INTEGRAL-BALANCE SOLUTION TO THE STOKES' FIRST PROBLEM OF A VISCOELASTIC GENERALIZED SECOND GRADE FLUID

Abstract

Integral balance solution employing entire domain approximation and the penetration dept concept to the Stokes' first problem of a viscoelastic generalized second grade fluid has been developed. The solution has been performed by a parabolic profile with an unspecified exponent allowing optimization through minimization of the 2 L norm over the domain of the penetration depth. The closed form solution explicitly defines two dimensionless similarity variables ξ = y/sqrt(νt) and D0 = Χ2 = sqrt(p/νtΒ), responsible for the viscous and the elastic responses of the fluid to the step jump at the boundary. The solution was developed with three forms of the governing equation through its two dimensional forms (the main solution and example 1) and the dimensionless version showing various sides of the flow field and how the dimensionless groups control it: mainly the effect of the Deborah number. Numerical simulations demonstrating the effect of the various operating parameter and fluid properties on the developed flow filed have been performed.

Keywords

Dates

  • Submission Date2011-04-01
  • Revision Date2011-06-30
  • Acceptance Date2011-07-18

DOI Reference

10.2298/TSCI110401077H

References

  1. Naraim, A., Joseph, D.D., Remarks about the interpretations of impulsive experiments in shear flows of viscoplastic liquids, Rheologica Acta, 22 (1983), 6, pp.528-538
  2. Labsi , N. Benkah;a, Y.K., Boutra, A., Brunier, E. , Simultaneous hydrodynamic and thermal flow development of a thermodependent viscoplastic fluid, Int. Rev. Chem. Eng. 2 (2010),1,pp. 31-39
  3. Tan, W., Xu, M., The impulsive motion of a flat plate in a generalized second grade fluid, Mech. Res. Com. 29 (2002a), 1, pp.3-9
  4. Tan, W., Xu, M., Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model, Acta Mech. Sin., 18 (2002b), 4, pp.342-349
  5. Jordan, P.M., A note on start-up, plane Couette flow involving second-grade fluids, Mathematical Problems in Engineering, 2005 (2005), 5, pp.539-545. doi: 10.1155/MPE.2005.539
  6. Momoniat, E., A point source solution for unidirectional flow of a viscoelastic fluid, Physics Letters A, 372 (2008), 22, pp.4041-4044. doi: 10.1016/j.physleta.2008.03.020
  7. Bandelli, R., Rajagopal, K. R., Start-up flows of second grade fluids in domains with one finite dimension, Int. J. Non-Linear Mechanics, 30 (1995), 6, pp. 817-839
  8. Derkach, S.R., Rheology on the Way from Dilute to Concentrated Emulsions, Int. Rev. Chem. Eng. 2 (2009),3, pp.465-472
  9. Siginer, D.A., Letelier,M.F., Laminar flow of non-linear viscoelastic fluids in straight tubes of arbitrary contour , Int J Heat Mass Trans,54 (2011), 9-10,pp.2188-2202. doi: 10.1016/j.ijheatmasstransfer.2010.11.041
  10. Pirkle Jr., J.C. , Braatz , R.D. Instabilities and multiplicities in non-isothermal blown film extrusion including the effects of crystallization, J Proc Cont, 21 (2011), 3, pp.405-414. doi: 10.1016/j.jprocont.2010.12.007
  11. Hsiao, K.L., Manufacturing Extrusion Process for Forced Convection Micropolar Fluids Flow with Magnetic Effect over a Stretching Sheet, Int. Rev. Chem. Eng. 1 (2009),3, pp. 272-276
  12. Hsiao, K.L., Manufacturing Extrusion Process for Magnetic Mixed Convection of an Incompressible Viscoelastic Fluid over a Stretching Sheet , Int. Rev. Chem. Eng. 1 (2009),2, pp.164-169
  13. Karimi S.,Dabir, B., Dadvar, M. , Non-Newtonian Effect of Blood in Physiologically Realistic Pulsatile Flow , Int. Rev. Chem. Eng. 2 (2009),7, pp. 805-810
  14. Schmitt, C., Henni, A.H., Cloutier, G. , Characterization of blood clot viscoelasticity by dynamic ultrasound elastography and modeling of the rheological behavior, J. Biomech.,44 (2011),4,pp.622-629. doi: 10.1016/j.jbiomech.2010.11.015
  15. Aiboud, S.,Saouli, S. , Thermodynamic Analysis of Viscoelastic Magnetohydrodynamic Flow over a Stretching Surface with Heat and Mass Transfer, Int. Rev. Chem. Eng. 3 (2009),3, pp.315-324
  16. Pfitzenreiter, T., A physical basis for fractional derivatives in constitutive equations, ZAMM, 84 (2004),4, pp. 284-287. doi 10.1002/zamm.200310112
  17. Kang, J., Xu, M., Exact solutions for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit, Acta Mech. Sin., 25 (2009), 2, pp. 181-186. doi 10.1007/s10409-008-0209-3
  18. Qi , H., Xu, M. , Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative, Appl. Math. Model., 33 (2009),11,pp.4184-4191. doi: 10.1016/j.apm.2009.03.002
  19. Hayat, T., Asghar, S., Siddiqui, A.M., Some unsteady unidirectional flows of a non-Newtonian fluid, Int. J. Eng. Sci, 38 (2000), 3, pp.337-346
  20. Tan, W., Xu, M. , Unsteady flows of a generalized second grade fluid with the fractional derivative model between two parallel plates, Acta Mech. Sin., 20 (2004), 5, pp. 471-476
  21. Tan W., Xian F., Wei L., An exact solution of unsteady Couette flow of generalized second grade fluid , Chinese Science Bulletin , 47 (2002), 21, pp.1783-1785
  22. Goodman, T.R., The heat balance integral and its application to problems involving a change of phase, Transactions of ASME, 80 (1958), 1-2, pp. 335-342
  23. Goodman T.R., Application of Integral Methods to Transient Nonlinear Heat Transfer, Advances in Heat Transfer, T. F. Irvine and J. P. Hartnett, eds., 1 (1964), Academic Press, San Diego, CA, pp. 51-122
  24. Hristov, J., The heat-balance integral method by a parabolic profile with unspecified exponent: Analysis and benchmark exercises, Thermal Science, 13 (2009), 2, pp.22-48
  25. Hristov, J., Research note on a parabolic heat-balance integral method with unspecified exponent: an entropy generation approach in optimal profile determination, Thermal Science, (2) (2009), 2, pp.27-48, doi: 10.2298/TSCI0902049H
  26. Muzychka Y.S., Yovanovich, M.M., Unsteady viscous flows and Stokes' first problem, Int. J. Therm.Sci 49 (2010),5, pp. 820-828. doi: 10.1016/j.ijthermalsci.2009.11.013
  27. Hristov J., Heat-Balance Integral to Fractional (Half-Time) Heat Diffusion Sub-Model, Thermal Science, 14 (2010), 2, pp. 291-316. doi: 10.2298/TSCI1002291H
  28. Hristov J., A Short-Distance Integral-Balance Solution to a Strong Subdiffusion Equation: A Weak Power-Law Profile, Int. Rev. Chem. Eng., 2 (2010), 5, pp. 555-563
  29. Hristov J., Approximate Solutions to Fractional Subdiffusion Equations: The heat-balance integral method, The European Physical Journal-Special Topics, 193(2011), 1, pp.229-243. doi: 10.1140/epjst/e2011-01394-2
  30. Hristov J., Starting radial subdiffusion from a central point through a diverging medium (a sphere): Heat-balance Integral Method, Thermal Science, 15 (2011), Supl. 1, pp.S5-S20 . doi: 10.2298/TSCI1101S5H
  31. Carslaw, H.S., Jaeger, J.C. (1986), Conduction of Heat in Solids, 2nd Ed., p. 230., Clarendon Press, Oxford, UK
  32. Hristov, J. The Heat-Balance Integral: 1. How to Calibrate the Parabolic Profile?, CR Mechanique (in Press)
  33. Myers, T.G., Optimal exponent heat balance and refined integral methods applied to Stefan problem, Int. J. Heat Mass Transfer, 53 (2010), 5-6, pp. 1119-1127
  34. Hristov, J. The Heat-Balance Integral: 2. A Parabolic profile with a variable exponent: the concept and numerical experiments, CR Mechanique (in Press)
  35. Langford D., The heat balance integral method, Int. J. Heat Mass Transfer, 16 (1973), 12, pp. 2424-2428
  36. Goodwin, J.W., Hughes, R.W., Rheology for chemists: An Introduction, 2nd ed, RSC Publishing, Cambridge, UK., 2008
  37. Pan, Y., Zhu, K.Q., Thermodynamic compatibility, Science China-Physics Mechanics & Astronomy, 54(2011), 4, pp.737-742. doi: 10.1007/s11433-011-4271-7
Volume 16, Issue 2, Pages395 -410