Some simple flows of a Johnson-Segalman fluid
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Summary. Unlike most other fluid models, the Johnson-Segalman fluid allows for a non-monotonic relationship between the shear stress and rate of shear in a simple shear flow for certain values of the material parameter. This has been used for explaining a phenomenon such as `spurt'. Here, we study three simple flows of a Johnson-Segalman fluid with a view towards understanding its response characteristics. We find that boundary conditions can have a very interesting effect on the regularity of the solution; changing them continuously leads to solutions that change their regularity. First, we consider the flow through a circular pipe and find solutions that have discontinuous velocity profiles which have been used to explain the phenomenon of `spurt'. Second, we consider the flow past an infinite porous plate and show that it will not admit solutions which have discontinuous velocity gradients, the solutions being necessarily smooth. Lastly, we study Poiseuille flow in a concentric annulus with porous boundaries. While `spurt' could be explained alternatively by allowing for `stick-slip' at the wall the Johnson-Segalman model seems particularly suited in describing the appearance of `shear-layers'.
