TABLE OF CONTENTS This section considers three special cases. The first is a flat velocity profile that can result from an extreme form of fluid rheology. The second is a linear profile that results from relative motion between adjacent solid surfaces. The third special case is for motionless mixers where the velocity profile is very complex, but its net effects can sometimes be approximated for reaction engineering purposes.
Flow in a Tube. Laminar flow with a flat velocity profile and slip at the walls can occur when a viscous fluid is strongly heated at the walls or is highly non-Newtonian. It is sometimes called toothpaste ftow. If you have ever used Stripe toothpaste, you will recognize that toothpaste flow is quite different than piston flow. Although 
and 
, there is little or no mixing in the radial direction, and what mixing there is occurs by diffusion. In this situation, the centerline is the critical location with respect to stability, and the stability criterion is
| (8.47) |  |
and ? z max varies as ? r 2. The flat velocity profile and Equation (8.47) apply to the packed-bed models treated in Chapter 9. The marching-ahead equations are unchanged from those presented in Section 8.3.1, although the coefficients must be evaluated using the flat profile.
Toothpaste flow is an extreme example of non-Newtonian flow. Problem 8.2 gives a more typical example. Molten polymers have velocity profiles that are flattened compared with the parabolic distribution.
