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.. vx ImV0e sinh(dH ) 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 t (v )t (v ) t t (v ) a 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 depth. 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 Future Work v 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 Acknowledgements 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 experimental data. As a result, these equations reduce to a single dimension: v t t vx p References t G(t t ' ) xy x 0 ( y, t ' )dt ' 0 t r y xy y y 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.
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