Polymer Physics

whereDis the tube diameter,vis the average flow velocity,ris the density of
the fluid,is the viscosity of the fluid. The Reynolds number actually reflects the
ratio of the inertial forces to the viscous drag forces under a given flow condition.
The flow with smaller Reynolds numbers (Re<2,000) normally contains parallel
layers without mixing, which are calledlaminar flow. In this case, the influence of
the drag forces is larger than that of the inertia forces. Since the disturbance of
flow velocity in the flow field decays with the drag force, the laminar flow
remains stable. With the increase of Reynolds numbers, the streamlines start to
oscillate, and the frequency and amplitude of oscillations increase with the flow
velocity. This type of flow is called transition-region flow. When the flow
velocity rises up to some certain values (Re>4,000), the streamlines are not
anymore stable. This is because many small vortexes are generated in the flow
field, which destroy the streamlines. In this case, the influence of the inertia
forces to the flow field is larger than that of the viscous drag forces, and a minute
change in the flow velocity is easy to be developed into chaos in an irregular flow
field. The drag force of the flow is then drastically enhanced. This type of flow is
calledturbulent flow.

7.1.3 Laminar Flow

Due to the high viscosity, the flow of polymer fluids conventionally appears as
laminar flow with a small flow velocity. There are two basic approaches to realize
the laminar flow.
The first approach is shear flow, commonly due to the frictional drag forces near
a solid substrate, where the flow velocity varies along the latter’s normal direction.
Depending on the profile of velocity gradients, there are two types of shear flow.
One is the drag flow, known as the Couette flow, which is similar to the shear flow
under the grindstone. A linear velocity gradient develops from one side to the other,
as illustrated in Fig.7.1a. The other is the pressure flow, known as the Poiseuille
flow, which is similar to the shear flow in the injection tube of syringes. Since the
tube boundaries make a frictional hindrance to the flow, the velocity distribution
appears as a parabolic curve, with the largest velocity gradient near the tube
boundaries and the smallest gradient in the middle, as illustrated in Fig.7.1b.
In the shear flow, the shear stresss(f/A) works on the shear plane alongxz
directions, as illustrated in Fig.7.2. When the shear stresssis proportional to the
velocity gradient dv/dy, the fluid can be called aNewtonian fluid, and the proportion
factor is defined asshear viscosity. The unit of viscosity is thus Pa·s or N·s/m^2 .In
the centimeter-gram-second (CGS) unit system, the unit of viscosity is Poise (P)
with 1 Pa·s¼10 P. The velocity gradient

dv

dy

¼

d

dy

ð

dx

dt

Þ¼

d

dt

ð

dx

dy

Þ¼

dg

dt

¼g^0 (7.2)

128 7 Polymer Flow