Article Contents
Article Contents

# 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

• 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.

Mathematics Subject Classification: Primary:93A30, 35Q35, 49-XX, 93A30, 65M32;Secondary:65M32.

 Citation:

• Figure 1.  Geometry of flow

Figure 2.  Velocity profile of Classical model with variation of $M$ and $\beta$ for $g(t)=sin(t)$

Figure 3.  Atangana-Baleanu velocity profile for $g(t)=sin(t)$ with variation of $M$ and $\beta$

Figure 4.  Caputo-Fabrizio velocity profile for $g(t)=sin(t)$ with variation of $M$ and $\beta$

Figure 5.  Velocity profile of Classical model with variation of $M$ and $\beta$ for $g(t)=H(t)$

Figure 6.  Atangana-Baleanu velocity profile for $g(t)=H(t)$ with variation of $M$ and $\beta$

Figure 7.  Caputo-Fabrizio velocity profile for $g(t)=H(t)$ with variation of $M$ and $\beta$

Figure 8.  Velocity profile of Classical model with variation of $M$ and $\beta$ for $g(t)=t$

Figure 9.  Atangana-Baleanu velocity profile for $g(t)=t$ with variation of $M$ and $\beta$

Figure 10.  Caputo-Fabrizio velocity profile for $g(t)=t$ with variation of $M$ and $\beta$

Figure 11.  Velocity profile of Classical model for $\lambda$ variation with $g(t)=sin(t)$

Figure 12.  Velocity profile of Atangana-Baleanu for $\lambda$ variation with $g(t)=sin(t)$

Figure 13.  Velocity profile of Caputo-Fabrizio for $\lambda$ with $g(t)=sin(t)$

Figure 14.  Velocity profile of Classical model for $\lambda$ variation with $g(t)=H(t)$

Figure 15.  Velocity profile of Atangana-Baleanu for $\lambda$ variation with $g(t)=H(t)$

Figure 16.  Velocity profile of Caputo-Fabrizio for $\lambda$ with $g(t)=H(t)$

Figure 17.  Velocity profile of Classical model for $\lambda$ variation with $g(t)=t$

Figure 18.  Velocity profile of Atangana-Baleanu for $\lambda$ variation with $g(t)=t$

Figure 19.  Velocity profile of Caputo-Fabrizio for $\lambda$ with $g(t)=t$

Figure 20.  Integer order velocity profile for $\lambda_{r}$ variation with $g(t)=sin(t)$

Figure 21.  Atangana-Baleanu velocity profile for $\lambda_{r}$ variation with $g(t)=sin(t)$

Figure 22.  Caputo-Fabrizio velocity profile for $\lambda_{r}$ variation with $g(t)=sin(t)$

Figure 23.  Integer order velocity profile for $\lambda_{r}$ variation with $g(t)=H(t)$

Figure 24.  Atangana-Baleanu velocity profile for $\lambda_{r}$ variation with $g(t)=H(t)$

Figure 25.  Caputo-Fabrizio velocity profile for $\lambda_{r}$ variation with $g(t)=H(t)$

Figure 26.  Integer order velocity profile for $\lambda_{r}$ variation with $g(t)=t$

Figure 27.  Atangana-Baleanu velocity profile for $\lambda_{r}$ variation with $g(t)=t$

Figure 28.  Caputo-Fabrizio velocity profile for $\lambda_{r}$ variation with $g(t)=t$

Figure 29.  Atangana-Baleanu velocity profile with $\alpha$ variation for $g(t)=sin(t)$

Figure 30.  Caputo-Fabrizio velocity profile with $\alpha$ variation for $g(t)=sin(t)$

Figure 31.  Atangana-Baleanu velocity profile with $\alpha$ variation for $g(t)=H(t)$

Figure 32.  Caputo-Fabrizio velocity profile with $\alpha$ variation for $g(t)=H(t)$

Figure 33.  Atangana-Baleanu velocity profile with $\alpha$ variation for $g(t)=t$

Figure 34.  Caputo-Fabrizio velocity profile with $\alpha$ variation for $g(t)=t$

Table 1.  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

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