# American Institute of Mathematical Sciences

June  2019, 12(3): 645-664. doi: 10.3934/dcdss.2019041

## Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel

 1 Department of Mathematics, School of Science, University of Management and Technology, C-Ⅱ Johar Town, Lahore 54770, Pakistan 2 Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, 9301, Bloemfontein, South Africa

* Corresponding author: abdonatangana@yahoo.fr, atanganaA@ufs.ac.za

Received  March 2017 Revised  July 2017 Published  September 2018

Couette flows of an incompressible viscous fluid with non-integer order derivative without singular kernel produced by the motion of a flat plate are analyzed under the slip condition at boundaries. An analytical transform approach is used to obtain the exact expressions for velocity and shear stress. Three particular cases from the general results with and without slip at the wall are obtained. These solutions, which are organized in simple forms in terms of exponential and trigonometric functions, can be conveniently engaged to obtain known solutions from the literature. The control of the new non-integer order derivative on the velocity of the fluid moreover a comparative study with an older model, is analyzed for some flows with practical applications. The non-integer order derivative with non-singular kernel is more appropriate for handling mathematical calculations of obtained solutions.

Citation: Muhammad Bilal Riaz, Naseer Ahmad Asif, Abdon Atangana, Muhammad Imran Asjad. Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 645-664. doi: 10.3934/dcdss.2019041
##### References:

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##### References:
Geometry of flow
Velocity profiles (VP) varus $y$, with $\beta = 0.4$, $t = 0.2$ for different values of $\alpha$ and when translation of the plate with a constant velocity $(g(t) = H(t))$
VP varus $y$, with $\beta = 0.4$, $t = 0.4$ for different values of $\alpha$ and $g(t) = H(t)$
VP varus $y$, with $\beta=0.4$, $t=0.8$ for different values of $\alpha$ and $g(t)=H(t)$
VP varus $y$, with $\beta = 0.7$, $t = 0.2$ for different values of $\alpha$ and $g(t) = H(t)$
VP varus $y$, with $\alpha = 0.3$, $t = 0.2$ for different values of $\beta$ and $g(t) = H(t)$
VP varus $y$, with $\alpha = 0.6$, $t = 0.2$ for different values of $\beta$ and $g(t) = H(t)$
VP varus $y$, with $\alpha = 0.9$, $t = 0.2$ for different values of $\beta$ and $g(t) = H(t)$
VP varus $t$, with $\beta = 0.0$(no-slip), $y = 0.2$ for different values of $\alpha$ and $g(t) = H(t)$
VP varus $t$, with $\beta = 0.4$, $y = 0.2$ for different values of $\alpha$ and $g(t) = H(t)$
VP varus $t$, with $\beta = 0.7$, $y = 0.2$ for different values of $\alpha$ and $g(t) = H(t)$
VP varus $y$, with $\beta = 0.4$, $t = 0.2$ for different values of $\alpha$ and when translation of the plate with a constant acceleration $(g(t) = t)$
VP varus $y$, with $\beta = 0.4$, $t = 0.4$ for different values of $\alpha$ and $g(t) = t$
VP varus $y$, with $\beta = 0.4$, $t = 0.8$ for different values of $\alpha$ and $g(t) = t$
VP varus $y$, with $\beta = 0.7$, $t = 0.2$ for different values of $\alpha$ and $g(t) = t$
VP varus $y$, with $\alpha = 0.3$, $t = 0.2$ for different values of $\beta$ and $g(t) = t$
VP varus $y$, with $\alpha = 0.6$, $t = 0.2$ for different values of $\beta$ and $g(t) = t$
VP varus $y$, with $\alpha = 0.9$, $t = 0.2$ for different values of $\beta$ and $g(t) = t$.
VP varus $t$, with $\beta = 0.0$(no-slip), $y = 0.2$ for different values of $\alpha$ and $g(t) = t$
VP varus $t$, with $\beta = 0.4$, $y = 0.2$ for different values of $\alpha$ and $g(t) = t$
VP varus $t$, with $\beta = 0.0$, $y = 0.2$ for different values of $\alpha$ and $g(t) = t$
VP varus $y$, with $\beta = 0.4$, $t = 0.2$ for different values of $\alpha$ and with the sinusoidal oscillations of the bottom plate $(g(t) = \sin t)$
VP varus $y$, with $\beta = 0.4$, $t = 0.4$ for different values of $\alpha$ and $g(t) = \sin t$
VP varus $y$, with $\beta = 0.4$, $t = 0.8$ for different values of $\alpha$ and $g(t) = \sin t$
VP varus $y$, with $\beta = 0.7$, $t = 0.2$ for different values of $\alpha$ and $g(t) = \sin t$
VP varus $y$, with $\alpha = 0.3$, $t = 0.2$ for different values of $\beta$ and $g(t) = \sin t$
VP varus $y$, with $\alpha = 0.6$, $t = 0.2$ for different values of $\beta$ and $g(t) = \sin t$
VP varus $y$, with $\alpha = 0.9$, $t = 0.2$ for different values of $\beta$ and $g(t) = \sin t$
VP varus $t$, with $\beta = 0.0$(no-slip), $y = 0.2$ for different values of $\alpha$ and $g(t) = \sin t$
VP varus $t$, with $\beta = 0.4$, $y = 0.2$ for different values of $\alpha$ and $g(t) = \sin t$
VP varus $t$, with $\beta = 0.7$, $y = 0.2$ for different values of $\alpha$ and $g(t) = \sin t$
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