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This thesis deals with the unsteady flow behavior of some rate type fluids under
different circumstances. Firstly, some basic definitions and concepts regarding fluid
motion and methods to solve the flow problems have been discussed. Then the motion
of ordinary Maxwell fluids and that of Oldroyd-B fluids with fractional derivatives
over an infinite plate is studied.
In chapter 2, we have studied the unsteady motion of a Maxwell fluid over an
infinite plate that applies an oscillating shear to the fluid which is the extension of
some previously obtained results. After time t = 0+ the fluid motion is produced by
applying an oscillating shear. Fourier and Laplace transforms are used to find exact
solutions that are presented as a sum of steady-state and transient solutions. They
describe the motion of the fluid some time after its initiation. After that time, when
the transients disappear, the motion of the fluid is described by the steady-state so-
lutions that are periodic in time and independent of initial conditions. Finally, the
time to reach the steady-state is determined. Similar solutions for Newtonian fluid
are obtained as particular cases of general solutions by making λ → 0.
The purpose of chapter 3, is to extend the first problem of Stokes to incom-
pressible Oldroyd-B fluids with fractional derivatives. The Fourier sine and Laplace
transforms are used. The solutions that have been obtained, are presented as a
sum between the Newtonian solutions and non-Newtonian contributions. The non-
Newtonian contributions, as expected, tend to zero for α = β and λ → λr . Fur-
thermore, the solutions for ordinary Oldroyd-B, fractional and ordinary Maxwell,
fractional and ordinary second grade fluid, performing the same motion, are obtained
as limiting cases of general solutions. The present solutions for ordinary Oldroyd-B
and second grade fluids are verified by comparison with previously known results.
Finally, the influence of material and fractional parameters on the fluid motion, as
well as a comparison among fractional and Newtonian fluids, is analyzed by graphical
illustrations.
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In chapter 4, our concern is to study the velocity field corresponding to the
Stokes’ problems for fluids of Brinkman type. The solutions that have been ob-
tained, are presented under suitable forms in terms of the classical solution of the
first problem of Stokes for Newtonian fluids or as a sum between the steady-state
and transient solutions. Furthermore, for α → 0 they are going to the well-known
solutions for Newtonian fluids. The required time to reach the steady-state, as well as
the temporal decay of the transients corresponding to the second problem of Stokes,
has been determined by graphical illustrations.
The aim of chapter 5, is to establish exact and approximate expressions for dissi-
pation, the power due to the shear stress at the wall and the boundary layer thickness
corresponding to the motion of an Oldroyd-B fluid induced by a constantly acceler-
ating plate. Similar expressions for Maxwell, second grade and Newtonian fluids,
performing the same motion, are obtained as limiting cases of general results. Some
specific features of the four modelss are emphasized by means of the asymptotic
approximations. |
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