# 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 and Continuous Dynamical Systems - S, 2019, 12 (3) : 645-664. doi: 10.3934/dcdss.2019041
##### References:
 [1] S. Abelman, E. Momoniat and T. Hayat, Couette flow of a third grade fluid with rotating frame and slip condition, Non-Linear Analysis: Real World Appl., 10 (2009), 3329-3334.  doi: 10.1016/j.nonrwa.2008.10.068. [2] A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reactiondiffusion equation, Appl. Math. Comput., 1 (2016), 948-956.  doi: 10.1016/j.amc.2015.10.021. [3] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A. [4] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85. [5] M. Caputo and M. Fabrizio, Damage and fatigue described by a fractional derivative model, J. Comput. Phys., 293 (2015), 400-408.  doi: 10.1016/j.jcp.2014.11.012. [6] M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11.  doi: 10.18576/pfda/020101. [7] M. A. Day, The non-slip boundary condition in fluid mechanics, Erkenntnis, 33 (1990), 285-296.  doi: 10.1007/BF00717588. [8] L. Debnath and D. Bhatta, Integral Transforms and Their Applications, second ed., Chapman and Hall/CRC Press, Boca-Raton, 2007. [9] A. Heibig and L. I. Palade, On the rest state stability of an objective fractional derivative, Journal of Mathematical Physics, 49 (2008), 043101, 22pp. doi: 10.1063/1.2907578. [10] A. R. A. Khaled and K. Vafai, The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions, Int. J. Non-Lin. Mech., 39 (2004), 795-809.  doi: 10.1016/S0020-7462(03)00043-X. [11] S. Kumar, A new fractional modeling arising in engineering sciences and its analytical approximate solution, Alexandria Eng. J., 52 (2013), 813-819.  doi: 10.1016/j.aej.2013.09.005. [12] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific Publishing, 2010. doi: 10.1142/9781848163300. [13] N. Makris, G. F. Dargush and M. C. Constantinou, Dynamic analysis of generalized viscoelastic fluids, J. Eng. Mech., 119 (1993), 1663-1679.  doi: 10.1061/(ASCE)0733-9399(1993)119:8(1663). [14] M. Mooney, Explicit formula for slip and fluidity, J. Rheol., 2 (1931), 210-222.  doi: 10.1122/1.2116364. [15] C. L. M. H. Navier, Sur les lois du movement des fluids, Mem. Acad. R. Sa: Inst. Fr., 6 (1827), 389-440. [16] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 2009. [17] I. J. Rao and K. Rajagopal, The effect of the slip boundary condition on the flow of fluids in a channel, Acta Mech, 135 (1999), 113-126.  doi: 10.1007/BF01305747. [18] I. Siddique and D. Vieru, Stokes flows of a Newtonian fluid with fractional derivatives and slip at the wall, Int. Rev. Chem. Eng. (IRECHE), 3 (2011), 822-826. [19] D. Vieru and A. A. Zafar, Some Couette flows of a Maxwell fluid with wall slip condition, Appl. Math. Inf. Sci., 7 (2013), 209-219.  doi: 10.12785/amis/070126. [20] D. Vieru and A. Rauf, Stokes Flows of a Maxwell fluid with wall slip condition, Can. J. Phys, 89 (2011), 1061-1071.  doi: 10.1139/p11-099.

show all references

##### References:
 [1] S. Abelman, E. Momoniat and T. Hayat, Couette flow of a third grade fluid with rotating frame and slip condition, Non-Linear Analysis: Real World Appl., 10 (2009), 3329-3334.  doi: 10.1016/j.nonrwa.2008.10.068. [2] A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reactiondiffusion equation, Appl. Math. Comput., 1 (2016), 948-956.  doi: 10.1016/j.amc.2015.10.021. [3] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A. [4] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85. [5] M. Caputo and M. Fabrizio, Damage and fatigue described by a fractional derivative model, J. Comput. Phys., 293 (2015), 400-408.  doi: 10.1016/j.jcp.2014.11.012. [6] M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11.  doi: 10.18576/pfda/020101. [7] M. A. Day, The non-slip boundary condition in fluid mechanics, Erkenntnis, 33 (1990), 285-296.  doi: 10.1007/BF00717588. [8] L. Debnath and D. Bhatta, Integral Transforms and Their Applications, second ed., Chapman and Hall/CRC Press, Boca-Raton, 2007. [9] A. Heibig and L. I. Palade, On the rest state stability of an objective fractional derivative, Journal of Mathematical Physics, 49 (2008), 043101, 22pp. doi: 10.1063/1.2907578. [10] A. R. A. Khaled and K. Vafai, The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions, Int. J. Non-Lin. Mech., 39 (2004), 795-809.  doi: 10.1016/S0020-7462(03)00043-X. [11] S. Kumar, A new fractional modeling arising in engineering sciences and its analytical approximate solution, Alexandria Eng. J., 52 (2013), 813-819.  doi: 10.1016/j.aej.2013.09.005. [12] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific Publishing, 2010. doi: 10.1142/9781848163300. [13] N. Makris, G. F. Dargush and M. C. Constantinou, Dynamic analysis of generalized viscoelastic fluids, J. Eng. Mech., 119 (1993), 1663-1679.  doi: 10.1061/(ASCE)0733-9399(1993)119:8(1663). [14] M. Mooney, Explicit formula for slip and fluidity, J. Rheol., 2 (1931), 210-222.  doi: 10.1122/1.2116364. [15] C. L. M. H. Navier, Sur les lois du movement des fluids, Mem. Acad. R. Sa: Inst. Fr., 6 (1827), 389-440. [16] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 2009. [17] I. J. Rao and K. Rajagopal, The effect of the slip boundary condition on the flow of fluids in a channel, Acta Mech, 135 (1999), 113-126.  doi: 10.1007/BF01305747. [18] I. Siddique and D. Vieru, Stokes flows of a Newtonian fluid with fractional derivatives and slip at the wall, Int. Rev. Chem. Eng. (IRECHE), 3 (2011), 822-826. [19] D. Vieru and A. A. Zafar, Some Couette flows of a Maxwell fluid with wall slip condition, Appl. Math. Inf. Sci., 7 (2013), 209-219.  doi: 10.12785/amis/070126. [20] D. Vieru and A. Rauf, Stokes Flows of a Maxwell fluid with wall slip condition, Can. J. Phys, 89 (2011), 1061-1071.  doi: 10.1139/p11-099.
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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