# Heat Transfer in a Fractional Nanofluid Flow through a Permeable Medium

Authors: Muhammad Shoaib Anwar; Muhammad Irfan; Majid Hussain; Taseer Muhammad; Zakir Hussain
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This study examines viscoelastic fractional nanofluid flow through Darcy medium. Memory characteristics due to elasticity are explored with noninteger time derivatives. The unsteady motion of MHD flow is modeled by nonlinear differential equations. Buoyancy forces are incorporated via convection parameters in the flow domain. Fractional relaxation time is considered to control the propagation speed of temperature. A finite difference, along with finite element, a numerical algorithm has been developed for the computation of governing flow equations. Friction coefficient, Sherwood numbers, and Nusselt numbers are computed for the noninteger derivative model. Simulations revealed that noninteger numbers have congruous behavior for concentration, temperature, and velocity fields. It is also noted that heat flux,δ1, and mass flux, δ2, numbers have contradictory effects on the friction coefficient. Various flows, particularly in polymer industries and electrospinning for the production of nanofibers, can be tackled in a comparable pattern.


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## Body

## 1. Introduction

In this communication, we have described the transport of momentum, heat, and concentration with the help of mathematical relations. Mathematical relations are formulated with constitutive expressions that handle fluxes of the above-prescribed quantities [1]. Formulated equations, remained helpful to analyze the transport of heat, diffusion of chemical species, movement of geological flows, engineering applications, meteorology, material science, and medicines [2, 3]. Molecular contact and Brownian motion [1] characterized the conductivity, diffusivity, and viscosity for the nanofluid in the flow domain. Particularly, transfer of heat is an analog of mass transfer in the constitutive expressions for fluxes. Fick’s law governs the diffusion in mass transport, and Fourier’s law handles the conduction mechanism in heat transfer. Here we have considered the generalized Fourier and Fick laws to incorporate relaxation times. Transport phenomena in Newtonian and non-Newtonian flows can be seen in heat exchangers, thermal devices, granular insulation, fiber technology, nuclear repositories, fermentation processes, geothermal extractions, and the production of crude oil [4–11]. In the literature, the transfer of heat and mass is widely reported by distinguished researchers. For example, Salama et al. [12] discussed the solution of a flow problem through porous media with flux approximations. In this communication, the viscoelastic fluid model of second grade is analyzed for the transport mechanism with suspended nanoparticles. Viscoelastic second-grade fluid model exhibits differential type nature. For differential fluids, stress is expressed by an explicit velocity gradient. Differential models describe the fluids in terms of their shear thickening and thinning effects, normal stress effects, thixotropy exhibition, and nonlinear behavior of creeping and yielding. Flow problems, which include polymer suspensions and slurries, in fiber handling can be discussed with a given mathematical model [13].Governing flow equations with noninteger time derivatives have been proven for useful analysis of viscoelastic trends [14]. Nowadays, mathematical modeling with noninteger derivatives have gained importance in various branches of science. For example, valuable impact can be seen in fluid mechanics. Moreover, the influence of noninteger modeling is seen in viscoelasticity, control theory, and electrochemistry [15–17]. Advancements in fractional modeling are carried out with the passage of time and the inclusion of new fractional derivatives in the literature, for example, Yang’s fractional derivative, Atangana–Baleanu derivatives, and Caputo derivatives [18, 19]. Both natural and artificial systems can be analyzed in a better way via noninteger time derivatives as they describe hereditary aspects of materials [20–24]. Simulations of anomalous nonlinear models can be used to approximate experimental data in a better way when compared to usual rheological models [25–30].Porous medium flows occur in soil mechanics, ceramics, and industrial and mineral processing. The complexity of the flow problem is increased with the presence of a porous medium. This is mainly due to the interface between packing particles and fluid molecules. Darcy flow is analyzed by various researchers. Nonlinear Darcy flow is examined by Ervin et al. [31]. The Lorentz force shows a significant role in viscoelastic problems such as in MHD generators and control theory. Magnetic field impacts can be seen in porous medium flows that are encountered in metallurgical systems, extrusion and penetration practices, etc. The flow domain is greatly influenced by convection as a transport mechanism, is greatly changed by both buoyancy and gravitational forces. Convection’s importance can be seen in boilers, nuclear reactors, and heat exchangers. Zhou and Liang [32] discussed convection in unsteady flow problems.The industrial importance of nanofluids is significant in lightweight materials production, the breakdown of organic pollutants, and in the production of nanofibers [33, 34]. Here, under consideration, flow is unidirectional through a channel of infinite extent. By this configuration, fluid velocity and its gradient are orthogonal to each other, which may lead to the addition of a nonlinear convection term in the governing equations. These flows are mostly observed in the long narrow channels. Many microchannel flows are described by these solutions, their superposition, and by the small perturbation of these flows [35, 36]. Viscoelastic flows can also be used in electrospinning for nanofibers production. Nanofibers are formulated by an electric charged jet of polymer solution that is of viscoelastic nature. An increase of yield stress along with a viscoelastic jet causes a decrease in fluid elongation [37, 38]. The structure of nanoparticles is between atomic and bulk. The possibility of nano particles’ inclusion is due to their strong interaction with base fluid. Inclusion of these particles effectively changes the characteristics of the base fluid. For instance, nanoparticle suspension increases the thermal conductivity of the base fluid. Wang and Zhang [39] analyzed heat transfer in nanofluid flow. With the passage of time, fractional nanofluid flows require more attention in literature. Solutions to fractional flow problems are usually discussed with integral transforms. But integral transforms are not helpful for solving nonlinear coupled fractional equations. Due to this, numerical solutions are proposed for flow problems(see [40–43]). In this article, we have tackled the problem with finite element and finite difference techniques to solve the described fractional flow configuration.Particularly in this communication, we have considered modeling with noninteger derivatives of nanofluid flow. Due to this, we have achieved more control over the flow simulations with the help of fractional derivatives when compared with the classical mathematical flow models. Flux conditions are imposed aty=0, while quadratic variations are observed at the fixed boundary. A Darcy flow medium is observed with an applied magnetic field to the flow domain. Moreover, here we have proposed a scheme with numerical discretization to obtain the stable results of the coupled nonlinear fractional equations. Space variable is discretized by finite element while time variable is discretized by finite difference scheme. The influence of the involved physical parameters is discussed appropriately. In Section 2, modeling with noninteger time derivatives is discussed. The flow field is approximated by the proposed scheme in Section 3. A numerical approximation of the flow problem is given in Section 4. Finally, key results are given in Section 5.


## 2. Mathematical Description

In this study, we considered heat transfer in the MHD flow of the nanoliquid with noninteger derivatives of time. Modeling with noninteger derivatives helps to achieve more control over the flow simulations when compared with the classical models that contain ordinary derivatives. A nanofluid is formed with nanoparticles and base liquid. The medium of fluid flowing is considered to be a Darcy porous medium. Moreover, Neumann boundary conditions are supposed to be in the heating domain. At the start, there is no movement in the medium. At that time, the whole configuration is at constant temperatureθ0 and concentration ϕ0. With the passage of time, disturbance in the liquid is observed due to the lower domain. We consider the velocity field to be a function of y and t only. Then we consider(1)U=uy,tex.For the Cauchy stress tensor, Rivlin–Ericksen tensors, Darcy law and thermodynamic stability conditions of differential type second grade fluid (see [21] and references therein). The fractional formalism of thermal and concentration gradients can be seen in [45, 46].Here is a brief description of the mathematical modeling. In order to formulate an energy equation with Caputo fractional derivativeα including nanoparticles, consider the classical energy equation for incompressible fluid of the Buongiorno article (see [46]).(2)ρfcf∂θ∂t=−∇⋅q+hp∇⋅Jp,where q, Jp denote fluxes of energy and diffusion and for definition (see [46]). Temperature and concentration propagation of infinite order can be seen with q, Jp in Buongiorno article (see [46]). We can overcome this situation with the introduction of fractional relaxation times using reference [45], so we define q, Jp as(3)1+τ1αΓ1+α∂α∂tαq=−k∇θ+hp1+τ1αΓ1+α∂α∂tαJp,(4)1+τ1αΓ1+α∂α∂tαJp=−ρpDB∇ϕ−ρpDθ∇θθ0.Now using (2) and (3) and ∇hp=cp∇θ [46], we obtained(5)ρfcf1+τ1αΓ1+α∂α∂tα∂θ∂t=k∇2θ+ρpcp1+τ1αΓ1+α∂α∂tαDB∇ϕ⋅∇θ+Dθθ0∇θ⋅∇θ.Similarly using the equation of concentration in Buongiorno model [46], we get(6)∂∂t+∇⋅Uϕ=−1ρp∇⋅Jp,reduces to(7)1+τ2βΓ1+β∂β∂tβ∂ϕ∂t=DB∇2ϕ+Dθθ0∇2θ.Finally, governing present flow equations are(8)∇⋅U=0,(9)dUdt=1ρf∇⋅S+r+b+gβ1θ−θ0+β2θ−θ02,(10)ddt1+τ1αΓ1+α∂α∂tαθ=α3∂2θ∂y2+τ1+τ1αΓ1+α∂α∂tαDB∂θ∂y∂ϕ∂y+Dθθ0∂θ∂y2,(11)ddt1+τ2βΓ1+β∂β∂tβϕ=DB∂2ϕ∂y2+Dθθ0∂2θ∂y2,where S stands for the Cauchy tensor of stress for fluid of second grade (see [21]), r is the Darcy resistance of porous medium (see [21]), and b denotes body force.The pressure gradient is assumed to be negligible.


### 2.1. Governing Equations

The equations that modeled the flow problem along with their conditions are given here. Continuity (8) will reduce to identity with velocity given in (1). After some mathematical simplifications, modeled equations reduce to(12)∂u∂t=ν+α1ρf∂∂t∂2u∂y2−ψKν+α1ρf∂∂tu−σB02ρfu+gβ1θ−θ0+β2θ−θ02,(13)∂∂t1+τ1αΓ1+α∂α∂tαθ=α3∂2θ∂y2+τ1+τ1αΓ1+α∂α∂tαDB∂θ∂y∂ϕ∂y+Dθθ0∂θ∂y2,(14)∂∂t1+τ2βΓ1+β∂β∂tβϕ=DB∂2ϕ∂y2+Dθθ0∂2θ∂y2.


#### 2.1.1. Flow Problem Conditions

Here we are considering the flow in a channel separated by a distanceL. At the start, the whole flow configuration is still with constant temperature θ0 and constant concentration ϕ0. Flow is generated by the movement of the domain at y=0, while the pressure gradient is assumed to be zero in that case. For temperature and concentration, domain at y=0 is connected to a source, and the change in temperature is also changing with time, while in the domain at y=L, only temperature changes with time. The case for concentration is similar. Flow conditions are given as(15)u0,t=Aν2L4t2,uL,t=0,(16)−k∂θ∂yy=0=qθν2L4t2andθL,t=θ01+ν2L4t2,t>0,(17)−DB∂ϕ∂yy=0=qϕν2L4t2andϕL,t=ϕ01+ν2L4t2,t>0,(18)uy,0=0,θy,0=θ0,ϕy,0=ϕ0and∂θ∂ty,0=0=∂ϕ∂ty,0,y≤L.A complete description of the flow domain is identified via equations (12)–(18).


#### 2.1.2. Skin Friction Coefficients

Friction among fluid and solid boundaries is of great importance in the analysis of flow domains bounded by solid boundaries. Coefficients of friction are mathematically defined as(19)Cf≔2τwρA2,where shear stress at the wall is(20)τw=μ+α1∂∂t∂u∂yy=0,L.


#### 2.1.3. Nusselt Numbers

Change of temperature aty=0 and y=L are determined by the dimensionless, Nusselt number. Present flow domain, Nusselt numbers are given as(21)Nu1=−L∂θ/∂yy=Lθs1−θ0,where θs1 is constant temperature, at y=L. At y=0, the Nusselt number is given by(22)Nu2=−L∂θ/∂yy=0θs2−θ0,here θs2 is the temperature, at y=0.


#### 2.1.4. Mass Transfer Nusselt Numbers

Change of concentration aty=0 and y=L is determined by mass transfer Nusselt, Sherwood numbers. The present flow domain and Sherwood numbers are given as(23)Sh1=−L∂ϕ/∂yy=Lϕs1−ϕ0,where ϕs1 is the constant concentration at y=L. At y=0, the Sherwood number is given by(24)Sh2=−L∂ϕ/∂yy=0ϕs2−ϕ0,here ϕs2 is the concentration at y=0.


#### 2.1.5. Problem in Nondimensionalization

We have defined the following dimensionless quantities to make mathematical problem (12)–(18) as nondimensionalized.(25)y^≔yL,t^≔νL2tu^≔uA,θ^≔θ−θ0θ0ϕ^≔ϕ−ϕ0ϕ0.Using, (25), with dimensionless quantities, the IBVP (12)–(18) considered as without hats(26)1+ϵ∂u∂t=1+γ∂∂t∂2u∂y2−λ+Hau+λ1+λ2θθ,Pr∂∂t1+δ3∂α∂tαθ=∂2θ∂y2+Pr1+δ3∂α∂tαNb∂θ∂y∂ϕ∂y+Nt∂θ∂y2,Sc∂∂t1+δ4∂β∂tβϕ=∂2ϕ∂y2+NtNb∂2θ∂y2,y<1,t>0,and problem conditions are given by(27)u0,t=t2,u1,t=0,∂θ∂yy=0=−δ1t2,∂ϕ∂yy=0=−δ2t2,θ1,t=t2=ϕ1,t,uy,0=0,θy,0=0=∂θ∂ty,0andϕy,0=0=∂ϕ∂ty,0.Here dimensionless numbers are given as(28)ϵ≔ϕα1ρK,γ≔α1ρL2,λ≔ψL2K,Ha≔σB02L2μ,Pr≔να3,Sc≔νDB,λ1≔gL2β1θ0Aν,λ2≔gL2β2θ02Aν,Nb≔τϕ0DBν,δ1≔qθLθ0k,δ2≔qϕLϕ0DB,Nt≔τDθν,δ3≔τ1αναΓ1+αL2α,δ4≔τ2βνβΓ1+βL2β.Moreover, the dimensionless coefficients of friction, Nusselt, and mass transfer Nusselt numbers are defined by(29)ReCf2=1+γ∂∂t∂u∂yy=0,1,Nu1Re=−∂θ∂yy=1andNu2Re2=δ1t2,Sh1Re=−∂ϕ∂yy=1andSh2Re2=δ2t2.


## 2.1. Governing Equations

The equations that modeled the flow problem along with their conditions are given here. Continuity (8) will reduce to identity with velocity given in (1). After some mathematical simplifications, modeled equations reduce to(12)∂u∂t=ν+α1ρf∂∂t∂2u∂y2−ψKν+α1ρf∂∂tu−σB02ρfu+gβ1θ−θ0+β2θ−θ02,(13)∂∂t1+τ1αΓ1+α∂α∂tαθ=α3∂2θ∂y2+τ1+τ1αΓ1+α∂α∂tαDB∂θ∂y∂ϕ∂y+Dθθ0∂θ∂y2,(14)∂∂t1+τ2βΓ1+β∂β∂tβϕ=DB∂2ϕ∂y2+Dθθ0∂2θ∂y2.


### 2.1.1. Flow Problem Conditions

Here we are considering the flow in a channel separated by a distanceL. At the start, the whole flow configuration is still with constant temperature θ0 and constant concentration ϕ0. Flow is generated by the movement of the domain at y=0, while the pressure gradient is assumed to be zero in that case. For temperature and concentration, domain at y=0 is connected to a source, and the change in temperature is also changing with time, while in the domain at y=L, only temperature changes with time. The case for concentration is similar. Flow conditions are given as(15)u0,t=Aν2L4t2,uL,t=0,(16)−k∂θ∂yy=0=qθν2L4t2andθL,t=θ01+ν2L4t2,t>0,(17)−DB∂ϕ∂yy=0=qϕν2L4t2andϕL,t=ϕ01+ν2L4t2,t>0,(18)uy,0=0,θy,0=θ0,ϕy,0=ϕ0and∂θ∂ty,0=0=∂ϕ∂ty,0,y≤L.A complete description of the flow domain is identified via equations (12)–(18).


### 2.1.2. Skin Friction Coefficients

Friction among fluid and solid boundaries is of great importance in the analysis of flow domains bounded by solid boundaries. Coefficients of friction are mathematically defined as(19)Cf≔2τwρA2,where shear stress at the wall is(20)τw=μ+α1∂∂t∂u∂yy=0,L.


### 2.1.3. Nusselt Numbers

Change of temperature aty=0 and y=L are determined by the dimensionless, Nusselt number. Present flow domain, Nusselt numbers are given as(21)Nu1=−L∂θ/∂yy=Lθs1−θ0,where θs1 is constant temperature, at y=L. At y=0, the Nusselt number is given by(22)Nu2=−L∂θ/∂yy=0θs2−θ0,here θs2 is the temperature, at y=0.


### 2.1.4. Mass Transfer Nusselt Numbers

Change of concentration aty=0 and y=L is determined by mass transfer Nusselt, Sherwood numbers. The present flow domain and Sherwood numbers are given as(23)Sh1=−L∂ϕ/∂yy=Lϕs1−ϕ0,where ϕs1 is the constant concentration at y=L. At y=0, the Sherwood number is given by(24)Sh2=−L∂ϕ/∂yy=0ϕs2−ϕ0,here ϕs2 is the concentration at y=0.


### 2.1.5. Problem in Nondimensionalization

We have defined the following dimensionless quantities to make mathematical problem (12)–(18) as nondimensionalized.(25)y^≔yL,t^≔νL2tu^≔uA,θ^≔θ−θ0θ0ϕ^≔ϕ−ϕ0ϕ0.Using, (25), with dimensionless quantities, the IBVP (12)–(18) considered as without hats(26)1+ϵ∂u∂t=1+γ∂∂t∂2u∂y2−λ+Hau+λ1+λ2θθ,Pr∂∂t1+δ3∂α∂tαθ=∂2θ∂y2+Pr1+δ3∂α∂tαNb∂θ∂y∂ϕ∂y+Nt∂θ∂y2,Sc∂∂t1+δ4∂β∂tβϕ=∂2ϕ∂y2+NtNb∂2θ∂y2,y<1,t>0,and problem conditions are given by(27)u0,t=t2,u1,t=0,∂θ∂yy=0=−δ1t2,∂ϕ∂yy=0=−δ2t2,θ1,t=t2=ϕ1,t,uy,0=0,θy,0=0=∂θ∂ty,0andϕy,0=0=∂ϕ∂ty,0.Here dimensionless numbers are given as(28)ϵ≔ϕα1ρK,γ≔α1ρL2,λ≔ψL2K,Ha≔σB02L2μ,Pr≔να3,Sc≔νDB,λ1≔gL2β1θ0Aν,λ2≔gL2β2θ02Aν,Nb≔τϕ0DBν,δ1≔qθLθ0k,δ2≔qϕLϕ0DB,Nt≔τDθν,δ3≔τ1αναΓ1+αL2α,δ4≔τ2βνβΓ1+βL2β.Moreover, the dimensionless coefficients of friction, Nusselt, and mass transfer Nusselt numbers are defined by(29)ReCf2=1+γ∂∂t∂u∂yy=0,1,Nu1Re=−∂θ∂yy=1andNu2Re2=δ1t2,Sh1Re=−∂ϕ∂yy=1andSh2Re2=δ2t2.


## 2.1.1. Flow Problem Conditions

Here we are considering the flow in a channel separated by a distanceL. At the start, the whole flow configuration is still with constant temperature θ0 and constant concentration ϕ0. Flow is generated by the movement of the domain at y=0, while the pressure gradient is assumed to be zero in that case. For temperature and concentration, domain at y=0 is connected to a source, and the change in temperature is also changing with time, while in the domain at y=L, only temperature changes with time. The case for concentration is similar. Flow conditions are given as(15)u0,t=Aν2L4t2,uL,t=0,(16)−k∂θ∂yy=0=qθν2L4t2andθL,t=θ01+ν2L4t2,t>0,(17)−DB∂ϕ∂yy=0=qϕν2L4t2andϕL,t=ϕ01+ν2L4t2,t>0,(18)uy,0=0,θy,0=θ0,ϕy,0=ϕ0and∂θ∂ty,0=0=∂ϕ∂ty,0,y≤L.A complete description of the flow domain is identified via equations (12)–(18).


## 2.1.2. Skin Friction Coefficients

Friction among fluid and solid boundaries is of great importance in the analysis of flow domains bounded by solid boundaries. Coefficients of friction are mathematically defined as(19)Cf≔2τwρA2,where shear stress at the wall is(20)τw=μ+α1∂∂t∂u∂yy=0,L.


## 2.1.3. Nusselt Numbers

Change of temperature aty=0 and y=L are determined by the dimensionless, Nusselt number. Present flow domain, Nusselt numbers are given as(21)Nu1=−L∂θ/∂yy=Lθs1−θ0,where θs1 is constant temperature, at y=L. At y=0, the Nusselt number is given by(22)Nu2=−L∂θ/∂yy=0θs2−θ0,here θs2 is the temperature, at y=0.


## 2.1.4. Mass Transfer Nusselt Numbers

Change of concentration aty=0 and y=L is determined by mass transfer Nusselt, Sherwood numbers. The present flow domain and Sherwood numbers are given as(23)Sh1=−L∂ϕ/∂yy=Lϕs1−ϕ0,where ϕs1 is the constant concentration at y=L. At y=0, the Sherwood number is given by(24)Sh2=−L∂ϕ/∂yy=0ϕs2−ϕ0,here ϕs2 is the concentration at y=0.


## 2.1.5. Problem in Nondimensionalization

We have defined the following dimensionless quantities to make mathematical problem (12)–(18) as nondimensionalized.(25)y^≔yL,t^≔νL2tu^≔uA,θ^≔θ−θ0θ0ϕ^≔ϕ−ϕ0ϕ0.Using, (25), with dimensionless quantities, the IBVP (12)–(18) considered as without hats(26)1+ϵ∂u∂t=1+γ∂∂t∂2u∂y2−λ+Hau+λ1+λ2θθ,Pr∂∂t1+δ3∂α∂tαθ=∂2θ∂y2+Pr1+δ3∂α∂tαNb∂θ∂y∂ϕ∂y+Nt∂θ∂y2,Sc∂∂t1+δ4∂β∂tβϕ=∂2ϕ∂y2+NtNb∂2θ∂y2,y<1,t>0,and problem conditions are given by(27)u0,t=t2,u1,t=0,∂θ∂yy=0=−δ1t2,∂ϕ∂yy=0=−δ2t2,θ1,t=t2=ϕ1,t,uy,0=0,θy,0=0=∂θ∂ty,0andϕy,0=0=∂ϕ∂ty,0.Here dimensionless numbers are given as(28)ϵ≔ϕα1ρK,γ≔α1ρL2,λ≔ψL2K,Ha≔σB02L2μ,Pr≔να3,Sc≔νDB,λ1≔gL2β1θ0Aν,λ2≔gL2β2θ02Aν,Nb≔τϕ0DBν,δ1≔qθLθ0k,δ2≔qϕLϕ0DB,Nt≔τDθν,δ3≔τ1αναΓ1+αL2α,δ4≔τ2βνβΓ1+βL2β.Moreover, the dimensionless coefficients of friction, Nusselt, and mass transfer Nusselt numbers are defined by(29)ReCf2=1+γ∂∂t∂u∂yy=0,1,Nu1Re=−∂θ∂yy=1andNu2Re2=δ1t2,Sh1Re=−∂ϕ∂yy=1andSh2Re2=δ2t2.


## 3. Numerical Discretization Scheme

Numerical discretization of the flow problems (26) and (27) is presented here. The finite difference scheme(FDS) is used to discretize the fractional time derivative as given by [15, 21, 43], while the derivative with respect to space variable is discretized with the finite element scheme(FES) proposed in [15, 21]. The well-posedness of the flow problem can be checked in appropriate spaces [43]. Precisely, we note no ill-posedness in the defined flow problem.Functional spaces are incorporated in the discretization of the model (26) and (27).We point out thatℒ2Ω, square-integrable space of functions on Ω=0,1 along with ℒ2, norm and inner space product. Further, ℋpΩ stands for the Sobolev space, with p>0, ℋ0pΩ is defined as the closure of C0∞Ω¯ in ℋpΩ and C0∞Ω¯ represents infinite differentiable continuous functions, with proper compact support, in Ω [44]. Also, we use the space(30)ℋ1pΩ=u∈ℋpΩuy=1=0,with L2Ω=ℒ2Ω×ℒ2Ω×ℒ2Ω and ℍΩ=ℋ0pΩ×ℋ1pΩ×ℋ1pΩ.Letℒ20,T;VΩ:0,tf⟶V equipped with(31)u,vℒ20,tf;VΩ≔∫0tfu,vVΩdtanduℒ20,T;VΩ≔∫0tfuVΩ2dt1/2.Furthermore,C00,tf;VΩ is the space of continuous functions u:0,tf⟶V with(32)uC00,tf;VΩ≔maxt∈0,tfuV.Analogously, fork∈ℕ,(33)Ck0,tf;VΩ≔u∈C00,tf;VΩ∂tju∈C00,T;VΩ,∀j≤k:j∈ℕ.and(34)uCk0,tf;VΩ≔maxj=0k∂tjuC00,tf;VΩ,and also, we denote by ℂk0,tf;VΩ=Ck0,tf;VΩ3.


### 3.1. Approximations using FDS

Fractional time derivative in (26) and (27) is discretized by FDS. The interval of time, 0,tf is partitioned by time step τ≔tf/m so that tk≔kτ, and k=0,1,2,…,m. Estimation of the derivative of time at some fixed, tk, 0<k<m(35)∂u∂ty,tk≃uy,tk+1−uy,tkτ,tk≤s≤tk+1,here, k=0,(36)∂u∂ty,t0≃uy,t1−uy,t0τ.Initial conditions of flow (27) give(37)uy,t1≃0anduy,t0≃0.Fractional derivative of time,∂tα0<α<1, 0≤k<m, estimated with the Caputo derivative [43].(38)∂tαϕt≔1Γm−α∫0tt−τm−α−1∂m∂τmϕτdτ,m−1<ℜeα<m,m∈ℕ,and the Gamma function Γ⋅ is given as(39)Γz≔∫ℝξz−1e−ξdξ,z∈ℂ,ℜz>0.Fractional derivative of time∂tαϕ agrees with ordinary derivative ∂tmϕ as α⟶m, m−1<α<m, for an integer m. Also, we specify the following operators(40)ℒtαθ⋅t≔∂∂t+δ2∂α+1∂tα+1θt,Qtαθ⋅t≔1+δ2∂α∂tαθt.Estimations with FDS of operatorsℒtαθ, Qtαθ can be taken as (please see [26])(41)ℒtαθtk+1=∂∂t+δ2∂α+1∂tα+1θtk+1,≃θtk+1−θtkτ+Cαθtk+1−2θtk+θtk−1+Cαψkαθ−ψk−1αθ,and(42)Qtαθtk+1=1+δ2∂α∂tαθtk+1,≃θtk+1+δ2Cαθtk+1−θtk+δ2Cαψkαθ,with cα=τ−α/Γ2−α and where bsα≔s+11−α−s1−α, for 0≤s≤m, with(43)ψkαθ≔∑s=1kbsαθy,tk+1−s−θy,tk−swithψ0αθ≔0.


### 3.2. Approximations using FES

In this section, the FES discretization of space variables is discussed in detail. It can be achieved by the partition of domainΩ=0,1 with n subdomains Ωi=yi,yi+1 such that i=1,2,…,n, satisfying(44)Ω¯=∪i=1nΩ¯iandΩi∩Ωj=∅,∀i≠j.The defined elementsΩi of width h are constant, i.e., h≔2/n≔yi+1−yi. Considering subspace of finite dimension V0hΩh>0 of ℋ01Ω and V1hΩh>0 of ℋ11Ω(45)V0hΩ≔ϕ∈ℋ01ΩϕΩi∈PrΩi,∀i=1,2,…,n,V1hΩ≔ϕ∈ℋ11ΩϕΩi∈PrΩi,∀i=1,2,…,n,PrΩi is the Lagrange polynomial space of degree less or equal to r over Ωi with i=1,2,…,n. Also, we note that VhΩ=V0hΩ×V1hΩ×V1hΩ.The weak form of the flow model (26) and (27) can be taken as.Weak form: We need to find u,θ,ϕ∈ℂ10,T;ℍΩ such that(46)1+ϵ∂∂tu,v+1+γ∂∂tu,v+λ+Hau,v−λ1+λ2θθ,v=0,Prℒtαθ,ζ+θ,ζ−PrQtαθ,ζNb∂θ∂y∂ϕ∂y,ζ+Nt∂θ∂y2,ζ=−δ1t,ζ0,Scℒtβϕ,ψ+ϕ,ψ+NtNbθ,ψ=−δ1t−δ2t,ψ0,uy,0=,θy,0=0=∂θ∂ty,0,andϕy,0=0=∂ϕ∂ty,0,for all v,ζ,ψ∈ℍΩ.Weak form (46) is used to incorporate, discrete weak form, at t=tk, 0<k<m(47)Finduh⋅,tk+1,θ⋅,tk+1,ϕ⋅,tk+1∈VhΩs.t.∀v,ς,ϕ∈VhΩ1+ϵ∂∂tuhy,tk+1,v+1+γ∂∂tuhy,tk+1,v+λ+Hauhy,tk+1,v−λ1+λ2θhy,tk+1θhy,tk+1,v=0,Prℒtαθhy,tk+1,ζ+θhy,tk+1,ζ−PrNbQtα∂θhy,tk+1∂y∂ϕhy,tk+1∂y,ζ−PrNtQtα∂θhy,tk+1∂y2,ζ=−δ1t,ζ0,Scℒtβϕhy,tk+1,ψ+ϕhy,tk+1,ψ+NtNbθhy,tk+1,ψ=−δ1t−δ2t,ψ0,uh0y=0,θh0y=0=θh1y,ϕh0y=0=ϕh1y,where uh0⋅=uh⋅,t0, θh1⋅=θh⋅,t1, ϕh1⋅=ϕh⋅,t1, θh0⋅=θ⋅,t0, and ϕh0⋅=ϕ⋅,t0. The numerical solution uh,θh,ϕh to (47) is given as(48)uhy,tk+1=∑p=1Nhuptk+1W0hpy,y∈Ω¯,θhy,tk+1=∑l=1Nhθltk+1W1hly,y∈Ω¯,ϕhy,tk+1=∑l=1Nhϕltk+1W1hly,y∈Ω¯,where W0h=W0hp|p=1,2,…,N0h defines a basis, V0hΩ along with N0h≔dimV0h, W1h=W1hl|l=1,2,…,N1h forms a basis of V1hΩ with N1h≔dimV1h and up,θl,ϕl are yet to be determined. Further, considering v as W0hq for various values of q as q=1,2,…,N0h, ζ , and ψ, W1hm for various values of m as m=1,2,…,N1h, consequently we obtain the following nonlinear algebraic system of equations:(49)1+ϵA0hddtUh+τB0h1+γddtUh+τλ+HaM0hUh+τλ1M0hΘh+τλ2ΘhtkM0hΘh=0,PrA1hℒk+1αΘh+τB1hΘh−τPrNbΦhtkC1hQk+1αΘh−τPrNtθhtkC1hQk+1αΘh=−δ1tI,ScA1hℒk+1βΦh+τB1hΦh+τNtNbB1hΘh=−δ1t−δ2tI,Uh0=0,Θh0=0=Θh1,Φh0=0=Φh1,where for all p,q=1,2,…,N0h and l,m=1,2,…,N1h.(50)Uhp≔up,Θhl≔θl,Φhl≔ϕl,A0hqp≔W0hp,W0hq,B0hqp≔W0hp,W0hq,M0hqp≔W0hp,W0hq,A1hlm≔W1hl,W1hm,B1hlm≔W1hl,W1hm,C1hlm≔W1hl,W1hmIm≔1,0,0,…,0†.The system of algebraic equations, (50) has been computed by Newton’s method. Linear Lagrange elements have been applied to get matrices in the defined system (50).


### 3.3. Convergence of Proposed Scheme

Here we have given the validation of the proposed numerical scheme so that one can confidently perform simulations of real scenarios. A comparison of numerical and theoretical error estimates is presented here. We postulate the proposed mathematical model will satisfy the given error estimates.(51)ζhtk−ζextkL2Ω≤D1hr+1+τα,ζhtk−ζextkH1Ω≤D2hr+τα,here constants D1>0 and D2>0 are not dependent of h, step size, and τ (see [43]).Theoretical and numerical error estimates are compared with the induction of source termsFfab1 in the momentum equation, Ffab2 in the energy equation, and Ffab3 in the concentration equation of model (26) to obtain an exact fabricated solution.(52)1+ϵ∂u∂t=1+γ∂∂t∂2u∂y2−λ+Hau+λ1+λ2θθ+Ffab1y,t,Pr∂∂t1+δ3∂α∂tαθ=∂2θ∂y2+Pr1+δ3∂α∂tαNb∂θ∂y∂ϕ∂y+Nt∂θ∂y2+Ffab2y,t,Sc∂∂t1+δ4∂β∂tβϕ=∂2ϕ∂y2+NtNb∂2θ∂y2+Ffab3y,t,with given conditions. The fabricated solutions completely satisfied the given equations, uexy,t, θexy,t, and ϕexy,t along with conditions. For the validation of the proposed scheme, we have inserted Ffab1, Ffab2, and Ffab3 in the system (50). Error curves have been plotted and they are shown in Figures 1(a), 1(b), and 2.Figure 1
Error estimates for temperature and velocity. (a) VelocityH1 and L2 error curves. (b) Temperature H1 and L2 error curves.
(a)(b)Figure 2
ConcentrationH1 and L2 error curves.Errors are presented in norms on log scales ofH1Ω and L2Ω for different values of h. The slopes of error curves are nearly equal to 2 in the case of H1Ω and 3 in case of L2Ω. Theoretically, in 3.15 and 3.16, Lagrange polynomials are of degrees r+1 and r, respectively. In the proposed scheme we have incorporated a second-degree Lagrange polynomial, so exact values must be equal to 2 in case of H1Ω and 3 in case of L2Ω. This shows the agreement between theoretical and numerical error estimates. So, the proposed scheme (50), is convergent and can efficiently handle real simulations of proposed model (26). The error estimates for velocity, temperature, and concentration are given in Tables 1 and 2.Table 1
Error estimates for velocity and temperature.
nhError estimates of velocityError estimates of temperatureu−uexL2Ωu−uexH1Ωθ−θexL2Ωθ−θexH1Ω70.30002.5816528×10−51.2760729×10−11.211254×10−41.4354533×10−1100.23601.7530926×10−51.0335894×10−13.809745×10−58.0745776×10−2130.18071.2588759×10−58.5420742×10−21.584725×10−55.1678524×10−2160.17289.5167115×10−67.1777154×10−27.961548×10−63.5888249×10−2190.15297.5460756×10−66.1157856×10−24.640422×10−62.6367821×10−2220.15026.2508125×10−65.2734257×10−23.080778×10−62.0187853×10−2250.14565.3821874×10−64.5938245×10−22.2959072×10−61.5949251×10−2280.13784.7890743×10−64.0375489×10−21.8817548×10−61.2918457×10−2Table 2
Error estimates for concentration.
nhError estimates of concentrationu−uexL2Ωu−uexH1Ω70.30003.6875492×10−42.2760729×10−2100.23601.7254876×10−41.0335894×10−2130.18071.4258149×10−49.5124783×10−3160.17288.1568473×10−58.2549678×10−3190.15296.4582167×10−56.3548962×10−3220.15024.8723459×10−55.7854216×10−3250.14563.1248753×10−54.3698524×10−3280.13782.4529751×10−53.9645871×10−3


## 3.1. Approximations using FDS

Fractional time derivative in (26) and (27) is discretized by FDS. The interval of time, 0,tf is partitioned by time step τ≔tf/m so that tk≔kτ, and k=0,1,2,…,m. Estimation of the derivative of time at some fixed, tk, 0<k<m(35)∂u∂ty,tk≃uy,tk+1−uy,tkτ,tk≤s≤tk+1,here, k=0,(36)∂u∂ty,t0≃uy,t1−uy,t0τ.Initial conditions of flow (27) give(37)uy,t1≃0anduy,t0≃0.Fractional derivative of time,∂tα0<α<1, 0≤k<m, estimated with the Caputo derivative [43].(38)∂tαϕt≔1Γm−α∫0tt−τm−α−1∂m∂τmϕτdτ,m−1<ℜeα<m,m∈ℕ,and the Gamma function Γ⋅ is given as(39)Γz≔∫ℝξz−1e−ξdξ,z∈ℂ,ℜz>0.Fractional derivative of time∂tαϕ agrees with ordinary derivative ∂tmϕ as α⟶m, m−1<α<m, for an integer m. Also, we specify the following operators(40)ℒtαθ⋅t≔∂∂t+δ2∂α+1∂tα+1θt,Qtαθ⋅t≔1+δ2∂α∂tαθt.Estimations with FDS of operatorsℒtαθ, Qtαθ can be taken as (please see [26])(41)ℒtαθtk+1=∂∂t+δ2∂α+1∂tα+1θtk+1,≃θtk+1−θtkτ+Cαθtk+1−2θtk+θtk−1+Cαψkαθ−ψk−1αθ,and(42)Qtαθtk+1=1+δ2∂α∂tαθtk+1,≃θtk+1+δ2Cαθtk+1−θtk+δ2Cαψkαθ,with cα=τ−α/Γ2−α and where bsα≔s+11−α−s1−α, for 0≤s≤m, with(43)ψkαθ≔∑s=1kbsαθy,tk+1−s−θy,tk−swithψ0αθ≔0.


## 3.2. Approximations using FES

In this section, the FES discretization of space variables is discussed in detail. It can be achieved by the partition of domainΩ=0,1 with n subdomains Ωi=yi,yi+1 such that i=1,2,…,n, satisfying(44)Ω¯=∪i=1nΩ¯iandΩi∩Ωj=∅,∀i≠j.The defined elementsΩi of width h are constant, i.e., h≔2/n≔yi+1−yi. Considering subspace of finite dimension V0hΩh>0 of ℋ01Ω and V1hΩh>0 of ℋ11Ω(45)V0hΩ≔ϕ∈ℋ01ΩϕΩi∈PrΩi,∀i=1,2,…,n,V1hΩ≔ϕ∈ℋ11ΩϕΩi∈PrΩi,∀i=1,2,…,n,PrΩi is the Lagrange polynomial space of degree less or equal to r over Ωi with i=1,2,…,n. Also, we note that VhΩ=V0hΩ×V1hΩ×V1hΩ.The weak form of the flow model (26) and (27) can be taken as.Weak form: We need to find u,θ,ϕ∈ℂ10,T;ℍΩ such that(46)1+ϵ∂∂tu,v+1+γ∂∂tu,v+λ+Hau,v−λ1+λ2θθ,v=0,Prℒtαθ,ζ+θ,ζ−PrQtαθ,ζNb∂θ∂y∂ϕ∂y,ζ+Nt∂θ∂y2,ζ=−δ1t,ζ0,Scℒtβϕ,ψ+ϕ,ψ+NtNbθ,ψ=−δ1t−δ2t,ψ0,uy,0=,θy,0=0=∂θ∂ty,0,andϕy,0=0=∂ϕ∂ty,0,for all v,ζ,ψ∈ℍΩ.Weak form (46) is used to incorporate, discrete weak form, at t=tk, 0<k<m(47)Finduh⋅,tk+1,θ⋅,tk+1,ϕ⋅,tk+1∈VhΩs.t.∀v,ς,ϕ∈VhΩ1+ϵ∂∂tuhy,tk+1,v+1+γ∂∂tuhy,tk+1,v+λ+Hauhy,tk+1,v−λ1+λ2θhy,tk+1θhy,tk+1,v=0,Prℒtαθhy,tk+1,ζ+θhy,tk+1,ζ−PrNbQtα∂θhy,tk+1∂y∂ϕhy,tk+1∂y,ζ−PrNtQtα∂θhy,tk+1∂y2,ζ=−δ1t,ζ0,Scℒtβϕhy,tk+1,ψ+ϕhy,tk+1,ψ+NtNbθhy,tk+1,ψ=−δ1t−δ2t,ψ0,uh0y=0,θh0y=0=θh1y,ϕh0y=0=ϕh1y,where uh0⋅=uh⋅,t0, θh1⋅=θh⋅,t1, ϕh1⋅=ϕh⋅,t1, θh0⋅=θ⋅,t0, and ϕh0⋅=ϕ⋅,t0. The numerical solution uh,θh,ϕh to (47) is given as(48)uhy,tk+1=∑p=1Nhuptk+1W0hpy,y∈Ω¯,θhy,tk+1=∑l=1Nhθltk+1W1hly,y∈Ω¯,ϕhy,tk+1=∑l=1Nhϕltk+1W1hly,y∈Ω¯,where W0h=W0hp|p=1,2,…,N0h defines a basis, V0hΩ along with N0h≔dimV0h, W1h=W1hl|l=1,2,…,N1h forms a basis of V1hΩ with N1h≔dimV1h and up,θl,ϕl are yet to be determined. Further, considering v as W0hq for various values of q as q=1,2,…,N0h, ζ , and ψ, W1hm for various values of m as m=1,2,…,N1h, consequently we obtain the following nonlinear algebraic system of equations:(49)1+ϵA0hddtUh+τB0h1+γddtUh+τλ+HaM0hUh+τλ1M0hΘh+τλ2ΘhtkM0hΘh=0,PrA1hℒk+1αΘh+τB1hΘh−τPrNbΦhtkC1hQk+1αΘh−τPrNtθhtkC1hQk+1αΘh=−δ1tI,ScA1hℒk+1βΦh+τB1hΦh+τNtNbB1hΘh=−δ1t−δ2tI,Uh0=0,Θh0=0=Θh1,Φh0=0=Φh1,where for all p,q=1,2,…,N0h and l,m=1,2,…,N1h.(50)Uhp≔up,Θhl≔θl,Φhl≔ϕl,A0hqp≔W0hp,W0hq,B0hqp≔W0hp,W0hq,M0hqp≔W0hp,W0hq,A1hlm≔W1hl,W1hm,B1hlm≔W1hl,W1hm,C1hlm≔W1hl,W1hmIm≔1,0,0,…,0†.The system of algebraic equations, (50) has been computed by Newton’s method. Linear Lagrange elements have been applied to get matrices in the defined system (50).


## 3.3. Convergence of Proposed Scheme

Here we have given the validation of the proposed numerical scheme so that one can confidently perform simulations of real scenarios. A comparison of numerical and theoretical error estimates is presented here. We postulate the proposed mathematical model will satisfy the given error estimates.(51)ζhtk−ζextkL2Ω≤D1hr+1+τα,ζhtk−ζextkH1Ω≤D2hr+τα,here constants D1>0 and D2>0 are not dependent of h, step size, and τ (see [43]).Theoretical and numerical error estimates are compared with the induction of source termsFfab1 in the momentum equation, Ffab2 in the energy equation, and Ffab3 in the concentration equation of model (26) to obtain an exact fabricated solution.(52)1+ϵ∂u∂t=1+γ∂∂t∂2u∂y2−λ+Hau+λ1+λ2θθ+Ffab1y,t,Pr∂∂t1+δ3∂α∂tαθ=∂2θ∂y2+Pr1+δ3∂α∂tαNb∂θ∂y∂ϕ∂y+Nt∂θ∂y2+Ffab2y,t,Sc∂∂t1+δ4∂β∂tβϕ=∂2ϕ∂y2+NtNb∂2θ∂y2+Ffab3y,t,with given conditions. The fabricated solutions completely satisfied the given equations, uexy,t, θexy,t, and ϕexy,t along with conditions. For the validation of the proposed scheme, we have inserted Ffab1, Ffab2, and Ffab3 in the system (50). Error curves have been plotted and they are shown in Figures 1(a), 1(b), and 2.Figure 1
Error estimates for temperature and velocity. (a) VelocityH1 and L2 error curves. (b) Temperature H1 and L2 error curves.
(a)(b)Figure 2
ConcentrationH1 and L2 error curves.Errors are presented in norms on log scales ofH1Ω and L2Ω for different values of h. The slopes of error curves are nearly equal to 2 in the case of H1Ω and 3 in case of L2Ω. Theoretically, in 3.15 and 3.16, Lagrange polynomials are of degrees r+1 and r, respectively. In the proposed scheme we have incorporated a second-degree Lagrange polynomial, so exact values must be equal to 2 in case of H1Ω and 3 in case of L2Ω. This shows the agreement between theoretical and numerical error estimates. So, the proposed scheme (50), is convergent and can efficiently handle real simulations of proposed model (26). The error estimates for velocity, temperature, and concentration are given in Tables 1 and 2.Table 1
Error estimates for velocity and temperature.
nhError estimates of velocityError estimates of temperatureu−uexL2Ωu−uexH1Ωθ−θexL2Ωθ−θexH1Ω70.30002.5816528×10−51.2760729×10−11.211254×10−41.4354533×10−1100.23601.7530926×10−51.0335894×10−13.809745×10−58.0745776×10−2130.18071.2588759×10−58.5420742×10−21.584725×10−55.1678524×10−2160.17289.5167115×10−67.1777154×10−27.961548×10−63.5888249×10−2190.15297.5460756×10−66.1157856×10−24.640422×10−62.6367821×10−2220.15026.2508125×10−65.2734257×10−23.080778×10−62.0187853×10−2250.14565.3821874×10−64.5938245×10−22.2959072×10−61.5949251×10−2280.13784.7890743×10−64.0375489×10−21.8817548×10−61.2918457×10−2Table 2
Error estimates for concentration.
nhError estimates of concentrationu−uexL2Ωu−uexH1Ω70.30003.6875492×10−42.2760729×10−2100.23601.7254876×10−41.0335894×10−2130.18071.4258149×10−49.5124783×10−3160.17288.1568473×10−58.2549678×10−3190.15296.4582167×10−56.3548962×10−3220.15024.8723459×10−55.7854216×10−3250.14563.1248753×10−54.3698524×10−3280.13782.4529751×10−53.9645871×10−3


## 4. Approximate Simulated Results

This section deals with the results of the velocity field, temperature field, and concentration of nanofluid with nonlinear convection and flux conditions using MATLAB R2017a. Brownian phenomena of motion and thermophoresis are analyzed while evaluating this study. Fluid is flowing through the Darcy porous medium. Flow and magnetic fields are orthogonal. The proposed algorithm is employed for numerical solutions of governing nonlinear fractional equations. Discretization of time and space variables is carried out by FDS and FES, respectively. We analyze the characteristics of physical numbers on flow field generation. Transfer of heat and mass are also discussed for various pertinent numbers. Variations of flow quantities with the variation of different nondimensional numbers help to understand the problem. Simulated results are seen over intervals of time [0, 2], and [0,2]. Figure 3, is used for demonstration of velocity behavior with the variation of 0<α<1. It is observed that for higher values of α, velocity increases (see Figure 3(a)). However, this occurrence is hereditary in character and cannot be considered to be other variations of pertinent parameters.Figure 3
Influence of noninteger number, Hartmann number, and viscoelastic number on velocity. (a) Change in velocity withα. (b) Change in velocity with Ha and ϒ.
(a)(b)Change in velocity with the variation of0.1≤Ha≤3, and 0.1≤γ≤0.8 are plotted in Figure 3(b). The plot showed that a decrease in velocity is noted for higher values of Ha and opposite behavior is obtained for the viscoelastic parameter, γ. As Ha increases, Lorentz force decreases, which slows down fluid motion. As a result, velocity decreases for higher values of Ha. Porosity parameter 0≤λ≤7 and convection number 0.1≤λ1≤3.5 influence on the velocity, which is plotted in Figure 4. It is noted that velocity decreases for higher values of λ, while an opposite trend is noted for λ1 in Figure 4(a). With the increase of λ, permeability decreases while porosity increases. So, the velocity profile decreases with the increase of λ. Force of inertia is inversely related to λ1 and direct influence is given for buoyancy force. As λ1>0, heat transfers from plates to fluid. So, θs1−θ0 and θs2−θ0 increase. Consequently, the increase in λ1 increases buoyancy forces, θs1−θ0 and θs2−θ0. So, increase in fluid velocity is noted in this case.Figure 4
Influence of porosity, Darcy resistance, and convection numbers on velocity. (a) Change in velocity withλ and λ1. (b) Change in velocity with λ2 and ε.
(a)(b)Similar behavior of nonlinear convection number0.1≤λ2≤6.5 is noted in Figure 4(b). When there is convection, it is nonlinear in nature, so as noted, the same results are observed for the convection parameter λ1. It is noted that an increase in the Darcy resistance number, 0.1≤ϵ≤6.5 reflects a decrease in velocity. This is because an increase in ϵ permeability decreases while porosity increases. Hence, a decrease in velocity with an increase in Darcy resistance number is noted.Figure5 is plotted to observe the temperature profile, for several values of α. Temperature profile, increases as α increases in Figure 5(a). Temperature profile behavior with the increase of pedesis number 0.1≤Nb≤4, Schmidt parameter 0.1≤Sc≤4 are observed via Figure 5(b). It is noted that temperature increases. as Sc, Nb increase. Momentum diffusivity enhances with the increase of Sc, which increases friction between different layers of the flow domain. As a result, an increase in temperature is seen with the increase of Sc. A decrease of temperature with the decrease of base fluid heat capacity is noted with an increase of Nb.Figure 5
Influence of noninteger number, pedesis parameter, and Schmidt number on temperature. (a) Change in temperature withα. (b) Change in temperature with Sc and Nb.
(a)(b)The influence of0.1≤Pr≤7, Prandtl number, 0.1≤Nt≤4 thermophoresis number, on temperature is plotted in Figure 6. With the increase of Nt, the temperature profile increases, while the opposite trend is observed in the case of Pr in Figure 6(a). With the increase of Nt, the heat capacity of the nanofluid decreases, as a result, the temperature profile increases, for higher values of Nt. With the increase of Pr, thermal diffusivity, decreases so temperature decreases with the increase of Pr. Figure 6(b) is plotted to show the influence of heat flux, 0.1≤δ1≤0.8 and mass flux, 0.1≤δ2≤0.8 numbers on temperature. It is noted temperature increases for higher values of δ1 while opposite trends are seen for the case of δ2. With the increase of δ1, thermal conductivity, the fluid decreases which decreases the heat passage rate through the base fluid, so temperature increases for higher values of δ1.Figure 6
Influence of Prandtl number, heat and mass fluxes, and thermophoresis numbers on temperature. (a) Change in temperature withPr and Nt. (b) Change in temperature with δ1 and δ2.
(a)(b)Figure7 is outlined to observe concentration change for various values of α. Concentration is at a higher level for higher values of α in Figure 7(a). Effects of thermophoresis, Nt, pedesis, Nb numbers on concentration are plotted in Figure 7(b). Concentration increases, for higher values of Nt, while the opposite behavior is noted for Nb. An increase in concentration, is because of an increase in the coefficient of thermophoretic diffusion.Figure 7
Influence of fractional number, thermophoresis, and pedesis numbers on concentration. (a) Change in concentration withα. (b) Change in concentration with Nt, and Nb.
(a)(b)The influence of PrandtlPr, and Schmidt Sc numbers on concentration can be seen via Figure 8. Concentration increases for higher values of Pr, while the opposite behavior is noted in the case of Sc as shown in Figure 8(a). Momentum diffusivity increases for higher values of Pr, as a result, concentration is at a greater level for higher values of Pr. The viscosity of fluid increases, and the Brownian diffusion coefficient decreases for higher values of Sc. So, the concentration remains at a lower level for higher values of Sc. Figure 8(b) is outlined to observe the impact of heat flux δ1, and mass flux δ2 numbers on concentration. Concentration increases for higher values of δ1 and δ2. The rate of transfer of concentration decreases for higher values of δ2. So, the concentration increases for higher values of δ2. At the end Figures 9(a) and 9(b)–12(a) and 12(b) are plotted for time dependent velocity, temperature, and concentration profiles. These figures showed anomalous character of noninteger nanofluid.Figure 8
Influence of Prandtl number, heat and mass fluxes numbers, and Schmidt number on concentration. (a) Change in concentration withPr, and Sc. (b) Change in concentration with δ1, and δ2.
(a)(b)Figure 9
Unsteady velocity for various values of physical parameters over the interval0,2.
(a)(b)Figure 10
Unsteady temperature and concentration for various values of physical parameters over the interval0,2.
(a)(b)Figure 11
Velocity profiles for various final times forNt=0.6, Sc=0.5, Nb=0.7, δ1=0.5, Pr=0.8, δ2=0.3, δ3=0.1, and δ4=0.1. (a) Variations of velocity profile when α = 0.5. (b) Variations of velocity profile when α = 0.1.
(a)(b)Figure 12
Temperature and concentration profiles for various final times forNt=0.9, Sc=0.3, Nb=0.8, δ1=0.7, Pr=1, δ2=0.8, δ3=0.1 and δ4=0.1. (a) Variations of temperature profile when α = 0.7. (b) Variations of temperature profile when β = 0.7.
(a)(b)Variations of skin friction with pertinent fractional model parameters are examined via Tables3 and 4. Skin friction magnitude increases with the increase of α, β, λ, δ2, Pr, and Ha , while it decreases with the increase of γ, λ1, λ2, δ1, Sc, Nt, and Nb.Table 3
Influence of fractional flow numbers on the coefficient of friction aty,t=0,0.1 for Sc=0.5, Nt=0.3, Pr=0.4, Nb=0.4, δ1=0.4, δ2=0.6, δ3=0.01, δ4=0.01, α=0.5, and β=0.5.
ϵγλλ1λ2HaReCf/20.10.20.10.20.10.2−2.3517330.2−2.4308000.30.2−2.5078210.3−2.3052090.40.1−2.1680510.2−2.1777600.30.2−2.1874450.3−2.1334040.40.1−2.0793620.2−2.0789980.30.2−2.0786330.3−2.0885480.4−2.098437Table 4
Influence of fractional flow numbers on the coefficient of friction aty,t=0,0.1 for ϵ=0.3, β=0.3, γ=0.4, λ=0.1, Ha=0.5, δ3=0.01, δ4=0.01, λ1=0.5, and λ2=0.7.
αδ1δ2PrScNtNbReCf/20.10.50.40.60.50.40.5−0.3143250.2−0.5673810.30.5−0.9365140.6−0.8905530.70.4−0.8444090.5−0.8445640.60.6−0.8447190.7−0.8703740.80.5−0.8929030.6−0.8928220.70.4−0.8927280.5−0.8924490.60.5−0.8921640.6−0.8919370.7−0.891685


## 5. Conclusion

The flow of a nanofluid with a fractional derivative of time and nonlinear convection is studied in this communication. The fractional derivative of time is used to analyze the memory characteristics. Applications of Darcy porous medium can be encountered in numerous industries. Brownian motion effects are also considered while formulating the flow domain. A numerical algorithm is incorporated to estimate variations in the flow field. Flow is carried out by variable acceleration of the lower boundary. The flow direction and the applied magnetic field are orthogonal. Conditions for heat and mass gradients are considered at the bottom boundary, while quadratic variation is seen at the upper boundary. Enhancement in velocity is noted, for higher values ofα, λ1, γ, andλ2, and opposite behavior is seen in case of λ, Ha, and β. Also, the enhancement of temperature profiles are observed for higher values of Nt, Nb, Sc, α, and δ1, and opposite behavior is noted in case of Pr and δ2. Concentration increases with the increase of α, Nt, Pr, δ1, and δ2, and decreasing behavior is observed for Nb and Sc. It is believed that fractional porous medium flows of nanofluids can be tackled with the current study. Flow, discussed in this communication, can be tackled in the manufacture of fiber and in geology. Moreover, the current study can be extended to Maxwell fluid flow. An analysis can also be performed to tackle nonlinear radiation effects in a fluid of differential type with fractional derivative.



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