Transport Phenomena In Viscoelastic Fluids
Brandon Lindley, Greg Forest, David Hill, Sorin Mitran, Lingxing Yao
University of North Carolina
Mathematics Departments and Cystic Fibrosis Center
Controlling the lower plate oscillations and imposing a fixed upper plate gives boundary conditions
Abstract Tracer Diffusion in Viscoelastic Shear Waves
The goal of this project is to assess the role of linear and non-linear viscoelasticity in flow and diffusive transport vx (0, t ) V0 sin(wt ) v x ( h, t ) 0
relevant to biology, in particular flow of mucus layers in the lung while tracers are simultaneously diffusing. We model Consider some passive tracer undergoing advection within a shear cell as described above. We might wonder what
shear wave propagation to characterize viscoelastic fluids, and then use those oscillatory flow profiles to simulate effect, if any, a viscoelastic shear wave has on the net diffusion of this tracer. Note: while this setup is much simpler than
coupled advection and diffusion in experimental shear cells. We quantify the increase in transit time of dyes to These equtions yield a long time exact solution: that of actual diffusion in the lung, it will give us insight into what type of non-linear velocity profiles may be important to
penetrate across a sheared viscoelastic layer, mimicking conditions in the lung. Further, improvements are made on a iwt sinh[d ( y H )] advection-diffusion in a viscoelastic media..
classical technique of Ferry for rheological characterization. The Ferry method of viscoelastic characterization involves
fitting the viscoelastic parameters of a linear viscoelastic model by finding the attenuation length and wavelength of
shear waves propagated in the fluid. This technique is generalize to fit rheological parameters in a finite channel depth
Where d is a complex number related to the parameters G’ and G’’. Further, it is possible to approach the same Here We use the advection diffusion equation for our velocity profiles obtained above.
and for Non-linear viscoelastic fluids including the Upper Convective Maxwell and Giesekus fluid. This work is in problem for a different constitutive law (namely the Upper Convective Maxwell or Giesekus) using a numerical
qt vx q q 0
collaboration with Greg Forest, David Hill, Sorin Mitran, and Lingxing Yao. approach. The constitutive equations here are nonlinear
(v )t (v ) t t (v )
t t t
And proceed numerically using operator splitting, solving the advection piece using the Lax/Wendroff method and the
Motivation and Applications t t 2 0 D where diffusion piece is approached using the semi-implicit Crank/Nicolson Algorithm with Successive Over-Relaxation. We can
G0 t then assess the role played by non-linear shear wave propagation on advection diffusion. By quantifying these effects, we
can get an idea of what effects linear and nonlinear viscoelasticity has on net transport.
Motivation: Inverse Characterization of Viscoelastic Fluids
This project is motivated by experiments on viscoelastic fluids at UNC. Because of the oscillatory nature of ciliary
forces acting on mucus within the lungs, we expect frequency dependent deformation to be of vital importance to
understanding the mechanics of mucus flow. To understand the effect imposed oscillatory strain has on viscoelastic
fluids, we revisit the classical Ferry model, where the storage and loss modulus are frequency dependent parameters One of our goals is to classify Viscoelastic fluids by observing the shear waves produced in the experiment
G’(w) and G’’(w). In this model, a sample of viscoelastic fluid is subjected to periodic oscillations of a lower plate, and discussed above. To do this, we use data obtained experimentally by tracking microscopic beads in a HA solution
the resulting shear wave is observed and the wavelength and attenuation length are measured. and fit the Ferry finite channel depth formula or the numerically obtained formula to the data. The technique used
here is to fit time series data for individual beads at various heights and to iterate by using the best fit at a given
height as a starting guess at another height. Performing several cycles of this should result in a fit of values for all
the heights observed. Note, we would have to image many more beads at many more heights in order to fit our
shear wave profile (y-axis data) to the experimental data for all heights. For now, the aforementioned approach
yields better results.
Fig 4 (a) and (b) show two concentration profiles for
advection diffusion on an initial Gaussian distribution one
under the influence of shear waves, the other in pure
diffusion. Figure 4(c) shows the difference in distribution and
Fig 1. A Ferry viscoelastic shear wave. The attenuation Fig 2. A Ferry Viscoelastic shear wave in a finite illustrates that enhanced diffusion is happening. Further
length can be thought of as the exponentially decaying channel. Here the upper plate is fixed at H=1 and
research in this area will classify what non-linear effects and
component of the wave. is stationary.
shear wave structures maximize this enhanced transport.
Generalizations of the Ferry Method for Fitting:
1. Inverse Viscoelastic Characterization: Experimentally determine values for G’, G’’ for samples in finite channel
2. Use a small number of microscopic beads at various heights to fit the Finite depth solution to the Ferry model
using a time series fit rather than imaging the entire wave profile.
3. Classify Nonlinear viscoelastic fluids using techniques developed below.
Linear and Nonlinear Viscoelastic Shear wave propogation
Fig3 (a) A snapshot of our Video bead tracking in action. (b) Comparing Experimentally “known”
Consider the Conservation equations for momentum in a viscoelastic fluid, values of the viscoelastic parameters to our solution
r (v )v ( p t )
t 1. Understand what nonlinear features are important in transport and advection diffusion.
Here r is the density, p the pressure and t the extra stress tensor. Assuming incompressibility & a linear viscoelastic 2. Consider 3-D advection-diffusion that simulates the biological situation with applications to drug therapies, and
fluid with modulus function G(t) pathogen/foreign body clearance..
t 3. Rheologically Characterize Mucus and other bio-fluids in a wide range of strains using a variety of non-linear
t G(t t ' ) D( y, t ' )dt ' models.
Where D is the rate of deformation tensor. Assuming that all the deformations are only in the x-direction, and that we
are controlling all externally applied forces gives conditions:
This work is supported by the National Science Foundation Research Training Group grant http://rtg.amath.unc.edu/
p p ( y, t ) v x v x ( y , t ) vy 0 vz 0 I’d also like to acknowledge they computer science department for their work on the bead tracking software used to obtain
As a result, these equations reduce to a single dimension:
v t t
vx p References
t G(t t ' )
( y, t ' )dt ' 0
t r y xy
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
Fig4: A Nonlinear regression fit for time series data. Here the parameters being fit are Stresses, , 593-611, (1947)
viscoelastic parameters G’ and G’’. 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
With this method well established, we can go forward comparing the values obtained with those in the literature from York, 1987.
other experiments, such as parallel plate rheometer data. Once the accuracy of this method is established, we can 5. R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworth, Guildford, UK, 1988.
proceed with using the parameter value obtained in further research in mucus transport.