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## 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  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, doi: 10.3934/dcdss.2020430
##### References:

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##### References:
Geometry of flow
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 Caputo-Fabrizio for $\lambda$ 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 Caputo-Fabrizio for $\lambda$ 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 $g(t)=t$
Velocity profile of Caputo-Fabrizio for $\lambda$ 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 $g(t)=H(t)$
Atangana-Baleanu velocity profile with $\alpha$ variation for $g(t)=t$
Caputo-Fabrizio velocity profile with $\alpha$ variation for $g(t)=t$
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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