# Mixed Convection Magnetohydrodynamics Flow of a Nanofluid with Heat Transfer: A Numerical Study

Authors: Abdul Quayam Khan; Amer Rasheed
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper we have studied the magnetohydrodynamic (MHD) mixed convection Maxwell flow of an incompressible nanofluid with magnetic field and heat transfer over a moving plate aligned horizontally. Thermal radiation has also been applied in order to investigate its effects on velocity and temperature variations in the fluid. The Caputo time derivative has been employed to derive the mathematical model. A numerical solution has been obtained using finite difference discretization along withL1-algorithm. Fractional and other pertinent physical fluid parameters like magnetic field parameter, thermal radiation, effect on velocity, and temperature distribution are analyzed and demonstrated through graphs.


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## Body

## 1. Introduction

Heat transfer and viscoelastic fluid flow gain much attention because it has widespread applications in several fields of science and engineering. Maxwell fluid model, also known as viscoelastic rate type fluid model, was introduced by James Clerk Maxwell [1] in his research article on the dynamical theory of gases. The Maxwell model has advantage of being expressed as a differential model in a basic form. This permits comparison between results of different methods. Much work has been done by many researchers in Maxwell fluids. Hayat et al. [2] studied Maxwell fluid flow of heterogeneous and homogeneous processes in the stagnation region over a stretched surface and analyzed the behavior of a flow with several set of physical parameters. Zhi Cao et al. [3] have investigated the effect of magnetic field on a fractional Maxwell nanofluid over a moving plate and also discussed the temperature variations due to nanoparticles present in the base fluid using L1 scheme along with finite difference approximations. Liu and Guo [4] studied MHD flow of Maxwell liquid with slip conditions pursued by a moving plate. Mashud et al. [5] examined a Maxwell fluid of unsteady flow with boundary layer approximation. Madhu et al. [6] analyzed MHD flow of a Maxwell nanofluid with thermal radiation and stretching surface and also discussed flow behavior of a Maxwell nanofluid using different values of physical parameters. Vieru and Rauf [7] investigated Maxwell flow in a channel with slip conditions and found exact as well as approximate solution of a model and discussed slip coefficient effects on velocity. Hsiao [8] studied the Maxwell fluid model of a thermal extrusion energy conversation problem and solved it by using numerical techniques. Hayat et al. [9] investigated the rotating flow of Maxwell liquid in three-dimensional nanofluid wherein the fluid was moving with the help of an exponentially stretching sheet and solved the problem by Homotopy Analysis Method. Ramesh et al. [10] investigated the Maxwell fluid model of stagnation point flow on a stretching sheet and found an approximate solution of the model by using Runge Kutta Fehlberg Method. Mukopadhyay [11] investigated the Maxwell fluid model of MHD over stretching sheet using temperature on the surface and numerically solved the problem by Shooting Method and also analyzed fluid behavior with different physical parameters.The enhancement of thermal conductivity with base fluid and heat transfer improvement of fluids is an important research topic due to its tremendous applications in industry and sciences. Heat transfer fluids have a new category known as nanofluids and were first suggested by Choi and Eastman [15] and refer to a liquid which is heated by diverged nanoparticles into a base fluid. Nanofluid is a fluid having nanometer (small diameters of nanoscale) size elements, known as nanoparticles. Nanoparticles can be metallic, for example, (Fe,Al,Ag,Au,Cu), or nonmetallic, for example, (Al2O3,Fe2O3,CuO,TiO2,SiO2), and could also be carbon nanotubes, for example, (CNT,MWCNT,SWCNT). Nanofluids enhance thermal properties compared to the base fluids and this fact makes nanofluids an alternative working fluids. More precisely, nanoparticles enhance thermal conductivity, viscosity, and density but decrease specific heat capacity. Prasannakumara et al. [16] studied radiative heat transfer of nanofluids using magnetic field over a plate. They noticed that the Nusselt number and Sherwood number are increased for nonlinear stretching sheet. Kumar et al. [17] investigated the Marangoni effects on nanofluid in the presence of heat. Sheikholeslami and Shehzad [18] studied heat transfer of nonequilibrium model for nanofluid with magnetic field and porous medium. Turkyilmazoglu and Pop [19] studied heat and mass transfer of free convection flow of nanofluid containing Al2O3,Cu,Ag,CuO, and TiO2 as nanoparticles. They conclude that the fluid having particles of kind TiO2 have low heat transfer and with Cu have maximum heat transfer. Recently various researchers studied heat transfer of nanofluids; see, for example, [20–24].Radiation for heat transfer is important because of its wide applications in nuclear power generation, reactor cooling, and combustion applications. The best possible comprehension of instrument of solar radiation has real significance in the plan of advance energy conversation system performed at high temperature. Some examples of these are space vehicle reentry, astrophysical flows, fossil fuel, solar power technology, and combustion energy. Such impacts usually happen when there is difference between surface and surrounding temperatures. Thermal radiation effects on micropolar fluid flow and heat transfer over a porous shrinking sheet is studied by Bhattacharyya et al. [25]. Bhattacharyya [26] analyzed MHD flow of Casson fluid in the presence of radiative stretching sheet in the stagnation region. Hayat et al. [27] investigated the Maxwell flow of mixed convection near a stagnation point along with thermal radiation and convective boundary conditions.It is now established experimentally as well as theoretically that the mathematical models derived with the help of fractional order derivatives simulate certain physical phenomenon more realistically, particularly for the systems wherein the hereditary effects are important as they depend on the past conditions. Recently fractional models have been developed and employed in many fields of science and engineering like fluid dynamics, electromagnetic, biopopulation models, viscoelasticity, optics, electro-chemistry, and signal processing in order to establish behavior of several physical quantities; see, for example, [28–31] and the references therein. For accurate modeling of damping, fractional models are used; the reader is referred to study the articles [32–35]. Physical phenomena of fluid mechanics, electricity, quantum mechanics models, etc. are controlled within their domain of validity by using integer order partial differential equations [36–38]. Many researchers are using fractional calculus in their research work such as Fetecau et al. [39] who investigated the fractional model of nanofluids with Caputo time derivative and discussed the influence of a fractional parameter on velocity and temperature. Shah et al. [40] studied the Caputo fractional derivative model of a blood flow in a circular cylinder with magnetic particles. Numerical simulations are used to study the behavior of a fractional parameter on fluid and particles velocity. Similarly many authors used fractional calculus in their research work [41–45].The aim of present study is to investigate the convection effects along with the magnetic field and thermal radiations on the boundary layer flow of fractional Maxwell fluid over a moving plate. The fluid motion is initiated by moving the plate impulsively along thex-axis. In order to capture memory effects during the motion of fluid, fractional calculus approach has been employed in order to derive the mathematical model which finally gives fractional coupled nonlinear partial differential equations with mixed time-space variables. The numerical simulations of the underlying model have been carried out by employing a newly developed finite difference method along with L1-algorithm. Various flow and fractional parameters effects on velocity and temperature profile are presented via graphs. Moreover, fractional and other physical parameters effects have also been discussed on average skin friction coefficient and average Nusselt number with the help of tables.


## 2. Mathematical Formulation of the Problem

The incompressible Navier-Stokes equations along with the energy conservation equation have been utilized in order to model the underlying physical phenomenon of two dimensional boundary layer Maxwell flow of nanofluid with heat transfer over a moving surface aligned withx-axis. Uniform magnetic field 0,B0,0 in the positive y- direction has been introduced in the model using Lorentz force and radiation qr has been applied externally while the fluid motion is considered along x coordinate. The velocity field is taken as(1)V=ux,y,t,vx,y,t,0 It is to be noted that the viscous dissipation effects are ignored in the derived model. Moreover, the fluid motion along horizontal direction is investigated; therefore the equation of velocity component v is ignored. Using the boundary layer assumptions and Boussinesq approximations, the equations for conservation of mass and momentum for nanofluid in the presence of magnetic field can be written as(2)∂u∂x+∂v∂y=0,(3)∂u∂t+u∂u∂x+v∂u∂y=νnf∂τxy∂y-σnfB02ρnfu-νnfknfu+βnfgT-T0.where nf stands for nanofluid, νnf is dynamic viscosity, ρnf is density, βnf is thermal expansion coefficient, knf is thermal conductivity, σnf is electrical conductivity of nanofluid, and g is the gravitational acceleration. The fractional order derivation is introduced into the constitutive equations for the Maxwell model as proposed by Friedrich [46](4)τxy+λ1α∂ατxy∂tα=μ∂u∂y,which leads to(5)1+λ1α∂α∂tατxy=μ∂u∂y,where λ1 is the relaxation time, α is fractional derivative such that 0<α<1, and τxy is shear stress. The operator ∂α/∂tα is Caputo time derivative of order α defined as [47](6)∂tαgt≔1Γn-α∫0tt-ξn-α-1∂ngξ∂ξndξ,n-1<Reα<n,n∈N,with Γ(·) denotes the classical Gamma function given by(7)Γz≔∫Rξz-1e-ξdξ,z∈C,Rz>0.In order to eliminate τxy from (3) and (5), applying the operator 1+λ1α∂α/∂tα on both sides of (3) and then using (5), we finally arrive at(8)∂u∂t+u∂u∂x+v∂u∂y+λ1α∂α+1u∂tα+1+λ1α∂α∂tαu∂u∂x+λ1α∂α∂tαv∂u∂y=νnf∂2u∂y2+βnfg1+λ1α∂α∂tαT-T0-1+λ1α∂α∂tασnfB02ρnf+νnfknfu,The temperature on the plate is presumed to be T0 and on the surface of the fluid the temperature is considered to be the ambient. The change of temperature within the fluid is modeled by using(9)∂T∂t+u∂T∂x+v∂T∂y=-∇·q-1ρcpnf∂qr∂y,where q is the heat flux, cpnf is heat capacity, and qr is the thermal radiation. Since the heat flux is supposed to be towards vertical direction, therefore the above equation can further be written as(10)∂T∂t+u∂T∂x+v∂T∂y=-∂q∂y-1ρcpnf∂qr∂y,The expression for q is given by the fractional form of generalized Fourier’s law introduced by Cattaneo [48](11)1+λ2γ∂γ∂tγq=-k∂T∂y,In order to eliminate q, first applying the operator 1+λ2γ∂γ/∂tα on both sides of (10) and then using (11), we have obtained(12)∂T∂t+u∂T∂x+v∂T∂y+λ2γ∂γ+1T∂tγ+1+λ2γ∂γ∂tγu∂T∂x+λ2γ∂γ∂tγv∂T∂y=knfρcpnf∂2T∂y2-1ρcpnf1+λ2α∂γ∂tγ∂qr∂y,where the expression qr defined by [49] (13)qr=-4σ3β∂T4∂y introduces the radiation contribution in the flow field. Expanding T4 by a Taylor series about T0 and neglecting higher orders of T, we can write T4=4T03T-3T04. Therefore qr can eventually be defined as(14)qr=-4σ3β∂T4∂y≅-16σT033β∂T∂yUsing (14) and (11), (12) can be written as(15)∂T∂t+u∂T∂x+v∂T∂y+λ2γ∂γ+1T∂tγ+1+λ2γ∂γ∂tγu∂T∂x+λ2γ∂γ∂tγv∂T∂y=knfρcpnf-16σT033βρcpnf∂2T∂y2.Finally, (2), (8), and (15) constitute the model problem which describes the underlying physical phenomenon addressed in this work(16)∂u∂x+∂v∂y=0,∂u∂t+u∂u∂x+v∂u∂y+λ1α∂α+1u∂tα+1+λ1α∂α∂tαu∂u∂x+λ1α∂α∂tαv∂u∂y=νnf∂2u∂y2+βnfg1+λ1α∂α∂tαT-T0-B121+λ1α∂α∂tαu,∂T∂t+u∂T∂x+v∂T∂y+λ2γ∂γ+1T∂tγ+1+λ2γ∂γ∂tγu∂T∂x+λ2γ∂γ∂tγv∂T∂y=Knf1∂2T∂y2.where (17)B12=σnfB02ρnf+νnfknf,Knf1=knfρcpnf-16σT033βρcpnfand ρnf is density of nanofluid, μnf is dynamic viscosity of nanofluid, and other parameters for the nanofluid are defined as [50](18)ρnf=1-ϕρnf+ϕρs,knfkf=ks+2kf-2ϕkf-ksks+2kf+ϕkf-ks,ρβnf=1-ϕρβf+ϕρβs,σnf=kf1+σ-1ϕ,ρcpnf=ϕρcps+1-ϕρcpf,μnf=μf1-ϕ2.5.In the above equations, ϕ denotes the volume fraction of the nanoparticles in nanofluid. ρs, ks, μs, and (cp)s denote density, thermal conductivity, viscosity, and specific heat capacities of surfactant respectively. Similarly ρf, kf, and (cp)f denote density, thermal conductivity, and specific heat capacities of base fluid, respectively.As described earlier, the fluid motion at timet=0 is initiated by an impulsive movement of the plate with the time dependent velocity tU0; the temperature of the plate is also dependent on time t; therefore the initial and boundary conditions which best suit the physical phenomenon can be given by(19)t⩽0:v=0,u=0,T=0,t>0:v=0,u=0,T=0atx=0,t>0:u=tU0,T=Td-T0T0taty=0,t>0:u→0,T→T0aty→∞.


### 2.1. Nondimensionalization

In order to understand physics of the proposed problem in an easy way we have nondimensionalized the governing problem. It helps to comprehend the underlying physical phenomenon with nondimensional parameters of interest without giving any weightage to units of involved quantities. Nondimensionalization of model (16) and (19) can be obtained by considering the following nondimensional variables and parameters: (20)t∗=tνd2,y∗=yd,x∗=xνd2U0,u∗=uU0,v∗=dvν,λ1∗=λ1νd2,λ2∗=λ2νd2,θ=T-T0Td-T0with the use of chain rule(21)∂u∂t=U0∂u∗∂t∗∂t∗∂t,∂u∂y=U0∂u∗∂y∗∂y∗∂y,∂αu∂tα=U0ναd2α∂αu∗∂t∗α.Using above expressions, model (16) and (19) takes the following form (after removing the asterisk (∗) sign):(22)∂u∂x+∂v∂y=0,∂u∂t+u∂u∂x+v∂u∂y+λ1α∂α+1u∂tα+1+λ1α∂α∂tαu∂u∂x+λ1α∂α∂tαv∂u∂y=∂2u∂y2+Grλ1α∂α∂tαθ+Grθ-M+Φ2u-M+Φ2λ1α∂αu∂tα,∂θ∂t+u∂θ∂x+v∂θ∂y+λ2γ∂γ+1θ∂tγ+1+λ2γ∂γ∂tγu∂θ∂x+λ2γ∂γ∂tγv∂θ∂y=1-Φ3Pr∂2θ∂y2.Φ2=d2/U0νknf, Φ3=-16T03(Td-T0)σ/3β(ρCp)nfknf, M=σnfB02d2/U0νμnf, Gr=βnfgTd-T0d2/U0ν Grashof number, and Pr=U0νcpμ/kd2 Prandtl number, respectively.Dimensionless form of boundary and initial conditions are(23)t⩽0:v=0,u=0,θ=0,t>0:v=0,u=0,θ=0atx=0,t>0:u=t,θ=taty=0,t>0:u→0,θ→0aty→∞.The subsequent section elucidates the numerical scheme employed to solve the nondimensional mathematical model derived in this section.


## 2.1. Nondimensionalization

In order to understand physics of the proposed problem in an easy way we have nondimensionalized the governing problem. It helps to comprehend the underlying physical phenomenon with nondimensional parameters of interest without giving any weightage to units of involved quantities. Nondimensionalization of model (16) and (19) can be obtained by considering the following nondimensional variables and parameters: (20)t∗=tνd2,y∗=yd,x∗=xνd2U0,u∗=uU0,v∗=dvν,λ1∗=λ1νd2,λ2∗=λ2νd2,θ=T-T0Td-T0with the use of chain rule(21)∂u∂t=U0∂u∗∂t∗∂t∗∂t,∂u∂y=U0∂u∗∂y∗∂y∗∂y,∂αu∂tα=U0ναd2α∂αu∗∂t∗α.Using above expressions, model (16) and (19) takes the following form (after removing the asterisk (∗) sign):(22)∂u∂x+∂v∂y=0,∂u∂t+u∂u∂x+v∂u∂y+λ1α∂α+1u∂tα+1+λ1α∂α∂tαu∂u∂x+λ1α∂α∂tαv∂u∂y=∂2u∂y2+Grλ1α∂α∂tαθ+Grθ-M+Φ2u-M+Φ2λ1α∂αu∂tα,∂θ∂t+u∂θ∂x+v∂θ∂y+λ2γ∂γ+1θ∂tγ+1+λ2γ∂γ∂tγu∂θ∂x+λ2γ∂γ∂tγv∂θ∂y=1-Φ3Pr∂2θ∂y2.Φ2=d2/U0νknf, Φ3=-16T03(Td-T0)σ/3β(ρCp)nfknf, M=σnfB02d2/U0νμnf, Gr=βnfgTd-T0d2/U0ν Grashof number, and Pr=U0νcpμ/kd2 Prandtl number, respectively.Dimensionless form of boundary and initial conditions are(23)t⩽0:v=0,u=0,θ=0,t>0:v=0,u=0,θ=0atx=0,t>0:u=t,θ=taty=0,t>0:u→0,θ→0aty→∞.The subsequent section elucidates the numerical scheme employed to solve the nondimensional mathematical model derived in this section.


## 3. Numerical Method

This section elucidates briefly the numerical scheme employed to solve the developed model (22)-(23). The finite difference approximations have been utilized along with L1 scheme in order to discretize the mixed fractional derivatives which appears in the model. Other numerical schemes can also be utilized in order to solve model (22)-(23); for example, V. E. Lynch et al. [51] have discussed two numerical schemes, namely, L2 and L2C, for solving partial differential equations of fractional order. The authors have investigated explicit and semi-implicit techniques along with two types of discretization schemes. It is found that these techniques depend on the correct choice of discretization methods as well as on the choice of α. The L2 scheme demonstrates good convergence results for α>1.5 whereas L2C shows adequate convergence for α<1.5.L 1 scheme is a newly introduced technique by F. Liu et. al. [52] wherein the authors have devised a method of discretizing the nonlinear convection terms as well as the terms which involve fractional order derivative of nonlinear terms using finite difference approximations. The L1 scheme does not depend on the particular choice of discretization or on the values of fractional order α. The technique has been successfully applied on various models and obtained significant results; see, for example, [3, 53] and references therein. The L1 scheme is best suited for the problems of the type considered in this manuscript. The scheme is described briefly in the subsequent section.


### 3.1. Discretization Method

Definetk=kΔt, k=0,1,2,3,…,r, time steps with Δt being a time step size and xi=iΔx, i=0,1,2,3,…,m; yj=jΔy, j=0,1,2,3,…,n, where Δx=Lx/m and Δy=Ly/n are space step sizes. Let uijk and θijk be numerical approximations of u(t,x,y) and θ(t,x,y) at point (xi,yj) and time tk in model (22)-(23).The Caputo time fractional derivative in (6) is discretized at time tk as follows:(24)∂tαutk=1Γ2-α∑s=0k∫tktk+1tk-ξ1-α-1∂uξ∂ξdξ,0<Reα<1,=Δt-αΓ2-αb0utk-bk-1ut0+∑s=1k-1bs-1-bsutk-s+OΔtwhere bs=(s)1-α-(s-1)1-α, s=0,1,2,…,r.The nonfractional derivatives are discretized using backward finite difference approximations as [52, 54](25)∂u∂tt=tk=uxi,yj,tk-uxi,yj,tk-1Δt+OΔt,u∂u∂xt=tk=uxi,yj,tk-1uxi,yj,tk-uxi-1,yj,tkΔx+OΔx,v∂u∂yt=tk=vxi,yj,tk-1uxi,yj,tk-uxi,yj-1,tkΔy+OΔy,∂2u∂y2=ui,j+1k-2ui,jk+ui,j-1k+ui,j+1k-1-2ui,jk-1+ui,j-1k-12Δy+OΔy2where ui,jk=u(xi,yj,tk) and the nonlinear terms have been linearized. Moreover the fractional derivatives of the above terms at time tk can be defined in the following way (see for details [52]):(26)∂α+1utk∂tα+1=Δt-1-αΓ2-αuijk-uijk-1-∑s=1k-1bs-1-bsuijk-s-uijk-s-1,∂α∂tαu∂u∂x=Δt-αΔxΓ2-αuijk-1uijk-ui-1jk-∑s=1k-1bs-1-bsuijk-s-1uijk-s-ui-1jk-s,∂α∂tαv∂u∂y=Δt-αΔyΓ2-αvijk-1uij+1k-uijk-∑s=1k-1bs-1-bsvijk-s-1uij+1k-s-uijk-s,Assume that(27)δ1=λαΔt-αΓ2-α,δ2=12Δy2,δ3=λγΔt-γΓ2-γ,and in order to simplify notations, we write(28)A1=∑s=1k-1bs-1-bsuijk-s-uijk-s-1,A2=∑s=1k-1bs-1-bsuijk-s-1uijk-s-ui-1jk-s,A3=∑s=1k-1bs-1-bsvijk-s-1uij+1k-s-uijk-s,A4=θijk-1-∑s=1k-1bs-1-bsθijk-s,A5=-∑s=1k-1bs-1-bsuijk-s.By using similar conventions, we can easily obtain the approximations for the fractional derivatives terms involved in the temperature equation given below(29)∂α+1θtk∂tα+1=Δt-1-αΓ2-αθijk-θijk-1-∑s=1k-1cs-1-csθijk-s-θijk-s-1,∂α∂tαu∂θ∂x=Δt-αΔxΓ2-αuijk-1θijk-θi-1jk-∑s=1k-1cs-1-csuijk-s-1θijk-s-θi-1jk-s,∂α∂tαv∂θ∂y=Δt-αΔyΓ2-αvijk-1θij+1k-θijk-∑s=1k-1cs-1-csvijk-s-1θij+1k-s-θijk-s,where cs=(s)1-α-(s-1)1-α, s=0,1,2,…,r. In order to simplify the notations, we assume that(30)B1=∑s=1k-1cs-1-csθijk-s-θijk-s-1,B2=∑s=1k-1cs-1-csuijk-s-1θijk-s-θi-1jk-s,B3=∑s=1k-1cs-1-csvijk-s-1θij+1k-s-θijk-s,Finally, by substituting approximations (25)-(30) in the model equations (22), we can obtain the following discretization scheme:(31)vijk=vij-1k+vij-1k-1-vijk-1+2hyhxui-1j-1k-1-uij-1k-1+ui-1jk-1-uijk-1+ui-1j-1k-uij-1k+ui-1jk-uijk,(32)uijk1Δt+uijk-1Δx+vijk-1Δy+δ1Δt+uijk-1δ1Δx-vijk-1δ1Δy+2δ2+M+Φ21+δ1Δt+ui-1jk-uijk-1Δx-uijk-1δ1Δx+uij-1k-vijk-1Δy-δ2+uij+1k-δ2+δ1vijk-1Δy=uijk-11Δt+δ1Δt-2δ2+δ1ΔtA1+δ1ΔxA2+δ1ΔyA3+δ2uij+1k-1+δ2uij-1k-1+δ1GrA4+Grθijk-1+M+ϕ2δ1A5,(33)θijk1Δt+uijk-11Δx+vijk-11Δy+δ31Δt+δ3uijk-11Δx+δ3vijk-11Δy+δ22pr1-ϕ3+θi-1jk-uijk-1Δx-δ3Δxuijk-1+θij-1k-vijk-1Δy-δ3Δyvijk-1-δ2Pr1-ϕ3-δ2Pr1-ϕ3θij+1k=θijk-11Δt+δ31Δt-2δ2Pr1-ϕ3+δ3B11Δt+B21Δx+B31Δy+δ2Pr1-ϕ3θij+1k-1+θij-1k-1,and the initial and boundary conditions can evidently be defined as (34)vij0=0,uij0=0,θij0=0,v0jk=0,u0jk=0,θ0jk=0,vi0k=0,ui0k=tk,θi0k=0,viLyk=0,uiLyk=0,θiLyk=0.


### 3.2. Solution Procedure

The computational domain is considered as a rectangle with sizesLx=1 and Ly=5 where Ly corresponds to y→∞ as the solution approaches to zeros in the vicinity of Ly=5; see Figure 5 for validation. The solution has been computed with mesh sizes Δx=0.01, Δy=0.01, and Δt=0.01. The iteration process in the numerical scheme (31)-(33) has been initiated by initial values uij0, vij0, and θij0 given by (34). Assuming that the earlier values uijk-1, vijk-1, and θijk-1 at time t=tk-1 are known for the k-level iteration. The memory terms A1,A2,A3 and B1,B2,B3 are also known as these terms are defined till the (k-1)th time step. Finally, the iteration at each kth time step can be written in the form of five diagonal linear systems by placing each term of (k-1)th step on the right hand side of the systems. The linear systems are then solved by using a classical direct method. A code has been developed in MATLAB in order to implement scheme (31)-(34).


## 3.1. Discretization Method

Definetk=kΔt, k=0,1,2,3,…,r, time steps with Δt being a time step size and xi=iΔx, i=0,1,2,3,…,m; yj=jΔy, j=0,1,2,3,…,n, where Δx=Lx/m and Δy=Ly/n are space step sizes. Let uijk and θijk be numerical approximations of u(t,x,y) and θ(t,x,y) at point (xi,yj) and time tk in model (22)-(23).The Caputo time fractional derivative in (6) is discretized at time tk as follows:(24)∂tαutk=1Γ2-α∑s=0k∫tktk+1tk-ξ1-α-1∂uξ∂ξdξ,0<Reα<1,=Δt-αΓ2-αb0utk-bk-1ut0+∑s=1k-1bs-1-bsutk-s+OΔtwhere bs=(s)1-α-(s-1)1-α, s=0,1,2,…,r.The nonfractional derivatives are discretized using backward finite difference approximations as [52, 54](25)∂u∂tt=tk=uxi,yj,tk-uxi,yj,tk-1Δt+OΔt,u∂u∂xt=tk=uxi,yj,tk-1uxi,yj,tk-uxi-1,yj,tkΔx+OΔx,v∂u∂yt=tk=vxi,yj,tk-1uxi,yj,tk-uxi,yj-1,tkΔy+OΔy,∂2u∂y2=ui,j+1k-2ui,jk+ui,j-1k+ui,j+1k-1-2ui,jk-1+ui,j-1k-12Δy+OΔy2where ui,jk=u(xi,yj,tk) and the nonlinear terms have been linearized. Moreover the fractional derivatives of the above terms at time tk can be defined in the following way (see for details [52]):(26)∂α+1utk∂tα+1=Δt-1-αΓ2-αuijk-uijk-1-∑s=1k-1bs-1-bsuijk-s-uijk-s-1,∂α∂tαu∂u∂x=Δt-αΔxΓ2-αuijk-1uijk-ui-1jk-∑s=1k-1bs-1-bsuijk-s-1uijk-s-ui-1jk-s,∂α∂tαv∂u∂y=Δt-αΔyΓ2-αvijk-1uij+1k-uijk-∑s=1k-1bs-1-bsvijk-s-1uij+1k-s-uijk-s,Assume that(27)δ1=λαΔt-αΓ2-α,δ2=12Δy2,δ3=λγΔt-γΓ2-γ,and in order to simplify notations, we write(28)A1=∑s=1k-1bs-1-bsuijk-s-uijk-s-1,A2=∑s=1k-1bs-1-bsuijk-s-1uijk-s-ui-1jk-s,A3=∑s=1k-1bs-1-bsvijk-s-1uij+1k-s-uijk-s,A4=θijk-1-∑s=1k-1bs-1-bsθijk-s,A5=-∑s=1k-1bs-1-bsuijk-s.By using similar conventions, we can easily obtain the approximations for the fractional derivatives terms involved in the temperature equation given below(29)∂α+1θtk∂tα+1=Δt-1-αΓ2-αθijk-θijk-1-∑s=1k-1cs-1-csθijk-s-θijk-s-1,∂α∂tαu∂θ∂x=Δt-αΔxΓ2-αuijk-1θijk-θi-1jk-∑s=1k-1cs-1-csuijk-s-1θijk-s-θi-1jk-s,∂α∂tαv∂θ∂y=Δt-αΔyΓ2-αvijk-1θij+1k-θijk-∑s=1k-1cs-1-csvijk-s-1θij+1k-s-θijk-s,where cs=(s)1-α-(s-1)1-α, s=0,1,2,…,r. In order to simplify the notations, we assume that(30)B1=∑s=1k-1cs-1-csθijk-s-θijk-s-1,B2=∑s=1k-1cs-1-csuijk-s-1θijk-s-θi-1jk-s,B3=∑s=1k-1cs-1-csvijk-s-1θij+1k-s-θijk-s,Finally, by substituting approximations (25)-(30) in the model equations (22), we can obtain the following discretization scheme:(31)vijk=vij-1k+vij-1k-1-vijk-1+2hyhxui-1j-1k-1-uij-1k-1+ui-1jk-1-uijk-1+ui-1j-1k-uij-1k+ui-1jk-uijk,(32)uijk1Δt+uijk-1Δx+vijk-1Δy+δ1Δt+uijk-1δ1Δx-vijk-1δ1Δy+2δ2+M+Φ21+δ1Δt+ui-1jk-uijk-1Δx-uijk-1δ1Δx+uij-1k-vijk-1Δy-δ2+uij+1k-δ2+δ1vijk-1Δy=uijk-11Δt+δ1Δt-2δ2+δ1ΔtA1+δ1ΔxA2+δ1ΔyA3+δ2uij+1k-1+δ2uij-1k-1+δ1GrA4+Grθijk-1+M+ϕ2δ1A5,(33)θijk1Δt+uijk-11Δx+vijk-11Δy+δ31Δt+δ3uijk-11Δx+δ3vijk-11Δy+δ22pr1-ϕ3+θi-1jk-uijk-1Δx-δ3Δxuijk-1+θij-1k-vijk-1Δy-δ3Δyvijk-1-δ2Pr1-ϕ3-δ2Pr1-ϕ3θij+1k=θijk-11Δt+δ31Δt-2δ2Pr1-ϕ3+δ3B11Δt+B21Δx+B31Δy+δ2Pr1-ϕ3θij+1k-1+θij-1k-1,and the initial and boundary conditions can evidently be defined as (34)vij0=0,uij0=0,θij0=0,v0jk=0,u0jk=0,θ0jk=0,vi0k=0,ui0k=tk,θi0k=0,viLyk=0,uiLyk=0,θiLyk=0.


## 3.2. Solution Procedure

The computational domain is considered as a rectangle with sizesLx=1 and Ly=5 where Ly corresponds to y→∞ as the solution approaches to zeros in the vicinity of Ly=5; see Figure 5 for validation. The solution has been computed with mesh sizes Δx=0.01, Δy=0.01, and Δt=0.01. The iteration process in the numerical scheme (31)-(33) has been initiated by initial values uij0, vij0, and θij0 given by (34). Assuming that the earlier values uijk-1, vijk-1, and θijk-1 at time t=tk-1 are known for the k-level iteration. The memory terms A1,A2,A3 and B1,B2,B3 are also known as these terms are defined till the (k-1)th time step. Finally, the iteration at each kth time step can be written in the form of five diagonal linear systems by placing each term of (k-1)th step on the right hand side of the systems. The linear systems are then solved by using a classical direct method. A code has been developed in MATLAB in order to implement scheme (31)-(34).


## 4. Skin Friction Coefficient

The fluid layer adjacent to the boundary (plate) is characterized by the skin friction coefficient or shear stress coefficient denoted asCf and defined by [53](35)Cf=μnfρfU02τw,with the stress at the boundary being defined by(36)τw=∂u∂yy=0,whereas the fractional form of the stress can be defined using (5) as(37)1+λ1α∂α∂tατw=∂u∂yy=0,Applying operator 1+λ1α(∂α/∂tα) on both sides of (35) and using (37), the skin friction coefficient Cf can finally be defined as(38)1+λ1α∂α∂tαCf=μnfρfU02∂u∂yy=0,Using the nondimensional quantities defined in Section 2.1, the dimensionless form of the skin friction is given by(39)1+λ1α∂α∂tαCf=1Re∂u∂yy=0,Using L1-discretization of Caputo derivative defined in (24), the skin friction coefficient Cf can be written as(40)Cf=δ1∑s=1k-1bs-1-bsCftk-s+1/Re∂u/∂yy=01+δ1 where Re=U0d/ν is the Reynolds number. The average skin friction coefficient can further be defined as(41)Cf¯=δ1∑s=1k-1bs-1-bsCf¯tk-s+1/Re∫01∂u/∂yy=0dx1+δ1The skin friction coefficient has been calculated for different values of physical parameters and the results have been shown in Table 1.Table 1
Average skin friction coefficient for different parameters.
α λ 1 M Φ 2 C f -  0.1 0.2 0.01 0.01 -8.2201 0.3 0.2 0.01 0.01 -9.3333 0.6 0.2 0.01 0.01 -10.5056 0.9 0.2 0.01 0.01 -11.1255  0.1 0.2 0.01 0.01 -8.2201 0.1 0.4 0.01 0.01 -8.0712 0.1 0.6 0.01 0.01 -7.9827 0.1 0.8 0.01 0.01 -7.9193  0.1 0.2 0.01 0.01 -8.2201 0.1 0.2 0.05 0.01 -8.2399 0.1 0.2 0.1 0.01 -8.2647 0.1 0.2 0.5 0.01 -8.4069  0.1 0.2 0.01 0.01 -8.2201 0.1 0.2 0.01 0.1 -8.2647 0.1 0.2 0.01 1.1 -8.7495 0.1 0.2 0.01 2.1 -9.2150


## 5. Local Nusselt Number

A ratio between convective and conductive heat transfer in the fluid is known as local Nusselt number denoted byNu. The large values of the Nusselt number show the dominated heat transfer by convection in the fluid. The Nusselt number for Maxwell fluid is [53](42)Nu=-dknfkfTd-T0qw,with the expression for qw defined by(43)qw=∂T∂yy=0and for the fractional Maxwell fluid, we can define (see for details [29])(44)1+λ2α∂α∂tαqw=∂T∂yy=0Applying operator 1+λ2γ∂γ/∂tγ on both sides of (42) and using above relations, the Nusselt number Nu can be written as(45)1+λ2α∂γ∂tγNu=-dknfkfTd-T0∂T∂yy=0 The dimensionless form of Nussult number can be defined using relations in Section 2.1 as(46)1+λ2γ∂γ∂tγNu=-knfkf∂θ∂yy=0Using L1-discretization of Caputo derivative defined in (24), the Nusselt number Nu can be discretized as(47)Nu=δ3∑s=1k-1cs-1-csNutk-s-knf/kf∂θ/∂yy=01+δ3Moreover, the average Nusselt number can be given as(48)Nu¯=δ3∑s=1k-1cs-1-csNu¯tk-s-knf/kf∫01∂θ/∂yy=0dx1+δ3The Nusselt number has been calculated for various values of physical quantities and the outcomes have been shown in Table 2.Table 2
Average Nusselt number for different parameters.
γ λ 2 P r Φ 3 N u -  0.1 0.2 0.01 0.01 -52.0576 0.3 0.2 0.01 0.01 -76.5177 0.6 0.2 0.01 0.01 -1.9658e+02 0.9 0.2 0.01 0.01 -1.1654e+03  0.1 0.2 0.01 0.01 -52.0576 0.1 0.4 0.01 0.01 -51.4028 0.1 0.6 0.01 0.01 -51.0632 0.1 01 0.01 0.01 -50.6861  0.1 0.2 0.01 0.01 -52.0576 0.1 0.2 0.05 0.01 -52.1020 0.1 0.2 0.09 0.01 –52.1483 0.1 0.2 0.1 0.01 –52.1594  0.1 0.2 0.01 0.01 -52.0576 0.1 0.2 0.01 0.05 -52.0518 0.1 0.2 0.01 0.1 -52.0466 0.1 0.2 0.01 0.5 -52.0193


## 6. Results and Discussion

In this section, the effects of heat transfer flow of fractional Maxwell fluid with magnetic field and radiation over a moving plate are studied. The nonlinear coupled equations are numerically solved by newly developedL1 technique along with the finite difference approximations as described earlier. The parameters involved in the model which are discussed are mainly the fractional parameters α, γ, Maxwell or the relaxation time parameters λ1, λ2, Prandtl number Pr, magnetic field M, radiation parameter Φ3, and nanofluid parameter Φ2. The effects of these parameters on the velocity and temperature profiles along with skin friction coefficient Cf and Nusselt number Nu have been investigated and explained via graphs and tables.


### 6.1. Effects on Velocity Field

Figure1(a) indicates that fractional parameter α effects on velocity profile with final time tf=1 and tf=2. For grater values of α, the velocity of the fluid increases. This implies that increase in the fractional parameter α gives rise in the thickness of velocity for the both short and long times. However this dependence is indeed nonmonotonic in behavior and cannot be generalized to all values of the involved parameters as for the cases discussed here. Change in behavior can be noted for some other values of involved parameters. Similar pattern can be seen for boundary layer thickness of the flow domain.Figure 1 (a)
Velocity distribution withα variation (b)
Velocity distribution forλ1Figure1(b) shows relaxation time λ1 effects on velocity profile. Velocity of the fluid decreases for larger values of λ1. For time tf=1, velocity of the fluid decreases quickly when we have larger values of λ1. It is perceived that velocity profile decreases while corresponding boundary layer thickness increases with the increase of λ1. The parameter λ1 appeared in nondimensional momentum equation due to dimensional relaxation time. The relaxation time is stronger for larger λ1. Hence velocity profile decreases by the increase of λ1. Influence of magnetic field on fluid velocity is shown in Figure 2(a). The increase of magnetic field parameter M tends to decrease velocity profile. The Lorentz force increases with the increase of magnetic number which offers resistance to the motion of fluid. Therefore magnitude of the skin friction coefficient and velocity profile decreases. Effects of nanoparticles volume ϕ on nanofluids are shown in Figure 2(b). The increase in the volumic concentration ϕ of nanoparticles decreases velocity which is physically adequate as observed in experimental findings.Figure 2 (a)
Velocity distribution forM (b)
Velocity distribution forΦ2


### 6.2. Effects on Temperature Field

Dimensionless form of a temperature is considered because of the fact that the natural convection is driven by temperature gradient. Figure3(a) demonstrates that fractional parameter effect of γ on temperature is the same like in α case on velocity profile. Decreasing γ values from 1 to 0 quickly decreases temperature. This shows that noninteger order derivatives weaken the effects of heat conduction. Relaxation time parameter λ2 effects on temperature field are shown in Figure 3(b). A small change occurs in temperature for small values of λ2. Petit increase in λ2 decreases temperature profile slightly. Radiation effects on temperature profile are shown in Figure 4(a). Radiation parameter Φ3 minor increment makes a larger change in temperature field. Prandtl number is defined as the ratio of momentum diffusivity by thermal diffusivity. Prandtl number Pr effects on temperature are shown in Figure 4(b). A petit increase in Prandtl number demonstrates decrease in temperature profile. Also note that thermal boundary layer thickness is at lower level for larger values of Pr. With the increase of Pr momentum diffusivity increases. Hence with the increase of Pr thermal diffusivity decreases while viscosity of fluid increases, as a result temperature and velocity profiles decrease. Relative thickness of thermal and momentum boundary layers is controlled by Pr. Heat diffuses gradually and thermal boundary layer is at lower level for larger values of Pr while opposite behavior is observed at smaller values of Pr.Figure 3 (a)
Temperature distributions withγ variation (b)
Temperature distribution forλ2Figure 4 (a)
Temperature distribution forΦ3 (b)
Temperature distribution forPrFigure 5 (a)
Three-dimensional velocity distribution (b)
Three-dimensional temperature distributionFigures5(a) and 5(b) show three-dimensional velocity and temperature distribution which demonstrate convergence and good stability in space and time variables. The particular values of α and γ are the Caputo fractional derivatives which vary from 0 to 1. As derivative of order zero makes no sense for the proposed physical problem and the derivative of order 1 will reduce the proposed model to the classical one. The values of λ1 and λ2 are the relaxation times for momentum and heat equations, respectively. We have to consider the values of these parameters greater than zero as for zero there is no contribution of relaxation times and the classical model is retrieved. The solution of proposed model remained convergent for any values of these parameters greater than or equal to zero. The values of the parameters in these simulations can be diverse; however the particular values have been selected from various sources already published in the literature; see, for example, [12–14, 53, 54].


### 6.3. Average Skin Friction Coefficient

The average skin friction coefficients for different parameters and for various values are shown in Table1. Skin friction plays a vital role in predicting the behavior of flow in the vicinity of the boundary. Effects of different parameters on the skin friction coefficient can be used to show the cumulative trend of friction in the flow domain. The average skin friction coefficient decreases with the increase of fractional parameter α which shows that the momentum boundary layer thickness reduces along the plate or wall boundary. Similar trends have been observed for the increasing values of nanofluid parameter Φ2 whereas opposite behavior has been observed while the values of magnetic field and relaxation parameter λ1 augment; that is, the momentum boundary layer thickness extends.


### 6.4. Average Nusselt Number

Table2 illustrates the influence of physical and fractional parameters on the average Nusselt number. Nusselt number is important for the calculation of thermal field variations along the wall or boundary. Realistic physical simulations can be executed if one takes into account the Nusselt number while calculating the thermal field. The numeric values of Nusselt number will be helpful to show the cumulative trend of temperature gradient in the flow domain. Average Nusselt number is comprehensive function of temperature gradient as shown in (48). For each increased value of γ, the average Nusselt number monotonically decreases which shows that less heat is transferred to the fluid from the plate. Similar behavior has been observed for the increasing values of the Prandtl number Pr, whereas increasing values of the thermal relaxation parameter λ2 and the radiation parameter Φ3 impose small increase on average Nusselt number which results in the greater heat exchange rate near the plate.In Table3, the comparison for different values of Prandtl number Pr has been presented with already published papers for the validation and accuracy of the numerical results. As it is the ratio of molecular diffusivity of momentum to the molecular diffusivity of heat, it shows the relative thickness of the velocity boundary layer to the thermal boundary layer. The increase in the value of Pr basically describes dominance of momentum diffusivity over the thermal diffusivity. Crepeau and Clarksean [12], Chamkha and khaled [13], and Chen [14] compared the value of rate of change of heat at the surface -∂θ/∂y at y=0 for various values of the Prandtl number Pr and other physical parameters. We have also compared the same for the similar values in order to validate our results.Table 3
Comparison of-∂θ/∂yy=0 at x=1, with α=1, γ=1, M=0, ϕ3=0, and λ1=λ2=0.
Pr Crepeau and clarksean [12] Chamkha and khaled [13] Chen [14] present work 0.01 0.0805 0.0600 0.0806 0.0585 0.1 0.2302 0.2119 0.23014 0.2460 1 0.5671 0.5649 0.5671 0.6631 10 1.1690 1.1720 1.1693 1.1412 100 2.1910 2.1943 2.1913 2.1126


## 6.1. Effects on Velocity Field

Figure1(a) indicates that fractional parameter α effects on velocity profile with final time tf=1 and tf=2. For grater values of α, the velocity of the fluid increases. This implies that increase in the fractional parameter α gives rise in the thickness of velocity for the both short and long times. However this dependence is indeed nonmonotonic in behavior and cannot be generalized to all values of the involved parameters as for the cases discussed here. Change in behavior can be noted for some other values of involved parameters. Similar pattern can be seen for boundary layer thickness of the flow domain.Figure 1 (a)
Velocity distribution withα variation (b)
Velocity distribution forλ1Figure1(b) shows relaxation time λ1 effects on velocity profile. Velocity of the fluid decreases for larger values of λ1. For time tf=1, velocity of the fluid decreases quickly when we have larger values of λ1. It is perceived that velocity profile decreases while corresponding boundary layer thickness increases with the increase of λ1. The parameter λ1 appeared in nondimensional momentum equation due to dimensional relaxation time. The relaxation time is stronger for larger λ1. Hence velocity profile decreases by the increase of λ1. Influence of magnetic field on fluid velocity is shown in Figure 2(a). The increase of magnetic field parameter M tends to decrease velocity profile. The Lorentz force increases with the increase of magnetic number which offers resistance to the motion of fluid. Therefore magnitude of the skin friction coefficient and velocity profile decreases. Effects of nanoparticles volume ϕ on nanofluids are shown in Figure 2(b). The increase in the volumic concentration ϕ of nanoparticles decreases velocity which is physically adequate as observed in experimental findings.Figure 2 (a)
Velocity distribution forM (b)
Velocity distribution forΦ2


## 6.2. Effects on Temperature Field

Dimensionless form of a temperature is considered because of the fact that the natural convection is driven by temperature gradient. Figure3(a) demonstrates that fractional parameter effect of γ on temperature is the same like in α case on velocity profile. Decreasing γ values from 1 to 0 quickly decreases temperature. This shows that noninteger order derivatives weaken the effects of heat conduction. Relaxation time parameter λ2 effects on temperature field are shown in Figure 3(b). A small change occurs in temperature for small values of λ2. Petit increase in λ2 decreases temperature profile slightly. Radiation effects on temperature profile are shown in Figure 4(a). Radiation parameter Φ3 minor increment makes a larger change in temperature field. Prandtl number is defined as the ratio of momentum diffusivity by thermal diffusivity. Prandtl number Pr effects on temperature are shown in Figure 4(b). A petit increase in Prandtl number demonstrates decrease in temperature profile. Also note that thermal boundary layer thickness is at lower level for larger values of Pr. With the increase of Pr momentum diffusivity increases. Hence with the increase of Pr thermal diffusivity decreases while viscosity of fluid increases, as a result temperature and velocity profiles decrease. Relative thickness of thermal and momentum boundary layers is controlled by Pr. Heat diffuses gradually and thermal boundary layer is at lower level for larger values of Pr while opposite behavior is observed at smaller values of Pr.Figure 3 (a)
Temperature distributions withγ variation (b)
Temperature distribution forλ2Figure 4 (a)
Temperature distribution forΦ3 (b)
Temperature distribution forPrFigure 5 (a)
Three-dimensional velocity distribution (b)
Three-dimensional temperature distributionFigures5(a) and 5(b) show three-dimensional velocity and temperature distribution which demonstrate convergence and good stability in space and time variables. The particular values of α and γ are the Caputo fractional derivatives which vary from 0 to 1. As derivative of order zero makes no sense for the proposed physical problem and the derivative of order 1 will reduce the proposed model to the classical one. The values of λ1 and λ2 are the relaxation times for momentum and heat equations, respectively. We have to consider the values of these parameters greater than zero as for zero there is no contribution of relaxation times and the classical model is retrieved. The solution of proposed model remained convergent for any values of these parameters greater than or equal to zero. The values of the parameters in these simulations can be diverse; however the particular values have been selected from various sources already published in the literature; see, for example, [12–14, 53, 54].


## 6.3. Average Skin Friction Coefficient

The average skin friction coefficients for different parameters and for various values are shown in Table1. Skin friction plays a vital role in predicting the behavior of flow in the vicinity of the boundary. Effects of different parameters on the skin friction coefficient can be used to show the cumulative trend of friction in the flow domain. The average skin friction coefficient decreases with the increase of fractional parameter α which shows that the momentum boundary layer thickness reduces along the plate or wall boundary. Similar trends have been observed for the increasing values of nanofluid parameter Φ2 whereas opposite behavior has been observed while the values of magnetic field and relaxation parameter λ1 augment; that is, the momentum boundary layer thickness extends.


## 6.4. Average Nusselt Number

Table2 illustrates the influence of physical and fractional parameters on the average Nusselt number. Nusselt number is important for the calculation of thermal field variations along the wall or boundary. Realistic physical simulations can be executed if one takes into account the Nusselt number while calculating the thermal field. The numeric values of Nusselt number will be helpful to show the cumulative trend of temperature gradient in the flow domain. Average Nusselt number is comprehensive function of temperature gradient as shown in (48). For each increased value of γ, the average Nusselt number monotonically decreases which shows that less heat is transferred to the fluid from the plate. Similar behavior has been observed for the increasing values of the Prandtl number Pr, whereas increasing values of the thermal relaxation parameter λ2 and the radiation parameter Φ3 impose small increase on average Nusselt number which results in the greater heat exchange rate near the plate.In Table3, the comparison for different values of Prandtl number Pr has been presented with already published papers for the validation and accuracy of the numerical results. As it is the ratio of molecular diffusivity of momentum to the molecular diffusivity of heat, it shows the relative thickness of the velocity boundary layer to the thermal boundary layer. The increase in the value of Pr basically describes dominance of momentum diffusivity over the thermal diffusivity. Crepeau and Clarksean [12], Chamkha and khaled [13], and Chen [14] compared the value of rate of change of heat at the surface -∂θ/∂y at y=0 for various values of the Prandtl number Pr and other physical parameters. We have also compared the same for the similar values in order to validate our results.Table 3
Comparison of-∂θ/∂yy=0 at x=1, with α=1, γ=1, M=0, ϕ3=0, and λ1=λ2=0.
Pr Crepeau and clarksean [12] Chamkha and khaled [13] Chen [14] present work 0.01 0.0805 0.0600 0.0806 0.0585 0.1 0.2302 0.2119 0.23014 0.2460 1 0.5671 0.5649 0.5671 0.6631 10 1.1690 1.1720 1.1693 1.1412 100 2.1910 2.1943 2.1913 2.1126


## 7. Conclusion

In this paper we consider heat transfer in unsteady mixed convection flow of the Maxwell nanofluids on a horizontal moving plate along with vertically applied magnetic field and radiations. The fractional mathematical model is derived by introducing fractional time derivative into Maxwell constitutive equations. The governing equations are numerically solved by employing a newly developed finite difference method (FDM) combined withL1-algorithm [52]. The effects of fractional parameters α and γ along with other pertinent physical parameters on the velocity and temperature profiles have been discussed with the help of graphs for various values. The skin friction coefficient (Cf) and Nusselt number (Nu) have also been discussed for different values of physical parameters. In graphical results, the fractional parameter α increases the thickness of velocity and γ increases the thermal boundary layer. The variations of parameters λ1, λ2, Pr, M, Φ2, and Φ3 change the course of velocity and temperature profiles significantly as discussed with details in section 6. Fractional parameter α has main role on average skin friction coefficient and γ on average Nusselt number. It is concluded that the fractional parameters play a vital role to capture the memory of the viscoelastic fluid for different values of fractional values instead of only one fixed value of derivative, i.e., α=γ=1. Momentum and thermal profiles are at lower level for all fractional values other than 1, which are quite helpful in predicting the realistic results.



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