INFLUENCE OF SLIP CONDITION ON PERISTALTIC TRANSPORT OF A VISCOELASTIC FLUID WITH FRACTIONAL BURGER'S MODEL

Abstract

The investigation is to explore the transportation of a viscoelastic fluid with fractional Burgers' model by peristalsis through a channel under the influence of wall slip condition. This analysis has been carried out under the assumption of long wavelength and low Reynolds number. An approximate analytical solution of the problem is obtained by using Homotopy Analysis method (HAM). It is assumed that the cross-section of the channel varies sinusoidally along the length of channel. The expressions for axial velocity, volume flow rate and pressure gradient are obtained. The effects of fractional parameters α and β, material constants λ1, λ2, λ3, slip parameter k and Φ amplitude on the pressure difference and friction force across one wavelength are discussed numerically and with the help of illustrations.

Dates

  • Submission Date2009-09-24
  • Revision Date2010-02-14
  • Acceptance Date2010-04-14

DOI Reference

10.2298/TSCI1102501T

References

  1. Latham, T.W., Fluid Motion in a Peristaltic Pump, M.Sc. Thesis, MIT, Cambridge, 1966
  2. Burns, J.C., Parkers, T., Peristaltic Motion, J. Fluid Mech., 29 (1970), pp. 731-743
  3. Shapiro, A.H., Jafferin, M.Y., Weinberg, S.L., Peristaltic pumping with long wavelengths at low Reynolds number, J. Fluid Mech., 35 (1969), pp. 669-675
  4. Ebaid, A., Effects of magnetic field and wall slip conditions on the peristaltic transport of a Newtonian fluid in an asymmetric channel, Physics Letters A, 372 (2008), pp. 4493-4499
  5. Ali, N., Hussain, Q., Hayat, T., Asghar, S., Slip effects on the peristaltic transport of MHD fluid with variable viscosity, Physics Letters A, 372 (2008), pp. 1477-1489
  6. Hayat, T., Qureshi, M.U., Ali, N., The influence of slip on the peristaltic motion of a third order fluid in an asymmetric channel, Physics Letters A, 372 (2008), pp. 2653-2664
  7. El-Shehawy, E.F., El-Dabe, N.T., El-Desoki, I.M., Slip effects on the peristaltic flow of a non-Newtonian Maxwellian fluid, Acta Mechanica, 186 (2006), pp. 141-159
  8. Tsiklauri, D., Beresnev, I., Non-Newtonian effects in the peristaltic flow of a Maxwell fluid, Physical Review E, 64 (2001), 036303
  9. Hayat, T., Ali, N., Asghar, S., Hall effects on the peristaltic flow of a Maxwell fluid in a porous medium, Physics Letter A, 363 (2007), pp. 397-403
  10. Ali, N., Hayat, T., Asghar, S., Peristaltic flow of a Maxwell fluid in a channel with compliant walls, Chaos, Solitons and Fractals, 39 (2009), pp. 407-416
  11. Hayat, T., Ali, N., Peristaltic motion of a Jeffrey fluid under the effect of a magnetic field in a tube, Communication in Nonlinear Science and Numerical Simulation, 13 (2008), pp. 1343-1352
  12. Hayat, T., Ali, N., Asghar, S., Siddiqui, A.M., Exact peristaltic flow in tubes with an endoscope, Applied Mathematics and Computation, 182 (2006), pp. 359-368
  13. Hayat, T., Ali, N., Asghar, S., An analysis of peristaltic transport for flow of a Jeffrey fluid, Acta Mechanica, 193 (2007), pp. 101-112
  14. Hayat, T., Nadeem, S., Asghar, S., Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model, Applied Mathematics and Computation, 151 (2004), pp. 153-161
  15. Wenchang, T., Wenxiao, P., Mingyu, X., A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates, Int. J. of Non-Linear Mechanics, 38 (2003), pp. 645-650
  16. Wenchang, T., Mingyu, X., Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model, Acta Mechanica Sinica, 18 (2002), pp. 342-349
  17. Friedrich, C., Relaxation and retardation functions of the Maxwell model with fractional derivatives, Rheologica Acta, 30 (1991), pp. 151-158
  18. Qi, H., Jin, H., Unsteady rotating flows of a viscoelastic fluid with the fractional Maxwell model between coaxial cylinders, Acta Mechanica Sinica, 22 (2006), pp. 301-305
  19. Qi, H., Xu, M., Unsteady flow of viscoelastic fluid with fractional Maxwell model in channel, Mechanics Research Communications, 34 (2007), pp. 210-212
  20. Khan, M., Ali, S.H., Fetecau, C., Qi, H., Decay of potential vortex for a viscoelastic fluid with fractional Maxwell model, Applied Mathematical Modelling, 33 (2009), pp. 2526-2533
  21. Vieru, D., Fetecau, C., Fetecau, C., Flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate, Applied Mathematics and Computation, 200 (2008), pp. 459-464
  22. Mahmood, A., Parveen, S., Ara, A., Khan, N.A., Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model, Communication in Nonlinear Science and Numerical Simulation, 14 (2009), pp. 3309-3319
  23. Wang, S., Xu, M., Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus, Nonlinear Analysis: Real World Applications, 10 (2009), pp. 1087-1096
  24. Khan, M., Ali, S.H., Qi, H., On accelerated flows of a viscoelastic fluid with the fractional Burgers' model, Nonlinear Analysis: Real World Applications, 10 (2009), pp. 2286-2296
  25. Qi, H., Xu, M., Some unsteady unidirectional flows of a generalized Oldroyd-B fluidwith fractional derivative, Applied Math. Modelling, 33 (2009), pp. 4184-4191
  26. Qi, H., Xu, M., Stokes' first problem for a viscoelastic fluid with the generalized Oldroyd-B model, Acta Mech Sin, 23 (2007), pp. 463-469
  27. Nadeem, S., General periodic flows of fractional Oldroyd-B fluid for an edge, Physics Letters A, 368 (2007), pp. 181-187
  28. Hayat, T., Khan, M., Asghar, S., On the MHD flow of fractional generalized Burgers' fluid with modified Darcy's law, Acta Mech Sin, 23 (2007), pp. 257-261
  29. Liao, S.J., Homotopy analysis method: A new analytic method for nonlinear problems, Applied Mathematics and Mechanics, 19 (1998), pp. 957-962
  30. Liao, S.J., On the proposed homotopy analysis technique for nonlinear problems and its applications, Ph.D. Dissertation, S. Jiao Tong University, Shanghai, 1992
  31. Das, S., Gupta, P.K., Application of homotopy perturbation method and homotopy analysis method to fractional vibration equation, International Journal of Computer Mathematics, Accepted (2009)
  32. Das, S., Gupta, P.K., Homotopy Analysis Method for Solving Fractional Hyperbolic Partial Differential Equations, International Journal of Computer Mathematics, Accepted (2010)
  33. Hayat, T., Khan, M., Asghar, S., Magneto hydrodynamic flow of an Oldroyd 6-constant fluid, Applied Mathematics and Computation, 155 (2004), pp. 417-425
  34. Liao, S.J., An analytical solution of unsteady boundary layer flows caused by an impulsively stretching plate. Comm. in Nonlinear Sci. and Numerical Simulation, 11 (2006), pp. 326-339
  35. Wu, W., and Liao, S.J., Solving solitary waves with discontinuity by means of the homotopy analysis method, Chaos, Solitons and Fractals, 23 (2004), pp. 1733-1740
  36. Liao, S.J., Beyond perturbation: Introduction to the homotopy analysis method, Boca Raton: CRC Press, Chapman and Hall, 2003
  37. Abbaoui, K., Cherruault, Y., New ideas for proving convergence of decom-position methods, Comp. & Mathe. with Applications, 29 (1995), pp. 103-108
Volume 15, Issue 2, Pages501 -515