Our preliminary research concentrated on the correct formulation of the problem and on the construction of the analogy of the Darcy's law for generalized Newtonian fluid.

• L. Kondic, Michael J. Shelley and Peter Palffy-Muhoray, On Models of Non-Newtonian Hele-Shaw Flow, Phys. Rev. E 54, R4536, 1996.
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We have developed full numerical simulation of the problem of an expanding bubble.

• L. Kondic, Michael J. Shelley and Peter Palffy-Muhoray, Non-Newtonian Hele-Shaw flow and the Saffman-Taylor instability, Phys. Rev. Lett. 80, 1433, 1998.
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The snapshots of the gas - fluid interface are given below. In Fig. 1 we give the results for the case of an bubble expanding into Newtonian fluid. Well known viscous splitting is obvious. Figures 2 and 3, where bubble expands into shear-thinning liquid, are very different. Finger tips are much more stable and their radius of curvature is decreased. (Two different models for the viscosity of the driven fluid were used in these two figures.)

Fig. 1: Newtonian fluid

Fig. 2: Shear-thinning fluid 1

Fig. 3: Shear-thinning fluid 2

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Currently, we are exploring rich parameter space which seems to govern the shape of the gas - fluid interface. For Newtonian fluids, there is just one parameter - modified capillary number (or "dimensionless surface tension"). But, non-Newtonian fluids make the story much more complicated (and interesting), since a number of parameters could be entering if complicated liquids like polymeric materials, liquid crystals, foams, etc., are considered. It is well known that the interface could develop complicated, dendritic-like structure, similar to the patterns which develop in the process of solidification. For the time being, we try to simplify matters by using simple model which specifies the non-Newtonian behavior of the fluid. In this model, elastic properties of the fluid are neglected. Motivation for these assumption comes from the fact that most of the experiments are performed in the flow regime there fluid is not driven too strongly, so the fluid has enough time to rearrange its structure. Quantity which measures the importance of elastic effects, Deborah number (De), naturally enters into our viscous model. Small value of De, which we use in our simulation, justifies the simplification from viscoelastic fluid to shear-thinning one.

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Of course, one would like to understand why the gas-fluid interface looks so differently for Newtonian and shear-thinning fluids. The preliminary understanding comes from the viscosity and pressure diagrams given below. While the pressure distribution in the driven fluid for both Newtonian and non_Newtonian case is similar, the higher pressure gradient at the tips leads to large decrease of the viscosity. Since the velocity of the interface is inversely proportional to the viscosity of the fluid, the tips "shoot out" and form long fingers, much more stable with respect to splitting, compared to the Newtonian case.

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We are working towards better understanding of the differences in morphology of the gas - fluid interfaces and its implications. One could hope that, based on this rather simpler fluid model which we are using, it is possible to get better understanding of pattern formation in much more complicated system, such as flow in porous media, directional solidification, and many others.

Most of our work is given in the following paper

• P. Fast, L. Kondic, Michael J. Shelley, and Peter Palffy-Muhoray, Pattern formation in non-Newtonian Hele-Shaw flow, to appear in Phys. Fluids (May 2001).
  • Abstract

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  Some of the results can be found here:

  • L. Kondic, P. Fast, and Michael J. Shelley, About Computations of Hele-Shaw flow of non-Newtonian Fluids, Dynamics in Small Confining Systems IV, eds. J. M. Drake, G. S. Grest, J. Klafter, and R. Kopelman, Materials Research Society Proceedings Series 543, 207 (1999).
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    Newtonian fluid: Pressure

    

    Non-Newtonian fluid: Pressure

    

    Non-Newtonian fluid: Viscosity

    

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