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October  2021, 14(10): 3719-3746. doi: 10.3934/dcdss.2020430

## Comprehensive analysis of integer-order, Caputo-Fabrizio (CF) and Atangana-Baleanu (ABC) fractional time derivative for MHD Oldroyd-B fluid with slip effect and time dependent boundary condition

 1 Institute for Groundwater Studies (IGS), University of the Free state Bloemfotain, South Africa, and, Department of Mathematics, School of Science, University of Management and Technology, C-Ⅱ Johar Town, Lahore 54770, Pakistan 2 Department of Sciences & Humanities, National University of computer & Emerging Sciences, Lahore Campus, BLOCK-B, Faisal Town, Lahore 54000, Pakistan

* Corresponding author: Bilalsehole@gmail.com

Received  November 2019 Revised  January 2020 Published  October 2021 Early access  September 2020

This article is focused on the slip effect in the unsteady flow of MHD Oldroyd-B fluid over a moving vertical plate with velocity $U_{o}f(t)$. The Laplace transformation and inversion algorithm are used to evaluate the expression for fluid velocity and shear stress. Fractional time derivatives are used to analyze the impact of fractional parameters (memory effect) on the dynamics of the fluid. While making a comparison, it is observed that the fractional-order model is best to explain the memory effect as compared to the classical model. The behavior of slip condition as well as no-slip condition is discussed with all physical quantities. The influence of dimensionless physical parameters like magnetic force $M$, retardation time $\lambda_{r}$, fractional parameter $\alpha$, and relaxation time $\lambda$ on fluid velocity has been discussed through graphical illustration. Our results suggest that the velocity field decreases by increasing the value of the magnetic field. In the absence of a slip parameter, the strength of the magnetic field is maximum. Furthermore, it is noted that the Atangana-Baleanu derivative in Caputo sense (ABC) is the best to highlight the dynamics of the fluid.

Citation: Muhammad Bilal Riaz, Syed Tauseef Saeed. Comprehensive analysis of integer-order, Caputo-Fabrizio (CF) and Atangana-Baleanu (ABC) fractional time derivative for MHD Oldroyd-B fluid with slip effect and time dependent boundary condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3719-3746. doi: 10.3934/dcdss.2020430
##### References:
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Arshad and D. Baleanu, A new fractional modelling and control strategy for the outbreak of dengue fever, Physica A: Statistical Mechanics and its Applications, 535 (2019), 122524. doi: 10.1016/j.physa.2019.122524.  Google Scholar  S. Jamshad, T. Gul, S. Islam, M. A. Khan, R. A. Shah, S. Nasir and H. Rasheed, Flow of unsteady second grade fluid between two vertical plates when one of the plate oscillating and other is stationary, J. Appl. Environ. Biol. Sci., 4 (2014), 41-52.   Google Scholar  M. A. Khan, Z. Hammouch and D. Baleanu, Modeling the dynamics of hepatitius-E via the Caputo-Fabrizio derivative, Math. Model. Nat. Phenom, 14 (2019), 311. doi: 10.1051/mmnp/2018074.  Google Scholar  Z. Li, Z. Liu and M. A. Khan, Fractional investigation of bank data with fractal-fractional Caputo derivative, Chaos, Solitons and Fractals, (2020), 109528. doi: 10.1016/j.chaos.2019.109528.  Google Scholar  T. Madeeha, M. A. Imran, N. Raza, M. Abdullah and A. Maryam, Wall slip and noninteger order derivative effects on the heat transfer flow of Maxwell fluid over an oscillating vertical plate with new definition of fractional Caputo-Fabrizio derivatives, Results in physics, 7 (2017), 1887-1898.   Google Scholar  M. Navier, Memoire sur les lois du mouvement des fluids, Mem. LAcad. Sci. LInst. France, 6 (1823), 389-440.   Google Scholar  K. R. Rajagopal and R. K. Bhatnagar, Exact solutions for some simple flows of an Oldroyd-B fluid, Acta Mechanica, 113 (1995), 233-239.  doi: 10.1007/BF01212645.  Google Scholar  M. B. Riaz, N. A. Asif, A. Atangana and M. I. Asjad, Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernal, Discrete and Continuous Dynamical System, 12 (2019), 645-664.  doi: 10.3934/dcdss.2019041.  Google Scholar  N. A. Shah, M. A. Imran and F. Miraj, Exact solutions of time fractional free convection flows of viscous fluid over an isothermal vertical plate with caputo and caputo-fabrizio derivatives, J. Prime Res. Math., 13 (2017), 56-74. Google Scholar  A. Shakeel, S. Ahmad, H. Khan, N. A. Shah and S. U. Haq, Flows with slip of Oldroyd-B fluids over a moving plate, Advances in Mathematical Physics, (2016), Art. ID 8619634, 9 pp. doi: 10.1155/2016/8619634.  Google Scholar  A. M. Siddiqui, T. Haroon, M. Zahid and A. Shahzad, Effect of slip condition on unsteady flows of an Oldroyd-B fluid between parallel plates, World Applied Sciences Journal, 13 (2011), 2282-2287.   Google Scholar  H. Stehfest Algorithm, Numerical inversion of Laplace transforms, Commun. ACM, 13 (1970), 9-47.   Google Scholar  A. Tassaddiq, MHD flow of a fractional second grade fluid over an inclined heated plate, Chaos, Solitons and Fractals, 123 (2019), 341-346.  doi: 10.1016/j.chaos.2019.04.029.  Google Scholar  DY. Tzou, Macro to Microscale Heat Transfer: The Lagging Behaviour, Washington: Taylor and Francis, (1970). Google Scholar  S. Ullah, M. A. Khan and M. Farooq, A new fractional model for the dynamics of the hepatitius-B virus using the Caputo-Fabrizio, Eur. Phys. J. Plus., 133 (2018), 237. Google Scholar  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.  Google Scholar  D. Vieru, C. Fetecau and C. Fetecau, Time-fractional free convection flow near a vertical plate with newtonian heating and mass diffusion, Therm. Sci., 19 (2015), 85-98.  doi: 10.2298/TSCI15S1S85V. Google Scholar  H. Zaman, Z. Ahmad and M. Ayub, A note on the unsteady incompressible MHD fluid flow with slip conditions and porous walls, ISRN Mathematical Physics, 1 (2013), 1-10.  doi: 10.1155/2013/705296. Google Scholar  Z. Zhang, C. Fu and W. Tan, Linear and nonlinear stability analyses of thermal convection for Oldroyd-B fluids in porous media heated from below, Phys. Fluids, 20 (2008), 84-103.   Google Scholar

show all references

##### References:
  T. Abdeljawad, Different type kernel h-fractional differences and their fractional h-sums, Chaos, Solitons and Fractals, 116 (2018), 146-156.  doi: 10.1016/j.chaos.2018.09.022.  Google Scholar  A. Asif, Z. Hammouch and M. B. Riaz, Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative, Eur. Phys. J. Plus, 133 (2018), 272. Google Scholar  A. Atangana, On the new fractional derivative and application to nonlinear fishers reaction-diffusion equation, Appl. Math. Comput, 273 (2016), 948-956.  doi: 10.1016/j.amc.2015.10.021.  Google Scholar  D. Avc, M. Yavuz and N. Zdemir, Fundamental Solutions to the Cauchy and Dirichlet Problems for a Heat Conduction Equation Equipped with the Caputo-Fabrizio Differentiation. Heat Conduction: Methods, Applications and Research, , Nova Science Publishers, (eds: Jordan Hristov, Rachid Bennacer) Book Chapter, ISBN: 978-1-53614-673-8 2019, 95–107. Google Scholar  D. Avc, M. Yavuz and N. Zdemir, Fractional Optimal Control of a Diffusive Transport Acting on a Spherical Region, Methods of Mathematical Modelling: Fractional Differential Equations, Taylor and Francis Group Publishing, (eds: Harendra Singh, Devendra Kumar, Dumitru Baleanu) Book Chapter, ISBN: 978-0-367-22008-2 (2019), 63–82. Google Scholar  D. Baleanu, A. Jajarmi, S. S. Sajjadi and D. Mozyrska, A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 083127, 15 pp. doi: 10.1063/1.5096159.  Google Scholar  M. A. Dokuyucu, E. Celik, H. Bulut and H. M. Baskonus, Cancer treatment model with the caputo-fabrizio fractional derivative, Eur. Phys. J. Plus, 133 (2018), 93. Google Scholar  C. Fetecau, D. Vieru, C. Fetecau and S. Akhter, General solutions for magnetohydrodynamic natural convection flow with radiative heat transfer and slip condition over a moving plate, Z. Naturforsch. A, 68 (2013), 659-667.   Google Scholar  M. Hajipour, A. Jajarmi and D. Baleanu, On the accurate discretization of a highly nonlinear boundary value problem, Numerical Algoritham, 79 (2018), 679-695.  doi: 10.1007/s11075-017-0455-1.  Google Scholar  T. Hayat, A. M. Siddiqui and S. Asghar, Some simple flows of an Oldroyd-B fluid, International Journal of Engineering Science, 39 (2001), 135-147.  doi: 10.1016/S0020-7225(00)00026-4. Google Scholar  M. A. Imran, M. B. Riaz, N. A. Shah and A. A. Zafar, Boundary layer flow of MHD generalized Maxwell fluid over an exponentially accelerated infinite vertical surface with slip and Newtonian heating at the boundary, Results in physics, 8 (2018), 1061-1067.  doi: 10.1016/j.rinp.2018.01.036. Google Scholar  A. Jajarmi, D. Baleanu, S. S. Sajjadi and J. H. Asad, A new feature of the fractional Euler-Lagrange equations for a coupled oscillator using a nonsingular operator approach, Frontiers in Physics, 196 (2019), 7. Google Scholar  A. Jajarmi, S. Arshad and D. Baleanu, A new fractional modelling and control strategy for the outbreak of dengue fever, Physica A: Statistical Mechanics and its Applications, 535 (2019), 122524. doi: 10.1016/j.physa.2019.122524.  Google Scholar  S. Jamshad, T. Gul, S. Islam, M. A. Khan, R. A. Shah, S. Nasir and H. Rasheed, Flow of unsteady second grade fluid between two vertical plates when one of the plate oscillating and other is stationary, J. Appl. Environ. Biol. Sci., 4 (2014), 41-52.   Google Scholar  M. A. Khan, Z. Hammouch and D. Baleanu, Modeling the dynamics of hepatitius-E via the Caputo-Fabrizio derivative, Math. Model. Nat. Phenom, 14 (2019), 311. doi: 10.1051/mmnp/2018074.  Google Scholar  Z. Li, Z. Liu and M. A. Khan, Fractional investigation of bank data with fractal-fractional Caputo derivative, Chaos, Solitons and Fractals, (2020), 109528. doi: 10.1016/j.chaos.2019.109528.  Google Scholar  T. Madeeha, M. A. Imran, N. Raza, M. Abdullah and A. Maryam, Wall slip and noninteger order derivative effects on the heat transfer flow of Maxwell fluid over an oscillating vertical plate with new definition of fractional Caputo-Fabrizio derivatives, Results in physics, 7 (2017), 1887-1898.   Google Scholar  M. Navier, Memoire sur les lois du mouvement des fluids, Mem. LAcad. Sci. LInst. France, 6 (1823), 389-440.   Google Scholar  K. R. Rajagopal and R. K. Bhatnagar, Exact solutions for some simple flows of an Oldroyd-B fluid, Acta Mechanica, 113 (1995), 233-239.  doi: 10.1007/BF01212645.  Google Scholar  M. B. Riaz, N. A. Asif, A. Atangana and M. I. Asjad, Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernal, Discrete and Continuous Dynamical System, 12 (2019), 645-664.  doi: 10.3934/dcdss.2019041.  Google Scholar  N. A. Shah, M. A. Imran and F. Miraj, Exact solutions of time fractional free convection flows of viscous fluid over an isothermal vertical plate with caputo and caputo-fabrizio derivatives, J. Prime Res. Math., 13 (2017), 56-74. Google Scholar  A. Shakeel, S. Ahmad, H. Khan, N. A. Shah and S. U. Haq, Flows with slip of Oldroyd-B fluids over a moving plate, Advances in Mathematical Physics, (2016), Art. ID 8619634, 9 pp. doi: 10.1155/2016/8619634.  Google Scholar  A. M. Siddiqui, T. Haroon, M. Zahid and A. Shahzad, Effect of slip condition on unsteady flows of an Oldroyd-B fluid between parallel plates, World Applied Sciences Journal, 13 (2011), 2282-2287.   Google Scholar  H. Stehfest Algorithm, Numerical inversion of Laplace transforms, Commun. ACM, 13 (1970), 9-47.   Google Scholar  A. Tassaddiq, MHD flow of a fractional second grade fluid over an inclined heated plate, Chaos, Solitons and Fractals, 123 (2019), 341-346.  doi: 10.1016/j.chaos.2019.04.029.  Google Scholar  DY. Tzou, Macro to Microscale Heat Transfer: The Lagging Behaviour, Washington: Taylor and Francis, (1970). Google Scholar  S. Ullah, M. A. Khan and M. Farooq, A new fractional model for the dynamics of the hepatitius-B virus using the Caputo-Fabrizio, Eur. Phys. J. Plus., 133 (2018), 237. Google Scholar  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.  Google Scholar  D. Vieru, C. Fetecau and C. Fetecau, Time-fractional free convection flow near a vertical plate with newtonian heating and mass diffusion, Therm. Sci., 19 (2015), 85-98.  doi: 10.2298/TSCI15S1S85V. Google Scholar  H. Zaman, Z. Ahmad and M. Ayub, A note on the unsteady incompressible MHD fluid flow with slip conditions and porous walls, ISRN Mathematical Physics, 1 (2013), 1-10.  doi: 10.1155/2013/705296. Google Scholar  Z. Zhang, C. Fu and W. Tan, Linear and nonlinear stability analyses of thermal convection for Oldroyd-B fluids in porous media heated from below, Phys. Fluids, 20 (2008), 84-103.   Google Scholar Velocity profile of Classical model with variation of $M$ and $\beta$ for $g(t)=sin(t)$ Atangana-Baleanu velocity profile for $g(t)=sin(t)$ with variation of $M$ and $\beta$ Caputo-Fabrizio velocity profile for $g(t)=sin(t)$ with variation of $M$ and $\beta$ Velocity profile of Classical model with variation of $M$ and $\beta$ for $g(t)=H(t)$ Atangana-Baleanu velocity profile for $g(t)=H(t)$ with variation of $M$ and $\beta$ Caputo-Fabrizio velocity profile for $g(t)=H(t)$ with variation of $M$ and $\beta$ Velocity profile of Classical model with variation of $M$ and $\beta$ for $g(t)=t$ Atangana-Baleanu velocity profile for $g(t)=t$ with variation of $M$ and $\beta$ Caputo-Fabrizio velocity profile for $g(t)=t$ with variation of $M$ and $\beta$ Velocity profile of Classical model for $\lambda$ variation with $g(t)=sin(t)$ Velocity profile of Atangana-Baleanu for $\lambda$ variation with $g(t)=sin(t)$ Velocity profile of Classical model for $\lambda$ variation with $g(t)=H(t)$ Velocity profile of Atangana-Baleanu for $\lambda$ variation with $g(t)=H(t)$ Velocity profile of Classical model for $\lambda$ variation with $g(t)=t$ Velocity profile of Atangana-Baleanu for $\lambda$ variation with $g(t)=t$ Integer order velocity profile for $\lambda_{r}$ variation with $g(t)=sin(t)$ Atangana-Baleanu velocity profile for $\lambda_{r}$ variation with $g(t)=sin(t)$ Caputo-Fabrizio velocity profile for $\lambda_{r}$ variation with $g(t)=sin(t)$ Integer order velocity profile for $\lambda_{r}$ variation with $g(t)=H(t)$ Atangana-Baleanu velocity profile for $\lambda_{r}$ variation with $g(t)=H(t)$ Caputo-Fabrizio velocity profile for $\lambda_{r}$ variation with $g(t)=H(t)$ Integer order velocity profile for $\lambda_{r}$ variation with $g(t)=t$ Atangana-Baleanu velocity profile for $\lambda_{r}$ variation with $g(t)=t$ Caputo-Fabrizio velocity profile for $\lambda_{r}$ variation with $g(t)=t$ Atangana-Baleanu velocity profile with $\alpha$ variation for $g(t)=sin(t)$ Caputo-Fabrizio velocity profile with $\alpha$ variation for $g(t)=sin(t)$ Atangana-Baleanu velocity profile with $\alpha$ variation for $g(t)=H(t)$ Caputo-Fabrizio velocity profile with $\alpha$ variation for $g(t)=H(t)$
Nomenclature
 Symbol Quantity $u$ Fluid velocity $B_{0}$ Magnitude of applied magnetic field $q$ Laplace transforms parameter $S$ Extra stress tensor $A$ Rivlin Ericken $L$ Velocity gradient $R$ Reynold number $\rho$ Fluid density $\lambda$ Relaxation time $\lambda_{r}$ Retardation time $\mu$ Dynamic viscosity $\upsilon$ Kinematic viscosity $\beta$ slip parameter $\nabla$ gradient operator $\tau$ shear stress
 Symbol Quantity $u$ Fluid velocity $B_{0}$ Magnitude of applied magnetic field $q$ Laplace transforms parameter $S$ Extra stress tensor $A$ Rivlin Ericken $L$ Velocity gradient $R$ Reynold number $\rho$ Fluid density $\lambda$ Relaxation time $\lambda_{r}$ Retardation time $\mu$ Dynamic viscosity $\upsilon$ Kinematic viscosity $\beta$ slip parameter $\nabla$ gradient operator $\tau$ shear stress
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