters on the fluid motion is brought to
light by graphical illustrations.
Chapter 5 deals with the motion of an ordinary Oldroyd-B fluid in an infinite
circular cylinder subject to a time-dependent couple. At time t = 0+ the cylinder is
set in rotation about its axis by a time-dependent torque per unit length. The exact
solutions for the velocity field and the shear stress are established by means of the
Hankel transform. The similar solutions for Maxwell, second grade and Newtonian
fluids are obtained as limiting cases of general solutions. The influence of the material
parameters on the velocity and the shear stress is spotlighted by means of graphical
illustrations.
Chapter 6 concerns with the unsteady flow of an Oldroyd-B fluid with fractional
derivative model through an infinite circular cylinder. The fluid motions is studied
by means of finite Hankel and Laplace transforms. The motion is produced by the
cylinder, that at time t = 0+ , is subject to a time-dependent rotational shear stress.
The solutions that have been obtained, presented under series form in terms of the
generalized Ga, b, c (·, t) functions, satisfy all imposed initial and boundary conditions.
The solutions for fractional Maxwell fluids as well as those for ordinary fluids are
obtained as limiting cases of general solutions. Finally, the influence of the mate-
rial constants and fractional parameters on the velocity and shear stress variations is
discussed by graphical illustrations.