This work is in collaboration with Greg Forest, David Hill, Sorin Mitran, and ... law (namely the Upper Convective Maxwell or Giesekus) using a numerical.
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 The goal of this project is to assess the role of linear and non-linear viscoelasticity in flow and diffusive transport relevant to biology, in particular flow of mucus layers in the lung while tracers are simultaneously diffusing. We model shear wave propagation to characterize viscoelastic fluids, and then use those oscillatory flow profiles to simulate coupled advection and diffusion in experimental shear cells. We quantify the increase in transit time of dyes to penetrate across a sheared viscoelastic layer, mimicking conditions in the lung. Further, improvements are made on a 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 and for Non-linear viscoelastic fluids including the Upper Convective Maxwell and Giesekus fluid. This work is in collaboration with Greg Forest, David Hill, Sorin Mitran, and Lingxing Yao.
These equtions yield a long time exact solution:
iωt sinh[δ ( y − H )] v x = ImV0 e sinh(δH )
Where δ is a complex number related to the parameters G’ and G’’. Further, it is possible to approach the same problem for a different constitutive law (namely the Upper Convective Maxwell or Giesekus) using a numerical approach. The constitutive equations here are nonlinear ∇
τ + λτ
Motivation and Applications Motivation: 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 G’(ω) and G’’(ω). In this model, a sample of viscoelastic fluid is subjected to periodic oscillations of a lower plate, and the resulting shear wave is observed and the wavelength and attenuation length are measured.
Fig 1. A Ferry viscoelastic shear wave. The attenuation length can be thought of as the exponentially decaying component of the wave.
v x (h, t ) = 0
v x (0, t ) = V0 sin(ωt )
a + τ ⋅τ = 2η 0 D G0
∇
where
τ
∂τ Τ = + (v ⋅ ∇ )τ − (∇v ) ⋅τ − τ ⋅ (∇v ) ∂t
Inverse Characterization of Viscoelastic Fluids
Tracer Diffusion in Viscoelastic Shear Waves Consider some passive tracer undergoing advection within a shear cell as described above. We might wonder what effect, if any, a viscoelastic shear wave has on the net diffusion of this tracer. Note: while this setup is much simpler than that of actual diffusion in the lung, it will give us insight into what type of non-linear velocity profiles may be important to advection-diffusion in a viscoelastic media..
Here We use the advection diffusion equation for our velocity profiles obtained above.
qt + v x ⋅ ∇q − υ∆q = 0 And proceed numerically using operator splitting, solving the advection piece using the Lax/Wendroff method and the diffusion piece is approached using the semi-implicit Crank/Nicolson Algorithm with Successive Over-Relaxation. We can 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.
One of our goals is to classify Viscoelastic fluids by observing the shear waves produced in the experiment discussed above. To do this, we use data obtained experimentally by tracking microscopic beads in a HA solution 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 illustrates that enhanced diffusion is happening. Further research in this area will classify what non-linear effects and shear wave structures maximize this enhanced transport.
Fig 2. A Ferry Viscoelastic shear wave in a finite channel. Here the upper plate is fixed at H=1 and is stationary.
Generalizations of the Ferry Method for Fitting: Ø1. Inverse Viscoelastic Characterization: Experimentally determine values for G’, G’’ for samples in finite channel depth. Ø2. 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. Ø3. 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” values of the viscoelastic parameters to our solution
Consider the Conservation equations for momentum in a viscoelastic fluid,
∂v ρ + (v ⋅ ∇ )v = ∇ ⋅ (− pΙ + τ ) ∂t
1. Understand what nonlinear features are important in transport and advection diffusion. 2. Consider 3-D advection-diffusion that simulates the biological situation with applications to drug therapies, and pathogen/foreign body clearance.. 3. Rheologically Characterize Mucus and other bio-fluids in a wide range of strains using a variety of non-linear models.
Here ρ is the density, p the pressure and τ the extra stress tensor. Assuming incompressibility & a linear viscoelastic fluid with modulus function G(t) t
τ = ∫ G (t − t ' ) D( y, t ' )dt '
−∞ 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:
p = p( y, t )
v x =v x ( y , t )
As a result, these equations reduce to a single dimension: t
vy= 0
∂v x η 0 ∂ τ xy ∂v x = τ xy = ∫ G (t − t ' ) ( y, t ' )dt ' ∂t ρ ∂y ∂y −∞
Future Work
Acknowledgements This work is supported by the National Science Foundation Research Training Group grant http://rtg.amath.unc.edu/
vz= 0
I’d also like to acknowledge they computer science department for their work on the bead tracking software used to obtain experimental data.
References
∂p =0 ∂y
1. John D. Ferry, Studies of the Mechanical Properties of Substances of High Molecular Weight I. A Photoelastic Method for
Fig4: A Nonlinear regression fit for time series data. Here the parameters being fit are viscoelastic parameters G’ and G’’.
With this method well established, we can go forward comparing the values obtained with those in the literature from other experiments, such as parallel plate rheometer data. Once the accuracy of this method is established, we can proceed with using the parameter value obtained in further research in mucus transport.
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.
