Open Problems in Non-Newtonian Fluids - The Details

As promised yesterday, here's a more detailed look at some open problems in non-Newtonian fluids.

1. The first involves a rather surprising result from a very simple set of experiments. For bubble of gas rising through a non-Newtonian fluid, a plot of terminal velocity vs. bubble size shows a large discontinuity:
   
   As the bubble size increases, the discontinuity is also associated with a change in the bubble shape and also the appearance of a negative wake behind the rising bubble. The free-form shape of the bubble adds to the challenge of modeling this. I was never aware of this behavior before reading the article, despite this behavior first being described back in 1965.
2. The next problem is much more of a modelling issue in my mind than an intellectual curiosity. The picture below shows the behavior of a polyacrylamide solution flowing downward and leaving through a small opening.
   
   These images are a series of stills taken through a plane in the flow channel. The recirculation pattern on the left moves later to the right and then later yet is not seen at all - the recirculation zone is rotating around the cylinder over time.
   
   One small bone to pick in this section of the paper was the introductory sentence:"Contraction flows occur in many polymer manufacturing processes, as molten polymer is extruded to form (ideally) a smooth, uniform thread." (Emphasis added). I would say that contraction flows occur in ALL or MOST polymer manufacturing processes. Any process that initially requires melting the polymer and then shaping it, such as extrusion (sheet, film, fiber...), or molding (blow, injection,...) will have contraction flows and the associated extensional components. Rotomolding (filling a hollow mold with polymer pellets, then tumbling the mold so that the molten polymer uniformly coats the walls of the mold) would be the only process that is on the edge of this definition. You still are melting the polymer in order to shape it, but the shaping is done only by gravity, not by any externally supplied pressure, a very unique situation for polymer processing.
3. Lastly, there is the asymmetric flow patterns that occur in cross-slot flows. In the diagram below, the fluid is pumped into the die in opposing directions (from the left and right in this case) and exit the die through the top and the bottom.
   
   In properly machined dies, the impinging flows should split and an equal amount should exit the die in each direction but that is clearly not the case. It is thought that this is occurring to minimize the energy in the experimental setup. Because of the non-Newtonian nature of the liquid, higher overall fluid flow is obtained by dividing each flow into a wide and a narrow channel rather than having two equally sized channels. The asymmetry only occurs at higher flow rates.
   
   This is not strictly an academic problem as cross-slot flows are used for extensional flow viscometry. Being able to better understand the origin of the breakdown in the flow field could lead to more robust instrumentation.

What all three of these problems have in common are elements of an extensional flow field, although that is almost a tautology as any flow problem beyond flow in a circular pipe (or between two parallel plates) has extensional flow characteristics. Just like spherical cows of uniform density in a vacuum, pure shear flow fields are the study of academics, not of the "real" world.