# Analysis of Heat and Mass Transfer of Fractionalized MHD Second-Grade Fluid over Nonlinearly Moving Porous Plate

Authors: Muhammad Jamil; Israr Ahmed; Ilyas Khan; Umar Faryaz; Abdulaziz H. Alghtani; Mulugeta Andualem
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This study investigates the heat and mass transfer in the MHD flow of fractionalized second-grade fluid induced by impulsively moved bottom porous plate with nonlinear velocity of the magnitude KTD. To acquire the fractionalized nondimensional set of flow administering differential equations, fractional calculus and dimensionless variables are considered. The solution process utilizes Laplace transform and results in the acquired outputs in terms of generalized functions. The exact solutions for concentration, temperature, velocity, and the shear stress are then reduced by certain limits into fractional/traditional second-grade and Newtonian fluids as per the special cases within and out of the magnetic and porous effects. It is observed that these special cases occur in the previous published literature which verify the results of this study. The results are pictorially visualized to perform the analysis for impacts of diverse physical parameters and dimensionless quantities on concentration, temperature, and velocity fields. It is learned from the analysis that magnitude of viscoelastic parameter is directly proportional to velocity whereas the porous and magmatic effects are inversely proportional. Increasing fractional parameter values reduce flow fields of velocity and temperature. Effects of dimensionless parameters for heat and mass transfer are analysed in detail.


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## Body

## 1. Introduction

A number of technological developments and industrial applications of fluids require the complete information about their complex rheological flow, due to which the study of non-Newtonian fluids is carried out at large scale with a variety of enclosure limitations in different situations. These situations practically appear in many electrochemical, geophysical, biorheological, petroleum engineering, metallurgical, and other industrial practices like plastic and polymer melts, pulps, oil, and greases, etc. A second-grade submodel of a non-Newtonian differential type model suitably describes the complex rheological behavior of the above-mentioned fluids [1]. The analytic solutions of this type of fluid model are found in literature serving as a solution to such existing flow problem, and it supports a verification of different numerical schemes for extremely complex flows. Haq [2] investigated solutions of different flow forms of second-grade fluid with MHD Darcy’s law using an Caputo-Fabrizio derivative approach. Salman [3] calculated the values for movement of second-grade fluid over the plane having constant acceleration and presented the exact solutions as the collection of high magnitude time and transient exact solutions simply reducible to the similar solution for Newtonian fluid there by concluding that non-Newtonian effects diminish by time.For a proper and accurate investigation of all the viscoelastic properties of a fluid, fractional calculus is adopted to acquire the fractionalized form of flow administering equations [4]. Like the rate type fluids, the second-grade fluid model is passed through a modification for the purpose of generalization, that is, roughly speaking, nothing but a replacement of integer order differential operator of time derivative with one of the broadly applied fractional order differential operators as here we use the Caputo [5]. Flow produced by a heated plane moving impulsively having fractionalized second grade fluid is studied by Tassaddiq [6]; he concluded that the fractional approach is more useful for theory analysis of viscoelasticity. Riaz and Iftikhar [7] compared local and nonlocal derivative study to analyse heat transfer in MHD Maxwell fluid. Khan and Wang [8] investigated the generalized second-grade fluid flow enclosed by two perpendicular walls induced through the impulsive movement of surface over which fluid is supposed to be, and they verified their results with previous literature by vanishing the perpendicular bounding walls. Fahad et al. [9] analysed MHD second grade fluid, Newtonian heating, and Dufour effect over an infinite vertical plate with fractional mass diffusion and thermal transports using noninteger-order derivative Caputo Fabrizio (CF) with nonsingular kernel.MHD flow incorporated with porous medium is reported in many processes of the biology and medicine industry. It is utilized in chemical industry for filtration, separation, and purification processes which involve the interaction of electric or magnetic field with hydrodynamic boundary layers. Analytical results including porous and MHD terms in second-grade fluid are acquired by Ali et al. [10], where there exists the magnetohydrodynamic fluctuating free convection flow of incompressible electrically conducted viscoelastic fluid in a porous medium in the presence of a pressure gradient. Salah [11] analysed the rotationally accelerated fluid and found that variable and constant accelerated MHD flow behave equivalently. Bajwa et al. [12] found that when the transpiration parameter approaches zero, the solution for the flow with transpiration tends to the solution corresponding to the case without perspiration. Influence of hall currents with MHD and porous effects is investigated in the second grade fluid with oscillations by Hussain et al. [13].Boundary layer flow of viscoelastic non-Newtonian fluids is of fundamental importance in industrial and applied sciences. Sticking to the problem under study, heat transfer is observed due to both the exponential motion of plate and the buoyancy force that is generated by the difference of temperatures between the fluid and moving bottom plate. In practical applications, reduction of drag due to friction; paper production processes; and the cooling mechanisms of electronic, nuclear, chemical, and industrial processes involve such issues. A lot of research material is available over the flow in touch with the first layer of boundary and heat transfer for being found in much application of industrial interest. The presence stretching sheet with fluid flow of second-grade type is effected by heat of friction, heat absorbed or generated internally, and the heat in terms of deformation work is dealt by Vajravelu and Roper [14]. Tan and Masuoka [15] solved Stokes’ first problem over the high-temperature surface in a semipermeable space. They obtained the steady-state solution in the form of damping exponential function of distance to a hot plate. Concentration and chemical reaction are considered by Hayat et al. [16] with heat transfer in second-grade fluid thereby taking into account the HAM method. Hayat and Abbas [17]also obtained the solutions to same problem in the presence of MHD and porous effects. The movement of fluid of second-grade type in the presence of stretching sheet in unsteady form is discussed by Sajid et al. [18]; they addressed both processes of heating and the prescribed surface temperature, as well as prescribed surface heat flux, and obtained the analytic solutions valid for all times by using HAM method.In recent years, Baoku et al. [19] adopted numerical approach to find the approximate solutions, where they applied the Runge-Kutta-Fehlberg of order five by shooting method strategy. Das [20] did the same in addition with heated sheet of stretchable form but utilized Nachtsheim-Swigert shooting method for sixth order Runge-Kutta scheme. El-Dabe et al. [21] analysed the second-grade fluid flow with nonlinear form of transfer of both the mass and heat and discussed the thermophoresis applications in detail. Khan [22] studied heat transfer in a thin film of the steady form of flow in touch with the first layer of boundary of porous material in the presence of second-grade fluid. Wakif analysed the analytical and numerical solution of fluid flow with convection heat transfer [23–25]. Bhatti et al. [26] analysed higher-order slip flow of Eyring-Powell nanofluid for Darcy-Forchheimer in the presence of bioconvection and nonlinear thermal radiation and proved that bioconvection Lewis number declines the microorganism profile while the increasing trend is noted for higher values of slip parameter. Bhatti et al. [27] also carried out group analysis and used a robust computational approach to examine mass transport and found successive linearization approach (SLM) more efficient.This study investigates the heat and mass transfer in the MHD flow of fractionalized second-grade fluid induced by impulsively moved bottom porous plate with exponential velocity of the magnitudeK⊺d. To acquire the fractionalized nondimensional set of flow administering differential equations, fractional calculus and dimensionless variables are considered. The solution process utilizes Laplace transform and results in the acquired outputs in terms of generalized functions. The solutions in the exact form for concentration, temperature, velocity, and stress are then reduced by certain limits into fractional second-grade and Newtonian fluids as per the special cases within and out of the magnetic and porous effects. The results are pictorially presented to perform the analysis for impacts of diverse physical parameters and dimensionless quantities. This study mainly focuses on the following research questions.What is the compact form of analytical solutions of flow field in the presence of MHD and porous medium in terms of summation style and newly definedM function?What are the effects of nonlinear movement of plate over the fluid velocity, heat, and mass transfer?How the shear stress profile behaves for nonlinear movement of porous surface in presence of MHD effect?Differentiate the responses of the flow field between the linear and nonlinear/exponential movement of the porous surface (Table1).Table 1
Nomenclature.
NomenclatureυVelocity fieldms−1ŁShear stressPaℷMass concentrationKgm−3ℸTemperature distributionKνKinematic viscositym2s−1μDynamic viscosityPa.sφPorosity constantσFluid electric conductivityΩ−1m−1kFluid thermal conductivityWm−1K−1ρFluid densityKgm−3ΨDimensionless porosity parameterκPermeability of the porous mediumm2gGravitational acceleration9.8ms−2υ¯Laplace transform ofυℷ¯Laplace transform ofℷB0Magnetic field strengthTB1Volumetric heat transfer coefficientK−1B2Volumetric mass transfer coefficientCpSpecific heat capacityJm−3K−1qrHeat flux radiationWm−2DMass diffusion coefficientm2s−1FThermal radiationJα1second grade fluid parameterMHartmann numberGrThermal Grashof numberGmMass Grashof numberPrPrandtl numberScSchmidt numberℸ¯Laplace transform ofℸD⊺αCaputo fractional operator


## 2. Flow Field Description and Its Solutions

The free convection heat and mass transfer in MHD unsteady flow of second-grade fluid over an infinite vertical porous plate inx,z plane is considered, while y-axis being normal to the plate. Species concentration ℷ, temperature distribution ℸ, velocity υ in x-direction, and shear stress are considered as function of only ⊺ and ξ. Initially the plate is at rest and the fluid too with ambient fluid temperature ℸ∞ (constant) and the constant concentration ℷ∞. At the moment ⊺=0+, plate starts to move in its own plane with the nonlinear velocity K⊺d there by raising the temperature and concentration level of the plate to ℸw and ℷw, respectively. A transverse magnetic field B0 with uniform strength is normally applied to the plate in the direction parallel to y-axis as shown in Figure 1, while assuming the negligible induced magnetic field as compared to transverse magnetic field due to very low Reynold’s number. The viscous dissipation, Soret and Duoffer effects are also neglected for low level of concentration.Figure 1
Geometric view of the problem.From the above assumptions with constant Boussinesq approximation, the set of flow administrative equations for heat and mass transfer in incompressible second-grade fluid is shown [28–30].(1)∂υξ,⊺∂⊺=ν+α∂∂⊺∂2υξ,⊺∂ξ2−σB02ρυξ,⊺−φκν+α∂∂⊺υξ,⊺+gB1ℸξ,⊺−ℸ∞+gB2ℷξ,⊺−ℷ∞;ξ,⊺>0,(2)Łξ,⊺=μ+α1∂∂⊺∂υξ,⊺∂ξ;ξ,⊺>0,(3)ρCp∂ℸξ,⊺∂⊺=k∂2ℸξ,⊺∂ξ2−∂qrξ,⊺∂ξ;ξ,⊺>0,(4)∂ℷξ,⊺∂⊺=D∂2ℷξ,⊺∂ξ2;ξ,⊺>0,subject to the following initial, boundary and natural conditions, respectively.(5)υξ,0=0,Łξ,0=0,ℸξ,0=ℸ∞,ℷξ,0=ℷ∞,ξ≥0,(6)υ0,⊺=K⊺d,ℸ0,⊺=ℸw,ℷ0,⊺=ℷw,⊺>0,(7)υξ,⊺⟶0,ℸξ,⊺⟶ℸ∞,ℷξ,⊺⟶ℷ∞,asξ⟶∞,where ν,μ,σ,k,ρ,φ,κ,g,B0,B1,B2,Cp,qr, D, and α1 are kinematic viscosity, dynamic viscosity, fluid electric conductivity, fluid thermal conductivity, fluid density, porosity parameter, permeability of the porous medium, gravitational acceleration, magnetic field strength, volumetric heat transfer coefficient, volumetric mass transfer coefficient, specific heat capacity, heat flux radiation, mass diffusion coefficient, and second grade fluid parameter, respectively, whereas α=α1/ρ. By introducing the following dimensionless variables,(8)υ∗=υKνd1/2d+1,ξ∗=ξKνd1/2d+1ν,⊺∗=⊺Kνd2/2d+1ν,Ł∗=ŁKνd2/2d+1ρ,ℸ∗=ℸ−ℸ∞ℸw−ℸ∞,ℷ∗=ℷ−ℷ∞ℷw−ℷ∞,and considering(9)λ=Kνd2/2d+1αν2,M=νσB02Kνd2/2d+1ρ,Ψ=ν2φKνd2/2d+1κ,Gr=νgB1ℸ−ℸ∞Kνd3/2d+1,Gm=νgB2ℷ−ℷ∞Kνd3/2d+1κPr=μCpκ,F=ν24IKνd2/2d+1,Sc=νD,where the dimensionless quantities M, Gr, Gm, Pr, Sc, Ψ, and F represent Hartmann number, thermal Grashof number, mass Grashof number, Prandtl number, Schmidt number, thermal radiation, and porosity parameter, respectively.For the sake of brevity, we omit “∗”; thus, the results are(10)∂υξ,⊺∂⊺=1+λ∂∂⊺∂2υξ,⊺∂ξ2−Muξ,⊺−Ψ1+λ∂∂⊺υξ,⊺+Grℸξ,⊺+Gmℷξ,⊺,(11)Lξ,⊺=1+λ∂∂⊺∂υξ,⊺∂ξ,(12)∂ℸξ,⊺∂⊺=1Pr∂2ℸξ,⊺∂ξ2−FPrℸξ,⊺,(13)∂ℷξ,⊺∂⊺=1Sc∂2ℷξ,⊺∂ξ2,with imposed conditions(14)υξ,0=0,Łξ,0=0,ℸξ,0=0,ℷξ,0=0,ξ≥0,(15)υ0,⊺=⊺d,ℸ0,⊺=1,ℷ0,⊺=1,⊺>0,(16)υξ,⊺⟶0,ℸξ,⊺⟶0,ℷξ,⊺⟶0,asξ⟶∞.The fractionalized form of the governing equations is(17)∂υξ,⊺∂⊺=1+λD⊺β∂2υξ,⊺∂ξ2−Mυξ,⊺−Ψ1+λD⊺βυξ,⊺+Grℸξ,⊺+Gmℷξ,⊺,(18)Łξ,⊺=1+λD⊺β∂υξ,⊺∂ξ,(19)D⊺γℸξ,⊺=1Pr∂2ℸξ,⊺∂ξ2−FPrℸξ,⊺,(20)D⊺ηℷξ,⊺=1Sc∂2ℷξ,⊺∂ξ2,where β, η, and γ are the fractional parameters, and D⊺α is the Caputo fractional operator defined by [23, 24].(21)D⊺αf⊺=11−α∫0⊺exp−α1−α⊺−sf′sds,α≥0;df⊺d⊺,α=1.As per the above system of fractional partial differential equations (17)–(20) with (14) in mind, we take Laplace of (15) and (16).(22)υ¯0,q=Γd+1qd+1,ℸ¯0,q=1q,ℷ¯0,q=1q,(23)υ¯0,q⟶0,ℸ¯0,q⟶0,ℷ¯⟶0,asξ⟶∞,where q stands for transform parameter while image function of υξ,⊺ is represented by υ¯ξ,q.


### 2.1. Calculation of the Mass Concentration

Taking the Laplace transform of (20) and using the initial condition (14), we get(24)∂2ℷ¯ξ,⊺∂ξ2−Scqηℷ¯ξ,⊺=0.Solving (24) using the natural condition (23) and boundary condition (22), we obtained(25)ℷ¯ξ,q=exp−Scqη1/2ξq.In terms of series, the above equation can be written as(26)ℷ¯ξ,q=1q+∑1=1∞−Scξ11!qη1/2−1.Applying the inverse discrete Laplace transform, we have(27)ℷξ,⊺=1+∑ϱ1=1∞−Scξϱ1ϱ1!⊺−ηϱ1/2Γ−ηϱ1/2+1,(28)ℷξ,⊺=∑ϱ1=0∞−Sc/⊺ηξϱ1ϱ1!Γ−ηϱ1/2+1.In the form of general Wright function [31],(29)ℷξ,⊺=W−η/2,1−Sc⊺ηξwhere the general Wright function is defined as(30)Wλ,μξ=∑k=0∞zkk!Γλk+μ,λ>−1.


### 2.2. Calculation of the Temperature Distribution

Taking the Laplace transform of (19) and using the initial condition (14), we have(31)∂2ℸ¯ξ,q∂ξ2−Prqγ+Fℸ¯ξ,q=0.Solving (31) using natural condition (23) and boundary condition (22), we get(32)ℸ¯ξ,q=exp−Prqγ+F1/2ξq.In terms of series, the above equation can be given as(33)ℸ¯ξ,q=1q+∑ϱ1=1∞−Prξϱ1ϱ1!∑ϱ2=0∞Γ−ϱ1/2+ϱ2Γ−ϱ1/2−FPrϱ2qγϱ1/2−ϱ2−1.Applying the inverse discrete Laplace transform, we have(34)ℸξ,⊺=1+∑ϱ1=1∞−Prξϱ1ϱ1!∑ϱ2=0∞−F⊺γ/Prϱ2Γ−ϱ1/2+ϱ2⊺−γϱ1/2ϱ1/2ϱ2!Γ−ϱ1/2Γ−γϱ1/2+γϱ2+1.Presenting the above equation in more generalized way ofM function,(35)ℸξ,⊺=1+∑ϱ1=1∞−Prξϱ1ϱ1!M1,31,1FtγPr0,1,1+ϱ1/2,0,γϱ1/2,γ1−γϱ1/2,1,where the newly formulated M-function with the help of Fox H-function [31–33] is explained by(36)⊺yn−1∑ϱ=0∞−zϱ∏j=1mΓxj+Xjϱϱ!∏j=1nΓyj+Yjϱ=⊺yn−1Hm,n+11,mz|=Mm,n+11,mz|.


### 2.3. Calculation of the Velocity

Initial conditions (14) are imposed after taking Laplace transform of (17), which yield the results(37)∂2υ¯ξ,q∂ξ2−q+M1+λβqβ+Ψυ¯ξ,q=−Grqexp−Prqγ+F1/2ξ1+λβqβ−Gmqexp−Scqη1/2ξ1+λβqβ,Solving (37) utilizing conditions (22) and (23), we get(38)υ¯ξ,q=Γd+1qd+1+Grq1+λβqβ−1Prqγ+F−q+M1+λβqβ−1+Ψ+Gmq+λβqβ−1Scqη−q+M1+λβqβ−1+Ψ×exp−q+M1+λβqβ−1+Ψ1/2ξ−Grq1+λβqβ−1Prqγ+F−q+M1+λβqβ−1+Ψexp−Prqγ+F1/2ξ−Gmq1+λβqβ−1Scqη−q+M1+λβqβ−1+Ψexp−Scqη1/2ξ.Presenting (38) in series form to easily produce υξ,⊺=L−1υ¯ξ,q without prolix computational complexities of residuals and contours integrals,(39)υ¯ξ,q=Γd+1qd+1+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2+ϱ3×∑ϱ6=0∞−1/λβϱ6Γ−ϱ1/2+ϱ3Γ−ϱ1/2+ϱ3+ϱ4Γϱ1/2ϱ1/2−ϱ3+ϱ6ϱ6!Γ−ϱ1/2Γ−ϱ1/2+ϱ3Γϱ1/2−ϱ3q1−βϱ1/2−3−ϱ4−βϱ6−b−1+Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1/2−ϱ2+ϱ3−1×∑ϱ6=0∞−1/λβ6Γ−ϱ1/2−ϱ2+ϱ3Γ−ϱ1/2−ϱ2+ϱ3+ϱ4Γϱ2+ϱ5+1Γϱ1/2+ϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ1/2−ϱ2Γ−ϱ1/2−ϱ2+ϱ3Γϱ2+1Γϱ1/2+ϱ2−ϱ3+1×q1−βϱ1/2+ϱ2−ϱ3−β−ϱ4−βϱ6−γϱ2+ϱ5+1−1+Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2−ϱ2+ϱ3−1×∑ϱ6=0∞−1/λβϱ6Γ−ϱ1/2−ϱ2+ϱ3Γ−ϱ1/2−ϱ2+ϱ3+ϱ4Γϱ1/2+ϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ1/2−ϱ2Γ−ϱ1/2−ϱ2+ϱ3Γϱϱ1/2+ϱ2−ϱ3+1×q1−βϱ1/2+ϱ2−ϱ3−β−ϱ4−βϱ6−ηϱ2+1−1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3−1×∑ϱ6=0∞−1/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γ−ϱ1/2+ϱ2+ϱ5+1Γϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γ−ϱ1/2+ϱ2+1Γϱ2−ϱ3+1×q1−βϱ2−ϱ3−β−ϱ4−βϱ6+γϱ1/2−ϱ2−ϱ5−1−1−Gm∑1,2,3,4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3−1×∑ϱ6=0∞−1/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γϱ2−ϱ3+1q1−βϱ2−ϱ3−β−ϱ4−βϱ6+ηϱ1/2−ϱ2−1−1.Practicing the discrete inverse Laplace transform, we obtain(40)υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2+3⊺β−1ϱ1/2−ϱ3+ϱ4+b×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ1/2+ϱ3Γ−ϱ1/2+ϱ3+ϱ4Γϱ1/2−ϱ3+ϱ6ϱ6!Γ−ϱ1/2Γ−ϱ1/2+ϱ3Γϱ1/2−ϱ3Γβ−1ϱ1/2−ϱ3+ϱ4+βϱ6+b+1+Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1/2−ϱ2+ϱ3−1×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ1/2−ϱ2+ϱ3Γ−ϱ1/2−ϱ2+ϱ3+ϱ4Γϱ2+ϱ5+1Γϱ1/2+ϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ1/2−ϱ2Γ−ϱ1/2−ϱ2+ϱ3Γϱ2+1Γϱ1/2+ϱ2−ϱ3+1×⊺β−1ϱ1/2+ϱ2−ϱ3+β+ϱ4+γϱ2+ϱ5+1Γβ−1ϱ1/2+ϱ2−ϱ3+β+ϱ4+βϱ6+γϱ2+ϱ5+1+1+Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2−ϱ2+ϱ3−1×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ1/2−ϱ2+ϱ3Γ−ϱ1/2−ϱ2+ϱ3+ϱ4Γϱ1/2+ϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ1/2−ϱ2Γ−ϱ1/2−ϱ2+ϱ3Γϱ1/2+ϱ2−ϱ3+1×⊺β−1ϱ1/2+ϱ2−ϱ3+β+ϱ4+ηϱ2+1Γβ−1ϱ1/2+ϱ2−ϱ3+β+ϱ4+βϱ6+ηϱ2+1+1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3−1×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γ−ϱ1/2+ϱ2+ϱ5+1Γϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γ−ϱ1/2+ϱ2+1Γϱ2−ϱ3+1×⊺β−1ϱ2−ϱ3+β+ϱ4−γϱ1/2−ϱ2−ϱ5−1Γβ−1ϱ2−ϱ3+β+ϱ4+βϱ6−γϱ1/2−ϱ2−ϱ5−1+1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3−1⊺β−1ϱ2−ϱ3+β+ϱ4−ηϱ1/2−ϱ2−1×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γϱ2−ϱ3+1Γβ−1ϱ2−ϱ3+β+ϱ4+βϱ6−ηϱ1/2−ϱ2−1+1.Rewriting the velocity expression using generalized M-function,(41)υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2+ϱ3∈×M3,51,3⊺βλβ|0,1,1+ϱ1/2,0,1+ϱ1/2−ϱ3,0,1−ϱ1/2+ϱ3,0,1−βϱ1/2−ϱ3−ϱ4−b,β1+ϱ1/2−ϱ3,0,1+ϱ1/2−ϱ3−ϱ4,0,1−ϱ1/2+ϱ3,1+Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1/2−ϱ2+ϱ3−1×M4,61,4⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,−ϱ2,0,−ϱ1/2−ϱ2+ϱ3,0,1−βϱ1/2+ϱ2−ϱ3−β−ϱ4−γϱ2+ϱ5+1,β1+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3−ϱ4,0,−ϱ2−ϱ5,0−ϱ1/2−ϱ2+ϱ3,1+Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,−ϱ1/2−ϱ2+ϱ3,0,1−βϱ1/2+ϱ2−ϱ3−β−ϱ4−ηϱ2+1,β1+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3−ϱ4,0,−ϱ1/2−ϱ2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3−1×M4,61,4⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,1+ϱ1/2−ϱ2,0,−ϱ2+ϱ3,0,1−βϱ2−ϱ5−β−ϱ4−γϱ1/2−ϱ2−ϱ5−1,β1+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1+ϱ1/2−ϱ2−ϱ5,0,−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,−k2+ϱ3,0,1−βϱ2−ϱ5−β−ϱ4−ηϱ1/2−ϱ2−1,β1+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,−ϱ2+ϱ3,1.


### 2.4. Formulating Shear Stress

Initial conditions (14) are imposed after taking Laplace transform of Equation (18) yielding the results(42)Ł¯ξ,q=1+λβqβ∂υ¯ξ,q∂ξ,where the Laplace transform of Łξ,⊺ is given as Ł¯ξ,q. With the help of (38), the given expression can be rearranged as(43)Ł¯ξ,q=Γd+1qd+11+λβqβ+Grq1Prqγ+F−q+M1+λβqβ−1+Ψ+Gmq1Scqη−q+M1+λβqβ−1+Ψ−q+M1+λβqβ−1+Ψ1/2×exp−q+M1+λβqβ−1+Ψ1/2ξ+GrqPrqγ+F1/2Prqγ+F−q+M1+λβqβ−1+Ψexp−Prqγ+F1/2ξ+GmqScqη1/2Scqη−q+M1+λβqβ−1+Ψexp−Scqη1/2ξ.Writing the given expression in the form of a series(44)Ł¯ξ,q=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1+1/2+ϱ3×∑ϱ6=0∞−1/λβϱ6Γ−ϱ1+1/2+ϱ3Γ−ϱ1+1/2+ϱ3+ϱ4Γϱ1+1/2−ϱ3+ϱ6ϱ6!Γ−ϱ1+1/2Γ−ϱ1+1/2+ϱ3Γϱ1+1/2−ϱ3q1−βϱ1/2−ϱ3+1/21−β−ϱ4−βϱ6−b−1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1+1/2−ϱ2+ϱ3×∑ϱ6=0∞−1/λβϱ6Γ−ϱ1+1/2−ϱ2+ϱ3Γ−ϱ1+1/2−ϱ2+ϱ3+ϱ4Γϱ2+ϱ5+1Γϱ1+1/2+ϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ1+1/2−ϱ2Γ−ϱ1+1/2−ϱ2+ϱ3Γϱ2+1Γϱ1+1/2+ϱ2−ϱ3q1−βϱ1+1/2+ϱ2−ϱ3−ϱ4−βϱ6−γϱ2+ϱ5+1−1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1+1/2−ϱ2+ϱ3×∑ϱ6=0∞−1/λβϱ6Γ−ϱ1+1/2−ϱ2+ϱ3Γ−ϱ1+1/2−ϱ2+ϱ3+ϱ4Γϱ1+1/2+ϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ1+1/2−ϱ2Γ−ϱ1+1/2−ϱ2+ϱ3Γϱ1+1/2+ϱ2−ϱ3q1−βϱ1+1/2+ϱ2−ϱ3−ϱ4−βϱ6−ηϱ2+1−1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3×∑ϱ6=0∞−1/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γ−ϱ1+1/2+ϱ2+ϱ5+1Γϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γ−ϱ1+1/2+ϱ2+1Γϱ2−ϱ3q1−βϱ2−ϱ3−ϱ4−βϱ6+γϱ1+1/2−ϱ2−ϱ5−1−1+GmSc∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3×∑ϱ6=0∞−1/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γϱ2−ϱ3q1−βϱ2−ϱ3−ϱ4−βϱ6+ηϱ1+1/2−ϱ2−1−1.Inverting the Laplace transform results,(45)Łξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1+1/2+ϱ3⊺β−1ϱ1/2−ϱ3−1/2β−1+ϱ4+b×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ1+1/2+ϱ3Γ−ϱ1+1/2+ϱ3+ϱ4Γϱ1+1/2−ϱ3+ϱ6ϱ6!Γ−ϱ1+1/2Γ−ϱ1+1/2+ϱ3Γϱ1+1/2−ϱ3Γβ−1ϱ1/2−ϱ3−1/2β+1+ϱ4+βϱ6+b+1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1+1/2−ϱ2+ϱ3×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ1+1/2−ϱ2+ϱ3Γ−ϱ1+1/2−ϱ2+ϱ3+ϱ4Γϱ2+ϱ5+1Γϱ1+1/2+ϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ1+1/2−ϱ2Γ−ϱ1+1/2−ϱ2+ϱ3Γϱ2+1Γϱ1+1/2+ϱ2−ϱ3×⊺β−1ϱ1+1/2+ϱ2−ϱ3−1/2β+1+ϱ4+γϱ2+ϱ5+1Γβ−1ϱ1/2+ϱ2−ϱ3+ϱ4+βϱ6+γϱ2+ϱ5+1+1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1+1/2−ϱ2+ϱ3×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ1+1/2−ϱ2+ϱ3Γ−ϱ1+1/2−ϱ2+ϱ3+ϱ4Γϱ1+1/2+ϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ1+1/2−ϱ2Γ−ϱ1+1/2−ϱ2+ϱ3Γϱ1+1/2+ϱ2−ϱ3×⊺β−1ϱ1/2+ϱ2+ϱ3+ϱ4+ηϱ2+1Γβ−1ϱ1/2+ϱ2+ϱ3+ϱ4+βϱ6+ηϱ2+1+1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γ−ϱ1+1/2+ϱ2+ϱ5+1Γϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γ−ϱ1+1/2+ϱ2+1Γϱ2−ϱ3×⊺β−1ϱ2−ϱ3+ϱ4−γϱ1+1/2−ϱ2−ϱ5−1Γβ−1ϱ2−ϱ3+ϱ4+βϱ6−γϱ1+1/2−ϱ2−ϱ5−1+1+GmSc∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3⊺β−1ϱ2−ϱ3+ϱ4−ηϱ1+1/2−ϱ2−1×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γϱ2−ϱ3Γβ−1ϱ2−ϱ3−β+ϱ4+βϱ6−ηϱ1+1/2−ϱ2−1+1.In terms of M-function(46)Łξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1+1/2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ1+1/2,0,1+ϱ1+1/2−ϱ3,0,1−ϱ1+1/2+ϱ3,0,1−βϱ1/2−ϱ3+1/21+β−ϱ4−b,β1+ϱ1+1/2−ϱ3,0,1+ϱ1+1/2−ϱ3−ϱ4,0,1−ϱ1+1/2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1+1/2−ϱ2+ϱ3×M4,61,4⊺βλβ|0,1,1+ϱ1+1/2+ϱ2,0,1+ϱ1+1/2+ϱ2−ϱ3,0,−ϱ2,0,1−ϱ1+1/2−ϱ2+ϱ3,0,1−βϱ1+1/2+ϱ2−ϱ3−ϱ4−γϱ2+ϱ5+1,β1+ϱ1+1/2+ϱ2−ϱ3,0,1+ϱ1+1/2+ϱ2−ϱ3−ϱ4,0,−ϱ2−ϱ5,01−ϱ1+1/2−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1+1/2−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ1+1/2+ϱ2,0,1+ϱ1+1/2+ϱ2−ϱ3,0,1−ϱ1+1/2−ϱ2+ϱ3,0,1−βϱ1+1/2+ϱ2−ϱ3−ϱ4−ηϱ2+1,β1+ϱ1+1/2+ϱ2−ϱ3,0,1+ϱ1+1/2+ϱ2−ϱ3−ϱ4,01−ϱ1+1/2−ϱ2+ϱ3,1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3×M4,61,4⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,ϱ1+1/2−ϱ2,0,1−ϱ2+ϱ3,0,1−βϱ2−ϱ3−ϱ4+γϱ1+1/2−ϱ2−ϱ5−1,β1+ϱ3−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1+ϱ1+1/2−ϱ2−ϱ5,0,1−ϱ2+ϱ3,1+GmSc∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,1−ϱ2+ϱ3,0,1−βϱ2−ϱ3−ϱ4+ηϱ1+1/2−ϱ2−1,β1+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1−ϱ2+ϱ3,1.


## 2.1. Calculation of the Mass Concentration

Taking the Laplace transform of (20) and using the initial condition (14), we get(24)∂2ℷ¯ξ,⊺∂ξ2−Scqηℷ¯ξ,⊺=0.Solving (24) using the natural condition (23) and boundary condition (22), we obtained(25)ℷ¯ξ,q=exp−Scqη1/2ξq.In terms of series, the above equation can be written as(26)ℷ¯ξ,q=1q+∑1=1∞−Scξ11!qη1/2−1.Applying the inverse discrete Laplace transform, we have(27)ℷξ,⊺=1+∑ϱ1=1∞−Scξϱ1ϱ1!⊺−ηϱ1/2Γ−ηϱ1/2+1,(28)ℷξ,⊺=∑ϱ1=0∞−Sc/⊺ηξϱ1ϱ1!Γ−ηϱ1/2+1.In the form of general Wright function [31],(29)ℷξ,⊺=W−η/2,1−Sc⊺ηξwhere the general Wright function is defined as(30)Wλ,μξ=∑k=0∞zkk!Γλk+μ,λ>−1.


## 2.2. Calculation of the Temperature Distribution

Taking the Laplace transform of (19) and using the initial condition (14), we have(31)∂2ℸ¯ξ,q∂ξ2−Prqγ+Fℸ¯ξ,q=0.Solving (31) using natural condition (23) and boundary condition (22), we get(32)ℸ¯ξ,q=exp−Prqγ+F1/2ξq.In terms of series, the above equation can be given as(33)ℸ¯ξ,q=1q+∑ϱ1=1∞−Prξϱ1ϱ1!∑ϱ2=0∞Γ−ϱ1/2+ϱ2Γ−ϱ1/2−FPrϱ2qγϱ1/2−ϱ2−1.Applying the inverse discrete Laplace transform, we have(34)ℸξ,⊺=1+∑ϱ1=1∞−Prξϱ1ϱ1!∑ϱ2=0∞−F⊺γ/Prϱ2Γ−ϱ1/2+ϱ2⊺−γϱ1/2ϱ1/2ϱ2!Γ−ϱ1/2Γ−γϱ1/2+γϱ2+1.Presenting the above equation in more generalized way ofM function,(35)ℸξ,⊺=1+∑ϱ1=1∞−Prξϱ1ϱ1!M1,31,1FtγPr0,1,1+ϱ1/2,0,γϱ1/2,γ1−γϱ1/2,1,where the newly formulated M-function with the help of Fox H-function [31–33] is explained by(36)⊺yn−1∑ϱ=0∞−zϱ∏j=1mΓxj+Xjϱϱ!∏j=1nΓyj+Yjϱ=⊺yn−1Hm,n+11,mz|=Mm,n+11,mz|.


## 2.3. Calculation of the Velocity

Initial conditions (14) are imposed after taking Laplace transform of (17), which yield the results(37)∂2υ¯ξ,q∂ξ2−q+M1+λβqβ+Ψυ¯ξ,q=−Grqexp−Prqγ+F1/2ξ1+λβqβ−Gmqexp−Scqη1/2ξ1+λβqβ,Solving (37) utilizing conditions (22) and (23), we get(38)υ¯ξ,q=Γd+1qd+1+Grq1+λβqβ−1Prqγ+F−q+M1+λβqβ−1+Ψ+Gmq+λβqβ−1Scqη−q+M1+λβqβ−1+Ψ×exp−q+M1+λβqβ−1+Ψ1/2ξ−Grq1+λβqβ−1Prqγ+F−q+M1+λβqβ−1+Ψexp−Prqγ+F1/2ξ−Gmq1+λβqβ−1Scqη−q+M1+λβqβ−1+Ψexp−Scqη1/2ξ.Presenting (38) in series form to easily produce υξ,⊺=L−1υ¯ξ,q without prolix computational complexities of residuals and contours integrals,(39)υ¯ξ,q=Γd+1qd+1+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2+ϱ3×∑ϱ6=0∞−1/λβϱ6Γ−ϱ1/2+ϱ3Γ−ϱ1/2+ϱ3+ϱ4Γϱ1/2ϱ1/2−ϱ3+ϱ6ϱ6!Γ−ϱ1/2Γ−ϱ1/2+ϱ3Γϱ1/2−ϱ3q1−βϱ1/2−3−ϱ4−βϱ6−b−1+Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1/2−ϱ2+ϱ3−1×∑ϱ6=0∞−1/λβ6Γ−ϱ1/2−ϱ2+ϱ3Γ−ϱ1/2−ϱ2+ϱ3+ϱ4Γϱ2+ϱ5+1Γϱ1/2+ϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ1/2−ϱ2Γ−ϱ1/2−ϱ2+ϱ3Γϱ2+1Γϱ1/2+ϱ2−ϱ3+1×q1−βϱ1/2+ϱ2−ϱ3−β−ϱ4−βϱ6−γϱ2+ϱ5+1−1+Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2−ϱ2+ϱ3−1×∑ϱ6=0∞−1/λβϱ6Γ−ϱ1/2−ϱ2+ϱ3Γ−ϱ1/2−ϱ2+ϱ3+ϱ4Γϱ1/2+ϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ1/2−ϱ2Γ−ϱ1/2−ϱ2+ϱ3Γϱϱ1/2+ϱ2−ϱ3+1×q1−βϱ1/2+ϱ2−ϱ3−β−ϱ4−βϱ6−ηϱ2+1−1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3−1×∑ϱ6=0∞−1/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γ−ϱ1/2+ϱ2+ϱ5+1Γϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γ−ϱ1/2+ϱ2+1Γϱ2−ϱ3+1×q1−βϱ2−ϱ3−β−ϱ4−βϱ6+γϱ1/2−ϱ2−ϱ5−1−1−Gm∑1,2,3,4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3−1×∑ϱ6=0∞−1/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γϱ2−ϱ3+1q1−βϱ2−ϱ3−β−ϱ4−βϱ6+ηϱ1/2−ϱ2−1−1.Practicing the discrete inverse Laplace transform, we obtain(40)υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2+3⊺β−1ϱ1/2−ϱ3+ϱ4+b×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ1/2+ϱ3Γ−ϱ1/2+ϱ3+ϱ4Γϱ1/2−ϱ3+ϱ6ϱ6!Γ−ϱ1/2Γ−ϱ1/2+ϱ3Γϱ1/2−ϱ3Γβ−1ϱ1/2−ϱ3+ϱ4+βϱ6+b+1+Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1/2−ϱ2+ϱ3−1×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ1/2−ϱ2+ϱ3Γ−ϱ1/2−ϱ2+ϱ3+ϱ4Γϱ2+ϱ5+1Γϱ1/2+ϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ1/2−ϱ2Γ−ϱ1/2−ϱ2+ϱ3Γϱ2+1Γϱ1/2+ϱ2−ϱ3+1×⊺β−1ϱ1/2+ϱ2−ϱ3+β+ϱ4+γϱ2+ϱ5+1Γβ−1ϱ1/2+ϱ2−ϱ3+β+ϱ4+βϱ6+γϱ2+ϱ5+1+1+Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2−ϱ2+ϱ3−1×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ1/2−ϱ2+ϱ3Γ−ϱ1/2−ϱ2+ϱ3+ϱ4Γϱ1/2+ϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ1/2−ϱ2Γ−ϱ1/2−ϱ2+ϱ3Γϱ1/2+ϱ2−ϱ3+1×⊺β−1ϱ1/2+ϱ2−ϱ3+β+ϱ4+ηϱ2+1Γβ−1ϱ1/2+ϱ2−ϱ3+β+ϱ4+βϱ6+ηϱ2+1+1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3−1×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γ−ϱ1/2+ϱ2+ϱ5+1Γϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γ−ϱ1/2+ϱ2+1Γϱ2−ϱ3+1×⊺β−1ϱ2−ϱ3+β+ϱ4−γϱ1/2−ϱ2−ϱ5−1Γβ−1ϱ2−ϱ3+β+ϱ4+βϱ6−γϱ1/2−ϱ2−ϱ5−1+1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3−1⊺β−1ϱ2−ϱ3+β+ϱ4−ηϱ1/2−ϱ2−1×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γϱ2−ϱ3+ϱ6+1ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γϱ2−ϱ3+1Γβ−1ϱ2−ϱ3+β+ϱ4+βϱ6−ηϱ1/2−ϱ2−1+1.Rewriting the velocity expression using generalized M-function,(41)υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2+ϱ3∈×M3,51,3⊺βλβ|0,1,1+ϱ1/2,0,1+ϱ1/2−ϱ3,0,1−ϱ1/2+ϱ3,0,1−βϱ1/2−ϱ3−ϱ4−b,β1+ϱ1/2−ϱ3,0,1+ϱ1/2−ϱ3−ϱ4,0,1−ϱ1/2+ϱ3,1+Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1/2−ϱ2+ϱ3−1×M4,61,4⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,−ϱ2,0,−ϱ1/2−ϱ2+ϱ3,0,1−βϱ1/2+ϱ2−ϱ3−β−ϱ4−γϱ2+ϱ5+1,β1+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3−ϱ4,0,−ϱ2−ϱ5,0−ϱ1/2−ϱ2+ϱ3,1+Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,−ϱ1/2−ϱ2+ϱ3,0,1−βϱ1/2+ϱ2−ϱ3−β−ϱ4−ηϱ2+1,β1+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3−ϱ4,0,−ϱ1/2−ϱ2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3−1×M4,61,4⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,1+ϱ1/2−ϱ2,0,−ϱ2+ϱ3,0,1−βϱ2−ϱ5−β−ϱ4−γϱ1/2−ϱ2−ϱ5−1,β1+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1+ϱ1/2−ϱ2−ϱ5,0,−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,−k2+ϱ3,0,1−βϱ2−ϱ5−β−ϱ4−ηϱ1/2−ϱ2−1,β1+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,−ϱ2+ϱ3,1.


## 2.4. Formulating Shear Stress

Initial conditions (14) are imposed after taking Laplace transform of Equation (18) yielding the results(42)Ł¯ξ,q=1+λβqβ∂υ¯ξ,q∂ξ,where the Laplace transform of Łξ,⊺ is given as Ł¯ξ,q. With the help of (38), the given expression can be rearranged as(43)Ł¯ξ,q=Γd+1qd+11+λβqβ+Grq1Prqγ+F−q+M1+λβqβ−1+Ψ+Gmq1Scqη−q+M1+λβqβ−1+Ψ−q+M1+λβqβ−1+Ψ1/2×exp−q+M1+λβqβ−1+Ψ1/2ξ+GrqPrqγ+F1/2Prqγ+F−q+M1+λβqβ−1+Ψexp−Prqγ+F1/2ξ+GmqScqη1/2Scqη−q+M1+λβqβ−1+Ψexp−Scqη1/2ξ.Writing the given expression in the form of a series(44)Ł¯ξ,q=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1+1/2+ϱ3×∑ϱ6=0∞−1/λβϱ6Γ−ϱ1+1/2+ϱ3Γ−ϱ1+1/2+ϱ3+ϱ4Γϱ1+1/2−ϱ3+ϱ6ϱ6!Γ−ϱ1+1/2Γ−ϱ1+1/2+ϱ3Γϱ1+1/2−ϱ3q1−βϱ1/2−ϱ3+1/21−β−ϱ4−βϱ6−b−1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1+1/2−ϱ2+ϱ3×∑ϱ6=0∞−1/λβϱ6Γ−ϱ1+1/2−ϱ2+ϱ3Γ−ϱ1+1/2−ϱ2+ϱ3+ϱ4Γϱ2+ϱ5+1Γϱ1+1/2+ϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ1+1/2−ϱ2Γ−ϱ1+1/2−ϱ2+ϱ3Γϱ2+1Γϱ1+1/2+ϱ2−ϱ3q1−βϱ1+1/2+ϱ2−ϱ3−ϱ4−βϱ6−γϱ2+ϱ5+1−1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1+1/2−ϱ2+ϱ3×∑ϱ6=0∞−1/λβϱ6Γ−ϱ1+1/2−ϱ2+ϱ3Γ−ϱ1+1/2−ϱ2+ϱ3+ϱ4Γϱ1+1/2+ϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ1+1/2−ϱ2Γ−ϱ1+1/2−ϱ2+ϱ3Γϱ1+1/2+ϱ2−ϱ3q1−βϱ1+1/2+ϱ2−ϱ3−ϱ4−βϱ6−ηϱ2+1−1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3×∑ϱ6=0∞−1/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γ−ϱ1+1/2+ϱ2+ϱ5+1Γϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γ−ϱ1+1/2+ϱ2+1Γϱ2−ϱ3q1−βϱ2−ϱ3−ϱ4−βϱ6+γϱ1+1/2−ϱ2−ϱ5−1−1+GmSc∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3×∑ϱ6=0∞−1/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γϱ2−ϱ3q1−βϱ2−ϱ3−ϱ4−βϱ6+ηϱ1+1/2−ϱ2−1−1.Inverting the Laplace transform results,(45)Łξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1+1/2+ϱ3⊺β−1ϱ1/2−ϱ3−1/2β−1+ϱ4+b×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ1+1/2+ϱ3Γ−ϱ1+1/2+ϱ3+ϱ4Γϱ1+1/2−ϱ3+ϱ6ϱ6!Γ−ϱ1+1/2Γ−ϱ1+1/2+ϱ3Γϱ1+1/2−ϱ3Γβ−1ϱ1/2−ϱ3−1/2β+1+ϱ4+βϱ6+b+1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1+1/2−ϱ2+ϱ3×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ1+1/2−ϱ2+ϱ3Γ−ϱ1+1/2−ϱ2+ϱ3+ϱ4Γϱ2+ϱ5+1Γϱ1+1/2+ϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ1+1/2−ϱ2Γ−ϱ1+1/2−ϱ2+ϱ3Γϱ2+1Γϱ1+1/2+ϱ2−ϱ3×⊺β−1ϱ1+1/2+ϱ2−ϱ3−1/2β+1+ϱ4+γϱ2+ϱ5+1Γβ−1ϱ1/2+ϱ2−ϱ3+ϱ4+βϱ6+γϱ2+ϱ5+1+1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1+1/2−ϱ2+ϱ3×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ1+1/2−ϱ2+ϱ3Γ−ϱ1+1/2−ϱ2+ϱ3+ϱ4Γϱ1+1/2+ϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ1+1/2−ϱ2Γ−ϱ1+1/2−ϱ2+ϱ3Γϱ1+1/2+ϱ2−ϱ3×⊺β−1ϱ1/2+ϱ2+ϱ3+ϱ4+ηϱ2+1Γβ−1ϱ1/2+ϱ2+ϱ3+ϱ4+βϱ6+ηϱ2+1+1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γ−ϱ1+1/2+ϱ2+ϱ5+1Γϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γ−ϱ1+1/2+ϱ2+1Γϱ2−ϱ3×⊺β−1ϱ2−ϱ3+ϱ4−γϱ1+1/2−ϱ2−ϱ5−1Γβ−1ϱ2−ϱ3+ϱ4+βϱ6−γϱ1+1/2−ϱ2−ϱ5−1+1+GmSc∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3⊺β−1ϱ2−ϱ3+ϱ4−ηϱ1+1/2−ϱ2−1×∑ϱ6=0∞−⊺β/λβϱ6Γ−ϱ2+ϱ3Γ−ϱ2+ϱ3+ϱ4Γϱ2−ϱ3+ϱ6ϱ6!Γ−ϱ2Γ−ϱ2+ϱ3Γϱ2−ϱ3Γβ−1ϱ2−ϱ3−β+ϱ4+βϱ6−ηϱ1+1/2−ϱ2−1+1.In terms of M-function(46)Łξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1+1/2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ1+1/2,0,1+ϱ1+1/2−ϱ3,0,1−ϱ1+1/2+ϱ3,0,1−βϱ1/2−ϱ3+1/21+β−ϱ4−b,β1+ϱ1+1/2−ϱ3,0,1+ϱ1+1/2−ϱ3−ϱ4,0,1−ϱ1+1/2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1+1/2−ϱ2+ϱ3×M4,61,4⊺βλβ|0,1,1+ϱ1+1/2+ϱ2,0,1+ϱ1+1/2+ϱ2−ϱ3,0,−ϱ2,0,1−ϱ1+1/2−ϱ2+ϱ3,0,1−βϱ1+1/2+ϱ2−ϱ3−ϱ4−γϱ2+ϱ5+1,β1+ϱ1+1/2+ϱ2−ϱ3,0,1+ϱ1+1/2+ϱ2−ϱ3−ϱ4,0,−ϱ2−ϱ5,01−ϱ1+1/2−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1+1/2−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ1+1/2+ϱ2,0,1+ϱ1+1/2+ϱ2−ϱ3,0,1−ϱ1+1/2−ϱ2+ϱ3,0,1−βϱ1+1/2+ϱ2−ϱ3−ϱ4−ηϱ2+1,β1+ϱ1+1/2+ϱ2−ϱ3,0,1+ϱ1+1/2+ϱ2−ϱ3−ϱ4,01−ϱ1+1/2−ϱ2+ϱ3,1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3×M4,61,4⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,ϱ1+1/2−ϱ2,0,1−ϱ2+ϱ3,0,1−βϱ2−ϱ3−ϱ4+γϱ1+1/2−ϱ2−ϱ5−1,β1+ϱ3−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1+ϱ1+1/2−ϱ2−ϱ5,0,1−ϱ2+ϱ3,1+GmSc∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,1−ϱ2+ϱ3,0,1−βϱ2−ϱ3−ϱ4+ηϱ1+1/2−ϱ2−1,β1+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1−ϱ2+ϱ3,1.


## 3. Some Interesting Particularized Cases

### 3.1. Ordinary MHD Second-Grade Fluid in Porous Material

Consideringβ⟶0,γ⟶0 and η⟶0, in equations (29), (35), (40), and (46), the obtained results are(47)ℷξ,⊺=W−1/2,1−Sc⊺ξ,ℸξ,⊺=1+∑ϱ1=1∞−Prξϱ1ϱ1!M1,31,1FtPr|0,1,1+ϱ1/2,0,ϱ1/2,11−ϱ1/2,1,υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λ−ϱ1/2+ϱ3×M3,51,3⊺λ|0,1,1+ϱ1/2,0,1+ϱ1/2−ϱ3,0,1−ϱ1/2+ϱ3,0,−ϱ4−b,11+ϱ1/2−ϱ3,0,1+ϱ1/2−ϱ3−ϱ4,0,1−ϱ1/2+ϱ3,1+Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λ−ϱ1/2−ϱ2+ϱ3−1×M4,61,4⊺λ|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,−ϱ2,0,−ϱ1/2−ϱ2+ϱ3,0,−1−ϱ4−ϱ2+ϱ5+1,11+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3−ϱ4,0,−ϱ2−ϱ5,0−ϱ1/2−ϱ2+ϱ3,1+Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λ−ϱ1/2−ϱ2+ϱ3−1×M3,51,3⊺λ|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,−ϱ1/2−ϱ2+ϱ3,0,−1−ϱ4−ϱ2+1,11+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3−ϱ4,0,−ϱ1/2−ϱ2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λ−ϱ2+ϱ3−1×M4,61,4⊺λ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,1+ϱ1/2−ϱ2,0,−ϱ2+ϱ3,0,−1−ϱ4−ϱ1/2−ϱ2−ϱ5−1,11+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1+ϱ1/2−ϱ2−ϱ5,0,−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λ−ϱ2+ϱ3−1×M3,51,3⊺λ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,−ϱ2+ϱ3,0,−1−ϱ4−ϱ1/2−ϱ2−1,11+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,−ϱ2+ϱ3,1,and shear stress is(48)Łξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λ−ϱ1−1/2+ϱ3×M3,51,3⊺λ|0,1,1+ϱ1−1/2,0,1+ϱ1−1/2−ϱ3,0,1−ϱ1−1/2+ϱ3,0,1−ϱ4−b,11+ϱ1−1/2−ϱ3,0,1+ϱ1−1/2−ϱ3−ϱ4,0,1−ϱ1−1/2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λ−ϱ1−1/2−ϱ2+ϱ3×M4,61,4⊺λ|0,1,1+ϱ1−1/2+ϱ2,0,1+ϱ1−1/2+ϱ2−ϱ3,0,−ϱ2,0,1−ϱ1−1/2−ϱ2+ϱ3,0,−ϱ4−ϱ2+ϱ5+1,11+ϱ1−1/2+ϱ2−ϱ3,0,1+ϱ1−1/2+ϱ2−ϱ3−ϱ4,0,−ϱ2−ϱ5,01−ϱ1−1/2−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λ−ϱ1−1/2−ϱ2+ϱ3×M3,51,3⊺λ|0,1,1+ϱ1−1/2+ϱ2,0,1+ϱ1−1/2+ϱ2−ϱ3,0,1−ϱ1−1/2−ϱ2+ϱ3,0,−ϱ4−ϱ2+1,11+ϱ1−1/2+ϱ2−ϱ3,0,1+ϱ1−1/2+ϱ2−ϱ3−ϱ4,01−ϱ1−1/2−ϱ2+ϱ3,1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λ−ϱ2+ϱ3×M4,61,4⊺λ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,ϱ1−1/2−ϱ2,0,1−ϱ2+ϱ3,0,−ϱ4+ϱ1−1/2−ϱ2−ϱ5−1,11+ϱ3−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1+ϱ1−1/2−ϱ2−ϱ5,0,1−ϱ2+ϱ3,1+GmSc∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λ−ϱ2+ϱ3×M3,51,3⊺λ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,1−ϱ2+ϱ3,0,−ϱ4+ϱ1−1/2−ϱ2−1,11+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1−ϱ2+ϱ3,1.


### 3.2. Fractionalized MHD Second-Grade Fluid in Porous Medium with Constant Radiative Heat Flux

For the constant radiative heat flux alongy-direction, qr= constant or F⟶0, the flow results are(49)ℷξ,⊺=W−η2,1−Sc⊺ηξ,ℸξ,⊺=W−γ2,1−Pr⊺γξ,with velocity(50)υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ12,0,1+ϱ12−ϱ3,0,1−ϱ12+ϱ3,0,1−βϱ1/2−ϱ3−ϱ4−b,β1+ϱ1/2−ϱ3,0,1+ϱ1/2−ϱ3−ϱ4,0,1−ϱ1/2+ϱ3,1+Gr∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,−ϱ1/2−ϱ2+ϱ3,0,1−βϱ1/2+ϱ2−ϱ3−β−ϱ4−γϱ2+1,β1+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3−ϱ4,0−ϱ1/2−ϱ2+ϱ3,1+Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,−ϱ1/2−ϱ2+ϱ3,0,1−βϱ1/2+ϱ2−ϱ3−β−ϱ4−ηϱ2+1,β1+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3−ϱ4,0,−ϱ1/2−ϱ2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,−ϱ2+ϱ3,0,1−βϱ2−ϱ3−β−ϱ4−γϱ1/2−ϱ2−1,β1+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,−k2+ϱ3,0,1−βϱ2−ϱ5−β−ϱ4−ηϱ1/2−ϱ2−1,β1+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,−ϱ2+ϱ3,1,and the respective shear stress is(51)Lξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1−1/2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ1−1/2,0,1+ϱ1−1/2−ϱ3,0,1−ϱ1−1/2+ϱ3,0,1−βϱ1/2−ϱ3+1/21+β−ϱ4−b,β1+ϱ1−1/2−ϱ3,0,1+ϱ1−1/2−ϱ3−ϱ4,0,1−ϱ1−1/2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1−1/2−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ1−1/2+ϱ2,0,1+ϱ1−1/2+ϱ2−ϱ3,0,1−ϱ1−1/2−ϱ2+ϱ3,0,1−βϱ1−1/2+ϱ2−ϱ3−ϱ4−γϱ2+ϱ5+1,β1+ϱ1−1/2+ϱ2−ϱ3,0,1+ϱ1−1/2+ϱ2−ϱ3−ϱ4,01−ϱ1−1/2−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1−1/2−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ1−1/2+ϱ2,0,1+ϱ1−1/2+ϱ2−ϱ3,0,1−ϱ1−1/2−ϱ2+ϱ3,0,1−βϱ1−1/2+ϱ2−ϱ3−ϱ4−ηϱ2+1,β1+ϱ1−1/2+ϱ2−ϱ3,0,1+ϱ1−1/2+ϱ2−ϱ3−ϱ4,01−ϱ1−1/2−ϱ2+ϱ3,1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,1−ϱ2+ϱ3,0,1−βϱ2−ϱ3−ϱ4+γϱ1−1/2−ϱ2−1,β1+ϱ3−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1−ϱ2+ϱ3,1+GmSc∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,1−ϱ2+ϱ3,0,1−βϱ2−ϱ3−ϱ4+ηϱ1−1/2−ϱ2−1,β1+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1−ϱ2+ϱ3,1.


### 3.3. Fractionalized Second-Grade Fluid in Porous Medium

ForM⟶0, in (41) and (46), the obtained velocity results are(52)υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3=0∞−Ψϱ3ϱ3!λβ−ϱ1/2+ϱ3×M2,41,2⊺βλβ|0,1,1+ϱ1/2,0,1−ϱ1/2+ϱ3,0,1−βϱ1/2−ϱ3−b,β1+ϱ1/2−ϱ3,0,1−ϱ1/2+ϱ3,1+Gr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!λβ−ϱ1/2−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,−ϱ2,0,−ϱ1/2−ϱ2+ϱ3,0,1−βϱ1/2+ϱ2−ϱ3−β−γϱ2+ϱ5+1,β1+ϱ1/2+ϱ2−ϱ3,0,−ϱ2−ϱ5,0−ϱ1/2−ϱ2+ϱ3,1+Gm∑ϱ1,ϱ2,ϱ3=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!λβ−ϱ1/2−ϱ2+ϱ3−1×M2,41,2⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,−ϱ1/2−ϱ2+ϱ3,0,1−βϱ1/2+ϱ2−ϱ3−β−ηϱ2+1,β1+ϱ1/2+ϱ2−ϱ3,0,−ϱ1/2−ϱ2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ1/2−ϱ2,0,−ϱ2+ϱ3,0,1−βϱ2−ϱ3−β−γϱ1/2−ϱ2−ϱ5−1,β1+ϱ2−ϱ3,0,1+ϱ1/2−ϱ2−ϱ5,0,−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!λβ−ϱ2+ϱ3−1×M2,41,2⊺βλβ|0,1,1+ϱ2,0,−ϱ2+ϱ3,0,1−βϱ2−ϱ5−β−ηϱ1/2−ϱ2−1,β1+ϱ2−ϱ3,0,−ϱ2+ϱ3,1,and the corresponding shear stress is(53)Łξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3=0∞−Ψϱ3ϱ3!λβ−ϱ1−1/2+ϱ3×M2,41,2⊺βλβ|0,1,1+ϱ1−1/2,0,1−ϱ1−1/2+ϱ3,0,1−βϱ1/2−ϱ3+1/21+β−b,β1+ϱ1−1/2−ϱ3,0,1−ϱ1−1/2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!λβ−ϱ1−1/2−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ1−1/2+ϱ2,0,−ϱ2,0,1−ϱ1−1/2−ϱ2+ϱ3,0,1−βϱ1−1/2+ϱ2−ϱ3−γϱ2+ϱ5+1,β1+ϱ1−1/2+ϱ2−ϱ3,0,−ϱ2−ϱ5,0,1−ϱ1−1/2−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!λβ−ϱ1+12−ϱ2+ϱ3×M2,41,2⊺βλβ|0,1,1+ϱ1−1/2+ϱ2,0,1−ϱ1−1/2−ϱ2+ϱ3,0,1−βϱ1−1/2+ϱ2−ϱ3−ηϱ2+1,β1+ϱ1−1/2+ϱ2−ϱ3,0,1−ϱ1−1/2−ϱ2+ϱ3,1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ2,0,ϱ1−1/2−ϱ2,0,1−ϱ2+ϱ3,0,1−βϱ2−ϱ3+γϱ1−1/2−ϱ2−ϱ5−1,β1+ϱ3−ϱ3,0,1+ϱ1−1/2−ϱ2−ϱ5,0,1−ϱ2+ϱ3,1+GmSc∑ϱ1,ϱ2,ϱ3=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!λβ−ϱ2+ϱ3×M2,41,2⊺βλβ|0,1,1+ϱ2,0,1−ϱ2+ϱ3,0,1−βϱ2−ϱ3+ηϱ1−1/2−ϱ2−1,β1+ϱ2−ϱ3,0,1−ϱ2+ϱ3,1.


### 3.4. Fractionalized MHD Second-Grade Fluid after Vanishing Porosity

SubstitutingΨ⟶0, in (41) and (46), the acquired velocity is(54)υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ4=0∞−Mϱ4ϱ4!λβ−ϱ1/2+ϱ3×M2,41,2⊺βλβ|0,1,1+ϱ1/2,0,1−ϱ1/2,0,1−βϱ1/2−ϱ4−b,β1+ϱ1/2−ϱ4,0,1−ϱ1/2,1+Gr∑ϱ1,ϱ2,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1/2−ϱ2−1×M3,51,3⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,−ϱ2,0,−ϱ1/2−ϱ2,0,1−βϱ1/2+ϱ2−β−ϱ4−γϱ2+ϱ5+1,β1+ϱ1/2+ϱ2−ϱ4,0,−ϱ2−ϱ5,0−ϱ1/2−ϱ2,1+Gm∑ϱ1,ϱ2,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Mϱ4ϱ4!λβ−ϱ1/2−ϱ2−1×M2,41,2⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,−ϱ1/2−ϱ2,0,1−βϱ1/2+ϱ2−β−ϱ4−ηϱ2+1,β1+ϱ1/2+ϱ2−ϱ4,0,−ϱ1/2−ϱ2,1−Gr∑ϱ1,ϱ2,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2−1×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ1/2−ϱ2,0,−ϱ2,0,1−βϱ2−β−ϱ4−γϱ1/2−ϱ2−ϱ5−1,β1+ϱ2−ϱ4,0,1+ϱ1/2−ϱ2−ϱ5,0,−ϱ2,1−Gm∑ϱ1,ϱ2,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Mϱ4ϱ4!λβ−ϱ2−1×M2,41,2⊺βλβ|0,1,1+ϱ2,0,−ϱ2,0,1−βϱ2−ϱ5−β−ϱ4−ηϱ1/2−ϱ2−1,β1+ϱ2−ϱ4,0,−ϱ2,1,whereas the related shear stress is(55)Łξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ4=0∞−Mϱ4ϱ4!λβ−ϱ1−1/2×M2,41,2|0,1,1+ϱ1+1/2,0,1−ϱ1−1/2,0,1−βϱ1/2+1/21+β−ϱ4−b,β1+ϱ1+1/2−ϱ4,0,1−ϱ1−1/2,1−Gr∑ϱ1,ϱ2,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1+1/2−ϱ2×M3,51,3⊺βλβ|0,1,1+ϱ1+1/2+ϱ2,0,−ϱ2,0,1−ϱ1+1/2−ϱ2,0,1−βϱ1+1/2+ϱ2−ϱ4−γϱ2+ϱ5+1,β1+ϱ1+1/2+ϱ2+ϱ4,0,−ϱ2−ϱ5,01−ϱ1+1/2−ϱ2,1−Gm∑ϱ1,ϱ2,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Mϱ4ϱ4!λβ−ϱ1+1/2−ϱ2×M2,41,2⊺βλβ|0,1,1+ϱ1+1/2+ϱ2,0,1−ϱ1+1/2−ϱ2,0,1−βϱ1+1/2+ϱ2−ϱ4−ηϱ2+1,β1+ϱ1+1/2+ϱ2,0,1+ϱ1+1/2+ϱ2−ϱ4,01−ϱ1+1/2−ϱ2,1+GrPr∑ϱ1,ϱ2,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2×M3,51,3⊺βλβ|0,1,1+ϱ2,0,ϱ1+1/2−ϱ2,0,1−ϱ2,0,1−βϱ2−ϱ4+γϱ1+1/2−ϱ2−ϱ5−1,β1+ϱ2−ϱ4,0,1+ϱ1+1/2−ϱ2−ϱ5,0,1−ϱ2,1+GmSc∑ϱ1,ϱ2,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Mϱ4ϱ4!λβ−ϱ2×M2,41,2⊺βλβ|0,1,1+ϱ2,0,1−ϱ2,0,1−βϱ2−ϱ4+ηϱ1+1/2−ϱ2−1,β1+ϱ2−ϱ4,0,1−ϱ2,1.


### 3.5. Fractionalized MHD Newtonian Fluid in Porous Medium

By makingλ⟶0, in (38) and (43), proceed as in the previous section, we obtain the velocity of fractionalized Newtonian fluid:(56)υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3=0∞−Ψϱ3ϱ3!λβ−ϱ1/2+ϱ3×M2,41,2M⊺|0,1,1+ϱ1/2,0,1+ϱ1/2−ϱ3,0,ϱ1/2−ϱ3−b,11+ϱ1/2−ϱ3,0,1+ϱ1/2−ϱ3,1+Gr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!×M3,51,3M⊺|0,1,1+ϱ1/2+ϱ2,0,−ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,ϱ1/2+ϱ2−ϱ3−γϱ2+ϱ5+1,11+ϱ1/2+ϱ2−ϱ3,0,−ϱ2−ϱ5,0,1+ϱ1/2+ϱ2−ϱ3,1+Gm∑ϱ1,ϱ2,ϱ3=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!×M2,41,2M⊺|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,ϱ1/2+ϱ2−ϱ3−ηϱ2+1,11+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!×M3,51,3M⊺|0,1,1+ϱ2,0,1+ϱ1/2−ϱ2,0,1+ϱ2−ϱ3,0,ϱ2−ϱ3−γϱ1/2−ϱ2−ϱ5−1,11+ϱ2−ϱ3,0,1+ϱ1/2−ϱ2−ϱ5,0,1+ϱ2−ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!×M2,41,2M⊺|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,ϱ2−ϱ5−ηϱ1/2−ϱ2−1,11+ϱ2−ϱ3,0,1+ϱ2−ϱ3,1,and the associated shear stress is(57)Łξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3=0∞−Ψϱ3ϱ3!×M2,41,2M⊺|0,1,1+ϱ1+1/2,0,1+ϱ1+1/2−ϱ3,0,ϱ1/2−ϱ3+1/2−b,11+ϱ1+1/2−ϱ3,0,1+ϱ1+1/2−ϱ3−ϱ4,0,1−ϱ1+1/2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!×M3,51,3M⊺|0,1,1+ϱ1+1/2+ϱ2,0,−ϱ2,0,1+ϱ1+1/2+ϱ2−ϱ3,0,ϱ1+1/2+ϱ2−ϱ3−γϱ2+ϱ5+1,11+ϱ1+1/2+ϱ2−ϱ3,0,−ϱ2−ϱ5,0,1+ϱ1+1/2+ϱ2−ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!×M2,41,2M⊺|0,1,1+ϱ1+1/2+ϱ2,0,1+ϱ1+1/2+ϱ2−ϱ3,0,ϱ1+1/2+ϱ2−ϱ3−ηϱ2+1,11+ϱ1+1/2+ϱ2−ϱ3,0,1+ϱ1+1/2+ϱ2−ϱ3,1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!×M3,51,3M⊺|0,1,1+ϱ2,0,ϱ1+1/2−ϱ2,0,1+ϱ2−ϱ3,0,ϱ2−ϱ3+γϱ1+1/2−ϱ2−ϱ5−1,11+ϱ3−ϱ3,0,1+ϱ1+1/2−ϱ2−ϱ5,0,1+ϱ2−ϱ3,1+GmSc∑ϱ1,ϱ2,ϱ3=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!×M2,41,2M⊺|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,ϱ2−ϱ3ηϱ1+1/2−ϱ2−1,11+ϱ2−ϱ3,0,1+ϱ2−ϱ3,1.


## 3.1. Ordinary MHD Second-Grade Fluid in Porous Material

Consideringβ⟶0,γ⟶0 and η⟶0, in equations (29), (35), (40), and (46), the obtained results are(47)ℷξ,⊺=W−1/2,1−Sc⊺ξ,ℸξ,⊺=1+∑ϱ1=1∞−Prξϱ1ϱ1!M1,31,1FtPr|0,1,1+ϱ1/2,0,ϱ1/2,11−ϱ1/2,1,υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λ−ϱ1/2+ϱ3×M3,51,3⊺λ|0,1,1+ϱ1/2,0,1+ϱ1/2−ϱ3,0,1−ϱ1/2+ϱ3,0,−ϱ4−b,11+ϱ1/2−ϱ3,0,1+ϱ1/2−ϱ3−ϱ4,0,1−ϱ1/2+ϱ3,1+Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λ−ϱ1/2−ϱ2+ϱ3−1×M4,61,4⊺λ|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,−ϱ2,0,−ϱ1/2−ϱ2+ϱ3,0,−1−ϱ4−ϱ2+ϱ5+1,11+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3−ϱ4,0,−ϱ2−ϱ5,0−ϱ1/2−ϱ2+ϱ3,1+Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λ−ϱ1/2−ϱ2+ϱ3−1×M3,51,3⊺λ|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,−ϱ1/2−ϱ2+ϱ3,0,−1−ϱ4−ϱ2+1,11+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3−ϱ4,0,−ϱ1/2−ϱ2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λ−ϱ2+ϱ3−1×M4,61,4⊺λ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,1+ϱ1/2−ϱ2,0,−ϱ2+ϱ3,0,−1−ϱ4−ϱ1/2−ϱ2−ϱ5−1,11+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1+ϱ1/2−ϱ2−ϱ5,0,−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λ−ϱ2+ϱ3−1×M3,51,3⊺λ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,−ϱ2+ϱ3,0,−1−ϱ4−ϱ1/2−ϱ2−1,11+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,−ϱ2+ϱ3,1,and shear stress is(48)Łξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λ−ϱ1−1/2+ϱ3×M3,51,3⊺λ|0,1,1+ϱ1−1/2,0,1+ϱ1−1/2−ϱ3,0,1−ϱ1−1/2+ϱ3,0,1−ϱ4−b,11+ϱ1−1/2−ϱ3,0,1+ϱ1−1/2−ϱ3−ϱ4,0,1−ϱ1−1/2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λ−ϱ1−1/2−ϱ2+ϱ3×M4,61,4⊺λ|0,1,1+ϱ1−1/2+ϱ2,0,1+ϱ1−1/2+ϱ2−ϱ3,0,−ϱ2,0,1−ϱ1−1/2−ϱ2+ϱ3,0,−ϱ4−ϱ2+ϱ5+1,11+ϱ1−1/2+ϱ2−ϱ3,0,1+ϱ1−1/2+ϱ2−ϱ3−ϱ4,0,−ϱ2−ϱ5,01−ϱ1−1/2−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λ−ϱ1−1/2−ϱ2+ϱ3×M3,51,3⊺λ|0,1,1+ϱ1−1/2+ϱ2,0,1+ϱ1−1/2+ϱ2−ϱ3,0,1−ϱ1−1/2−ϱ2+ϱ3,0,−ϱ4−ϱ2+1,11+ϱ1−1/2+ϱ2−ϱ3,0,1+ϱ1−1/2+ϱ2−ϱ3−ϱ4,01−ϱ1−1/2−ϱ2+ϱ3,1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!−F/Prϱ5ϱ5!λ−ϱ2+ϱ3×M4,61,4⊺λ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,ϱ1−1/2−ϱ2,0,1−ϱ2+ϱ3,0,−ϱ4+ϱ1−1/2−ϱ2−ϱ5−1,11+ϱ3−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1+ϱ1−1/2−ϱ2−ϱ5,0,1−ϱ2+ϱ3,1+GmSc∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λ−ϱ2+ϱ3×M3,51,3⊺λ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,1−ϱ2+ϱ3,0,−ϱ4+ϱ1−1/2−ϱ2−1,11+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1−ϱ2+ϱ3,1.


## 3.2. Fractionalized MHD Second-Grade Fluid in Porous Medium with Constant Radiative Heat Flux

For the constant radiative heat flux alongy-direction, qr= constant or F⟶0, the flow results are(49)ℷξ,⊺=W−η2,1−Sc⊺ηξ,ℸξ,⊺=W−γ2,1−Pr⊺γξ,with velocity(50)υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ12,0,1+ϱ12−ϱ3,0,1−ϱ12+ϱ3,0,1−βϱ1/2−ϱ3−ϱ4−b,β1+ϱ1/2−ϱ3,0,1+ϱ1/2−ϱ3−ϱ4,0,1−ϱ1/2+ϱ3,1+Gr∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,−ϱ1/2−ϱ2+ϱ3,0,1−βϱ1/2+ϱ2−ϱ3−β−ϱ4−γϱ2+1,β1+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3−ϱ4,0−ϱ1/2−ϱ2+ϱ3,1+Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1/2−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,−ϱ1/2−ϱ2+ϱ3,0,1−βϱ1/2+ϱ2−ϱ3−β−ϱ4−ηϱ2+1,β1+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3−ϱ4,0,−ϱ1/2−ϱ2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,−ϱ2+ϱ3,0,1−βϱ2−ϱ3−β−ϱ4−γϱ1/2−ϱ2−1,β1+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,−k2+ϱ3,0,1−βϱ2−ϱ5−β−ϱ4−ηϱ1/2−ϱ2−1,β1+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,−ϱ2+ϱ3,1,and the respective shear stress is(51)Lξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3,ϱ4=0∞−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1−1/2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ1−1/2,0,1+ϱ1−1/2−ϱ3,0,1−ϱ1−1/2+ϱ3,0,1−βϱ1/2−ϱ3+1/21+β−ϱ4−b,β1+ϱ1−1/2−ϱ3,0,1+ϱ1−1/2−ϱ3−ϱ4,0,1−ϱ1−1/2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1−1/2−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ1−1/2+ϱ2,0,1+ϱ1−1/2+ϱ2−ϱ3,0,1−ϱ1−1/2−ϱ2+ϱ3,0,1−βϱ1−1/2+ϱ2−ϱ3−ϱ4−γϱ2+ϱ5+1,β1+ϱ1−1/2+ϱ2−ϱ3,0,1+ϱ1−1/2+ϱ2−ϱ3−ϱ4,01−ϱ1−1/2−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ1−1/2−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ1−1/2+ϱ2,0,1+ϱ1−1/2+ϱ2−ϱ3,0,1−ϱ1−1/2−ϱ2+ϱ3,0,1−βϱ1−1/2+ϱ2−ϱ3−ϱ4−ηϱ2+1,β1+ϱ1−1/2+ϱ2−ϱ3,0,1+ϱ1−1/2+ϱ2−ϱ3−ϱ4,01−ϱ1−1/2−ϱ2+ϱ3,1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,1−ϱ2+ϱ3,0,1−βϱ2−ϱ3−ϱ4+γϱ1−1/2−ϱ2−1,β1+ϱ3−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1−ϱ2+ϱ3,1+GmSc∑ϱ1,ϱ2,ϱ3,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!−Mϱ4ϱ4!λβ−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,1−ϱ2+ϱ3,0,1−βϱ2−ϱ3−ϱ4+ηϱ1−1/2−ϱ2−1,β1+ϱ2−ϱ3,0,1+ϱ2−ϱ3−ϱ4,0,1−ϱ2+ϱ3,1.


## 3.3. Fractionalized Second-Grade Fluid in Porous Medium

ForM⟶0, in (41) and (46), the obtained velocity results are(52)υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3=0∞−Ψϱ3ϱ3!λβ−ϱ1/2+ϱ3×M2,41,2⊺βλβ|0,1,1+ϱ1/2,0,1−ϱ1/2+ϱ3,0,1−βϱ1/2−ϱ3−b,β1+ϱ1/2−ϱ3,0,1−ϱ1/2+ϱ3,1+Gr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!λβ−ϱ1/2−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,−ϱ2,0,−ϱ1/2−ϱ2+ϱ3,0,1−βϱ1/2+ϱ2−ϱ3−β−γϱ2+ϱ5+1,β1+ϱ1/2+ϱ2−ϱ3,0,−ϱ2−ϱ5,0−ϱ1/2−ϱ2+ϱ3,1+Gm∑ϱ1,ϱ2,ϱ3=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!λβ−ϱ1/2−ϱ2+ϱ3−1×M2,41,2⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,−ϱ1/2−ϱ2+ϱ3,0,1−βϱ1/2+ϱ2−ϱ3−β−ηϱ2+1,β1+ϱ1/2+ϱ2−ϱ3,0,−ϱ1/2−ϱ2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3−1×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ1/2−ϱ2,0,−ϱ2+ϱ3,0,1−βϱ2−ϱ3−β−γϱ1/2−ϱ2−ϱ5−1,β1+ϱ2−ϱ3,0,1+ϱ1/2−ϱ2−ϱ5,0,−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!λβ−ϱ2+ϱ3−1×M2,41,2⊺βλβ|0,1,1+ϱ2,0,−ϱ2+ϱ3,0,1−βϱ2−ϱ5−β−ηϱ1/2−ϱ2−1,β1+ϱ2−ϱ3,0,−ϱ2+ϱ3,1,and the corresponding shear stress is(53)Łξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3=0∞−Ψϱ3ϱ3!λβ−ϱ1−1/2+ϱ3×M2,41,2⊺βλβ|0,1,1+ϱ1−1/2,0,1−ϱ1−1/2+ϱ3,0,1−βϱ1/2−ϱ3+1/21+β−b,β1+ϱ1−1/2−ϱ3,0,1−ϱ1−1/2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!λβ−ϱ1−1/2−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ1−1/2+ϱ2,0,−ϱ2,0,1−ϱ1−1/2−ϱ2+ϱ3,0,1−βϱ1−1/2+ϱ2−ϱ3−γϱ2+ϱ5+1,β1+ϱ1−1/2+ϱ2−ϱ3,0,−ϱ2−ϱ5,0,1−ϱ1−1/2−ϱ2+ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!λβ−ϱ1+12−ϱ2+ϱ3×M2,41,2⊺βλβ|0,1,1+ϱ1−1/2+ϱ2,0,1−ϱ1−1/2−ϱ2+ϱ3,0,1−βϱ1−1/2+ϱ2−ϱ3−ηϱ2+1,β1+ϱ1−1/2+ϱ2−ϱ3,0,1−ϱ1−1/2−ϱ2+ϱ3,1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!λβ−ϱ2+ϱ3×M3,51,3⊺βλβ|0,1,1+ϱ2,0,ϱ1−1/2−ϱ2,0,1−ϱ2+ϱ3,0,1−βϱ2−ϱ3+γϱ1−1/2−ϱ2−ϱ5−1,β1+ϱ3−ϱ3,0,1+ϱ1−1/2−ϱ2−ϱ5,0,1−ϱ2+ϱ3,1+GmSc∑ϱ1,ϱ2,ϱ3=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!λβ−ϱ2+ϱ3×M2,41,2⊺βλβ|0,1,1+ϱ2,0,1−ϱ2+ϱ3,0,1−βϱ2−ϱ3+ηϱ1−1/2−ϱ2−1,β1+ϱ2−ϱ3,0,1−ϱ2+ϱ3,1.


## 3.4. Fractionalized MHD Second-Grade Fluid after Vanishing Porosity

SubstitutingΨ⟶0, in (41) and (46), the acquired velocity is(54)υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ4=0∞−Mϱ4ϱ4!λβ−ϱ1/2+ϱ3×M2,41,2⊺βλβ|0,1,1+ϱ1/2,0,1−ϱ1/2,0,1−βϱ1/2−ϱ4−b,β1+ϱ1/2−ϱ4,0,1−ϱ1/2,1+Gr∑ϱ1,ϱ2,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1/2−ϱ2−1×M3,51,3⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,−ϱ2,0,−ϱ1/2−ϱ2,0,1−βϱ1/2+ϱ2−β−ϱ4−γϱ2+ϱ5+1,β1+ϱ1/2+ϱ2−ϱ4,0,−ϱ2−ϱ5,0−ϱ1/2−ϱ2,1+Gm∑ϱ1,ϱ2,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Mϱ4ϱ4!λβ−ϱ1/2−ϱ2−1×M2,41,2⊺βλβ|0,1,1+ϱ1/2+ϱ2,0,−ϱ1/2−ϱ2,0,1−βϱ1/2+ϱ2−β−ϱ4−ηϱ2+1,β1+ϱ1/2+ϱ2−ϱ4,0,−ϱ1/2−ϱ2,1−Gr∑ϱ1,ϱ2,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2−1×M3,51,3⊺βλβ|0,1,1+ϱ2,0,1+ϱ1/2−ϱ2,0,−ϱ2,0,1−βϱ2−β−ϱ4−γϱ1/2−ϱ2−ϱ5−1,β1+ϱ2−ϱ4,0,1+ϱ1/2−ϱ2−ϱ5,0,−ϱ2,1−Gm∑ϱ1,ϱ2,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Mϱ4ϱ4!λβ−ϱ2−1×M2,41,2⊺βλβ|0,1,1+ϱ2,0,−ϱ2,0,1−βϱ2−ϱ5−β−ϱ4−ηϱ1/2−ϱ2−1,β1+ϱ2−ϱ4,0,−ϱ2,1,whereas the related shear stress is(55)Łξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ4=0∞−Mϱ4ϱ4!λβ−ϱ1−1/2×M2,41,2|0,1,1+ϱ1+1/2,0,1−ϱ1−1/2,0,1−βϱ1/2+1/21+β−ϱ4−b,β1+ϱ1+1/2−ϱ4,0,1−ϱ1−1/2,1−Gr∑ϱ1,ϱ2,ϱ4,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ1+1/2−ϱ2×M3,51,3⊺βλβ|0,1,1+ϱ1+1/2+ϱ2,0,−ϱ2,0,1−ϱ1+1/2−ϱ2,0,1−βϱ1+1/2+ϱ2−ϱ4−γϱ2+ϱ5+1,β1+ϱ1+1/2+ϱ2+ϱ4,0,−ϱ2−ϱ5,01−ϱ1+1/2−ϱ2,1−Gm∑ϱ1,ϱ2,ϱ4=0∞−ξϱ1ϱ1!1Scϱ2+1−Mϱ4ϱ4!λβ−ϱ1+1/2−ϱ2×M2,41,2⊺βλβ|0,1,1+ϱ1+1/2+ϱ2,0,1−ϱ1+1/2−ϱ2,0,1−βϱ1+1/2+ϱ2−ϱ4−ηϱ2+1,β1+ϱ1+1/2+ϱ2,0,1+ϱ1+1/2+ϱ2−ϱ4,01−ϱ1+1/2−ϱ2,1+GrPr∑ϱ1,ϱ2,ϱ4,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Mϱ4ϱ4!−F/Prϱ5ϱ5!λβ−ϱ2×M3,51,3⊺βλβ|0,1,1+ϱ2,0,ϱ1+1/2−ϱ2,0,1−ϱ2,0,1−βϱ2−ϱ4+γϱ1+1/2−ϱ2−ϱ5−1,β1+ϱ2−ϱ4,0,1+ϱ1+1/2−ϱ2−ϱ5,0,1−ϱ2,1+GmSc∑ϱ1,ϱ2,ϱ4=0∞−Scξϱ1ϱ1!1Scϱ2+1−Mϱ4ϱ4!λβ−ϱ2×M2,41,2⊺βλβ|0,1,1+ϱ2,0,1−ϱ2,0,1−βϱ2−ϱ4+ηϱ1+1/2−ϱ2−1,β1+ϱ2−ϱ4,0,1−ϱ2,1.


## 3.5. Fractionalized MHD Newtonian Fluid in Porous Medium

By makingλ⟶0, in (38) and (43), proceed as in the previous section, we obtain the velocity of fractionalized Newtonian fluid:(56)υξ,⊺=⊺b+Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3=0∞−Ψϱ3ϱ3!λβ−ϱ1/2+ϱ3×M2,41,2M⊺|0,1,1+ϱ1/2,0,1+ϱ1/2−ϱ3,0,ϱ1/2−ϱ3−b,11+ϱ1/2−ϱ3,0,1+ϱ1/2−ϱ3,1+Gr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!×M3,51,3M⊺|0,1,1+ϱ1/2+ϱ2,0,−ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,ϱ1/2+ϱ2−ϱ3−γϱ2+ϱ5+1,11+ϱ1/2+ϱ2−ϱ3,0,−ϱ2−ϱ5,0,1+ϱ1/2+ϱ2−ϱ3,1+Gm∑ϱ1,ϱ2,ϱ3=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!×M2,41,2M⊺|0,1,1+ϱ1/2+ϱ2,0,1+ϱ1/2+ϱ2−ϱ3,0,ϱ1/2+ϱ2−ϱ3−ηϱ2+1,11+ϱ1/2+ϱ2−ϱ3,0,1+ϱ1/2+ϱ2−ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!×M3,51,3M⊺|0,1,1+ϱ2,0,1+ϱ1/2−ϱ2,0,1+ϱ2−ϱ3,0,ϱ2−ϱ3−γϱ1/2−ϱ2−ϱ5−1,11+ϱ2−ϱ3,0,1+ϱ1/2−ϱ2−ϱ5,0,1+ϱ2−ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!×M2,41,2M⊺|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,ϱ2−ϱ5−ηϱ1/2−ϱ2−1,11+ϱ2−ϱ3,0,1+ϱ2−ϱ3,1,and the associated shear stress is(57)Łξ,⊺=−Γd+1∑ϱ1=1∞−ξϱ1ϱ1!∑ϱ3=0∞−Ψϱ3ϱ3!×M2,41,2M⊺|0,1,1+ϱ1+1/2,0,1+ϱ1+1/2−ϱ3,0,ϱ1/2−ϱ3+1/2−b,11+ϱ1+1/2−ϱ3,0,1+ϱ1+1/2−ϱ3−ϱ4,0,1−ϱ1+1/2+ϱ3,1−Gr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−ξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!×M3,51,3M⊺|0,1,1+ϱ1+1/2+ϱ2,0,−ϱ2,0,1+ϱ1+1/2+ϱ2−ϱ3,0,ϱ1+1/2+ϱ2−ϱ3−γϱ2+ϱ5+1,11+ϱ1+1/2+ϱ2−ϱ3,0,−ϱ2−ϱ5,0,1+ϱ1+1/2+ϱ2−ϱ3,1−Gm∑ϱ1,ϱ2,ϱ3=0∞−ξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!×M2,41,2M⊺|0,1,1+ϱ1+1/2+ϱ2,0,1+ϱ1+1/2+ϱ2−ϱ3,0,ϱ1+1/2+ϱ2−ϱ3−ηϱ2+1,11+ϱ1+1/2+ϱ2−ϱ3,0,1+ϱ1+1/2+ϱ2−ϱ3,1+GrPr∑ϱ1,ϱ2,ϱ3,ϱ5=0∞−Prξϱ1ϱ1!1Prϱ2+1−Ψϱ3ϱ3!−F/Prϱ5ϱ5!×M3,51,3M⊺|0,1,1+ϱ2,0,ϱ1+1/2−ϱ2,0,1+ϱ2−ϱ3,0,ϱ2−ϱ3+γϱ1+1/2−ϱ2−ϱ5−1,11+ϱ3−ϱ3,0,1+ϱ1+1/2−ϱ2−ϱ5,0,1+ϱ2−ϱ3,1+GmSc∑ϱ1,ϱ2,ϱ3=0∞−Scξϱ1ϱ1!1Scϱ2+1−Ψϱ3ϱ3!×M2,41,2M⊺|0,1,1+ϱ2,0,1+ϱ2−ϱ3,0,ϱ2−ϱ3ηϱ1+1/2−ϱ2−1,11+ϱ2−ϱ3,0,1+ϱ2−ϱ3,1.


## 4. Results and Discussions

Here, we analyse the flow behavior with the help of graphical illustrations after finding the exact solutions of temperature, concentration, velocity, and shear stress in the previous sections. All graphs are made in Mathcad software with SI units. The values of parameters of interest that are used in the study are provided in respective figures. All the imposed and boundary conditions mentioned in (14)–(16) are satisfied. It can be observed in Figures 2 and 3 that natural and boundary conditions for mass concentration ℷξ,0=0 and ℷ0,⊺=1 are, respectively, satisfied. From Figures 4–6 it can be easily analysed that natural and boundary conditions for temperature ℸξ,0=0 and ℷ0,⊺=1 are satisfied. Natural and boundary conditions for velocity and shear stress υξ,0=0,Łξ,0=0 and υ0,⊺=⊺d are well visualized in Figures 7 and 8 , respectively.Figure 2
Descriptions of the mass concentrationℷξ,⊺ given by (29), for Sc=0.6,η=0.5, and different values of ξ and ⊺.
(a)(b)Figure 3
Descriptions of the mass concentrationℷξ,⊺ given by (29), for ⊺=2s, and different values of Sc and η.
(a)(b)Figure 4
Descriptions of the temperature distributionℷξ,⊺ given by (35), for Pr=0.6,F=0.2, and different values of ξ and ⊺.
(a)(b)Figure 5
Descriptions of the temperature distributionℷξ,⊺ given by (35), for F=0.2, for variational points of ⊺ and Pr.
(a)(b)Figure 6
Descriptions of the temperature distributionℷξ,⊺ given by (35), for Pr=2, for variational points of F and η.
(a)(b)Figure 7
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for λ=3,M=0.5,Ψ=0.2,Gr=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1 for variational points of ξ.
(a)(b)Figure 8
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for λ=3,M=0.5,Ψ=0.2,Gr=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1 for variational points of ⊺.
(a)(b)Assurance of present work is manifested by constructing comparison with pervious published literature by limiting cases in “section3” of this study. These special cases are compared to those of Amanet al. [29] and Shahid [30]. Flow field results with ordinary derivatives and flectional derivatives are compared graphically, and the analytical results are the same as those of the previously published literature by vanishing few parameters.In Figure2 we have discussed the variations in mass concentration at different values of time and space above the plate. A natural behavior of decay in mass concentration is observed for higher position of the fluid over the plate. The significance of fractional parameter η and the effects of variations in Schmidt number over the dimensionless mass concentration are analysed in Figure 3. This reveals the facts of increase in mass concentration with increase in fractional parameter values. On the other hand, the opposite behavior is observed for increase in Schmidt number. Figure 4 reflects the temperature distribution with respect to increasing time and vertical space values. The thermal boundary layer thickens by increasing time while temperature values decrease by considering the increasing vertical position of fluid over plate. The effects of fractional parameter and Prandtl number on temperature of fluid over the moving plate are portrayed in Figure 5. Naturally, the viscosity of fluid gets minimized by increasing Prandtl number that ultimately yields the reduction in thermal boundary layer. Figure 6 depicts the influence of fractional parameter and thermal diffusivity on temperature distribution which implies that thermal boundary layer gets thinner with increasing thermal diffusivity whereas fractional parameter acts opposite to that of thermal diffusivity on temperature distribution.Velocity fields along with shear stress behavior are given in Figure7 for vertical space values above the moving plate which shows that velocity of the particles at the moving surface is higher to that of the particles away from plate, whereas shear stress profile behaves numerically opposite to it. It can be noted from Figure 8 that as the time passes, velocity of the fluid increases and the stress too, while the impact gets more intense with passage of time. Figure 9 shows the effects of increasing second-grade parameter λ; it reduces the flow velocity and the stress numerically which is due to the characteristics of second-grade fluid. The influence of magnetic and porous parameters is displayed in Figures 10 and 11, where both parameters have similar effect, and the speed and stress increase numerically by raising either the magnetic or porous parameter values.Figure 9
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for M=0.5,Ψ=0.2,Gr=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.5s for variational points of λ.
(a)(b)Figure 10
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for M=0.5,Ψ=0.2,Gr=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.5s for variational points of M.
(a)(b)Figure 11
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for M=0.5,Ψ=0.2,Gr=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.5s for variational points of Ψ.
(a)(b)To discuss the impacts of thermal Grashof number and modified Grashof number, we plot Figures12 and 13, where velocity and stress mount numerically for the dimensionless numbers individually since it is buoyancy force that flows the fluid, whereas the impacts of modified Grashof number are more intense than those of the thermal Grashof number. The importance of Prandtl and Schmidt number can be observed in Figures 14 and 15; both the dimensionless numbers endorse their definition there by increasing the stress and velocity of the particles for increasing values of these numbers. Figures 16 points out the relation of thermal diffusivity to the stress and velocity of the particles. Increasing the thermal diffusivity parameter, velocity of the fluid reduces and the stress reduces too. Figures 17–19 are prepared to analyse the significance of fractional parameters on the flow field. It can be noted that the increase in fractional parameters reduces the stress and velocity of the particles. Thus, fractional derivatives are essential in studying the fluid flows more significantly and accurately. Figure 20 displays increasing magnitude of stress and velocity of the particles in relation with increasing values of the exponent of the time factor in the boundary condition for velocity. A comparison of model flow is displayed graphically in Figure 21 which assures the influence of fractional parameters over the stress and velocity of the particles.Figure 12
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for λ=3,M=0.5,Ψ=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.2s for variational points of Gr.
(a)(b)Figure 13
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for λ=3,M=0.5,Ψ=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.5s for variational points of Gm.
(a)(b)Figure 14
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for λ=3,M=0.5,Ψ=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.5s for variational points of Sc.
(a)(b)Figure 15
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for λ=3,M=0.5,Ψ=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.5s for variational points of Pr.
(a)(b)Figure 16
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for λ=3,M=0.5,Ψ=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.5s for variational points of F.
(a)(b)Figure 17
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for λ=3,M=0.5,Ψ=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.2s for variational points of η.
(a)(b)Figure 18
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for λ=3,M=0.5,Ψ=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.5s for variational points of β.
(a)(b)Figure 19
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for λ=3,M=0.5,Ψ=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.5s for variational points of γ.
(a)(b)Figure 20
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ given by (41) and (46), for λ=3,M=0.5,Ψ=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=2s for variational points of d.
(a)(b)Figure 21
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ for MHD fractionalized second grade with porous given by (41) and (46), MHD ordinary second grade with porous given by (48) and (49), MHD fractionalized Newtonian with porous given by (56) and (57) and MHD ordinary Newtonian with porous for λ=3,M=0.5,Ψ=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.5s.
(a)(b)Flow characteristics of fractional model are higher in magnitude than those of the ordinary model. Stress and velocity of the particles second-grade particles are numerically higher in comparison with Newtonian fluid. Figure22 is plotted to analyse the significance of fractional parameter. In comparison with Figure 21, it can be observed that increasing the fractional parameters values, flow characteristics get closer to those of the ordinary fluid flow for both the second-grade and Newtonian-type fluids. The obtained results are compared with the analytical solutions of the flow field presented by the recent studies [23–25] which are found to be more compact and simplified. The accuracy of the flow behavior is graphically analysed by setting three decimal places in numerical calculations. However, solutions are analytical, thus the numerical solutions and their comparison are beyond the scope of this study.Figure 22
Descriptions of the velocityυξ,⊺ and the stress Łξ,⊺ for MHD fractionalized second grade with porous given by (41) and (46), MHD ordinary second-grade with porous given by (47) and (48), MHD fractionalized Newtonian with porous given by (56), (57) and MHD ordinary Newtonian with porous for λ=3,M=0.5,Ψ=0.2,Gm=0.6,Sc=0.6,Pr=0.6,F=0.2,η=β=γ=0.5,d=1,⊺=0.5s.
(a)(b)


## 5. Conclusions

In this article, transfer and flow characteristics of fractional MHD second-grade fluid on the porous plate moving with nonlinear velocity is analysed. Solutions for velocity, temperature distribution, and mass concentration are acquired using Laplace transforms after nondimensionalising the system of differential equations. Obtained results satisfying all imposed natural initial and boundary limitations are graphically analysed for transfer and rheological parameters at different times and positions over the plate. The investigation reveals the following results⋆. All solutions are represented in simpler forms in terms of new summation style and natural generalized function M for such flows.(i)
Mass concentration rate increases with increasing Schmidt number and passage of time while it decreases with increasing fractional parameter values and position of fluid over the plate.(ii)
Temperature distribution of the fluid is the decreasing function of Prandtl number, thermal radiation parameter, and position of fluid over the plate whereas it increases with increasing time and fractional parameter values.(iii)
Velocity field and stress gets higher magnitude for increasing thermal Grashof number, mass Grashof number, Prandtl number, Schmidt number, time, second grade, magnetic (Hartmann), and porous and thermal radiation parameters.(iv)
Fractional operators significantly affect the stress and velocity of the particles there by reducing their magnitude for increasing fractional parameters.(v)
The nonlinearity parameterd has direct effects on fluid motion. The large values of d enhance the motion of the fluid.(vi)
It is also verified from graphical results that when fractional parameters approach to 1, the behavior of fractionalized second-grade fluid turns into the usual fluid.



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