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Parts:
=> Temperature Dependence of Viscosity
=> Shear Rate Dependence of Viscosity
=> Constutative Equations for Polymer Flow
=> Simplest Assumption
----Link to Lodge Liquid.html (Lodge Liquid.pdf)
=> General for History of a Fluid Element
=> CEF Equation
=> LVE Equation
=> GNF Equations
=> Application of GNF Equations
=> Viscometric Flows
=> Elongational Flow


Chapter 3 Polymer Melt Rheology
(Tadmor Chapter 6)

Tadmor's Chapter 6, is an overview of Non-Newtonian Rheology, which is basically taken from Bird, Armstrong, Hassager, "Dynamics of Polymeric Liquids" Volume 1 (of 2) Fluid Mechanics. (Volume 2 of this set deals with theory of melt viscosity and is a common reference but of little use for processing.)

The most important non-Newtonian effects in polymer melt flow are the A) temperature and B) shear rate dependence of viscosity.

A) First issue in chapter 6 is to compare the temperature dependence of Newtonian vs. polymeric fluids (See homework problem). A comparison of the Arrhenius behavior in eqn. 6.1.1 (pp. 147) and the WLF behavior of

h=h0 exp[-17.444(T-Tg)/{51.6 +(T-Tg)}]

shows that the equations are similar (Arrhenius and WLF Functions). The thermal behavior is dramatically different as seen below in a semi-log plot.

B) The second issue in chapter 6 of Tadmor involves changes in the viscosity with shear rate (usually shear thinning behavior, see Chapter 1) and related issues of normal forces (Wiessenberg Effect and die swell). These issues are also related to the appearance of solid-like features (elastic component) to polymeric fluids including self-siphoning behavior, bubble shape, flow stabilization, fibrillation (ability to form fibers) and fluid memory effects. Several examples mostly from Bird Armstrong Hassager are given below. Most of these examples can be duplicated with common "structured" fluids such as molasses, shampoo or motor oil.

From Bird, Armstrong, Hassager, "Dynamics of Polymeric Liquids, Vol. 1"

From Bird, Armstrong, Hassager, "Dynamics of Polymeric Liquids, Vol. 1"

From Bird, Armstrong, Hassager, "Dynamics of Polymeric Liquids, Vol. 1"

From Bird, Armstrong, Hassager, "Dynamics of Polymeric Liquids, Vol. 1"

From Bird, Armstrong, Hassager, "Dynamics of Polymeric Liquids, Vol. 1"

From Bird, Armstrong, Hassager, "Dynamics of Polymeric Liquids, Vol. 1"

From Bird, Armstrong, Hassager, "Dynamics of Polymeric Liquids, Vol. 1"

From Bird, Armstrong, Hassager, "Dynamics of Polymeric Liquids, Vol. 1"

From Bird, Armstrong, Hassager, "Dynamics of Polymeric Liquids, Vol. 1"

From Bird, Armstrong, Hassager, "Dynamics of Polymeric Liquids, Vol. 1"

From Bird, Armstrong, Hassager, "Dynamics of Polymeric Liquids, Vol. 1"

Constutive Equations for Polymer Flow

In order to deal with these effects a number of equations have been developed which describe some of these features. The primary reason for the vast diversity of equations which have been generated is that rheologists have developed different frames of reference to account for fluid elements which can be deformed in a flow. For example, if you consider a polymer chain as being deformed in a flow you will need a reference frame which follows a fluid element and describes its deformation. The main complication in dealing with these new frames of reference is converting from the machine or lab frame of reference to the new frame, doing calculations in the new frame and then converting back to the machine frame. These conversions are fairly complicated and we will not deal in detail with how they are carried out. (Bird Armstrong Hassager is a good source for reference frame conversions as is Christensen's "Theory of Viscoelasticity" 1982).

We do need to consider the generalities of these approaches in order to understand the source of the various equations used to describe non-Newtonian flow. To describe the flows shown in class we will need:

  1. Tensors (we have normal forces)
  2. A description of fluid memory (memory of stress/strain history)
  3. Description of viscoelasticity.
  4. Definition of various Reference Frames useful for the description of 2 and 3.

From an engineer's perspective the golden rule is "If it ain't broke, don't fix it" which translates into always use the simplest equations that "work". If the Newtonian fluid equations are good enough, then use them. If you can live with a small modification of Newtonian fluids such as a power-law fluid, then do it.

Below is a list of terms useful in dealing with polymer flow and non-Newtonian rheology: