Page 479 - Introduction to Continuum Mechanics

Non-Newtonian Fluids 463

For a steady laminar flow of water placed between two very long coaxial cylinders of radii
TI and r2, if the inner cylinder is at rest while the outer one is rotating with an angular velocity
Q, the theory of Newtonian fluid gives the result agreeing with experimental observations that
the torque per unit height which must be applied to the cylinders to maintain the flow is
proportional to Q. In fact [see Eq. (6.15.5)]

However, for those fluids which do not obey Eq. (8.0.1), it is found that they do not obey
Eq. (8.0.2) either. Furthermore, for water in this flow, the normal stress exerted on the outer
cylinder is always larger than that on the inner cylinder due to the effect of centrifugal forces.
However, for those fluids which do not obey Eq. (8.0.1), the compressive normal stress on the
inner cylinder can be larger than that on the outer cylinder. Fig. 8.2 is a schematic diagram
showing a higher fluid level in the center tube than in the outer tube for a non-Newtonian fluid
in spite of the centrifugal forces due to the rotations of the cylinders. Other manifestations of
the non-Newtonian behaviors include the ability of the fluids to store elastic energy and the
occurrence of non-zero stress relaxation time when the fluid is suddenly given a constant shear
deformation. (For Newtonian fluids, relaxation is instantaneous).

Fig. 8.2