Partial slip effects on the peristaltic motion of an upper-convected Maxwell fluid through an irregular channel
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Partial slip effects on the peristaltic motion of an upper‑convected Maxwell fluid through an irregular channel Musharafa Saleem1,3 · Qasem M. Al‑Mdallal6 · Qasim Ali Chaudhry2,3,6 · Saima Noreen4,5 · Aun Haider1 Received: 28 October 2019 / Accepted: 9 March 2020 / Published online: 28 April 2020 © Springer Nature Switzerland AG 2020
Abstract A theoretical investigation was carried out in this paper by taking the partial slip result in an irregular wavy channel for the incompressible upper-convected Maxwell fluid. Due to peristaltic motion, asymmetric waves with different amplitudes are produced. This flow is driven in an irregular channel due to the pressure gradient, where the perturbation technique applied to tackle the stream function and the pressure gradient. A numerical integration technique was used to find out the different expressions of the frictional rise per wavelength and pressure rise per wavelength and presented their graphs. The graphical results for the partial slip parameter, small wave number, phase difference, Reynolds number, Weissenberg number, wave amplitudes a and b, and channel width d are included. The pressure gradient is an increasing function of the wave number, but the slip parameter is vice versa. The velocity profile u is increased by a small increase in the wave number while it is decreased by a rise in the slip parameter. The frictional forces have the same behavior for the lower and upper wall. According to the slip effects, the bolus has improved behavior. Moreover, the relaxation parameter enhanced the strength of the bolus. Keywords Irregular channel · Non-Newtonian fluid model · Weissenberg number · Slip parameter List of symbols a1 Upper wave amplitude (m) a2 Lower wave amplitude (m) d1 Upper channel width (m) d2 Lower channel width (m) c Wave speed (m/s) h1 Upper wall h2 Lower wall 𝐒 Extra stress tensor t ′ Transpose t̄ Time (s) A1 First Rivlin–Erickson tensor X x component in laboratory frame Y y component in laboratory frame
̄ V̄ Axial and normal components of velocity in laboU, ratory frame ̄u, v̄ Axial and normal components of velocity in wave frame (m/s) x x component in the wave frame y y component in the wave frame a Ratio of upper wave amplitude to width b Ratio of lower wave amplitude to width d Ratio of channel width Re Reynolds number Greek symbols 𝜆1 Wavelength (m) 𝜆 Weissenberg number 𝜇 Viscosity (Pa s)
* Musharafa Saleem, [email protected] | 1University of Management and Technology Lahore, Sialkot Campus, Sialkot 51310, Pakistan. 2Department of Mathematics, College of Science, University of Ha’il, Ha’il, Saudi Arabia. 3Department of Mathematics, University of Engineering and Technology, Lahore 54890, Pakistan. 4Department of Mathematics, Comsats Institute of Information Technology 45550, Tarlai Kalan Park Road, Islamabad 44000, Pakistan. 5Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang 212013, China. 6Department of Mathematical Sciences, College of Science, UAE University, Al‑Ain 15551,
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