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New concept about existence of Hartmann boundary layer in peristalsis through curved channel-asymptotic solution

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Abstract

The main objective of this paper is to investigate boundary layer character of the velocity in peristaltic flow of a Sisko fluid in a curved channel under the influence of strong imposed radial magnetic field. The Sisko fluid model falls in the category of generalized Newtonian fluid models. The constitutive equation of Sisko model is described in terms of three material constants namely; power-law index (n), infinite shear rate viscosity (a) and consistency index (b). This model is capable of predicting shear-thinning and shear-thickening effects for n < 1 and n > 1, respectively. The equation governing the flow is first derived under the assumptions of long wavelength and low Reynolds number, and then made dimensionless by defining appropriate parameters. In dimensionless form it contains three dimensionless parameters namely; generalized ratio of infinite-shear rate viscosity to consistency index, power-law index and Hartmann number characterizing strength of the imposed magnetic field. It is found that the governing equation of flow becomes singular for large values of Hartmann number. Asymptotic solutions representing flow velocity at large values of Hartmann number are reported for two specific values of power-law index (namely n = 1 and n = 1/2) using singular perturbation technique. The flow velocity in either case exhibits qualitatively similar behavior. In fact, it exhibits boundary layer character i.e., it varies sharply in thin layer near the walls and varies linearly over rest of the cross-sections. This is contrary to what that is observed for flow velocity in straight channel (where except in thin layer near the channel walls the velocity over rest of the cross-section is uniform). The estimates of boundary layer thickness at upper and lower walls in either case are different. Moreover, the boundary layer thickness in either case is found to be inversely proportional to the Hartmann number.

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Acknowledgments

The second author is grateful to the Higher Education Commission (HEC) Pakistan for award of indigenous scholarship for his Ph.D. studies. We further thank the anonymous reviewer for his useful suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan

    N. Ali & K. Javid

  2. Theoretical Physics Division, PINSTECH, P.O. Nilore, Islamabad, 44000, Pakistan

    M. Sajid

  3. Department of Mathematics, Quaid-I-Azam University, Islamabad, 44000, Pakistan

    T. Hayat

  4. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21598, Saudi Arabia

    T. Hayat

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  1. N. Ali
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  4. T. Hayat
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Correspondence to K. Javid.

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Ali, N., Javid, K., Sajid, M. et al. New concept about existence of Hartmann boundary layer in peristalsis through curved channel-asymptotic solution. Meccanica 51, 1783–1795 (2016). https://doi.org/10.1007/s11012-015-0346-2

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  • DOI: https://doi.org/10.1007/s11012-015-0346-2

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