Heat Transfer in a Fractional Nanofluid Flow through a Permeable Medium

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, δ 1 , and mass ﬂux, δ 2 , 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.


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][5][6][7][8][9][10][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][16][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][21][22][23][24]. Simulations of anomalous nonlinear models can be used to approximate experimental data in a better way when compared to usual rheological models [25][26][27][28][29][30].
Porous medium flows occur in soil mechanics, ceramics, and industrial and mineral processing. e complexity of the flow problem is increased with the presence of a porous medium. is 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]. e 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. e 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. e 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. ese 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]. e structure of nanoparticles is between atomic and bulk. e 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][41][42][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 at y � 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. e influence of the involved physical parameters is discussed appropriately. In Section 2, modeling with noninteger time derivatives is discussed. e 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.

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. e 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 temperature θ 0 and concentration ϕ 0 . 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.
en we consider 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). e 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 derivative α including nanoparticles, consider the classical energy equation for incompressible fluid of the Buongiorno article (see [46]).

2
Mathematical Problems in Engineering where q, J p denote fluxes of energy and diffusion and for definition (see [46]). Temperature and concentration propagation of infinite order can be seen with q, J p in Buongiorno article (see [46]). We can overcome this situation with the introduction of fractional relaxation times using reference [45], so we define q, J p as Here we are considering the flow in a channel separated by a distance L. At the start, the whole flow configuration is still with constant temperature θ 0 and constant concentration ϕ 0 . 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. e case for concentration is similar. Flow conditions are given as A complete description of the flow domain is identified via equations (12)-(18).

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 where shear stress at the wall is

Nusselt Numbers.
Change of temperature at y � 0 and y � L are determined by the dimensionless, Nusselt number. Present flow domain, Nusselt numbers are given as where θ s 1 is constant temperature, at y � L. At y � 0, the Nusselt number is given by here θ s 2 is the temperature, at y � 0.

Mass Transfer Nusselt Numbers.
Change of concentration at y � 0 and y � L is determined by mass transfer Nusselt, Sherwood numbers. e present flow domain and Sherwood numbers are given as where ϕ s 1 is the constant concentration at y � L. At y � 0, the Sherwood number is given by here ϕ s 2 is the concentration at y � 0.

Problem in Nondimensionalization.
We have defined the following dimensionless quantities to make mathematical problem (12)-(18) as nondimensionalized.
Using, (25), with dimensionless quantities, the IBVP (12)-(18) considered as without hats Here dimensionless numbers are given as Moreover, the dimensionless coefficients of friction, Nusselt, and mass transfer Nusselt numbers are defined by

Numerical Discretization Scheme
Numerical discretization of the flow problems (26) and (27) is presented here. e 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]. e 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 that L 2 (Ω), square-integrable space of functions on Ω � (0, 1) along with L 2 , norm and inner space product. Further, H p (Ω) stands for the Sobolev space, represents infinite differentiable continuous functions, with proper compact support, in Ω [44]. Also, we use the space Analogously, for k ∈ N, Mathematical Problems in Engineering 5 and (34) and also, we denote by 3.1. Approximations using FDS. Fractional time derivative in (26) and (27) is discretized by FDS. e interval of time, here, k � 0, Initial conditions of flow (27) give u y, t 1 ≃0 and u y, t 0 ≃0.
and the Gamma function Γ(·) is given as Fractional derivative of time z α t ϕ agrees with ordinary derivative z m t ϕ as α ⟶ m, m − 1 < α < m, for an integer m. Also, we specify the following operators can be taken as (please see [26]) and

Approximations using FES.
In this section, the FES discretization of space variables is discussed in detail. It can be achieved by the partition of domain Ω � [0, 1] with n . e weak form of the flow model (26) and (27) can be taken as.
e 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).

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.
here constants D 1 > 0 and D 2 > 0 are not dependent of h, step size, and τ (see [43]). eoretical and numerical error estimates are compared with the induction of source terms F fab1 in the momentum equation, F fab2 in the energy equation, and F fab3 in the concentration equation of model (26) to obtain an exact fabricated solution. 8 Mathematical Problems in Engineering with given conditions. e fabricated solutions completely satisfied the given equations, u ex (y, t), θ ex (y, t), and ϕ ex (y, t) along with conditions. For the validation of the proposed scheme, we have inserted F fab1 , F fab2 , and F fab3 in the system (50). Error curves have been plotted and they are shown in Figures 1(a), 1(b), and 2.
Errors are presented in norms on log scales of H 1 (Ω) and L 2 (Ω) for different values of h. e slopes of error curves are nearly equal to 2 in the case of H 1 (Ω) and 3 in case of L 2 (Ω). eoretically, 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 H 1 (Ω) and 3 in case of L 2 (Ω). is 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). e error estimates for velocity, temperature, and concentration are given in Tables 1 and 2.

Approximate Simulated Results
is 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. e 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, 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.
Change in velocity with the variation of 0.1 ≤ Ha ≤ 3, and 0.1 ≤ c ≤ 0.8 are plotted in Figure 3(b). e plot showed that a decrease in velocity is noted for higher values of Ha and opposite behavior is obtained for the viscoelastic parameter, c. 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 ≤ λ 1 ≤ 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 λ 1 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 λ 1 and direct influence is given for buoyancy force. As λ 1 > 0, heat transfers from plates to fluid. So, (θ s 1 − θ 0 ) and (θ s 2 − θ 0 ) increase. Consequently, the increase in λ 1 increases buoyancy forces, (θ s 1 − θ 0 ) and (θ s 2 − θ 0 ). So, increase in fluid velocity is noted in this case.
Similar behavior of nonlinear convection number 0.1 ≤ λ 2 ≤ 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 λ 1 . It is noted that an increase in the Darcy resistance number, 0.1 ≤ ϵ ≤ 6.5 reflects a decrease in velocity. is 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. Figure 5 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. e influence of 0.1 ≤ 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(       temperature increases for higher values of δ 1 while opposite trends are seen for the case of δ 2 . With the increase of δ 1 , thermal conductivity, the fluid decreases which decreases the heat passage rate through the base fluid, so temperature increases for higher values of δ 1 . Figure 7 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. e influence of Prandtl Pr, 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

Conclusion
e flow of a nanofluid with a fractional derivative of time and nonlinear convection is studied in this communication.
e 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. e 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 of α, λ 1 , c, andλ 2 , 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 δ 1 , and opposite behavior is noted in case of Pr and δ 2 . Concentration increases with the increase of α, Nt, Pr, δ 1 , and δ 2 , 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. Fluid concentration ϕ 0 : Initial concentration c p : Nanoparticles specific heat h p : Nanoparticles enthalpy α, β: Fractional derivatives α 3 : ermal diffusivity g: Gravitational acceleration τ: Heat capacities ratio σ: Electrical conductivity ψ: Porosity α 1 : Second-grade fluid material parameter k: ermal conductivity q θ : Heat flux ϵ: Darcy resistance parameter λ: Porosity parameter λ 1 , λ 2 : Convection parameters Nt: ermophoresis parameter δ 1 : Heat flux parameter δ 3 , δ 4 : Relaxation time parameters Pr: Prandtl parameter y: Space coordinate θ: Fluid temperature θ 0 : Initial temperature c f : Specific heat