## THERMAL SCIENCE

International Scientific Journal

### 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 amplitude φ on the pressure difference and friction force across one wavelength are discussed numerically and with the help of illustrations.

**KEYWORDS**

PAPER SUBMITTED: 2009-09-24

PAPER REVISED: 2010-02-14

PAPER ACCEPTED: 2010-04-14

**THERMAL SCIENCE** YEAR

**2011**, VOLUME

**15**, ISSUE

**2**, PAGES [501 - 515]

- Latham, T.W., Fluid Motion in a Peristaltic Pump, M.Sc. Thesis, MIT, Cambridge, 1966
- Burns, J.C., Parkers, T., Peristaltic Motion, J. Fluid Mech., 29 (1970), pp. 731-743
- 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
- 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
- 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
- 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
- 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
- Tsiklauri, D., Beresnev, I., Non-Newtonian effects in the peristaltic flow of a Maxwell fluid, Physical Review E, 64 (2001), 036303
- 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
- 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
- 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
- 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
- Hayat, T., Ali, N., Asghar, S., An analysis of peristaltic transport for flow of a Jeffrey fluid, Acta Mechanica, 193 (2007), pp. 101-112
- 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
- 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
- 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
- Friedrich, C., Relaxation and retardation functions of the Maxwell model with fractional derivatives, Rheologica Acta, 30 (1991), pp. 151-158
- 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
- Qi, H., Xu, M., Unsteady flow of viscoelastic fluid with fractional Maxwell model in channel, Mechanics Research Communications, 34 (2007), pp. 210-212
- 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
- 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
- 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
- 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
- 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
- Qi, H., Xu, M., Some unsteady unidirectional flows of a generalized Oldroyd-B fluidwith fractional derivative, Applied Math. Modelling, 33 (2009), pp. 4184-4191
- 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
- Nadeem, S., General periodic flows of fractional Oldroyd-B fluid for an edge, Physics Letters A, 368 (2007), pp. 181-187
- 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
- Liao, S.J., Homotopy analysis method: A new analytic method for nonlinear problems, Applied Mathematics and Mechanics, 19 (1998), pp. 957-962
- Liao, S.J., On the proposed homotopy analysis technique for nonlinear problems and its applications, Ph.D. Dissertation, S. Jiao Tong University, Shanghai, 1992
- 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)
- Das, S., Gupta, P.K., Homotopy Analysis Method for Solving Fractional Hyperbolic Partial Differential Equations, International Journal of Computer Mathematics, Accepted (2010)
- Hayat, T., Khan, M., Asghar, S., Magneto hydrodynamic flow of an Oldroyd 6-constant fluid, Applied Mathematics and Computation, 155 (2004), pp. 417-425
- 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
- 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
- Liao, S.J., Beyond perturbation: Introduction to the homotopy analysis method, Boca Raton: CRC Press, Chapman and Hall, 2003
- Abbaoui, K., Cherruault, Y., New ideas for proving convergence of decom-position methods, Comp. & Mathe. with Applications, 29 (1995), pp. 103-108