In the previous work, Part I; [1], the steady flow of an incompressible Oldroyd
8-constant fluid in the annular region between two concentric spheres is investigated
up to the second-order approximation. Hence, the normalized second-order velocity
field, V shows to be
) 1 ( ) 1 (
) 1 (
) 1 ( 3 ) 2 ( ) 2 ( ) 1 ( ) 0 ( ˆ ˆ
sin
); ( ˆ ˆ V U r
r
O W W V
U U
The leading velocity term represents the Newtonian flow in the -direction, while the
first-order term denoted by the stream–function ) , r ( ) 1 ( , produces a secondary flow
field that divides the flow region into four parts symmetric about the z-axis which is
the axis of rotation. The second-order approximation gives a viscoelastic
contribution ) , r ( W ) 2 ( in the -direction. is the retardation time parameter.
The present work is devoted to the solution of the third-order approximation of
the same problem treated in [1]. The solution produces a stream–function ) , r ( ) 3 (
which is being a secondary flow field divides the domain of flow into two similar
regions symmetric about an axis perpendicular to the axis of rotation. The streamlines
. const ) , r ( ) 3 ( are sketched for Maxwell, Oldroyd-B and Oldroyd 8-constant
model fluids; respectively. The results show that the distribution of flow for these
fields are mainly affected by the values of their elastic parameters.
|
