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Numerical Modelling

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Fluid Mechanics of Viscoplasticity

Abstract

In this Chapter, we shall summarise and apply two powerful techniques for the solution of flow problems in Bingham and other viscoplastic fluids.

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Notes

  1. 1.

    Uzawa’s papers deal with finding the maximum of concave functions subject to constraints; see Chaps. 3, 5, 7 and 10 in [6]. The algorithm for solving the saddle point problem appears in Chap. 10.

  2. 2.

    A very readable introduction to these ideas is contained in Nocedal and Wright [7]. See Chaps. 12 and 17 for the quoted results.

  3. 3.

    The proof given here follows that in Chap. 4, Sect. 19 in [3].

  4. 4.

    In viscoelastic fluid mechanics, splitting the constitutive equation into a viscous and an elastic part was conceived and applied to Oldroyd-B fluids in 1977; see [16].

References

  1. Glowinski R (1984) Numerical methods for nonlinear variational problems. Springer, New York

    Google Scholar 

  2. Glowinski R, Le Tallec P (1989) Augmented Lagrangian and operator-splitting methods in nonlinear mechanics. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  3. Glowinski R (2003) Finite element methods for incompressible viscous flow. In: Ciarlet PG, Lions J-L (eds) Handbook of numerical analysis, vol IX. North-Holland, Amsterdam, pp 3–1176

    Google Scholar 

  4. Glowinski R, Wachs A (2011) On the numerical simulation of viscoplastic fluid flow. In: Handbook of numerical analysis, vol XVI. Ciarlet PG (General Ed), Glowinski R, Xu J (Guest eds.) North-Holland, Amsterdam, pp 483–717

    Google Scholar 

  5. Cea J, Glowinski R (1972) Methodes numeriques pour l’ecoulement laminaire d’un fluide rigide viscoplastique incompressible. Int J Comp Math Sec B 3:225–255

    Article  MATH  MathSciNet  Google Scholar 

  6. Arrow KJ, Hurwicz L, Uzawa H (1958) Studies in linear and nonlinear programming. Stanford University Press, Stanford

    Google Scholar 

  7. Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, New York

    MATH  Google Scholar 

  8. Apostol TM (1974) Mathematical analysis, 2nd edn. Addison-Wesley, Reading

    MATH  Google Scholar 

  9. Huilgol RR, Panizza MP (1995) On the determination of the plug flow region in Bingham fluids through the application of variational inequalities. J Non-Newton Fluid Mech 58:207–217

    Article  Google Scholar 

  10. Huilgol RR, You Z (2005) Application of the augmented Lagrangian method to steady pipe flows of Bingham, Casson and Herschel-Bulkley fluids. J Non-Newton Fluid Mech 128:126–143

    Google Scholar 

  11. Vinay G, Wachs A, Agassant J-F (2005) Numerical simulation of non-isothermal viscoplastic waxy crude oil flows. J Non-Newton Fluid Mech 128:144–162

    Article  MATH  Google Scholar 

  12. Putz A, Frigaard IA (2010) Creeping flow around particles in a Bingham fluid. J Non-Newton Fluid Mech 165:263–280

    Article  MATH  Google Scholar 

  13. Yu Z, Wachs A (2007) A fictitious domain method for dynamic simulation of particle sedimentation in Bingham fluids. J Non-Newton Fluid Mech 145:78–91

    Article  MATH  Google Scholar 

  14. Li CH (1991) Numerical solution of Navier-Stokes equations by operator splitting. In: Proceedings sixth international conference in Australia on finite element methods, vol 1. pp 205–214

    Google Scholar 

  15. Glowinski R, Pironneau O (1992) Finite element methods for Navier-Stokes equations. Ann Rev Fluid Mech 24:167–204

    Article  MathSciNet  Google Scholar 

  16. Perera MGN, Walters K (1977) Long-range memory effects in flows involving abrupt changes in geometry. Part I : flows associated with L-shaped and T-shaped geometries. J Non-Newton Fluid Mech 2:49–81

    Article  MATH  Google Scholar 

  17. Sánchez FJ (1998) Application of a first-order operator splitting method to Bingham fluid flow simulation. Comp Math Appl 36:71–86

    Article  MATH  Google Scholar 

  18. Dean EJ, Glowinski R (2002) Operator-splitting methods for the simulation of Bingham visco-plastic flow. Chin Ann Math 23B:187–204

    Article  MathSciNet  Google Scholar 

  19. Duvaut G, Lions JL (1972) Transfert de chaleur dans un fluide de Bingham dont la viscosité dépend de la température. J Funct Anal 11:93–110

    Article  MATH  MathSciNet  Google Scholar 

  20. Kato Y (1992) On a Bingham fluid whose viscosity and yield limit depend on the temperature. Nagoya Math J 128:1–14

    MATH  MathSciNet  Google Scholar 

  21. Li C-H, Glowinski R (1996) Modelling and numerical simulation of low-mach-number compressible flows. Int J Numer Methods Fluids 23:77–103

    Article  MATH  MathSciNet  Google Scholar 

  22. Duvaut G, Lions JL (1972) Les Inéquations en Physique et en Mécanique. Dunod, Paris

    MATH  Google Scholar 

  23. Duvaut G, Lions JL (1976) Inequalities in mechanics and physics. Springer, New York

    Book  MATH  Google Scholar 

  24. Huilgol RR, You Z (2009) Prolegomena to variational inequalities and numerical schemes for compressible viscoplastic fluids. J Non-Newton Fluid Mech 158:113–126

    Article  MATH  Google Scholar 

  25. Huilgol RR, Kefayati GHR (2014) Natural convection problem in a Bingham fluid using the operator-splitting method. J Non-Newton Fluid Mech. doi:10.1016/j.jnnfm.2014.06.005

  26. Vola D, Boscardin L, Latché JC (2003) Laminar unsteady flows of Bingham fluids: a numerical strategy and some benchmark results. J Comp Phys 187:441–456

    Article  MATH  Google Scholar 

  27. Lions P-L (1996) Mathematical topics in fluid mechanics, vol 1. Incompressible models, Clarendon Press, Oxford

    Google Scholar 

  28. Lions P-L (1998) Mathematical topics in fluid mechanics, vol 2. Compressible models, Clarendon Press, Oxford

    Google Scholar 

  29. Vinay G, Wachs A, Agassant J-F (2006) Numerical simulation of weakly compressible Bingham flows: the restart of pipeline flows of waxy crude oils. J Non-Newton Fluid Mech 136:93–105

    Google Scholar 

  30. Frigaard I, Vinay G, Wachs A (2007) Compressible displacement of waxy crude oils in long pipeline startup flows. J Non-Newton Fluid Mech 147:45–64

    Article  MATH  Google Scholar 

  31. Horibata Y (1992) Numerical simulation of low-Mach-number flow with a large temperature variation. Comp Fluids 21:185–200

    Article  MATH  Google Scholar 

  32. Frigaard IA, Nouar C (2005) On the usage of viscosity regularisation methods for visco-plastic fluid flow computation. J Non-Newton Fluid Mech 127:1–26

    Article  MATH  Google Scholar 

  33. Chatzimina M, Georgiou GC, Argyropaidas I, Mitsoulis E, Huilgol RR (2005) Cessation of Couette and Poiseuille flows of a Bingham plastic and finite stopping times. J Non-Newton Fluid Mech 129:117–127

    Article  MATH  Google Scholar 

  34. Dimakopoulos Y, Pavlidis M, Tsamopoulos J (2013) Steady bubble rise in Herschel-Bulkley fluids and comparison of predictions via the Augmented Lagrangian Method with those via the Papanastasiou model. J Non-Newton Fluid Mech 200:34–51

    Article  Google Scholar 

  35. Beris AN, Tsamopoulos JA, Armstrong RC, Brown RA (1985) Creeping motion of a sphere through Bingham plastic. J Fluid Mech 158:219–244

    Article  MATH  MathSciNet  Google Scholar 

  36. Blackery J, Mitsoulis E (1997) Creeping motion of a sphere in tubes filled with a Bingham plastic material. J Non-Newton Fluid Mech 70:59–77

    Article  Google Scholar 

  37. Beaulne M, Mitsoulis E (1997) Creeping motion of a sphere in tubes filled with Herschel-Bulkley fluids. J Non-Newton Fluid Mech 72:55–71

    Article  Google Scholar 

  38. Patel SA, Chhabra RP (2014) Steady flow of Bingham plastic fluids past an elliptical cylinder. J Non-Newton Fluid Mech 202:32–53

    Article  Google Scholar 

  39. Mitsoulis E, Huilgol RR (2004) Entry flows of Bingham plastics in expansions. J Non-Newton Fluid Mech 122:45–54

    Article  MATH  Google Scholar 

  40. Mitsoulis E (2007) Annular extrudate swell of pseudoplastic and viscoplastic fluids. J Non-Newton Fluid Mech 141:138–147

    Article  MATH  Google Scholar 

  41. Mitsoulis E (2010) Fountain flow of pseudoplastic and viscoplastic fluids. J Non-Newton Fluid Mech 165:45–55

    Article  MATH  Google Scholar 

  42. Chatzimina M, Xenopohontos C, Georgiou G, Argyropaidas A, Mitsoulis E (2007) Cessation of annular Poiseuille flows of Bingham plastics. J Non-Newton Fluid Mech 142:135–142

    Article  MATH  Google Scholar 

  43. 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. J Fluid Mech 601:123–164

    Article  MATH  MathSciNet  Google Scholar 

  44. Turan O, Chakraborty N, Poole RJ (2010) Laminar natural convection of Bingham fluids in a square enclosure with differentially heated side walls. J Non-Newton Fluid Mech 165:901–913

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Flinders Mathematical Sciences Laboratory, School of Computer Science, Engineering and Mathematics, Flinders University of South Australia, Adelaide, Australia

    Raja R. Huilgol

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  1. Raja R. Huilgol
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Correspondence to Raja R. Huilgol .

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Huilgol, R.R. (2015). Numerical Modelling. In: Fluid Mechanics of Viscoplasticity. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45617-0_10

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  • DOI: https://doi.org/10.1007/978-3-662-45617-0_10

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