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Research Papers

Slip Effects on the Peristaltic Flow of a Third Grade Fluid in a Circular Cylindrical Tube

[+] Author and Article Information
N. Ali1

Chair of Fluid Dynamics, Department of Mechanical Engineering, Darmstadt University of Technology, Hochschulstrasse 1, 64289 Darmstadt, Germanynasirali_qau@yahoo.com

Y. Wang, M. Oberlack

Chair of Fluid Dynamics, Department of Mechanical Engineering, Darmstadt University of Technology, Hochschulstrasse 1, 64289 Darmstadt, Germany

T. Hayat

Department of Mathematics, Quaid-i-Azam University, Islamabad, 45320, Pakistan

1

Corresponding author. Permanent address: Department of Mathematics, Faculty of Basic and Applied Sciences, International Islamic University Islamabad, Pakistan.

J. Appl. Mech 76(1), 011006 (Oct 31, 2008) (10 pages) doi:10.1115/1.2998761 History: Received December 12, 2007; Revised August 29, 2008; Published October 31, 2008

Peristaltic flow of a third grade fluid in a circular cylindrical tube is undertaken when the no-slip condition at the tube wall is no longer valid. The governing nonlinear equation together with nonlinear boundary conditions is solved analytically by means of the perturbation method for small values of the non-Newtonian parameter, the Debroah number. A numerical solution is also obtained for which no restriction is imposed on the non-Newtonian parameter involved in the governing equation and the boundary conditions. A comparison of the series solution and the numerical solution is presented. Furthermore, the effects of slip and non-Newtonian parameters on the axial velocity and stream function are discussed in detail. The salient features of pumping and trapping are discussed with particular focus on the effects of slip and non-Newtonian parameters. It is observed that an increase in the slip parameter decreases the peristaltic pumping rate for a given pressure rise. On the contrary, the peristaltic pumping rate increases with an increase in the slip parameter for a given pressure drop (copumping). The size of the trapped bolus decreases and finally vanishes for large values of the slip parameter.

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Figures

Grahic Jump Location
Figure 1

Plot showing profiles for the stream function ψ(r) (left panels) and axial velocity w(r) (right panels) for different values of F and γ and Γ. Solid lines indicate the numerical solution while dotted lines indicate the perturbation solution of the problem. (a) and (b) correspond to F=−0.2, γ=0.1, and Γ=0.1; (c) and (d) correspond to F=−0.2, γ=0.1, and Γ=0.2; (e) and (f) correspond to F=−0.2, γ=0.01, and Γ=0.2; (g) and (h) correspond to F=−1.5, γ=0.1, and Γ=0.2. The other parameters chosen are z=π/2 and a=0.2.

Grahic Jump Location
Figure 2

Transverse profiles of axial velocity w(r) (left panels) and stream function ψ(r) (right panels) for different values of Γ(γ=0.05) ((a) and (b)) and γ(Γ=0.2) ((c) and (d)). The other parameters chosen are z=−π, F=−0.8, and a=0.2.

Grahic Jump Location
Figure 3

Axial distribution of axial pressure gradient dp/dz within a wavelength z∊[−π,π] for various values of Γ(γ=0.1) (left panels) and γ(Γ=0.1) (right panels) for three flow rates F=−0.8(Θ=−0.29) ((a) and (b)), F=−0.5(Θ=0.01) ((c) and (d)), and F=−0.2(Θ=0.31) ((e) and (f)). The value of ϕ is chosen to be equal to 0.2.

Grahic Jump Location
Figure 4

Pressure rise per wavelength Δp versus flow rate Θ for various values of Γ(γ=0.05) (a) and γ(Γ=0.05) (b). Here ϕ is chosen to be equal to 0.2.

Grahic Jump Location
Figure 5

Streamlines for different values of Γ. (a)–(h) correspond to the values of Γ=(0,0.02,0.06,0.1,1,5,10,20), respectively. The other parameters chosen are γ=0.0, ϕ=0.2, and Θ=0.31.

Grahic Jump Location
Figure 6

Streamlines for different values of γ. (a)–(f) correspond to the values of γ=(0,0.02,0.05,0.08,0.1,0.5), respectively. The other parameter chosen are Γ=0.1, ϕ=0.2, and Θ=0.31.

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