This thesis investigates helical flow for generalized Newtonian yield stress fluids in the region between two infinitely long concentric coaxial cylinders. Such a flow may arise either by superimposing an axial flow driven by applying an axial pressure gr

Helical flow of yield stress fluids.

In the Casson model, a fluid behaves as a solid for low stress but yields to flow as a viscoplastic fluid as the stress increases beyond a yield stress. The Casson model is often used to model the flow of blood or the flows occurring in food processing. We analyze the flow of a Casson fluid between infinitely long coaxial cylinders, with the inner cylinder rotating and the outer cylinder stationary, so that only the fluid adjacent to the inner cylinder yields. Simultaneously, axial flow arising from a small axial pressure gradient causes the flow to transform from a circular to a helical flow. Such flow is relevant to a number of applications, particularly rheometry. A perturbation analysis based on the pressure gradient provides explicit approximations for the fluid velocity profiles, as well as the change in location of the solid-fluid boundary. These approximations show the dependence of flow quantities on a range of fluid parameters, not just for specific parameter values, as occurs when numerical calculations are used.

 References E. O. Afoakwa, A. Paterson and M. Fowler. Factors influencing rheological and textural qualities in chocolate–-a review. Trends Food Sci. Tech. , 18(6):290–298, 2007. doi:10.1016/j.tifs.2007.02.002 E. C. Bingham. Fluidity and plasticity . McGraw-Hill, New York 1922. R. B. Bird, R. C. Armstrong and O. Hassager. Dynamics of Polymeric Liquids, Volume 1, Fluid Mechanics . John Wiley and Sons New York, 1987. http://au.wiley.com/WileyCDA/WileyTitle/productCd-047180245X.html G. W. Scott Blair. An equation for the flow of blood, plasma and serum through glass capillaries. Nature , 183:613–614, 1959. doi:10.1038/183613a0 C. Chiera, H. J. Connell and J. J. Shepherd. A perturbation approach to the analysis of a Casson model fluid, in Proceedings of the Second Biennial Australian Engineering Mathematics Conference: 1996; Engineering Mathematics; Research, Education and Industry Linkage . Institution of Engineers, Australia, pp. 315, 1996. http://search.informit.com.au/documentSummary;dn=718952602745151;res=IELENG B. D. Coleman and W. Noll. Helical flow of general fluids, J. Appl. Phys. , 30:1508–1512, 1959. doi:10.1063/1.1734990 M. T. Farrugia, J. J. Shepherd and A. J. Stacey. A perturbation analysis of the flow of a Powell–Eyring fluid between coaxial cylinders, CTAC2010, ANZIAM J. , 52:C257–C270, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3748 W. E. Langlois. Slow viscous flow . Macmillan New York, 1964. doi:10.1007/978-3-319-03835-3 J. J. Shepherd, C. Chiera and H. J. Connell. Perturbation analysis of the helical flow of non-Newtonian fluids with application to a recirculating coaxial cylinder rheometer, Math. Comput. Model. , 18(10):131–140, 1993. doi:10.1016/0895-7177(93)90222-K J. J. Shepherd, A. J. Stacey, and A. A. Khan, Helical flow arising from the yielded annular flow of a Bingham fluid, Appl. Math. Model. , 38(23):5382–5391, 2014. doi:10.1016/j.apm.2014.04.031 R. I. Tanner. Engineering Rheology , Oxford University Press, New York, 1988. https://global.oup.com/academic/product/engineering-rheology-9780198564737?cc=au&lang=en&

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Transition of a partially yielded Casson fluid from circular to helical flow

The rheological behaviour of non-Newtonian fluids with yield stress has been extensively studied, but the task of constructing analytic solutions of the governing equations has proved extremely difficult. The so-called Casson fluid is a known to display a yield stress; and, moreover, its behaviour is described by a relatively simple two-parameter constitutive equation. This model has several applications, of which the modelling of blood flow is the most significant. This study examines the case of helical flow of a Casson fluid between coaxial cylinders with given constant rotation of the inner cylinder and axial flow rate. Such helical flow constitutes one of the simplest three-dimensional flows for which an analytic solution might be sought. For the particular case where the intercylindrical gap width is small, a perturbation approach based on a scaled version of the gap width as a perturbation parameter is applied to yield expressions for describing the fluid flow, together with the fundamental relationship linking the angular velocity of the inner cylinder, the torque experienced there, and the given axial flow rate. The accuracy of these expressions will be tested by comparison with solutions generated numerically.

HJ Connell

Papers

A Perturbation Approach to the Analysis of a Casson Model Fluid