NonNewtonian Fluid Mechanics
Elastic liquids, such as polymeric liquids and food being processed,
exhibit many strange phenomena.
Strong vortices form upstream of an orifice.
Two of my young colleagues and I have developed novel
finite difference and finite element codes, which function with
considerably higher elastic effects than others, and
these computations have demonstrated that the vortices
and their associated pressure drops cannot be predicted by standard
rheological models, requiring instead the additional anisotropic
viscosity suggested by the molecular simulations mentioned above.
One technical problem in computations is the singularity in the
stress at any sharp corner.
There have been many speculations that this singularity might be
Newtonian, like r^{0.455}, or, as a consequence of the nonlinearity
of the nonNewtonian fluid, more singular, perhaps nonintegrably so.
Recently I obtained a closed form analytic solution 61 with a
r^{2/3} singularity in the stress and a r^{5/9}
variation in the velocity.
This also shows that the upstream vortices are not generated by
singularities at sharp corners.
In flows with stagnation points, a small part of the fluid resides for a
long time in a stretching region, and thus emerges from a
stagnation point as a thin strand of highly stressed
material, which may be observed by birefringence.
A boundarylayer theory has been developed for the analysis of such
flows.
Sometimes the stresses are sufficiently strong to inhibit the
stretching flow, giving rise to `birefringent
pipes'.
It has been found possible to predict these pipes when
the hysteresis in the polymer rheology described above is included.
NonNewtonian flows suffer from instabilities.
The coextrusion of two different elastic liquids is used industrially
to place an expensive coat on a cheap recycled core.
The isolation of the mechanism of the instability of this flow
permitted generalisations to other rheological
equations, generalisations now being tested experimentally
in Grenoble.
A recent study of the useful stabilising effect of fluid
elasticity on jet flows required first an
explanation of why Newtonian jets are unstable before the stabilisation
mechanism could be discussed.
