Fluid Layers in LAOS: Stress Communication and

1
Fluid Layers in LAOS Stress Communication and
Peak Selection Criteria Brandon Lindley, Eddie
Howell, Breannan Smith Greg Forest, Sorin
Mitran, David Hill University of North
Carolina Mathematics Department
With boundary conditions, These equtions yield
a frequency locked solution Mitran Where, A
nd, Which gives an integral formulation for
the remaining stress term,
Abstract
In this poster, we explore linear and nonlinear
stress communication through viscoelastic layers
in large amplitude oscillatory shear (LAOS)
imposed at one interface. Motivated by questions
concerning biological layers, we are led to focus
on stress signals arriving at either interface,
the driven or opposing. In doing so, we identify
a remarkable structure in boundary normal and
shear stress signals, presented first in terms of
a frequency sweep and then generalized to a
scaling law with respect to all control
parameters (material properties and experimental
controls). The structure consists of peaks in the
maximum stress signals versus any parameter
sweep, indicating a clear selection mechanism
with robust redundancy for communicating shear
and normal stress across biological layers. In
this poster, we restrict to a single mode upper
convective Maxwell model, where analysis and
simulations reveal a transparent explanation of
the phenomenon. The upper convected nonlinearity
is common to all the nonlinear continuum
mechanical models, and thus the normal stress
communication mechanism revealed here is
fundamental
The degree of redundancy in the occurrence of
peak stresses allows for a very robust
understanding of the phenomenon. For example, we
can consider where the peaks and valleys occur as
a function of multiple parameters such as
viscosity and frequency of shear For
the purposes of biological applications, we can
consider a range of values for all of the
parameters, and find out which parameters can be
perturbed to increase or decrease stress
communication. For instance, in the case of
cystic fibrosis (CF), mucus depth and viscosity
both differ from healthy patients, so one might
be interested in how stress signals are
transmitted as a function of these
parameters. Currently, the models
used rely on exact formulas for one particular
model (UCM), and yet the exact scaling of the
peak heights with respect to all parameters is
not known analytically, though it has been
classified numerically. This same selection
criteria is also present in other nonlinear
constitutive models, which allow for shear
thinning or other non-Newtonian phenomena, and
this is currently being studied using numerical
methods.
Motivation and Applications
Motivation The behavior of viscoelastic layers
in large amplitude oscillatory shear (LAOS) has
been studied in depth in the rheology literature.
The methods of inquiry are varied, ranging from
presumed homogeneous deformations where the
problem analytically reduces to a dynamical
system of ordinary differential equations
Giacomin to presumed one-dimensional
heterogeneous deformations where the models are
coupled with systems of partial differential
equations Mitran to two-dimensional
heterogeneity and quite demanding numerical
solver technology Keunings. The phenomena
of interest for this poster are motivated by a
biological query. There are countless examples in
biology where a viscoelastic layer plays a vital
mechanistic function. Our focus arises from lung
biology, where mucus layers line pulmonary
pathways and serve as the medium between air from
the external environment and the cilia-epithelium
complex. There is definitive evidence that shear
stress signals propagate through mucus layers and
regulate biochemical release rates of epithelial
cells. This discovery raises a host of
fundamental questions about the stress signals
arriving at the epithelial cells from a sheared
mucus layer.
Stress Transfer and Peak Selection Criteria
Based on our motivation from stress communication
in strain-driven biological layers, we organize
the response functions of interest in terms of
transfer functions. Namely, for a given
realization of the experiment or model
description of it, we extract the maximum
boundary shear and normal stress signals arriving
at either the driven interface or the opposing
stationary interface. For the purposes of this
poster, we will focus on one transfer function on
interest, namely To begin, we give the graph of
the frequency sweep dependence of our transfer
function for some prescribed material
parameters We immediately observe the
existence of peaks and valleys in the transfer
function, and note that they appear to be a
regularly spaced phenomenon. They are also
increasing as the frequency (and consequently
shear rate) are increased, which is to be
expected. This implies a frequency selection
mechanism in in stress communication across the
layer modulation of the driving frequency can be
used to either minimize or maximize the signal.
The question then, what is this selection factor?
For the UCM Model we are studying, the answer is
definitive by a straightforward analysis of the
characteristics of the hyperbolic system. Namely,
the local wave speeds of the counter propagating
waves and the travel time of a stress wave across
the layer are respectively Where the second
expression in fact defines the fundamental
frequency. Essentially, if the lower boundary is
driven at precisely the roundtrip travel time of
a propagating wave (or any integer multiple), the
shear stress is maximized. Remarkably, the
formula above defines a natural, non-dimensional
scaling parameter for the whole problem (the
ratio of the travel time to the height times the
frequency), from which we can generalize the peak
selection criteria to ANY of the control
parameters! The remarkable consequence of
this scaling behavior of peaks in boundary normal
stress communication is the degree of redundancy
in the phenomenon. Namely, peaks or valleys in
normal stress communication at either boundary of
the driven layer can be tuned by any material
property of the fluid or by the geometry of the
layer or the frequency of the driving mechanism.

• Goals and Applications
• 1. Formulate a criteria for peak/valley stress
  transfer.
• 2. Make predictions about what can be done to
  enhance or reduce stress transfer in biological
  systems by virtue of changing the rheology, or
  the driving conditions,
• 3. Make predictions for the forces acting on
  epithelial cells, and understanding their
  response to imposed stress.

Future Work
Linear and Nonlinear Viscoelastic Shear Wave
Propagation

• 1. Couple biological feedback into the system.
  The lung receives stress signals and responds by
  changing the fluid chemistry or depth, or perhaps
  even shear rate.
• 2. Mimic biological conditions to make
  predictions about mucus and the peak stress
  signals one could expect from mucus.
• 3. We know this phenomenon appears in higher
  order nonlinear models, in fact, it was
  discovered there! Now we want to give laws for
  the scaling and separation of peaks in these
  models, which may more accurately model mucus

Consider the Conservation equations for momentum
in a viscoelastic fluid, between two surfaces,
one driven at height y0, the other fixed at
height yH. Here r is the density, p the
pressure and t the extra stress tensor. Assuming
incompressibility an upper convected Maxwell
(UCM) constitutive equation for the extra
stress, Where D is the rate of deformation
tensor, ? the relaxation parameter and?the
viscosity. Assuming that all the deformations are
only in the x-direction, and all other externally
applied forces are allowed to decay to 0
gives, As a result, these equations reduce to
a single dimension, and a system of hyperbolic,
quasi-linear equations
Acknowledgements
This work is supported by a SAMSI graduate
research fellowship. http//www.samsi.info
References
1. John D. Ferry, Studies of the Mechanical
Properties of Substances of High Molecular Weight
I. A Photoelastic Method for Study of Transverse
Vibrations in Gels, Rev. Sci. Inst., 12, 79-82,
(1941) 2. John D. Ferry, W.M. Sawyer, and J. N.
Ashworth, Behavior of Concentrated Polymer
Solutions under Periodic Stresses, , 593-611,
(1947) 3. J.D. Ferry, Viscoelastic Properties of
Polymers , John Wiley, New York, 1980. 4. R.B.
Bird, R.C. Armstrong, O. Hassager, Dynamics of
Polymeric Liquids, Fluid Mechanics Vol. 1,
Wiley, New York, 1987. 5. R.G. Larson,
Constitutive Equations for Polymer Melts and
Solutions, Butterworth, Guildford, UK, 1988. 6.
S.M Mitran, G.F. Forest, Lingxing Yao, B.
Lindley, D.B. Hill, Extensions of the Ferry Shear
Wave Model for Active Linear and Nonlinear
Microrheology, J. Non-Newtonian Fluid Mech
(Accepted), 2007 7. Jeyaseelan \ A.J. Giacomin,
Network theory for polymer solutions in large
amplitude oscillatory shear, J. Non-Newtonian
Fluid Mech., Accepted, 2007 8. R. Keunings \ K.
Atalik On the occurrence of even harmonics in the
shear stress response of viscoelastic fluids in
large amplitude oscillatory shear, J.
Non-Newtonian Fluid Mech., 122, 107-116, 2004.