# Unsteady MHD Flow for Fractional Casson Channel Fluid in a Porous Medium: An Application of the Caputo-Fabrizio Time-Fractional Derivative

Authors: Pongsakorn Sunthrayuth; A. A. Alderremy; Fazal Ghani; Ayékotan M. J. Tchalla; Shaban Aly; Yasser Elmasry
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Theoretically, this work describes the exact solutions of fractional Casson fluid through a channel under the effect of MHD and porous medium. The unsteady fluid motion of the bottom plate, which is confined by parallel but perpendicular sidewalls, supports the flow. By introducing the dimensionless parameters and variables, the momentum equation, as well as the initial and boundary conditions, has been transformed to a dimensionless form. A mix of Laplace and Fourier transformations is used to get the exact solution for the momentum equation. The constitutive equations for Caputo-Fabrizio’s time-fractional derivative are also incorporated for recovering the exact solutions of the flow problem under consideration. After recovering the exact solutions for flow characteristics, three different cases at the surface of the bottom plate are discussed, by addressing the limiting cases under the influence of the side walls. Moreover, these solutions are captured graphically, and the effects of the Reynolds numberRe, fractional parameter α, effective permeability Keff, and dimensionless parameter for Casson fluid β on the fluid’s motion are observed.


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## Body

## 1. Introduction

Fractional calculus plays a critical part in the solving of complicated engineering issues. Because of its significance, a fractional model solution for flow issues is preferred by many scientists and researchers. A fraction model correctly depicts the motion of a flow issue when compared to ordinary differential equations (ODEs). It recovers an ODE’s solution, which explains minute flow system fluctuations. Even for Newtonian fluids, the older scientists’ operators including the Caputo operator employed a solitary kernel that resulted in complicated series solutions. In 2015, Caputo and Fabrizio [1] suggested a new fractional operator that may be employed in simple ways to solve this problem. Following that, a lot of scholars have applied this notion to numerous sorts of flows using varied geometries. Alshabanat et al. [2] proposed a new fractional derivative utilizing a nonsingular form kernel of exponential and trigonometric functions. Singh et al. [3] investigated a fractional epidemiological model for viral determination in the computer utilizing fractional derivatives and numerically solved the modeled issue using the iterative technique. Shah and Khan [4] have discussed heat transmission for an oscillating second-grade fluid upon a vertical surface by employing the Caputo-Fabrizio derivative. In this work, the authors have determined the exact solutions for flow and thermal characteristics by applying Laplace transform. The authors of this investigation have also carried out a comparative study for the time derivative of fractional and integral order both for Newtonian and second-grade fluids and have highlighted that fractional parameter enhanced the flow characteristics due to augmented velocities of fractional fluid.Researchers have been studying mathematical models for non-Newtonian fluids because of the growing trend in technological and industrial applications. Because of their relevance at the industrial level, research into these fluids is desirable. Non-Newtonian fluids, for example, lubricant production for a variety of vehicles, spinning of metal, metal extrusion, removing nonmetallic inclusions from molten metal, food, shoe manufacturing (the shoe must be filled with a non-Newtonian fluid to protect the feet from damage), and industries that deal with medicine and coolant, have a wide range of uses. Casson fluid is known as Non-Newtonian fluid because of its rheological features. Casson’s model, introduced in 1959, was shown to be extremely flexible and to best reflect the curves of silicon suspension, among other things [5]. Hussain et al. [6] used the shooting method to find the numerical solution of a Casson fluid by changing viscosity flows near a shrinking/extending sheet by slip effects in MHD stagnation point flow. Casson nanofluid flows hydromagnetically through a porous stretched cylinder under Newtonian heat and mass conditions which were discussed by Naqvi et al. [7]. Rao et al. [8] found the exact solution of Casson fluid near a plate which is infinite, exponentially accelerated, and vertical with the effect of MHD and porosity. An overview of numerical approaches for heat and mass transfer in Casson fluids is presented by Verma and Mondal [9]. Sheikh et al. [10] obtained exact solutions of free convection MHD flow of Casson fluid in a channel by using Laplace transform. A fractional model with the Mittag-Leffler memory for generalized Casson MHD fluid by Newtonian heating was discussed by Tassaddiq et al. [11]. Goud et al. [12] calculated the exact solution of natural convection MHD flow of Casson fluid near a perpendicular plate through a porous medium by finite element method. The behavior of a non-Newtonian micropolar-Casson fluid pulsatile flow in a restricted channel influenced by Lorentz force according to Darcy’s law is investigated by Ali et al. [13].The flow in the channels is also more important. Because of its relevance, numerous academics have been drawn to explore channel flows during the previous few decades. Using the Dufour effect, Jha and Ajibade [14] examined heat and mass transfer for free convective fluid flow along a vertical channel. Free convection of transient flow on a flat surface was studied by Ingham [15]. The authors of this study focused on a vertically oriented flat plate. Free convective flow with MHD effect on a flat plate was examined by Raptis and Singh [16]. The authors employed an accelerated vertical plate in this investigation. Fluid flow near an exponential plate has been explored by Singh and Kumar [17]. MHD fluid flow across a flat plate was studied by Khan et al. [18]. The authors of this study looked at the effect of sidewalls on fluid flow. Haq et al. [19] explored the flow of MHD fluid on a porous sheet. Fetecau [20] has solved the fluid flow via a pipe analytically. The author used the Steklov expansion theorem to find the exact answer in this study. Furthermore, the same flow has been explored under the impact of side walls in this work by addressing the limiting instances. Using the Fourier transform, Fetecǎu and Zierep [21] examined a set of accurate solutions for the second-grade fluid modeled issue. An abrupt jerk was applied to the fluid in this experiment to get fluid motion.This work investigates a time-dependent fractional Casson fluid through a channel with porosity and MHD effect; the flow is caused by the bottom plate’s unstable motion, which is confined by sidewalls that are parallel to one another but normal to the bottom plate. For the aim of creating a dimensionless form of governing equations, a group of nondimensional variables has been used in the momentum equation and applied boundary conditions. The exact solutions for the momentum equation were then obtained using integral transforms [22, 23] such as Laplace, finite Fourier, and Fourier transforms, as well as the Caputo-Fabrizio fractional derivative. These ideas were then addressed for various bottom plate instances. We utilized Mathcad 15 to explain the graphical results for the modeled issue after recovering the exact answers for various scenarios. The impact of different parameters involved in the solution of the flow problem has been discussed upon flow characteristics.


## 2. Physical Description of Problem

Assume that a fractional unsteady Casson fluid flows on an infinite plate. The plate is limited by two sidewalls that are parallel to each other but normal to the bottom plate and are separated by a distance ofd. The flow is caused by the bottom plate because when t=0, both the plate and the fluid are at rest, and the bottom plate begins to move at an unsteady velocity U0ft, which satisfies the condition ft=0 at t=0 and piecewise continuous and exponential order at infinity. Because the motion is unidirectional, the equation describes the velocity of the flow system V=uy,z,ti such that v=w=0.Assumed flow system’s governing equations [24] are as follows:
(1)ρ∂u∂t=1+1ξ∂τxy∂y+∂τxz∂z−σBo2+μφku,y>0,0≤z≤d,where
(2)τxy=μ∂u∂y,τxz=μ∂u∂z.The initial and boundary conditions for the problem are as follows:(3)τxyy,z,0=τxzy,z,0=0,uy,z,0=0,u0,z,t=U0ft,uy,0,t=uy,d,t=0,although
(4)uy,z,t⟶0,∂uy,z,t∂y⟶0,fory⟶∞,where μ represents the dynamic viscosity and ρ is the density. Also, ft=0 at t=0 and ft≠0 for t>0.Consider the collection of dimensionless variables to transform the governing equation and its initial boundary conditions for the postulated flow problem into nondimensional form.(5)y∗=yd0,z∗=zd0,τxy∗,τxz∗=τxy,τxzρU02,u∗=uU0,t∗=tU0d0,L∗=dd0.By plugging Equation (5) into Equations (1) and (2), we get the following:
(6)∂u∂t=1β∂τxy∂y+∂τxz∂z−Keffu,y>0,0≤z≤L,(7)τxy=1Re∂u∂y,τxz=1Re∂u∂z.Here,Keff=M+1/K is the effective permeability; moreover, 1/K=νφd0/kU0 is the porous medium inverse permeability, and M=σBo2d0/ρU0 is the magnetic parameter.In a dimensionless form, the initial and boundary conditions are as follows:(8)uy,z,t=0,t=0,uy,z,t=fd0tU0=ft,y=0,uy,0,t=uy,L,t=0,uy,z,t⟶0,∂uy,z,t∂y⟶0,y⟶∞.The length characteristicsd0, Reynolds number Re=U0d0/v, and dimensionless Casson fluid parameters are depicted above.In a generalized form, the fractional constitutive equations are written as follows:(9)τxy=1ReDCFtα∂u∂y,τxz=1ReDCFtα∂u∂z.


## 3. Problem Solution

Using the Laplace transform to solve Equations (6) and (7) and incorporating Equation (9), we have the following:
(10)su¯=1β∂τ¯xy∂y+∂τ¯xz∂z−Keffu¯,τ¯xy=1Res1−αs+α∂u¯∂y,τ¯xz=1Res1−αs+α∂u¯∂z.By simplifying the above equations, we have the following:(11)∂2u¯∂y2+∂2u¯∂z2=Reβs+Keff1−αs+αsu¯.After using the Laplace transformation, the initial and boundary conditions were reduced to(12)u¯y,z,s=0,s=0,u¯y,z,s=F¯s,y=0,u¯y,z,s=u¯y,z,s=0,z=0,z=L,u¯y,z,s⟶0,∂u¯y,z,s∂y⟶0,y⟶∞.After that, multiply Equation (11) by 2/π∫0∞∫0Lsinψnzsinyχdzdy and using Equation (12), in which ζn=nπ/L and the resulting equation was rewritten as follows:
(13)u¯snχ,s=2π1−−1nζnχsFsReβ1−αs+αs+Keff+sχ2+ζn2.Rewrite Equation (13) in the following form:
(14)u¯snχ,s=2π1−−1nζnχFsχ2+ζn2−2π1−−1nζnχFsχ2+ζn21−αs+αs+Keff1−αs+αs+Keff+sχ2+ζn2/Reβ.Making use of Laplace inverse transform along with convolution theorem, we have the following equation:(15)u¯snχ,t=2π1−−1nζnχftχ2+ζn2+2π1−−1nζnf″t+F1f′t+F2ftχ2A1nχχ2+ζn2∗e−D1nχ−e−D2nχ.In Equation (15), we have the following:
(16)F1=1−αKeff+α1−α,F2=αKeff1−α,D1nχ=C1nχ2+A1nχ,D2nχ=C1nχ2−A1nχ,A1nχ=C1n2χ4−E2,C1nχ=Reβ1−αKeff+Reβα+χ2+ζn2Reβ1−α.After using Fourier inversion methods to solve Equation (15) and simplifying the resulting equation,
(17)uy,z,t=4ftL∑n=1∞e−ζmysinζmzζm+4πLf′′t+F1f′t+F2ft∑n=1∞sinζmzζm∗∫0∞χe−D11mχ−e−D22mχsinyξA11mχχ2+ζm2dχ.In Equation (17), m=2n−1, while
(18)A11mχ=C11m2χ4−F2,C11mχ=Reβ1−αKeff+Reβα+χ2+ζm2Reβ1−α,D11mχ=C11mχ2+A11mχ,D22mχ=C11mχ2−A11mχ,A11mχ=C11m2χ4−F2.TakeL=2h by translating system of the coordinate axis by putting z=z∗+h and ξm=2n−1/2hπ with ξm=ζm in Equation (17); after that, we have the following:
(19)uy,z,t=2fth∑n=1∞e−ξmy−1n+1cosξmzξm+2πhf″t+F1f′t+F2ft×∑n=1∞−1n+1cosξmzξm∗∫0∞χe−D1mχ−e−D2mχsinyχA1mχχ2+ξm2dχ,where
(20)A1mχ=C1m2χ4−F2,C1mχ=Reβ1−αKeff+Reβα+χ2+ξm2Reβ1−α,D11mχ=C11mχ2+A11mχ,D22mχ=C11mχ2−A11mχ.


## 4. Special Cases

### 4.1.h⟶∞ Means That Distance among the Sidewalls Will Be Maximized

When the distance between the sidewalls of a flow system is increased, the flow is no longer impacted by these sidewalls and Equation (19) is reduced to the following:
(21)uy,t=ft+1πf″t+F1f′t+F2ft∗∫0∞e−L1χt−e−L2χtsinyχχc1χdχ.In Equation (21),
(22)A2χ=C22χ4−F2,C2χ=Reβ1−αKeff+Reβα+χ2Reβ1−α,L1χ=C2χ2+A2χ,L2χ=C2χ2−A2χ.


### 4.2. Stokes’ First Problem Whenft=Ht

By choosingft=Htin Equation (19) (where Htis Heaviside’s unit step function), we will get the following:
(23)uy,z,t=2hHt∑n=1∞−1n+1e−ξmycosξmzξm+2hπ∑n=1∞−1n+1cosξmzξm×∫0∞χD2mχD1mχF1−D1m2χ−F2e−D1mχt−D1mχD2mχF1−D2m2χ−F2e−D2mχtsinyχD1mχD2mχA1mχχ2+ξm2dχ.If the side walls are ignored, Equation (23) becomes the following:
(24)uy,t=Ht+1π∫0∞L2χL1χF1−L21χ−F2e−L1χt−L1χL2χF1−L22χ−F2e−L2χtχL1χL2χc1χ⋅sinyχdχ.


### 4.3. Ifft=ta

By putting this case in Equation (19), after simplification, we will obtain the following:
(25)uy,z,t=2hta∑n=1∞−1n+1e−ξmycosξmzξm+2hπd2tadt2+F1dtadt+F2ta×∑n=1∞−1n+1cosξmzξm∗∫0∞χe−D1mχt−e−D2mχtsinyχc1mχχ2+ξm2dχ.The solution for an accelerating fractional Casson fluid is given by Equation (25). If the bottom plate is constantly accelerated, then ft=t; we get from Equation (19) that we obtain after simplification.
(26)uy,z,t=2ht∑n=1∞−1n+1e−ξmycosξmzξm+2hπ∑n=1∞−1n+1cosξmzξm×∫0∞χsinyχc1mχχ2+ξm2F1D1mχD2mχD1mχe−D2mχt−D2mχe−D1mχt+F2D21mχD22mχ−2c1mχD1mχD2mχt−A3mχ+D22mχe−D1mχt−D21mχe−D2mχtdχ.Again, by neglecting the effects of sidewalls that areh⟶∞ on fractional fluid flow, the velocity field takes the following form.
(27)uy,t=t+1π∫0∞sinyχχc1χF1L1χL2χL1χe−L2χt−L2χe−L1χt+F2L21χL22χ−2c1χL1χL2χt−A3χ+L22χe−L1χt−L21χe−L2χtdξ.


### 4.4.ft=Hteiωt

After the oscillation of the bottom plate, the fractional Casson fluid has oscillated with velocitiesusy,z,t and ucy,z,t,, respectively, corresponding to sine and cosine oscillations. This case gives the following:
(28)ucsy,z,t=2hcosωt∑n=1∞−1n+1e−ξmycosξmzξm+2hπ∑n=1∞−1n+1cosξmzξm∫0∞χsinyχA1mχχ2+ξm2×D1mχD2mχ−ω2ω2−F2−F1ω2C1mχcosωt+C1mχω2−F2+F1D1mχD2mχ−F1ω2sinωt×−2A1mχdχD1m2χ+ω2D2m2χ+ω2,ucty,z,t=2hπ∑n=1∞−1n+1cosξmzξm∫0∞χsinyχA1mχχ2+ξm2×D1mχF2−ω2+F1ω2D1m2χ+ω2e−D1mχt+D2mχF2−ω2−F1ω2D2m2χ+ω2e−D2mχtdχ,ussy,z,t=2hsinωt∑n=1∞−1n+1e−ξmycosξmzξm+2hπ∑n=1∞−1n+1cosξmzξm∫0∞χsinyχA1mχχ2+ξm2×D1mχD2mχ−ω2ω2−F2−F1ω2C1mχsinωt+F1ωω2−D1mχD2mχ−ωC1mχω2+F2cosωt×−2A1mχdχD1m2χ+ω2D2m2χ+ω2,usty,z,t=2hπ∑n=1∞−1n+1cosξmzξm∫0∞χsinyχA1mχχ2+ξm2×ωω2+F1D1mχ−F2D1m2χ+ω2e−D1mχt−ωω2+F1D2mχ−F2D2m2χ+ω2e−D2mχtdχ.Now, if we neglect the effects of sidewalls that areh⟶∞ on fractional fluid flow as a limiting case, then the velocity fields for such fluid flow caused by an oscillating bottom plate are as follows.
(29)ucsy,z,t=cosωt+1π∫0∞sinyχχA2χL1χL2χ−ω2ω2−F2−F1ω2C2χcosωt+C2χω2−F2+F1L1χL2χ−F1ω2sinωt×−2A2χdχL12χ+ω2L22χ+ω2,ucty,z,t=1π∫0∞sinyχχA2χL1χF2−ω2+F1ω2L12χ+ω2e−L1χt+L2χF2−ω2−F1ω2L22χ+ω2e−L2χtdχ,ussy,z,t=sinωt+1π∫0∞sinyχχA2χL1χL2χ−ω2ω2−F2−E1ω2C2χsinωt+F1ωω2−L1χL2χ−ωC2χω2+F2cosωt×−2A2χdχL12χ+ω2L22χ+ω2,usty,z,t=1π∫0∞sinyχχA2χωω2+F1L1χ−F2L12χ+ω2e−L1χt−ωω2+F1L2χ−F2L22χ+ω2e−L2χtdχ.


## 4.1.h⟶∞ Means That Distance among the Sidewalls Will Be Maximized

When the distance between the sidewalls of a flow system is increased, the flow is no longer impacted by these sidewalls and Equation (19) is reduced to the following:
(21)uy,t=ft+1πf″t+F1f′t+F2ft∗∫0∞e−L1χt−e−L2χtsinyχχc1χdχ.In Equation (21),
(22)A2χ=C22χ4−F2,C2χ=Reβ1−αKeff+Reβα+χ2Reβ1−α,L1χ=C2χ2+A2χ,L2χ=C2χ2−A2χ.


## 4.2. Stokes’ First Problem Whenft=Ht

By choosingft=Htin Equation (19) (where Htis Heaviside’s unit step function), we will get the following:
(23)uy,z,t=2hHt∑n=1∞−1n+1e−ξmycosξmzξm+2hπ∑n=1∞−1n+1cosξmzξm×∫0∞χD2mχD1mχF1−D1m2χ−F2e−D1mχt−D1mχD2mχF1−D2m2χ−F2e−D2mχtsinyχD1mχD2mχA1mχχ2+ξm2dχ.If the side walls are ignored, Equation (23) becomes the following:
(24)uy,t=Ht+1π∫0∞L2χL1χF1−L21χ−F2e−L1χt−L1χL2χF1−L22χ−F2e−L2χtχL1χL2χc1χ⋅sinyχdχ.


## 4.3. Ifft=ta

By putting this case in Equation (19), after simplification, we will obtain the following:
(25)uy,z,t=2hta∑n=1∞−1n+1e−ξmycosξmzξm+2hπd2tadt2+F1dtadt+F2ta×∑n=1∞−1n+1cosξmzξm∗∫0∞χe−D1mχt−e−D2mχtsinyχc1mχχ2+ξm2dχ.The solution for an accelerating fractional Casson fluid is given by Equation (25). If the bottom plate is constantly accelerated, then ft=t; we get from Equation (19) that we obtain after simplification.
(26)uy,z,t=2ht∑n=1∞−1n+1e−ξmycosξmzξm+2hπ∑n=1∞−1n+1cosξmzξm×∫0∞χsinyχc1mχχ2+ξm2F1D1mχD2mχD1mχe−D2mχt−D2mχe−D1mχt+F2D21mχD22mχ−2c1mχD1mχD2mχt−A3mχ+D22mχe−D1mχt−D21mχe−D2mχtdχ.Again, by neglecting the effects of sidewalls that areh⟶∞ on fractional fluid flow, the velocity field takes the following form.
(27)uy,t=t+1π∫0∞sinyχχc1χF1L1χL2χL1χe−L2χt−L2χe−L1χt+F2L21χL22χ−2c1χL1χL2χt−A3χ+L22χe−L1χt−L21χe−L2χtdξ.


## 4.4.ft=Hteiωt

After the oscillation of the bottom plate, the fractional Casson fluid has oscillated with velocitiesusy,z,t and ucy,z,t,, respectively, corresponding to sine and cosine oscillations. This case gives the following:
(28)ucsy,z,t=2hcosωt∑n=1∞−1n+1e−ξmycosξmzξm+2hπ∑n=1∞−1n+1cosξmzξm∫0∞χsinyχA1mχχ2+ξm2×D1mχD2mχ−ω2ω2−F2−F1ω2C1mχcosωt+C1mχω2−F2+F1D1mχD2mχ−F1ω2sinωt×−2A1mχdχD1m2χ+ω2D2m2χ+ω2,ucty,z,t=2hπ∑n=1∞−1n+1cosξmzξm∫0∞χsinyχA1mχχ2+ξm2×D1mχF2−ω2+F1ω2D1m2χ+ω2e−D1mχt+D2mχF2−ω2−F1ω2D2m2χ+ω2e−D2mχtdχ,ussy,z,t=2hsinωt∑n=1∞−1n+1e−ξmycosξmzξm+2hπ∑n=1∞−1n+1cosξmzξm∫0∞χsinyχA1mχχ2+ξm2×D1mχD2mχ−ω2ω2−F2−F1ω2C1mχsinωt+F1ωω2−D1mχD2mχ−ωC1mχω2+F2cosωt×−2A1mχdχD1m2χ+ω2D2m2χ+ω2,usty,z,t=2hπ∑n=1∞−1n+1cosξmzξm∫0∞χsinyχA1mχχ2+ξm2×ωω2+F1D1mχ−F2D1m2χ+ω2e−D1mχt−ωω2+F1D2mχ−F2D2m2χ+ω2e−D2mχtdχ.Now, if we neglect the effects of sidewalls that areh⟶∞ on fractional fluid flow as a limiting case, then the velocity fields for such fluid flow caused by an oscillating bottom plate are as follows.
(29)ucsy,z,t=cosωt+1π∫0∞sinyχχA2χL1χL2χ−ω2ω2−F2−F1ω2C2χcosωt+C2χω2−F2+F1L1χL2χ−F1ω2sinωt×−2A2χdχL12χ+ω2L22χ+ω2,ucty,z,t=1π∫0∞sinyχχA2χL1χF2−ω2+F1ω2L12χ+ω2e−L1χt+L2χF2−ω2−F1ω2L22χ+ω2e−L2χtdχ,ussy,z,t=sinωt+1π∫0∞sinyχχA2χL1χL2χ−ω2ω2−F2−E1ω2C2χsinωt+F1ωω2−L1χL2χ−ωC2χω2+F2cosωt×−2A2χdχL12χ+ω2L22χ+ω2,usty,z,t=1π∫0∞sinyχχA2χωω2+F1L1χ−F2L12χ+ω2e−L1χt−ωω2+F1L2χ−F2L22χ+ω2e−L2χtdχ.


## 5. Numerical Results and Discussions

With the influence of MHD and porosity, three distinct forms of motion have been examined for fractional Casson fluid in a channel passing from a horizontal, unsteadily, and infinite moving plate that is confined by two parallel side walls separated by a distanced. For the objective of obtaining an exact solution, joint Laplace and Fourier transforms were used. For the problem under examination, many situations at the bottom plate have been studied. The same flow has been studied under the impact of side walls by addressing the limiting instances. After determining general solutions for any arbitrary flow, solution for impulsive, solution for constantly accelerated, and solution for the sinusoidal fluid motion of the bottom plate were retrieved.Flow characteristics for Stokes’ first issue are depicted in Figures1(a)–1(d). It can be seen from these graphs that velocity decreases as the fractional parameter α increases because when the fractional parameter is increased, the fluid velocity decreases. An opposite impact is also observed for the dimensionless parameter for Casson fluid β upon flow characteristics. It is also worth noting that velocity is a decreasing function of effective permeability Keff, in these graphs. On the other hand, Reynolds number Re has the opposite effect on flow properties. The velocity of the fluid rises as the Reynolds number increases.Figure 1
Flow characteristics for the impulsive type motion of fractional fluid whenz∈0,1.
(a)(b)(c)(d)Figures2(a)–2(d) show velocity profiles for fractional fluid flow with a ramping type. Figure 2(a) indicates that when Keff=0.7, β=1, and Re=2 then with decreasing values of fractional parameter α, the characteristics of the flow are also improved. The velocity profile rises with increasing Casson fluid values, as seen in Figure 2(b). In Figure 2(c), the velocity decreases as the effective permeability increases. Similarly, when the Reynolds number Re increases, the flow properties improve. As a result, the velocity profile is a decreasing function of the fractional and effective permeability parameters.Figure 2
Flow characteristics for constant acceleration of fractional fluid whenz∈0,1.
(a)(b)(c)(d)The effects of cosine oscillations on flow characteristics are depicted in Figures3(a)–3(d). In the instance of cosine oscillations, we can see that velocity decreases as the fractional parameter α and effective permeability Keff increase, and that flow profiles decrease as these two components increase which is observed from Figures 3(a) and 3(c). The flow profile is rising with greater values of the dimensionless parameter for Casson fluid parameter β and Reynolds number Re as seen in Figures 3(b) and 3(d). The effect of sine oscillations on flow characteristics is seen in Figures 4(a)–4(d). For sine oscillations on flow characteristics, a similar effect has been seen for various values of significant parameters.Figure 3
Flow characteristics for cosine oscillations of fractional fluid whenz∈0,1.
(a)(b)(c)(d)Figure 4
Flow characteristics for sine oscillations of fractional fluid whenz∈0,1.
(a)(b)(c)(d)


## 6. Conclusions

The precise solutions for a time-dependent fractional Casson fluid traversing a channel under MHD and porosity effects are described in this paper. The flow is caused by the bottom plate’s unstable motion, which is restricted by parallel but perpendicular sidewalls. By using an appropriate set of dimensionless variables, the momentum equation, as well as the initial and boundary conditions, has been changed to a dimensionless form. A mix of Laplace and Fourier transformations is used to get the exact solution of the momentum equation. For the problem under examination, many situations at the bottom plate have been studied. The same flow has been studied under the impact of side walls by addressing the limiting instances. After determining general solutions for any arbitrary flow, solutions for impulsive, solutions for constant accelerated, and solutions for the sinusoidal fluid motion of the bottom plate were retrieved. These precise solutions are also visually depicted in support of our effort and then theoretically presented. The following points are emphasized after a thorough evaluation of the work.(i)
In this study, it was discovered that Reynolds number has a significant influence on flow characteristics since the velocity of the fluid increases as the Reynolds number grows in all limiting/special instances(ii)
Whether the bottom plate is at rest or moving by constant acceleration, the velocity of the fluid decreases as the fractional parameter and effective permeability increase for long periods of time(iii)
The velocity of cosine/sine oscillations increases with the rising Reynolds number and Casson parameter, since flow profiles increase with rising values of these two parameters, as seen in this work. The flow profile, on the other hand, decreases when the fractional parameter and effective permeability increase



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*Source: 2765924-2022-03-02.xml*