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Background Lectures on Ideal Visco-Plastic Fluid Flows

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Book cover Lectures on Visco-Plastic Fluid Mechanics

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 583))

Abstract

These notes are intended to supplement a short lecture course covering the theoretical background of the dynamics of ideal visco-plastic fluids, e.g. Bingham fluids, Herschel-Bulkley fluids. They are targeted at an applied mathematics or engineering audience. The intention is to give a non-rigorous introduction to those parts of the theory that: (a) appear to have use in applications; (b) are needed for computational methods; (c) mark out visco-plastic fluids from purely viscous generalised Newtonian fluids.

This research is funded by the NSERC Discovery grant programme which is gratefully acknowledged. Parts of these notes contain results from ongoing and recent work with my group. I would like to thank Emad Chaparian, Ida Karimfazli and Ali Roustaei for their help with computed examples.

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References

  • Adachi, K., & Yoshioka, N. (1973). On creeping flow of a visco-plastic fluid past a circular cylinder. Chemical Engineering Science, 28, 215–226.

    Article  Google Scholar 

  • Adams, R. A. (1975). Sobolev spaces. New York: Academic.

    MATH  Google Scholar 

  • Beris, A. N., Tsamopoulos, J. A., Armstrong, R. C., & Brown, R. A. (1985). Creeping motion of a sphere through a Bingham plastic. Journal of Fluid Mechanics, 158, 219–244.

    Article  MathSciNet  Google Scholar 

  • Bristeau, M. O. (1975). Application de la mthode des lments finis la rsolution numrique d’inquations variationnelles de type Bingham. These de 3me cycle, Universite de Paris VI, Juin.

    Google Scholar 

  • Chakrabarty, J. (2012). Theory of plasticity. Butterworth-Heinemann.

    Google Scholar 

  • Chaparian, E., Balmforth, N., & Frigaard, I. A. (2017). Yield limit analysis of symmetric particle sedimentation in a bingham fluid. preprint.

    Google Scholar 

  • Chatzimina, M., Georgiou, G. C., Argyropaidas, I., Mitsoulis, E., & Huilgol, R. R. (2005). Cessation of Couette and Poiseuille flows of a Bingham plastic and finite stopping times. Journal of Non-Newtonian Fluid Mechanics, 129, 117–127.

    Article  Google Scholar 

  • Cioranescu, D. (1976). Sur une classe de fluides non-newtoniens. Applied Mathematics and Optimization, 3, 263–282.

    Article  MathSciNet  Google Scholar 

  • Dacorogna, B. (2008). Applied mathematical sciences series. In Direct methods in the calculus of variations (Vol. 78). Springer.

    Google Scholar 

  • Dubash, N., & Frigaard, I. A. (2004). Conditions for static bubbles in viscoplastic fluids. Physics of Fluids, 16, 4319–4330.

    Article  Google Scholar 

  • Duvaut, G., & Lions, J. L. (1976). Inequalities in mechanics and physics. Springer.

    Book  Google Scholar 

  • Frigaard, I. A. (1998). Stratified exchange flows of two Bingham fluids in an inclined slot. Journal of Non-Newtonian Fluid Mechanics, 78, 61–87.

    Article  Google Scholar 

  • Frigaard, I. A., & Scherzer, O. (1998). Uniaxial exchange flows of two Bingham fluids in a cylindrical duct. IMA Journal of Applied Mathematics, 61, 237–266.

    Article  MathSciNet  Google Scholar 

  • Frigaard, I. A., & Scherzer, O. (2000). The effects of yield stress variation in uniaxial exchange flows of two Bingham fluids in a pipe. SIAM Journal on Applied Mathematics, 60, 1950–1976.

    Article  MathSciNet  Google Scholar 

  • Frigaard, I. A., Scherzer, O., & Sona, G. (2001). Uniqueness and non-uniqueness in the steady displacement of two visco-plastic fluids. ZAMM, 81, 99–118.

    Article  MathSciNet  Google Scholar 

  • Fuchs, M., & Seregin, G. (2000). Lecture notes in mathematics. In Variational methods for problems from plasticity theory and for generalized newtonian fluids (Vol. 1749). Springer

    Book  Google Scholar 

  • Glowinski, R. (1984). Numerical methods for nonlinear variational problems. Springer.

    Book  Google Scholar 

  • Glowinski, R., Lions, J. L., & Trémolières, R. (1981). Numerical analysis of variational inequalities. Studies in mathematics and its applications. (trans: from French version of 1976). North-Holland.

    Chapter  Google Scholar 

  • Hassani, R., Ionescu, I. R., & Lachand-Robert, T. (2005). Shape optimization and supremal minimization approaches in landslides modelling. Applied Mathematics and Optimization, 52, 349–364.

    Article  MathSciNet  Google Scholar 

  • Hild, P., Ionescu, I. R., Lachand-Robert, T., & Rosca, I. (2002). The blocking property of an inhomogeneous Bingham fluid. applications to landslides. Mathematical Modelling and Numerical Analysis (M2AN), 36, 1013–1026.

    Google Scholar 

  • Hill, R. (1950). The mathematical theory of plasticity. Oxford University Press.

    Google Scholar 

  • Huilgol, R. R. (2006). A systematic procedure to determine the minimum pressure gradient required for the flow of viscoplastic fluids in pipes of symmetric cross-section. Journal of Non-Newtonian Fluid Mechanics, 136, 140–146.

    Article  Google Scholar 

  • Huilgol, R. R. (2015). Fluid mechanics of viscoplasticity. Springer.

    Book  Google Scholar 

  • Ionescu, I. R., & Lachand-Robert, T. (2005). Generalized cheeger’s sets related to landslides. Calculus of Variations and PDEs, 23, 227–249.

    Article  MathSciNet  Google Scholar 

  • Johnson, M. W. (1960). Some variational theorems for non-newtonian flow. Physics of Fluids, 3, 871–878.

    Article  MathSciNet  Google Scholar 

  • Johnson, M. W. (1961). On variational principle for non-newtonian fluids. Transactions. Society of Rheology, 5, 9–21.

    Article  MathSciNet  Google Scholar 

  • Joseph, D. D. (1976a). Springer tracts in natural philosophy. Stability of fluid motions II. Springer, Heidelberg.

    MATH  Google Scholar 

  • Joseph, D. D. (1976b). Springer tracts in natural philosophy. Stability of fluid motions I. Springer, Heidelberg.

    MATH  Google Scholar 

  • Karimfazli, I., & Frigaard, I. A. (2013). Natural convection flows of a bingham fluid in a long vertical channel. Journal of Non-Newtonian Fluid Mechanics, 201, 39–55.

    Article  Google Scholar 

  • Karimfazli, I., Frigaard, I. A., & Wachs, A. (2015). A novel heat transfer switch using the yield stress. Journal of Fluid Mechanics, 783, 526–566.

    Article  MathSciNet  Google Scholar 

  • Malek, J., Ruzicka, M., & Shelukhin, V. V. (2005). Herschel-Bulkley fluids: existence and regularity of steady flows. Mathematical Models and Methods in Applied Sciences, 15, 1845–1861.

    Article  MathSciNet  Google Scholar 

  • Mosolov, P. P., & Miasnikov, V. P. (1965). Variational methods in the theory of the fluidity of a viscous-plastic medium. PPM. Journal of Mechanics and Applied Mathematics, 29, 468–492.

    Article  Google Scholar 

  • Mosolov, P. P., & Miasnikov, V. P. (1966). On stagnant flow regions of a viscous-plastic medium in pipes. PPM. Journal of Mechanics and Applied Mathematics, 30, 705–717.

    Article  Google Scholar 

  • Mosolov, P. P., & Miasnikov, V. P. (1967). On qualitative singularities of the flow of a viscoplastic medium in pipes. PPM. Journal of Mechanics and Applied Mathematics, 31, 581–585.

    Article  Google Scholar 

  • Moyers-Gonzalez, M. A., & Frigaard, I. A. (2004). Numerical solution of duct flows of multiple visco-plastic fluids. Journal of Non-Newtonian Fluid Mechanics, 122, 227–241.

    Article  Google Scholar 

  • Nouar, C., & Frigaard, I. A. (2001). Nonlinear stability of Poiseuille flow of a Bingham fluid: Theoretical results and comparison with phenomenological criteria. Journal of Non-Newtonian Fluid Mechanics, 100, 127–149.

    Article  Google Scholar 

  • Prager, W. (1954). Studies in mathematics and mechanics. In On slow visco-plastic flow (pp. 208–216). New York: Academic Press Inc. Presented to Richard von Mises by Friends, Colleagues, and Pupils.

    Chapter  Google Scholar 

  • Putz, A., & Frigaard, I. A. (2010). Creeping flow around particles in a Bingham fluid. Journal of Non-Newtonian Fluid Mechanics, 165, 263–280.

    Article  Google Scholar 

  • Randolph, M. F., & Houlsby, G. T. (1984). The limiting pressure on a circular pile loaded laterally in cohesive soil. Géotechnique, 34, 613–623.

    Article  Google Scholar 

  • Roustaei, A., Chevalier, T., Talon, L., & Frigaard, I. A. (2016). Non-darcy effects in fracture flows of a yield stress fluid. submitted to Journal of Fluid Mechanics.

    Article  MathSciNet  Google Scholar 

  • Serrin, J. (1959). On the stability of viscous fluid motions. Archive for Rational Mechanics and Analysis, 3, 1–13.

    Article  MathSciNet  Google Scholar 

  • Temam, R., & Strang, G. (1980). Functions of bounded deformation. Archive for Rational Mechanics and Analysis, 75, 7–21.

    Article  MathSciNet  Google Scholar 

  • Tokpavi, D., Magnin, A., & Jay, P. (2008). Very slow flow of Bingham viscoplastic fluid around a circular cylinder. Journal of Non-Newtonian Fluid Mechanics, 154, 65–76.

    Article  Google Scholar 

  • Tsamopoulos, J., Dimakopoulos, Y., Chatzidai, N., Karapetsas, G., & Pavlidis, M. (2008). Steady bubble rise and deformation in Newtonian and viscoplastic fluids and conditions for bubble entrapment. Journal of Fluid Mechanics, 601, 123–164.

    Article  MathSciNet  Google Scholar 

  • Wachs, A., & Frigaard, I. A. (2016). Particle settling in yield stress fluids: limiting time, distance and applications. submitted to Journal of Non-Newtonian Fluid Mechanics.

    Article  MathSciNet  Google Scholar 

  • Yoshioka, N., & Adachi, K. (1971a) On variational principles for a non-newtonian fluid. Journal of Chemical Engineering of Japan, 4, 217–220.

    Article  Google Scholar 

  • Yoshioka, N., & Adachi, K. (1971b). Applications of the extremum principles for non-newtonian fluids. Journal of Chemical Engineering of Japan, 4, 221–226.

    Article  Google Scholar 

  • Yoshioka, N., & Adachi, K. (1973). Some deductions from the extremum principles for non-newtonian fluids. Journal of Chemical Engineering of Japan, 6, 134–140.

    Article  Google Scholar 

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Frigaard, I.A. (2019). Background Lectures on Ideal Visco-Plastic Fluid Flows. In: Ovarlez, G., Hormozi, S. (eds) Lectures on Visco-Plastic Fluid Mechanics. CISM International Centre for Mechanical Sciences, vol 583. Springer, Cham. https://doi.org/10.1007/978-3-319-89438-6_1

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  • DOI: https://doi.org/10.1007/978-3-319-89438-6_1

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