Viscoelastic Fluids

This is the second of four conversations Gudrun had during the
British Applied Mathematics Colloquium which took place 5th – 8th
of April 2016 in Oxford.


Helen Wilson always wanted to do maths and had imagined herself
becoming a mathematician from a very young age. But after
graduation she did not have any road map ready in her mind. So
she applied for jobs which - due to a recession - did not exist.
Today she considers herself lucky for that since she took a
Master's course instead (at Cambridge University), which hooked
her to mathematical research in the field of viscoelastic fluids.
She stayed for a PhD and after that for postdoctoral work in the
States and then did lecturing at Leeds University. Today she is a
Reader in the Department of Mathematics at University College
London.


So what are viscoelastic fluids? If we consider everyday fluids
like water or honey, it is a safe assumption that their viscosity
does not change much - it is a material constant. Those fluids
are called Newtonian fluids. All other fluids, i.e. fluids with
non-constant viscosity or even more complex behaviours, are
called non-Newtonian and viscoelastic fluids are a large group
among them.


Already the name suggests, that viscoelastic fluids combine
viscous and elastic behaviour. Elastic effects in fluids often
stem from clusters of particles or long polymers in the fluid,
which align with the flow. It takes them a while to come back
when the flow pattern changes. We can consider that as keeping a
memory of what happened before. This behaviour can be observed,
e.g., when stirring tinned tomato soup and then waiting for it to
go to rest again. Shortly before it finally enters the rest state
one sees it springing back a bit before coming to a halt. This is
a motion necessary to complete the relaxation of the soup.


Another surprising behaviour is the so-called Weissenberg effect,
where in a rotation of elastic fluid the stretched out polymer
chains drag the fluid into the center of the rotation. This leads
to a peak in the center, instead of a funnel which we expect from
experiences stirring tea or coffee.


The big challenge with all non-Newtonian fluids is that we do not
have equations which we know are the right model. It is mostly
guess work and we definitely have to be content with
approximations.
And so it is a compromise of fitting what we can model and
measure to the easiest predictions possible. Of course, slow flow
often can be considered to be Newtonian whatever the material is.


The simplest models then take the so-called retarded fluid
assumption, i.e. the elastic properties are considered to be only
weak. Then, one can expand around the Newtonian model as a base
state.
The first non-linear model which is constructed in that way is
that of second-order fluids. They have two more parameters than
the Newtonian model, which are called normal stress coefficients.
The next step leads to third-order fluids etc. In practice no
higher than third-order fluids are investigated.


Of course there are a plethora of interesting questions connected
to complex fluids. The main question in the work of Helen Wilson
is the stability of the flow of those fluids in channels, i.e.
how does it react to small perturbations? Do they vanish in time
or could they build up to completely new flow patterns? In 1999,
she published results of her PhD thesis and predicted a new type
of instability for a shear-thinning material model. It was to her
great joy when in 2013 experimentalists found flow behaviour
which could be explained by her predicted instability.


More precisely, in the 2013 experiments a dilute polymer solution
was sent through a microchannel. The material model for the fluid
is shear thinning as in Helen Wilson's thesis. They observed
oscillations from side to side of the channel and surprising
noise in the maximum flow rate. This could only be explained by
an instability which they did not know about at that moment. In a
microchannel inertia is negligible and the very low Reynolds
number of suggested that the instability must be caused by the
non-Newtonian material properties since for Newtonian fluids
instabilities can only be observed if the flow configuration
exeeds a critical Reynolds number. Fortunately, the answer was
found in the 1999 paper.


Of course, even for the easiest non-linear models one arrives at
highly non-linear equations. In order to analyse stability of
solutions to them one firstly needs to know the corresponding
steady flow. Fortunately, if starting with the easiest non-linear
models in a channel one can still find the steady flow as an
analytic solution with paper and pencil since one arrives at a 1D
ODE, which is independent of time and one of the two space
variables.


The next question then is: How does it respond to small
perturbation? The classical procedure is to linearize around the
steady flow which leads to a linear problem to solve in order to
know the stability properties. The basic (steady) flow allows for
Fourier transformation which leads to a problem with two scalar
parameters - one real and one complex. The general structure is
an eigenvalue problem which can only be solved numerically. After
we know the eigenvalues we know about the (so-called linear)
stability of the solution.


An even more interesting research area is so-called non-linear
stability. But it is still an open field of research since it has
to keep the non-linear terms. The difference between the two
strategies (i.e. linear and non-linear stability) is that the
linear theory predicts instability to the smallest perturbations
but the non-linear theory describes what happens after
finite-amplitude instability has begun, and can find larger
instability regions. Sometimes (but unfortunately quite rarely)
both theories find the same point and we get a complete picture
of when a stable region changes into an unstable one.


One other really interesting field of research for Helen Wilson
is to find better constitutive relations. Especially since the
often used power law has inbuilt unphysical behaviour (which
means it is probably too simple). For example, taking a power law
with negative exponent says that In the middle of the flow there
is a singularity (we would divide by zero) and perturbations are
not able to cross the center line of a channel.


Also, it is unphysical that according to the usual models the
shear-thinning fluid should be instantly back to a state of high
viscosity after switching off the force. For example most ketchup
gets liquid enough to serve it only when we shake it. But it is
not instantly thick after the shaking stops - it takes a moment
to solidify. This behaviour is called thixotropy.

Literature and additional material

H. Wilson: UCL Lunch Hour Lectures, Feb. 2016.

H.J. Wilson and J.M. Rallison: Instability of channel flow of
a shear-thinning White–Metzner fluid, Journal of Non-Newtonian
Fluid Mechanics 87 (1999) 75–96.

Hugues Bodiguel, Julien Beaumont, Anaïs Machado, Laetitia
Martinie, Hamid Kellay, and Annie Colin: Flow Enhancement due to
Elastic Turbulence in Channel Flows of Shear Thinning Fluids,
Physical Review Letters 114 (2015) 028302.

Non-Newtonian Fluids Explained, Science Learning.