# Magnetic Field Effect on Heat and Momentum of Fractional Maxwell Nanofluid within a Channel by Power Law Kernel Using Finite Difference Method

Authors: Maha M. A. Lashin; Muhammad Usman; Muhammad Imran Asjad; Arfan Ali; Fahd Jarad; Taseer Muhammad
Journal: Complexity
 (2022)

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## Abstract

The mathematical model of physical problems interprets physical phenomena closely. This research work is focused on numerical solution of a nonlinear mathematical model of fractional Maxwell nanofluid with the finite difference element method. Addition of nanoparticles in base fluids such as water, sodium alginate, kerosene oil, and engine oil is observed, and velocity profile and heat transfer energy profile of solutions are investigated. The finite difference method involving the discretization of time and distance parameters is applied for numerical results by using the Caputo time fractional operator. These results are plotted against different physical parameters under the effects of magnetic field. These results depicts that a slight decrease occurs for velocity for a high value of Reynolds number, while a small value ofRe provides more dominant effects on velocity and temperature profile. It is observed that fractional parameters αandβ show inverse behavior against uy,t and θy,t. An increase in volumetric fraction of nanoparticles in base fluids decreases the temperature profile of fractional Maxwell nanofluids. Using mathematical software of MAPLE, codes are developed and executed to obtain these results.


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## Body

## 1. Introduction

Partial differential equations (PDEs) are the best way to express physical phenomena mathematically. PDEs are widely used in many fields of engineering like bioengineering, chemical engineering, and oceanography. Few years earlier, the main focus of researchers was the integral order of these PDEs. But, for the last few decades, the fractional order of PDEs is a hot topic among scientists. This is because the fractional modeling of natural phenomena gave a new direction to solutions of real-world problems, including diffusion, chaos, chemical reactions, dynamics, and viscoelasticity [1–3]. Approximately, all the polymeric matters have a viscoelastic behavior and conventional derivatives do not interpret such trend. Most of the fractional fluid problems are solved analytically due to the linearity of the problems. But, for the nonlinear problem, analytical techniques are complex to use. Fractional modeling of such physical problems can describe the heredity aspects and memory effect of problems. Nowadays, this idea of fractional modeling has been published in several articles of applied mathematics, fluid dynamics, and thermal engineering. These models are formulated by using various differentiation operators such as Caputo, Caputo-Fabrizio, and Atangana–Baleanu derivatives [4, 5]. An analytical solution has been obtained via Laplace transformation, and it is concluded that fractional results are better rather than using classical derivation for temperature and velocity profile [6, 7]. These operators are used to investigate mass concentration, heat flow, and momentum along different geometries. These theories have been applied to various fluids including Cassin fluids, Brickman type fluids, Oldroyd-B fluids, and Maxwell fluids as well [8, 9]. Recently, the Maxwell models have gained much attention from researchers as it is the first and one of the simplest rate type models (RTMs). The Maxwell model is widely used to represent the response of polymeric liquids. But, this model does not express the typical relation between shear strain and shear stress [10, 11]. The research work which has already been done for fractional Maxwell fluid (FMF) modeling (particularly on analytical side) has various bounds for momentum transfer only [12–17]. An investigation has been done for FMF flow, by introducing some suitable variables to make the irregular boundary of the stretching sheet and the regular one in [18]. It can be seen [19] that Brownian motion, mass concentration, and temperature profile as well are studied for FMF flow near a moving plate by using L1-algorithm i.e., numerically. By applying Laplace and Henkel transformation jointly, flow of FMF was investigated in [20]. The recent development in modeling of FMF rather than that of simple Maxwell fluids may be seen in [21–23]. In recent days, fractional modeling of Maxwell fluids with nanomaterials is the hot issue in nanotechnology. Nanomaterials are the nanoparticles of size range from 1 nm to 100 nm. These nanosized particles are helpful to enhance the thermal conductivity of base fluids (water, sodium alginate, kerosene oil, engine oil, etc.). This idea was given for the first time by Choi and Eastman in [24], and later on, the size and shapes of different nanoparticles were investigated in a square cavity in [25]. Since the addition of nanoparticles in base fluids increases the surface area of the fluid, it consequently enhances the heat conduction of the system, .i.e., control the entropy generation of heat. Analytical study has been done using Laplace transform for Caputo time derivatives of convective flow. Under the effects of magnetic field, exact solutions were obtained in [26]. Shamushuddin and Eid [27] examined heat transfer in water-based nanofluids containing ferromagnetic nanoparticles flowing between parallel stretchable spinning discs with variable viscosity influences and variable conductivities through the Chebyshev spectral collocation procedure. Unsteady flow was investigated under the effect of pressure gradient and magnetic field by using Laplace transformation as in [28]. Developing a fractional, coupled but linear PDEs model, the results were plotted against different physical parameters in [29, 30]. Similarly, it can be seen that solutions of many PDEs models are obtained analytically. After many assumptions, the models are turned into linear ones for simplicity of the problems. In [31–33], the analytical approach is used to find the solutions of mathematical models. Also, mostly results are driven by analytical technique by many assumptions to make the model a linear one for simplicity.The research work which has already been discussed has various research gaps in the field of nanofluids. As numerical study had not been performed, fractional behavior of mathematical models was not discussed properly with the basic tensor form. Therefore, this article deals with numerical solutions of unsteady flow of MHD-based fractional Maxwell nanofluids. This will provide the basis for further in-depth study while investigating the dynamics of FMF within a bounded channel instead of other geometric properties. Rather than the analytical technique, the strong numerical technique of the finite difference method FDM is applied to obtain solution of the FMF which involves discretization of spatial and time derivatives. The velocity profile and temperature profile have been plotted against various physical parameters by using MAPLE software. By developing and executing MAPLE coding against different physical parameters, results are obtained graphically.


## 2. Mathematical Modeling

The boundary layer flow within a channel is considered in this article, taking water-based nanofluids (Cu and Al2O3) in a vertical channel. Both the plates are separated by a distance d. One of the plates is fixed along the x-axis, vertically upward, i.e., x-axis is parallel to the plates and y-axis is normal to the plates, with B0 strength of magnetic field. At the start, for t=0, plates as well as fluids are supposed to have temperature θd. For some time t>0, the temperature is raised to θ0, causing the free convection flow as illustrated in Figure 1.Figure 1
Geometry of the problem.Hence, the velocity field is of the formVx,y,t=Vuy,t,0,0. Considering the unsteady flow of water-based nanofluid in this vertical channel, the assumptions for the mathematical formulation of PDEs of the coupled and nonlinear fractional Maxwell nanofluid model is as follows:(i)
Flow is incompressible, viscoelastic, and nonlinear(ii)
Flow is unsteady(iii)
Pressure gradient is neglected, i.e.,∂P/∂x=0(iv)
A uniform magnetic field is applied along the vertical directionalongy−direction, neglecting induced magnetic field(v)
Viscous dissipation is absentWe know that the tensor for the Maxwell fluid given in [34] is(1)T=−pI+S,S+λ1δSδt=μA1,where S,I,p,T,λ1,andA1 are the extra stress tensor, identity tensor (matrix tensor), dynamic pressure, Cauchy stress tensor, time relaxation, and first Rivlin–Ericksen tensor, respectively. And, DS/Dt is given in [35] and defined as(2)δSδt=DSDt−LS−SLT,where D/Dt is the material time derivative and A1 is the first Rivlin–Ericksen tensor defined as(3)A1=gradV+gradVT.Using all of the above-discussed results, the constitutive relation for Maxwell fluid model is obtained [15].(4)1+λ1αDtαSxy=μ∂u∂ywith0<α<1,where Sxy is the nonzero component of extra stress tensor, μ is the coefficient of viscosity, λ1 is the time relaxation, and Dtα is Caputo time fractional differentiation operator of order α, defined in [36].(5)Dtα0Cft=1Γ1−α∫0tt−η−α∂fη∂ηdη,0<α<1,where Γ. is the Gamma function defined in [36].(6)Γz=∫ηz−1e−ηdη,zϵℂ,ℝez>0.Under the aforementioned assumptions, the mathematical model of this problem is as follows. The equation of continuity [37] is(7)∂ρ∂t+ρ∇.V=0,where ρ is the density, ∇ is the gradient operator, and V is a velocity field.Here, we neglectv component of velocity along y−direction, in both momentum and energy equations. Also, taking into account the Boussinesq approximation, the momentum equation is given as in [38](8)ρnf∂u∂t=∂Sxy∂y+gρβθnfθ−θ0−σnfB02u,where ρnf is the dynamic viscosity of the nanofluid and g,βθnf,σnf,andB0 are acceleration due to gravity, coefficient of thermal expansion of nanofluid, and coefficient of electrical conductivity for nanofluids, and magnetic field strength, respectively.Multiplying1+λ1αDtα on both sides of (7),(9)1+λ1αDtαρnf∂u∂t=1+λ1αDtα∂Sxy∂y+gρβθnf1+λ1αDtαθ−θ0−σnfB021+λ1αDtαu.Using1+λ1αDtαSxy=μnf∂u/∂y, the constitutive relation for Maxwell fluid in [39] is(10)1+λ1αDtαρnf∂u∂t=μnf∂2u∂y2+1+λ1αDtαgρβθnfθ−θ0−1+λ1αDtασnfB02u.Also, the energy equation in the presence of Joule’s heating effect in [38] is(11)ρCpnf∂θ∂t=−∂q∂y+σnfB02u2.Applying1+λ2βDtβ on both sides of (10),(12)1+λ2βDtβρCpnf∂θ∂t=−∂∂y1+λ2βDtβq+σnfB021+λ2βDtβu2.But, by fractional Cattaneo’s Law [40],(13)1+λ2βDtβq=−knf∂θ∂y.Hence, (11) becomes(14)1+λ2βDtβρCpnf∂θ∂t=knf∂θ2∂y2+σnfB021+λ2βDtβu2.It has the following initial and boundary conditions:(15)uy,0=0,u0,t=0,ud,t=0,θy,0=θ0,θ0,t=θ0,θd,t=θd.Employ the following transformation for the channel flow:(16)u∗=dνfu,x∗=xd,t∗=νfd2t,θ∗=θ−θ0θd−θ0,λ1∗=νfd2λ1,λ2∗=νfd2λ2,y∗=yd,M∗=σfB02νfρCpfθd−θ0,Ha2=M=d2σfB02μf,Pr=μCpfKf,Gr=d3gβθfθd−θ0νf2,1Re=μρU0d.Here,Ha=M,Pr,M∗, and Gr given in [41] are the square of Hartmann number, Prandtl number, Joule’s heating parameter, and Grashof number, respectively. The following governing equations for velocity and temperature profile are obtained after omitting “∗” notation for the sack of brevity of mathematical modeling:(17)1+λ1αDtα∂u∂t=b1∂2u∂y2+1+λ1αDtαb2Grθ−b3M1+λ1αDtαu,(18)1+λ2βDtβ∂θ∂t=b41Pr∂2θ∂y2+b5M∗1+λ2βDtβu2.Here,b1=a3/a1,b2=a2/a1,b3=a6/a1,b4=a5/a4,andb5=a6/a4. But, thermophysical properties for nanofluids in [42, 43] are known.(19)ρnfρf=a1=1−ϕ+ϕρsρf,ρβθnfρβθf=a2=1−ϕ+ϕρβθsρβθfμnfμf=a3=11−ϕ2.5,ρCpnfρCpf=a4=1−ϕ+ϕρCpsρCpfknfkf=a5=ks+2kf−2ϕkf−ksks+2kf+ϕkf−ks,σnfσf=a6=1+3σs/σf−1ϕσs/σf−2−σs/σf−1ϕ.By using nondimensional parameters, the initial and boundary conditions are as follows:(20)uy,0=0,u0,t=0,u1,t=0,(21)θy,0=0,θ0,t=0,θ1,t=1.Thus, governing equations (17) and (18) for the velocity and temperature profile of fractional Maxwell nanofluid with initial and boundary conditions represented in (20)-(21) express the physical phenomena of the coupled nonlinear model. Also, physical properties of nanoparticles presented in (17) with some thermophysical properties of base fluids and nanoparticles given in Table 1 are used for numerical results.Table 1
Thermophysical properties of some base fluids and nanoparticles.
Materialsρkgm−3CpJkg−1k−1kWm−1k−1β∗10−5k−1σΩm−1Water99741970.613210.05Copper89333854001.675.96∗ 107Alumina3970765400.852.6∗ 106


## 3. Skin Friction and Nusselt Number

For measuring shear stress and heat transfer effects in an ordinary integer order system, local skin friction and Nusselt number are defined in [44] as follows:(22)Cf=μρU02∂u∂yy=0,Nu=−kdθw−θ0∂θ∂yy=0.The skin friction coefficient and local Nusselt number for (FMF) can be written by using (3) that is the fractional stress tensor for Maxwell fluid on the plate with fractional time Caputo derivative (details can be seen in [45]).(23)Cf+λ1α∂αSf∂tα=μρU02∂u∂yy=0,(24)Nu+λ1β∂βNu∂tβ=−kdθd−θ0∂θ∂yy=0.The nondimensional form of (23)-(24) is given as(25)Cf+λ1α∂αSf∂tα=1Re2∂u∂yy=0.(26)Nu+λ1β∂βNu∂tβ=−knfkf∂θ∂yy=0.


## 4. Numerical Procedure

The discretization of the method for fractional-order model,Dtα0Cu,Dt1+α0Cu when 0<α≤1,ut and uyy, is specified as follows:(27)Dtj+1α0Cuyi,tj+1=λ1αt−αΓ2−αuij+1−uij+λ1αt−αΓ2−α∑l=1juij−l+1−uij−ldlα,(28)Dtj+11+α0Cuyi,tj+1=λ1αt−1+αΓ2−αuij+1−2uij+uij−1+λ1αt−1+αΓ2−α×∑l=1juij−l+1−2uij−l+uij−l−1dlα,(29)∂∂tuyi,tj+1|t=tj+1=1Δtuij+1−uij,∂2∂y2uyi+1,tj|y=yi+1=1Δy2ui+1j+1−2uij+1+ui−1j+1.The nonlinear term is approximated by means of the following concept:(30)u2yi,tj=uyi,tj+1uyi,tj.In (27)-(28), dlα=−l1−α+1+l1−α when l=1,2,3,…,j. A rectilinear grid is considered for investigating the numerical solution of the deliberated fluid model through grid spacing Δt>0 and Δy>0 in time and space directions separately; here, Δy=L/M and Δt=T/N where Δy,Δt∈Z+. The inner grid points yi,tj in the considered domain Ω=0,T×0,L are defined as iΔy=yi and jΔt=tj. Discretization of the discussed problem at each inner grid point is given as(31)1/Δtuij+1−uij+λ1αΔt−1+αΓ2−αuij+1−2uij+uij−1+λ1αΔt−1+αΓ2−α×∑l=1juij−l+1−2uij−l+uij−l−1blα=b1Δy2ui+1j+1−2uij+1+ui−1j+1+b2Grθij+1+b2GrΔt−αΓ2−αθij+1−θij+b2GrΔt−αΓ2−α∑l=1jθij−l+1−θij−lblα−b3Ha2Δt−αΓ2−αuij+1−uij+b3Ha2Δt−αΓ2−α∑l=1juij−l+1−uij−lblα.Also,(32)1Δtθij+1−θij+λ1βΔt−1+βΓ2−βθij+1−2θij+θij−1+λ1βΔt−1+βΓ2−β∑l=1jθij−l+1−2θij−l+θij−l−1blβ=b4PrΔy2θi+1j+1−2θij+1+θi−1j+1+b5M∗uijuij+1+λ1βΔt−βΓ2−βuij+1−uij−∑l=1juij−l+1−uij−lblα,for j=1,2,3,…,N−1andi=1,2,3,…,N−1.The simplest form of the above discretization is given as(33)−b1Δy2ui+1j+1+1Δt−λ1αΔt−1−αΓ2−α+2b1Δy2+b3Ha2Δt−αΓ2−αuij+1−b1Δy2ui−1j+1−b2Gr1+Δt−αΓ2−αθij+1=1Δt+2λ1αΔt−1−αΓ2−α+b3Ha2Δt−αΓ2−αuij−b2GrΔt−αΓ2−αθij−λ1αΔt−1−αΓ2−αuij−1=Fi,j,−b4PrΔy2θi+1j+1+1Δt+λ1βΔt−1−βΓ2−β+2b4PrΔy2θij+1−b4PrΔy2θi−1j+1=1/Δt−2λ1βΔt−1−βΓ2−βθij−λ1βΔt−1−βΓ2−βθij−1+Gi,j+Ni,j,with the following initial and boundary conditions(34)ui0=0,ui1=ui−1,θi0=0,θi1=θi−1,fori=0,1,2,3,…,M,u0j=0,uMj=0,θ0j=0,θMj=1,forj=0,1,2,3,…,N−1,where(35)Fi,j=−λ1αΔt−1+αΓ2−α∑l=1juij−l+1−2uij−l+uij−l−1blα+b2GrΔt−αΓ2−α∑l=1jθij−l+1−θij−lblα+b3Ha2Δt−αΓ2−α∑l=1juij−l+1−uij−lblα,Gi,j=−λ1βΔt−1+βΓ2−β∑l=1jθij−l+1−2θij−l+θij−l−1blβ,Ni,j=b5M∗uijuij+1+λ1βΔt−βΓ2−βuij+1−uij−∑l=1juij−l+1−uij−lblα.


## 5. Numerical Analysis and Discussion

### 5.1. Test Problem

Consider the following problem:(36)Dtα0CUy,t=∂2∂y2Uy,t−∂∂yUy,t+hy,t.Here, the conditions are given as follows and source term can be selected against the choice of fractional-order derivative:(37)Uy,0=U0,t=Ud,t=0.The exact solution of this problem isUy,t=yy−tt2. Various simulations have been performed to check the accuracy of the proposed scheme. Figures 1(a) and 1(b) are plotted for maximum absolute error (MAE) and computational order of convergence (COC) given as follows when N = 10, 20, 40, 80, 160, 320, 640:(38)MAE=max1≤i≤M1≤j≤NUyi,tj−Uij,COC=logMAEk/MAEk+1logNk+1/Nk.It is noted that the scheme is convergent against the selection of each fractional-order derivative and its convergence order increases asα⟶1. Figures 2(c) and 2(b) contain the L∞-norm between consecutive solutions, i.e., Uj+1−Uj∞ and Ui+1−Ui∞ when 0 ≤ i, j ≤ N, and M = 500. Again, it is found that the proposed scheme is very efficient, accurate, and reliable for this problem. It is also demonstrated that the solution is stable against the selection of fractional-order parameters and mesh parameters.Figure 2
Code validation of the proposed scheme and varying time mesh sizes against (a) computational order of convergence (COC) and (b) maximum absolute error (MAE) and varying mesh sizes for (c) time and (d) space againstL∞-norm between consecutive solutions.
(a)(b)(c)(d)


## 5.1. Test Problem

Consider the following problem:(36)Dtα0CUy,t=∂2∂y2Uy,t−∂∂yUy,t+hy,t.Here, the conditions are given as follows and source term can be selected against the choice of fractional-order derivative:(37)Uy,0=U0,t=Ud,t=0.The exact solution of this problem isUy,t=yy−tt2. Various simulations have been performed to check the accuracy of the proposed scheme. Figures 1(a) and 1(b) are plotted for maximum absolute error (MAE) and computational order of convergence (COC) given as follows when N = 10, 20, 40, 80, 160, 320, 640:(38)MAE=max1≤i≤M1≤j≤NUyi,tj−Uij,COC=logMAEk/MAEk+1logNk+1/Nk.It is noted that the scheme is convergent against the selection of each fractional-order derivative and its convergence order increases asα⟶1. Figures 2(c) and 2(b) contain the L∞-norm between consecutive solutions, i.e., Uj+1−Uj∞ and Ui+1−Ui∞ when 0 ≤ i, j ≤ N, and M = 500. Again, it is found that the proposed scheme is very efficient, accurate, and reliable for this problem. It is also demonstrated that the solution is stable against the selection of fractional-order parameters and mesh parameters.Figure 2
Code validation of the proposed scheme and varying time mesh sizes against (a) computational order of convergence (COC) and (b) maximum absolute error (MAE) and varying mesh sizes for (c) time and (d) space againstL∞-norm between consecutive solutions.
(a)(b)(c)(d)


## 6. Results and Discussion

This section of our research work deals with a detailed overview of the key numerical findings and physical interpretations of different emerging parameters such asPr,M,α,β,M∗, andϕ which are the Prandtl number, magnetic field parameter, fractional parameters, Joule’s heating parameter, and volumetric fraction of nanoparticles, respectively. The behavior of the velocity profile uy,t and temperature profile θy,t and the effects of aforementioned physical parameters are deliberated, as well as graphical illustration is made via MAPLE. Discretization of time and spatial derivatives is done using finite difference methods. The coupled, nonlinear, and fractional model has been solved numerically by using finite difference method (FDM) which is a dominant tool to deal with such kind of problems.Results are obtained by solving (17)-(18) with initial and boundary conditions illustrated in (20)-(21) and physical properties of nanoparticle in (17) and Table 1. Various suitable ranges of physical parameters 0.01≤λ1α≤0.5,0≤Ha=M≤5,1≤Gr≤5,0.01≤ϕ≤0.2,and0.2≤ν≤1 for dimensionless velocity profile and 0.01≤λ1β≤0.2,0≤M∗≤2,6.2≤Pr≤35, 0.01≤ϕ≤0.2, and 0.2≤ν≤1 for heat transport are considered, and also particular exertion has been given on the effects of these parameters on the velocity and temperature profile.Figure3 depicts the impact of time relaxation parameter λ1α on momentum uy,t of the fractional Maxwell fluids. With increase in fractional parameter α, momentum and thermal boundary layers decrease and even become their thinnest for α=1. Therefore, increasing relaxation parameters with range (0.01≤λ1α≤0.5) has inverse impact on the velocity profile of the system, i.e., decrease occurs in the velocity profile.Figure 3
Influence ofλ1α on u when α=β=1,Gr=5,Ha=5,λ2β=0.1,Pr=6.2,M∗=0.5, and ϕ=0.1.Figure4 shows the influence of magnetic field Ha=M (the square of Hartmann Number) parameters on velocity profile uy,t. Both are inversely related, i.e., increasing value of Hartmann number Ha decreases the velocity profile. Since the increase in magnetic field parameter Ha gives hype to a well-known Lorentz force as this is the resistive force which works against the flow direction, consequently it shows decrease in all the velocity components.Figure 4
Influence ofHa on u when α=β=1,λ1α=0.5,Gr=5,λ2β=0.1,Pr=6.2,M∗=0.5, and ϕ=0.1.Figure5 displays the behavior of Grashof number Gr on velocity profile uy,t of fractional Maxwell fluids (FMFs) under the effects of magnetic field. Since Grashof number Gr is the ratio of buoyancy force to viscous force and is alsoknown as buoyancy parameter, motion is resisted by the viscous force. So it was expected that an increase in Gr leads to an increase in the velocity profile of the bounded system, specifically near the wall of the bounded channel.Figure 5
Influence ofGr on u when α=β=1,λ1α=0.5,Ha=5,λ2β=0.1,Pr=6.2,M∗=0.5, and ϕ=0.1.In Figure6, results are drawn for volumetric fraction of nanoparticles against flow of fractional Maxwell fluids (FMFs). Addition of nanoparticles in base fluids increases their thickness (viscosity) which causes the internal resistance between the layers of flowing fluids, consequently decreasing the velocity uy,t of the fluid. This is clearly deliberated in Figure 6.Figure 6
Influence ofϕ on u when α=β=1,λ1α=0.5,Gr=5,Ha=5,λ2β=0.1,Pr=6.2,andM∗=0.5.Finally, the velocity profile againstα=β=ν (fractional parameters) is plotted in Figure 7, and results are verified as expected. The consequences of fractional order on fluid motion have an inverse relation. That is, for increasing values of fractional parameter α, the velocity profile decreases. However, uy,t decreases for increasing values of α and attains its peak at α=1.Figure 7
Influence ofα=β=ν on u when λ1α=0.5,Gr=5,Ha=5,λ2β=0.1,Pr=6.2,M∗=0.5, and ϕ=0.1.The heat transfer capability of the coupled and nonlinear model is illustrated in Figure8. Here, the results for the temperature profile against time relaxation parameter λ2β are drawn and found as expected. Time relaxation is the key parameter used for characterization of the viscoelastic fluids, and it is the time in which a system relaxes under certain external conditions. Therefore, by the increase in λ2β, there results a decrease in the collision of particles within the fluids. This decreases the temperature profile θy,t of fractional Maxwell nanofluids.Figure 8
Influence ofλ2β on θ when α=β=1,λ1α=0.5,Gr=5,Ha=5,Pr=6.2,M∗=0.5, and ϕ=0.1.Figure9 displays that magnetic field parameter impacts directly the temperature of the system because the enhancement in magnetic field parameter Ha=M∗ gives rise to a Lorentz force. This results in increase in the temperature profile θy,t of the system.Figure 9
Influence ofM∗ on θ when α=β=1,λ1α=0.5,Gr=5,Ha=5,λ2β=0.1,Pr=6.2, and ϕ=0.1.Since Prandtl numberPr is the dimensionless number and is the ratio of momentum to thermal diffusivity. Since it is a fluid property, it does not have any dependence on flow type, as viscous forces exert a uniform effect on heat transfer for the whole of location of the channel. So, increase in Pr means heat transfer is favored to occur by momentum, not conduction. Therefore, increase in Pr decreases the temperature profile θy,t of fractional Maxwell fluids (FMFs) as expressed in Figure 10.Figure 10
Influence ofPr on θ when α=β=1,λ1α=0.5,Gr=5,Ha=5,λ2β=0.1,M∗=0.5, and ϕ=0.1.Figure11 gives the graphical results for influence of volume fraction of nanoparticles in base fluids on heat transfer capability of the system. Addition of nanoparticles in base fluids has a direct impact on enthalpy of the system. This results in entropy control of fluids during flow that is enhancement of thermal conductivity of fluids. The figure shows that increasing volume fraction ϕ enhances the thermal conductivity of the FMF with decrease in the temperature profile θy,t.Figure 11
Influence ofϕ on θ when α=β=1,λ1α=0.5,Gr=5,Ha=5,λ2β=0.1,Pr=6.2,andM∗=0.5.Finally, Figure12 depicts the effect of α=β=ν on the temperature profile of the fractional Maxwell fluids. The fractional-order parameter and temperature profile are inversely proportional. It was expected and we obtained that increase in α=β=ν decreases the heat transfer capability of the system.Figure 12
Influence ofα=β=ν on θ when λ1α=0.5,Gr=5,Ha=5,λ2β=0.1,Pr=6.2,M∗=0.5, and ϕ=0.1.The variations of skin friction coefficient and local Nusselt number are deliberated in Tables2 and 3. It is noted that the coefficient of skin friction increases with the increase in the physical parameters Gr, λ1α, and ϕ . The reverse behavior is observed against the variation of Hartmann number. Nusselt number impact against Pr, M∗, λ2β, and ϕ seems increasing. On the other hand, dominant impact of the fractional-order parameters α=β=ν can be seen in Tables 2 and 3.Table 2
Skin friction analysis against different physical parameters whenλ2β=0.1,Pr=6.8,M∗=0.5,andRe=10.
GrMλ1αϕν=0.4ν=0.7ν=10.1100.50.10.00470.00510.005820.09340.10160.116050.23160.25190.287701.09660.68190.405821.07300.70470.434851.06000.75000.48820.10.41770.41190.40580.30.45130.44450.43480.50.51660.50440.48820.010.36420.41050.37780.150.37530.44290.40490.250.40630.49100.4304Table 3
Nusselt number analysis against different physical parameters whenλ1α=0.5,M=10,Gr=5,andRe=10.
PrM∗λ2βϕν=0.4ν=0.7ν=13.940.50.10.10.91420.75110.21346.20.97100.81290.2105150.99990.88170.20383.940.10.70870.67990.638420.76870.73630.689450.84580.80700.75050.50.010.75430.70240.65520.10.76010.76170.75580.20.76720.83741.31880.010.45820.79670.93350.150.49280.85820.98230.250.58540.91400.9957


## 7. Conclusion

An unsteady flow and heat transfer for the coupled and nonlinear model of fractional Maxwell fluids is solved by using power law kernel. Findings are done under the effects of magnetic fields within a channel. Codes are developed and executed to obtain the numerical results by applying the finite difference method for discretization of spatial and time derivatives. Some key finding are illustrated as follows:(a)
Fractional-order parametersαandβ have a direct impact on velocity profile and inverse impact on temperature profile.(b)
The velocity and temperature are enhanced for a high value of the unsteadiness parameter. Velocity is slightly decreasing for higher values of Reynolds numberRe, while a smaller value of Reynolds number has more prominent impact on velocity and temperature.(c)
Addition of the nanoparticles to base fluids enhances the thermal conductivity by increasing the surface. Consequently, volumetric concentration of nanoparticlesϕ in base fluids results in decrease in the temperature profile of the FMF.(d)
Finally, the chosen numerical technique of the finite difference method shows stable results and gives new direction to such investigation.(e)
This method can be extended for more numerous types of physical sciences with complex geometries.This simplified research problem can be generalized to express the effects of viscosity (viscous dissipation), variable thermal conductivity, and multidimensional MHD flow regime and temperature profile of non-Newtonian nanofluids. Many opportunities for further investigation exist in this direction for detailed study.



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