The main focus of this study is to investigate the impact of heat generation/absorption with ramp velocity and ramp temperature on magnetohydrodynamic (MHD) time-dependent Maxwell fluid over an unbounded plate embedded in a permeable medium. Non-dimensional parameters along with Laplace transformation and inversion algorithms are used to find the solution of shear stress, energy, and velocity profile. Recently, new fractional differential operators are used to define ramped temperature and ramped velocity. The obtained analytical solutions are plotted for different values of emerging parameters. 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 classical models. Our results suggest that the velocity profile decrease by increasing the effective Prandtl number. The existence of an effective Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity. The incremental value of the M is observed for a decrease in the velocity field, which reflects to control resistive force. Further, it is noted that the Atangana-Baleanu derivative in Caputo sense (ABC) is the best to highlight the dynamics of the fluid. The influence of pertinent parameters is analyzed graphically for velocity and energy profile. Expressions for skin friction and Nusselt number are also derived for fractional differential operators.

Viscoelasticity has important implications due to the characterization of viscoelastic parameters (relaxation and retardation phenomenon), elastic shearing strain, thermal relaxation, time-dependent an elastic aspect, and other rheological properties [

Firstly, Maxwell [

The technique of fractional calculus has been used to formulate mathematical modeling in various technological development, engineering applications, and industrial sciences. Different valuable work has been discussed for modeling fluid dynamics, signal processing, viscoelasticity, electrochemistry, and biological structure through fractional time derivatives. This fractional differential operator found useful conclusions for experts to treat cancer cells with a suitable amount of heat source and have compared the results to see the memory effect of temperature function. As compared to classical models, the memory effect is much stronger in fractional derivatives. From the past to the present, modeling of the different processes is handled through various types of fractional derivatives and fractal-fractional differential operators, such that Caputo (Power law), Atangana–Baleanu (Mittage–Leffler law), Caputo–Fabrizio (exponential law), Riemann–Liouville, modified Riemann–Liouville (Power law with boundaries) and few others [

The main objective of this paper is to investigate the MHD Maxwell model with the definition of fractional order derivative in a Darcy medium. The solution of fluid velocity, energy, and shear stress are obtained by Caputo-Fabrizio (

Consider the MHD time-dependent Maxwell model over an unbounded vertical plate immersed in a permeable surface. The plate is along the x-axis, while the _{0} and

Introduce dimensionless elements to form the problem free from geometric

After simplification, we have the set of dimensionless governing equations:

with corresponding conditions

Fractional operators are quite flexible for describing the behaviors of energy transfer of MHD Maxwell fluid through the characterization of governing equations. Generating fractional governing equation of temperature

where,

Solving the uncoupled and fractionalized governing equation of temperature

employing Laplace transformation on

More suitable form of temperature field is

The solution of homogenous part of second order partial differential equation say

with the help of _{1} and _{2} for temperature equation

where

The expression of Nusselt number

Generating a fractional governing equation of temperature

where

where

Solving governing equation of temperature

employing Laplace transformation on

More suitable form of temperature field is,

The solution of homogenous part of second order partial differential equation say

with the help of _{1} and _{2} for temperature equation:

where

The expression of Nusselt number

Generating a fractional governing equation of velocity

The required homogeneous part of the

and particular solution can be give as follow after making use of

and solution of

using conditions given in

The suitable and simplified form for inversion algorithm, we have

and

where, letting parameters are described as

Differentiate

where

Plugging

The expression of skin friction _{f}

Generating a fractional governing equation of velocity

The homogeneous part of the

and particular solution can be give as follow after making use of

using conditions given in

The suitable and simplified form for inversion algorithm, we have

and

where, letting parameters are described as

Differentiate

where

Plugging

The expression of skin friction _{f}

In our flow models we use classical computational technique (Laplace transform) to solve the given models using different definitions of fractional derivatives. There are many algorithms for the numerical calculation of the inverse Laplace transform. The Stehfest’s formula, which approximates the inverse Laplace transform is simple, easy to use compared with other algorithms. In this paper we use Stehfest’s algorithm and also give comparison with other in tabular form. Tzou’s calculation for approval of our numerical inverse Laplace
_{1} is a natural number [

This section is dedicated to present physical interpretation of the obtained results via CF and AB differential operators under heat generation, ramp velocity, and ramp temperature on the MHD Maxwell model. Results are investigated via Laplace transformation with an inversion algorithm for velocity, energy, and shear stress based on singular verses non-singular and local versus non-local kernels. The graphical representations are depicted for showing the influences of different physical parameters such as effective Prandtl number _{reff}_{r}

_{reff}_{reff}_{reff}_{reff}_{reff}

_{reff}_{reff}_{reff}

_{r}_{r}

The influence of

_{reff}_{reff}_{reff}

_{reff} |
||
---|---|---|

1 | 0.074 | 0.071 |

2 | 0.068 | 0.063 |

3 | 0.055 | 0.051 |

4 | 0.044 | 0.040 |

5 | 0.039 | 0.037 |

6 | 0.035 | 0.033 |

7 | 0.026 | 0.023 |

8 | 0.024 | 0.021 |

9 | 0.016 | 0.017 |

10 | 0.014 | 0.013 |

_{reff} |
||
---|---|---|

1 | 0.088 | 0.083 |

2 | 0.069 | 0.065 |

3 | 0.067 | 0.061 |

4 | 0.057 | 0.054 |

5 | 0.045 | 0.041 |

6 | 0.036 | 0.034 |

7 | 0.029 | 0.027 |

8 | 0.025 | 0.022 |

9 | 0.019 | 0.018 |

10 | 0.015 | 0.012 |

Velocity (CF) [Stehfest’s] | Velocity (CF) [Tzou’s] | |
---|---|---|

0.2 | 0.978 | 0.965 |

0.4 | 0.921 | 0.925 |

0.6 | 0.876 | 0.879 |

0.8 | 0.833 | 0.821 |

1.0 | 0.791 | 0.782 |

1.2 | 0.752 | 0.731 |

1.4 | 0.713 | 0.708 |

1.6 | 0.677 | 0.663 |

1.8 | 0.641 | 0.628 |

2.0 | 0.617 | 0.609 |

Velocity (ABC) [Stehfest’s] | Velocity (ABC) [Tzou’s] | |
---|---|---|

0.2 | 0.957 | 0.946 |

0.4 | 0.924 | 0.913 |

0.6 | 0.883 | 0.881 |

0.8 | 0.843 | 0.839 |

1.0 | 0.805 | 0.799 |

1.2 | 0.768 | 0.759 |

1.4 | 0.732 | 0.724 |

1.6 | 0.698 | 0.685 |

1.8 | 0.665 | 0.657 |

2.0 | 0.634 | 0.621 |

_{reff} |
||
---|---|---|

1 | 0.335 | 0.331 |

2 | 0.502 | 0.487 |

3 | 0.614 | 0.603 |

4 | 0.709 | 0.756 |

5 | 0.793 | 0.721 |

6 | 0.869 | 0.833 |

7 | 0.938 | 0.913 |

8 | 1.003 | 1.034 |

9 | 1,064 | 1.078 |

10 | 1.122 | 1.176 |

_{reff} |
||
---|---|---|

1 | 0.478 | 0.452 |

2 | 0.544 | 0.523 |

3 | 0.678 | 0.634 |

4 | 0.739 | 0.765 |

5 | 0.799 | 0.732 |

6 | 0.876 | 0.821 |

7 | 0.959 | 0.926 |

8 | 1.009 | 1.052 |

9 | 1.078 | 1.098 |

10 | 1.129 | 1.189 |

The basic purpose of this article was to investigate the effect of the simultaneous use of ramped velocity and ramped temperature conditions on MHD Maxwell fluid. It is difficult to calculate the solutions of MHD Maxwell fluid using both ramp conditions. Fractional differential operators are used to finding solutions using Laplace transformation and inversion algorithm. Some comparisons have been drawn and they are in good agreement with the results published in [

The velocity decreases by magnifying the value of the magnetic profile.

The velocity increases with increasing values of _{r}

The Nusselt number describes that the heat transfer rate enhances with increasing thermal diffusivity.

The velocity decreases by magnifying the value of the _{reff}

The authors are highly thankful and grateful for generous support and facilities of this research work.