الفهرس 
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Abstract
In the present thesis, we consider the flow field about a sphere of
radius ”R” which rotates with a uniform angular velocity nabout its axis
in an infinite general Oldroyd-fluid. Morever, the problem is extended to
the case when a uniform translational motion in the direction of the axis
of rotation is superimposed onto the rotational motion of the sphere.
Throughout the present investigation, the viscoelasticity of the fluid
is assumed to dominate the inertia such that the latter can be neglected in
the momentum equation.
The velocity, stress and pressure fields are expanded in powers of a
dimensionless retardation parameter, A = A2a , which for moderate rate of
rotation (a Ad 02) is of order A’” 10-2 . Hence, a set of successive partial
differential equations are obtained which govern the successive
approximations of the velocity field determined by the <p-component
W(r,8)and the stream function ’¥(r,8).
In the first boundary value problem which concerns the rotational
flow only, the leading term, W(O)(r,8),is the Newtonian flow. The firstorder
approximation produces a stream function, ,¥(I)(r,8), which
describes a secondary flow in the plane <p=constant.This includes the
material constants of the Oldroyd-B fluid such that the general fluid up to
this order is approximated by the special fluid. The second-order
approximation leads to a viscoelastic contribution W(2)(r,l3’) which
depends on all the material parameters of the general model . The
contribution W(2)(r,9),which is superimposed onto the Newtonian flow
W(O)(r,9), can either enhance or oppose the flowW(O\r,9) according to
the values of the material parameters .
The stream function ’I’(3)(r, 9), which is obtained from the thirdorder
approximation, includes, as expected, all the material parameters of
the general fluid. Due to the complicated form of this term, it is discussed
for some special cases as upper-convected Maxwell fluid and Oldroyd-B
fluids besides the general Oldroyd fluid under consideration.
The torque calculated up to third-order approximation is composed
of a viscoelastic contribution M2 added to the viscous contribution Mo
due to the primary flow. The M2 field may enhance or oppose the Mo
field depending on the values of the material parameters. However, if
OIdroyd-B fluid is considered, the viscoelastic contribution adds
negatively to the viscous contribution.
In the second part of this thesis, where the fluid is assumed to
translate with velocity V0 in the direction of the axis of rotation of the
sphere, the approximation procedure leads to more elaborate results.
The zero-order approximation produces Newtonian flow, i.e. the <pcomponent
W(0)(r,9)and the stream function ’I’(O)(r,9) which agrees
iv
with results calculated for Newtonian fluids. The first-order term is
composed of W(l)(r,9) and ’I’(I)(r, 9) , which are superposed,
respectively, onto W(O)(r,9)and ,¥(O)(r,9). The two functions W(I)(r,9)
and ,¥(l)(r,9) depend on the material parameters of the Oldroyd-B fluid
only. The second-order solution leads to a viscoelastic contribution
W(2)(r,9), as well as the stream functiofiljl(2)(r,9), where both fields
depend on all the material parameters of the general Oldroyd fluid.
Further steps of approximation are practically intractable.
Each of the torque and drag fields due to combined rotational and
translational flows, when calculated up to the second-order, are
composed of the sum of a viscous (or Newtonian ) term and an elastic
term. The elastic contribution in both fields, which depends on all the
material parameters of the general Oldroyd model, may add positively or
negatively to the viscous term according to the numerical values of the
material parameters of the fluid.
On the basis of this theoretical calculations a rheometer may be
designed in order to determine the material parameters ”lo’ A.I”.t” A”.