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Flow of fluids with pressure dependent viscosities in an orthogonal rheometer subject to slip boundary conditions

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Abstract

We consider slow steady flows of Navier–Stokes-like fluids with pressure dependent viscosities between rotating infinite parallel plates with Navier slip boundary conditions. We derive exact solutions which correspond to flows in orthogonal and torsional rheometers, and investigate the effect of the slip coefficient and the material parameters on the solutions. We find that even when inertial effects are ignored, vorticity boundary layers develop at the upper boundary due to the pressure dependence of the viscosity. These boundary layers diminish and eventually disappear with increased slippage.

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References

  1. Stokes GG (1845) On the theories of the internal friction of fluids, and of the equilibrium and motion of elastic solids. Trans Camb Phil Soc 8:287–305

    Google Scholar 

  2. Andrade EC (1930) Viscosity of liquids. Nature 125:309–310.

    Article  MATH  ADS  Google Scholar 

  3. Bridgman PW (1931) The physics of high pressure. MacMillan, New York

    Google Scholar 

  4. Rajagopal KR (2006) On implicit constitutive theories for fluids. J Fluid Mech 550:243–249

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Dowson D, Higginson GR (1966) Elastohydrodynamic lubrication, the fundamentals of roller and gear lubrication. Pergamon, Elmsford

    Google Scholar 

  6. Szeri AZ (1998) Fluid film lubrication: theory and design. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  7. Hamrock BJ, Schmid SR, Jacobson BO (2004) Fundamentals of fluid film lubrication, 2nd edn. Marcel Dekker, New York

    Google Scholar 

  8. Renardy M (1986) Some remarks on the Navier–Stokes equations with a pressure-dependent viscosity. Commun PDEs 11:779–793

    MATH  MathSciNet  Google Scholar 

  9. Gazzola F (1997) A note on the evolution Navier–Stokes equations with pressure-dependent viscosity. Z Angew Math Phys 48:760–773

    Article  MATH  MathSciNet  Google Scholar 

  10. Gazzola F, Secchi P (1998) Some results about stationary Navier–Stokes equations with a pressure-dependent viscosity. In: Salvi R (ed) Navier–Stokes equations: theory and numerical methods, Varenna, 1997. Pitman Res Notes Math Ser, vol 388. Longman, Harlow, pp 31–37

    Google Scholar 

  11. Málek J, Nec̆as J, Rajagopal KR (2002) Global analysis of the flows of fluids with pressure-dependent viscosities. Arch Rational Mech Anal 165:243–269

    Article  MATH  ADS  Google Scholar 

  12. Málek J, Nec̆as J, Rajagopal KR (2002) Global existence of solutions for flows of fluids with pressure and shear dependent viscosities. Appl Math Lett 15:961–967

    Article  MATH  MathSciNet  Google Scholar 

  13. Málek J, Nec̆as J, Rajagopal KR (2003) Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure- and shear-dependent viscosities. Math Comput Simulation 61:297–315

    Article  MATH  MathSciNet  Google Scholar 

  14. Franta M, Málek J, Rajagopal KR (2005) On steady flows of fluids with pressure- and shear-dependent viscosities. Proc R Soc Lond Ser A 461:651–670

    Article  MATH  ADS  Google Scholar 

  15. Málek J, Rajagopal KR (2007) Incompressible rate type fluids with pressure and shear-rate dependent material moduli. Nonlin Anal Real World Appl 8:156–164

    Article  MATH  Google Scholar 

  16. Bulíček M, Málek J, Rajagopal KR (2007) Navier’s slip and evolutionary Navier–Stokes-like systems with pressure and shear-rate dependent viscosity. Indiana Univ Math J 56:51–85

    Article  MATH  MathSciNet  Google Scholar 

  17. Hron J, Málek J, Rajagopal KR (2001) Simple flows of fluids with pressure-dependent viscosities. Proc R Soc Lond Ser A 457:1603–1622

    Article  MATH  ADS  Google Scholar 

  18. Rajagopal KR (2004) Couette flows of fluids with pressure dependent viscosity. Int J Appl Mech Eng 9:573–585

    Google Scholar 

  19. Rajagopal KR, Kannan K (2004) Flows of a fluid with pressure dependent viscosities between rotating parallel plates. In: Fergola P et al. (eds) New trends in mathematical physics. World Scientific, Hackensack, pp 172–183

    Google Scholar 

  20. Vasudevaiah M, Rajagopal KR (2005) On fully developed flows of fluids with a pressure dependent viscosity in a pipe. Appl Math 50:341–353

    Article  MATH  MathSciNet  Google Scholar 

  21. Prasad SC, Rajagopal KR (2006) Flow of a fluid with pressure dependent viscosity due to a boundary that is being stretched. Appl Math Comput 173:50–68

    Article  MATH  MathSciNet  Google Scholar 

  22. Rajagopal KR, Saccomandi G (2006) Unsteady exact solution for flows with pressure-dependent viscosities. Math Proc Roy Irish Acad 106A:115–130

    Article  MATH  MathSciNet  Google Scholar 

  23. Málek J, Rajagopal KR (2007) Mathematical properties of the solutions to the equations governing the flow of fluids with pressure and shear dependent viscosities. In: Friedlander S, Serre D (eds) Handbook of mathematical fluid dynamics, vol 4. North-Holland, Amsterdam, pp 407–444

    Chapter  Google Scholar 

  24. Maxwell B, Chartoff RP (1965) Studies of a polymer melt in an orthogonal rheometer. Trans Soc Rheol 9:41–52

    Article  Google Scholar 

  25. Rajagopal KR (1992) Flow of viscoelastic fluids between rotating disks. Theor Comp Fluid Dynam 3:185–206

    Article  MATH  Google Scholar 

  26. Sirivat A, Rajagopal KR, Szeri AZ (1988) An experimental investigation of the flow of non-Newtonian fluids between rotating disks. J Fluid Mech 186:243–256

    Article  ADS  Google Scholar 

  27. Baek S, Rajagopal KR, Srinivasa AR (2001) Measurements related to the flow of a granular material in a torsional rheometer. Part Sci Tech 19:175–186

    Article  Google Scholar 

  28. Le Roux C (1999) Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions. Arch Rational Mech Anal 148:309–356

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. Navier CLMH (1823) Mémoire sur les lois du mouvement des fluides. Mémoires de L’Académie des Sciences de L’Institut de France 6:389–440

    Google Scholar 

  30. Le Roux C (2005) Steady Stokes flows with threshold slip boundary conditions. Math Models Methods Appl Sci 15:1141–1168

    Article  MATH  MathSciNet  Google Scholar 

  31. Berker R (1963) Intégration de équations du mouvement d’un fluide visqueux incompressible. Handbuch der Physik VIII/2. Springer, Berlin

    Google Scholar 

  32. Berker R (1979) A new solution of the Navier–Stokes equation for the motion of a fluid contained between two parallel planes rotating about the same axis. Arch Mech Stos 31:265–280

    MATH  MathSciNet  Google Scholar 

  33. Huilgol RR (1969) On the properties of the motion with constant stretch history occurring in the Maxwell rheometer. Trans Soc Rheol 13:513–526

    Article  Google Scholar 

  34. Huilgol RR (1971) A class of motions with constant stretch history. Quart Appl Math 29:1–15

    MATH  Google Scholar 

  35. Truesdell C, Rajagopal KR (2000) An introduction to the mechanics of fluids. Birkhäuser, Boston

    MATH  Google Scholar 

  36. Noll W (1962) Motions with constant stretch history. Arch Rational Mech Anal 11:97–105

    Article  MATH  ADS  MathSciNet  Google Scholar 

  37. Rajagopal KR (1982) On the flow of a simple fluid in an orthogonal rheometer. Arch Rational Mech Anal 79:39–47

    Article  MATH  ADS  MathSciNet  Google Scholar 

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  1. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa

    C. Le Roux

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  1. C. Le Roux
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Le Roux, C. Flow of fluids with pressure dependent viscosities in an orthogonal rheometer subject to slip boundary conditions. Meccanica 44, 71–83 (2009). https://doi.org/10.1007/s11012-008-9151-5

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  • DOI: https://doi.org/10.1007/s11012-008-9151-5

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