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It is looked into how fractional MHD and viscous fluid naturally flow across a vertical plate that oscillates with a steady heat flux in a porous medium. The analytical solution for non-dimensional momentum was generated using the Laplace transform and the Boussinesq approximation. Comparisons are presented between Fourier’s law-described classical thermal transport and Prabhakar-like fractional time free convection flows in terms of temperature and velocity. By using fractional viscous fluid, we were able to produce conventional viscous fluid solutions and complete their objective. Finally, diagrams are used to show how different parameters such as Prandtl number, fractional parameters, Grashof number, effective permeability, time on temperature and velocity, and graphically recovered the solution available in the literature.

The free convection flow of Prabhakar fractional Jeffrey fluid on an oscillated vertical plate with homogenous heat flux is investigated. With the help of the Laplace transform and the Boussinesq's approximation, precise solutions for dimensionless momentum may be found. The temperature and velocity of Prabhakar fractional time free convection flows are compared to conventional thermal transport, as shown by Fourier's law. They met all of the requirements and recovered Newtonian and ordinary Jeffrey fluid solutions from fractional Jeffrey fluid. Finally, graphs show the effect of various physical parameters such as fractional parameters, Grashof number, Prandtl number and Jeffrey parameters on both temperature and velocity.

The purpose of this paper is to evaluate the exact solution of the unsteady flow of a generalized Brinkman type fluid under the effect of MHD in a channel. The classical Brinkman model reduced to non-dimensional form by using appropriate dimensionless variables. Furthermore, the non-dimensional Brinkman model is transformed to a generalize Brinkman model with Caputo-Fabrizio fractional derivative. The dimensionless Brinkman model has been solved with applicable conditions by integral transforms techniques that is Fourier and Laplace. The effect of different physical parameters and fractional order on fluid velocity and shear stress are illustrated graphically. Moreover, through this recent work, the recovery of classical Brinkman type fluid is possible through graphs.

The objective of this article is to explore the unsteady natural convection flows of Prabhakar-like non integer Maxwell fluid. Moreover, wall slip condition on temperature and Newtonian effects on heating are also studied. The generalized memory effects are illustrated with fractional time Prabhakar derivative. Dimensionless temperature and velocity are calculated analytically with the help of Laplace transform technique. A comparison among Prabhakar fractional natural convection flows and classical thermal transport which, illustrated by the Fourier's law. As a limiting case, we recovered the solution of ordinary Maxwell and Newtonian fluids from fractional Maxwell fluids with slip and no slip conditions. The results of fractional and important physical parameters are graphically covered. Accordingly, by comparing Maxwell fluids to viscous fluids, we found out that Maxwell fluids are move rapidly than viscous fluids. Moreover, the ordinary fluids moving fast than fractional fluids.

Natural convection flow of unsteady Prabhakar-like fractional Brinkman type fluids over an infinite vertical heated wall with uniform heat flux is studied. Exact dimensionless solutions of momentum and energy equations, under Boussinesq approximation, are obtained using Laplace transforms. They satisfy all imposed conditions and reduce to known solutions from the literature as special cases. The velocity and heat transfer of Prabhakar-like fractional Brinkman type fluids with generalized thermal transport are compared with ordinary Brinkman type fluids with generalized thermal transport and with the ordinary viscoelastic fluids with classical Fourier thermal flux. Finally, the influence of different parameters like fractional parameters, Prandtl number, Grashof number, time and Brinkman parameter on temperature and velocity is shown by graphs.

In this paper, we calculated the velocity of the unsteady flow of a Brinkman MHD type viscous fluid which is enclosed between two parallel side walls perpendicular to a plate by applying the Fourier transformation. The movement of the fluid is caused by the plate which at time t=0+, exerts shear stress to the fluid. We presented the solutions into two categories i.e. steady state and transient state, which satisfy the given equation as well as boundary and initial conditions. Furthermore, by making h→∞, we recover the solutions obtained in the literature corresponding to the motion over an infinite plate. The effect of the side walls and the time required to reach the steady state are discussed through graphs. Also we checked the effect of different parameters by graphical representations. When Keff and Reynolds number become zero and one respectively, the general solution obtained in this manuscript reduce to special cases in the literature. Keywords: MHD, Brinkman type fluid, Shear stress, Side walls, Exact solution, Fourier transformation

In this article, unidirectional flows of fractional viscous fluids in a rectangular channel are studied. The flow is generated by the shear stress given on the bottom plate of the channel. The authors have developed a generalized model on the basis of constitutive equations described by the time-fractional Caputo–Fabrizio derivative. Many authors have published different results by applying the time-fractional derivative to the local part of acceleration in the momentum equation. This approach of the fractional models does not have sufficient physical background. By using fractional generalized constitutive equations, we have developed a proper model to investigate exact analytical solutions corresponding to the channel flow of a generalized viscous fluid. The exact solutions for velocity field and shear stress are obtained by using Laplace transform and Fourier integral transformation, for three different cases namely (i) constant shear, (ii) ramped type shear and (iii) oscillating shear. The results are plotted and discussed.