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7.3 Viscoelastic Fluid and Flow
Note that for simplicity asterisk * in Eqs. (7.3.117) and (7.3.118) is
dropped in the resultant finite difference equations of Eqs. (7.3.125) and
(7.3.126).
For given initial and boundary conditions, i.e. Eqs. (7.3.119) to
(7.3.122), repeated use of Eqs. (7.3.125) and (7.3.126) generates the nu-
merical solution at all interior grid points
i
r at time level 1k . That is,
incrementing
k for time and substituting the known values of
1ki
u
,
and
1ki,
W
into the right hand side of Eqs. (7.3.125) and (7.3.126) allows the
discrete solution to be marched toward in time. A typical numerical solu-
tion of
k
u
,1
, the time development of the axial velocity at the center of
pipe, is displayed in Fig. 7.22. As seen in the figure, an overshooting and
damping oscillation of the velocity are displayed for a step change of the
pressure gradient. It should be mentioned here that there might be some
difficulties for obtaining convergence of numerical solutions, depending
upon
and other geometric parameters.
art-up condition. Etter and Schowalter
(1965) calculated the transient behavior of viscoelastic flow of the same
problem, using Oldroyd-B model and showed similar overshooting charac-
ter of the flow parameter.
Fig. 7.22 Overshooting of flow parameter at start-up flow
The next important problem in viscoelastic fluid flow is the boundary
layer problem. In many engineering flow problems with viscoelastic fluids,
the pressure loss along the channel is a major concern, which is also di-
rectly connected with the formation of a boundary layer on the solid wall
of the channel. Let us examine the two dimensional boundary layer on a
flat plate (see Section 6.5.1 and refer Fig. 6.18). The effect of fluid elasticity
in viscoelastic fluid flow at sudden st
The overshooting of flow parameters is typically observed phenomena
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