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Analytical Approximation Techniques

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Fluid Mechanics of Viscoplasticity

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Abstract

Convened by The Institution of Mechanical Engineers, London, a committee headed by Beauchamp Tower was asked to investigate the lubrication of journal bearings.

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Notes

  1. 1.

    This equation has received much attention in the literature. For the first solution of the problem of full film lubrication in a bearing of finite length, see Tao [2]. For a solution including cavitation effects and employing a variational inequality, see Cimatti [3].

  2. 2.

    For experimental results on viscous fingering in a yield stress fluid, see [10].

  3. 3.

    The material in the remainder of this section is derived from [13], with minor modifications and additions.

  4. 4.

    An excellent introduction to linearised hydrodynamic stability theory of the flows of Newtonian fluids has been written by Lin [20].

References

  1. Reynolds O (1886) On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiments, including an experimental determination of the viscosity of olive oil. Philos Trans R Soc Lond 177:157–234

    Article  Google Scholar 

  2. Tao LN (1959) General solution of the Reynolds equation for a journal bearing of finite width. Q Appl Math 17:129–136

    MATH  Google Scholar 

  3. Cimatti G (1977) On a problem of the theory of lubrication governed by a variational inequality. Appl Math Optim 3:227–242

    Article  MATH  MathSciNet  Google Scholar 

  4. Huilgol RR (1994) Some comments on the paper by Lipscomb and Denn. Department of Mathematics and Statistics, Flinders University, Research Report

    Google Scholar 

  5. Lipscomb GG, Denn MM (1984) Flow of Bingham fluid in complex geometries. J Non-Newton Fluid Mech 14:337–346

    Article  MATH  Google Scholar 

  6. Roustaei A, Frigaard IA (2013) The occurrence of fouling layers in the flow of a yield stress fluid along a wavy-walled channel. J Non-Newton Fluid Mech 198:109–124

    Article  Google Scholar 

  7. Frigaard IA, Ryan DP (2004) Flow of a visco-plastic fluid in a channel of slowly varying width. J Non-Newton Fluid Mech 123:67–83

    Article  MATH  Google Scholar 

  8. Putz A, Frigaard IA, Martinez DM (2009) On the lubrication paradox and the use of the regularisation methods for lubrication flows. J Non-Newton Fluid Mech 163:62–77

    Article  MATH  Google Scholar 

  9. Saffman PG, Taylor GI (1958) The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc R Soc Lond A 245:312–329

    Article  MATH  MathSciNet  Google Scholar 

  10. Lindner A, Coussot P, Bonn D (2000) Viscous fingering in a yield stress fluid. Phys Rev Lett 85:314–317

    Article  Google Scholar 

  11. Pipkin AC, Tanner RI (1972) A survey of theory and experiment in viscometric flows of viscoelastic liquids. In: Nemat-Nasser S (ed), Mechanics Today, vol 1, pp 262–321

    Google Scholar 

  12. Huilgol RR, Phan-Thien N (1997) Fluid mechanics of viscoelasticity. Elsevier, Amsterdam

    Google Scholar 

  13. Huilgol RR (2006) On the derivation of the symmetric and asymmetric Hele-Shaw flow equations for viscous and viscoplastic fluids using the viscometric fluidity function. J Non-Newton Fluid Mech 138:209–213

    Article  MATH  Google Scholar 

  14. Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE (1996) On the Lambert W function. Adv Comput Math 5:329–359

    Article  MATH  MathSciNet  Google Scholar 

  15. You Z, Huilgol RR, Mitsoulis E (2008) Application of the Lambert W function to steady shearing flows of the Papanastasiou model. Int J Eng Sci 46:799–808

    Article  MATH  MathSciNet  Google Scholar 

  16. Alexandrou AN, Entov V (1997) On the steady-state advancement of fingers and bubbles in a Hele-Shaw cell filled by a non-Newtonian fluid. Eur J Appl Math 8:73–87

    MATH  MathSciNet  Google Scholar 

  17. Aronsson G, Janfalk U (1992) On Hele-Shaw flow of power-law fluids. Eur J Appl Math 3:343–366

    Article  MATH  MathSciNet  Google Scholar 

  18. Kennedy P (1995) Flow analysis of injection molds. Hanser Verlag, Munich

    Google Scholar 

  19. Zheng R, Tanner RI, Fan X-J (2011) Injection molding: integration of theory and modeling methods. Springer, Heidelberg

    Book  Google Scholar 

  20. Lin CC (1966) The theory of hydrodynamic stability. Cambridge University Press, Cambridge

    Google Scholar 

  21. Squire HB (1933) On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc R Soc Lond, Ser A 142:621–628

    Article  MATH  Google Scholar 

  22. Orszag SA (1971) Accurate solution of the Orr-Sommerfeld stability equation. J Fluid Mech 50:689–703

    Article  MATH  Google Scholar 

  23. Chapman SJ (2002) Subcritical transition in channel flows. J Fluid Mech 451:35–97

    Article  MATH  MathSciNet  Google Scholar 

  24. Frigaard IA, Howison SD, Sobey IJ (1994) On the stability of Poiseuille flow of a Bingham fluid. J Fluid Mech 263:133–150

    Article  MATH  MathSciNet  Google Scholar 

  25. Nouar C, Frigaard IA (2001) Nonlinear stability of Poiseuille flow of a Bingham fluid: theoretical results and comparison with phenomenological criteria. J Non-Newton Fluid Mech 100:127–149

    Article  MATH  Google Scholar 

  26. Frigaard I, Nouar C (2003) On three-dimensional linear stability of Poiseuille flow of Bingham fluids. Phys Fluids 15:2843–2851

    Article  MathSciNet  Google Scholar 

  27. Nouar C, Kabouya N, Dusek J, Mamou M (2007) Modal and non-modal linear stability of the plane Bingham-Poiseuille flow. J Fluid Mech 577:211–239

    Article  MATH  MathSciNet  Google Scholar 

  28. Peng J, Zhu K-Q (2004) Linear stability of Bingham fluids in spiral Couette flow. J Fluid Mech 512:21–45

    Article  MATH  Google Scholar 

  29. Landry MP, Frigaard IA, Martinez DM (2006) Stability and instability of Taylor-Couette flows of a Bingham fluid. J Fluid Mech 560:321–353

    Article  MATH  MathSciNet  Google Scholar 

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  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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Huilgol, R.R. (2015). Analytical Approximation Techniques. In: Fluid Mechanics of Viscoplasticity. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45617-0_7

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  • DOI: https://doi.org/10.1007/978-3-662-45617-0_7

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