# An Application of the Finite Element Method for Heat and Mass Transfer of Boundary Layer Flow Using Variable Thermal Conductivity and Mass Diffusivity

Authors: Muhammad Shoaib Arif; Kamaleldin Abodayeh; Yasir Nawaz
Journal: Journal of Mathematics
 (2022)

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## Abstract

Since mass diffusivity and thermal conductivity cannot be considered constants in practical analysis, an existing heat and mass transfer model of unsteady mixed convection flow over a stretching sheet is modified by incorporating the effects of nonlinear mixed convection variable thermal conductivity and mass diffusivity and solved its dimensionless form by applying the finite element method which is the main novelty of this work. Nonlinear governing equations are created because of changing thermal conductivity and mass diffusivity. Nonlinear solutions can be achieved by employing the Galerkin finite element approach. Temperature and concentration-based thermal conductivity and mass diffusivity are considered. The model is expressed as a set of partial differential equations, and furthermore, it is reduced to a system of dimensionless ordinary differential equations. These obtained equations are solved with the finite element method with linear interpolating polynomials and numerical integration. In addition, a Matlab solver bvp4c is also considered for comparison purposes or validation of the computed results. The graphs depict the effect of various parameters on the velocity, temperature, and concentration curves. The results show that thermal and diffusive wave propagation is significantly affected by thermal conductivity and mass diffusivity changes. Results show that flow velocity escalates by rising values of thermal and solutal Grashof numbers.


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## Body

## 1. Introduction

A study related to non-Newtonian fluids has gained exquisite attraction due to its massive applications in food and energy, specifically in the petroleum industry. The roots of plastic processing industries peculiarly relied on fluid dynamics in polymer melt and polymer solutions. Industrial slurry multiphase mixers, pharmaceutical formulations, cosmetics and toiletries, paints, biofluids, and food items have become more diverse because of the widespread usage of non-Newtonian fluids. Such studies follow Ostwald-de Waele’s power-law model and the boundary layer concept, which was first proposed by Schowalter [1]. Acrivos [2] formulated an extension for the boundary layer non-Newtonian fluid flow in 1960 and leveled the ground for upcoming scholars. Several investigations on chemicals, polymers, and molten plastics have been conducted since then. Many previous investigations of natural convection with transparent fluid media are considered Newtonian fluids by the thermo fluid community. A theoretical analysis of how the shear rate affects convective flow patterns and heat transfer rates in non-Newtonian fluids can be used in papermaking, oil drilling, slurry transport, food processing, and polymer engineering.Ice formation damage of organs preserves cells and tissues through freezing on the principle of mass and heat transfer. Such studies are essential as non-Newtonian fluids and have robust interference in industries such as petroleum storage, ground-water hydrology, nuclear waste disposal, geothermal energy formation, cooling, and design of solid matrix heat exchange and packed-bed chemical catalytic reactors. Authors such as Smith et al. [3] were interested in determining mass transfer in rat prostate tumor tissues. In contrast, El-Hakiem and El-Amin [4] first researched the same transfer in a porous material with nonuniform surface heat flow through a vertical plate. An infinite insulated flat plate near a laminar flow of viscoelastic incompressible fluid of low electrical conductivity was solved by Eldabe and Hassan [5]. Non-Newtonian elastic, viscous fluid flow past an infinite flat plate with changing suction was studied by Soundalgekar and Puri [6].The theory of non-Newtonian fluid presents the greatest challenge to mathematicians and engineers in developing analytical and numerical solutions for highly nonlinear governing equations. The study for hearing transfer of boundary layer flow over a stretching sheet has been given by Elbashbeshy et al. [7]. A numerical Runge–Kutta method has been considered to handle boundary value problems by Elbashbeshy et al. [7]. The flow was studied with heat generation, chemical reaction, and thermal radiation. The radiation effect was also considered in the boundary layer flow over the exponentially stretching sheet by Nadeem et al. [8]. Two types of effects for heat transfer have been studied: the prescribed exponential order surface temperature and the prescribed exponential order heat flux. An analytical method, namely homotopy analysis, has been utilized to solve differential equations in the radiative flow. It was concluded that the temperature profile was escalated by the growth of the radiation parameter. Both effects of prescribed surface temperature and prescribed heat flux were also studied by Nadeem et al. [9] in the stagnation point flow of third-order fluids.A shrinking sheet was deliberated with two problems: two-dimensional stagnation point flow over a shrinking sheet and axisymmetric stagnation point flow over axisymmetric shrinking. An analytical technique, homotopy analysis, was adopted to solve the ordinary differential equations. The study of stagnation point flow for a viscous fluid has been given by Nadeem et al. [10]. For the validity of the computed solutions obtained by the homotopy analysis method for solving the boundary layer equation, a comparison with the results obtained by another research was made and found good agreement. Nadeem and Akbar [11] also found an analytical solution for the peristaltic motion of the non-Newtonian incompressible Johnson Segalman fluid. The perturbation and homotopy analysis methods were adopted to tackle the differential equations arising in the studied flow. Nadeem and Ali [12] also employed the homotopy analysis method to solve flow phenomena of the steady incompressible fourth-grade fluid down a vertical cylinder. Variable viscosity and heat transport were also taken into account in the study. Buongiorno [13] used Brownian diffusion and thermophoresis slip mechanisms to develop the nonhomogeneous equilibrium model.A bounded domain with two-dimensional flows was investigated. Global in-time solutions exist and are unique according to Lukaszewicz [14] and their convergence to the stationary solution for viscous flows. Shenoy [15] presented many exciting applications of non-Newtonian power-law fluids. Together with streamlines and turbulence, mathematical models were studied by Astarita and Marrucci [16] and Bohme [17]. The non-Newtonian power-law fluid with heat transfer over a continuously moving flat porous plate has been analyzed by Kishan and Reddy [18]. The numerical solution was found by employing the implicit finite difference method. The solution was found to be dependent on involved dimensionless parameters. The effects of parameters on velocity and temperature profiles were carried out. Kavitha and Kishan have investigated a radiative boundary layer flow with heat transfer characteristics over a vertical stretching sheet under the effect of viscous dissipation [19]. Some obtained results were compared with existing results and found good agreement.Nanotechnology dealt with materials of a nanometer in size and gravitated attention due to their unique physical and chemical properties. Fluids comprising small-scaled unit particles are called nanofluids, and heat transfer through such fluids could enhance conductivity. It is the most relevant, innovative technology dealing with the suspension of solid nanoparticles ranging from 1 to 100 nm in diameter within common liquids such as water, oil, and ethylene glycol. A rise in thermal conductivity by 40% can be obtained by decreasing the number of solid nanoparticles in suspension by 1–5% by volume. It depends on solid nanoparticles’ size, shape, and thermal properties. The composition of these nanoparticles used in nanofluids is metals, oxides, carbides, and nanotubes. Water, ethylene glycol, and oil are all examples of base liquids. To reduce boiler flue gas temperatures and microelectronic noise, nanofluid technology can be used in various heat transfer applications, such as nuclear reactor coolants, pharmaceutical processes, and boiler flue gas cooling. Such fluids raised the convection and conductivity of heat transmission counterbalance with base fluids. The application of these fluids as heat exchanger plants and automotive cooling systems has been reported by Nadeem et al. [20].The term nanofluid was first proposed by Choi [21], while its properties such as stability, pressure drop, and passage through nanochannels were given by Zhou [22]. Xuan and Li [23] suggested a mechanism to reduce heat transmission among nanofluids by increasing volume fractions and resultant viscosity. Therefore, selecting a particular nanoparticle regarding its fractional volume substantially enhances conductivity. In addition, Buongiorno [13] introduced the term absolute velocity for nanoparticles which he reckoned by the sum of base fluid velocity and relative velocity (slip velocity). All seven mechanisms he studied included inertia, Brownian diffusion, thermophoresis, the Magnus effect, and fluid drainage and gravitational settling.(Pak and Cho [24], Wen and Ding [25], Ding et al. [26]) Utilizing nanofluids to promote forced convective heat transfer has been suggested. Alternatively, the mechanism behind this forced convective heat transfer and its characteristics were not elaborated. Due to its efficient convection, the topic of nanofluid has attracted a lot of attention in recent years [27]. Recent research [28, 29] on nanofluids has shown that the heat conductivity rises with decreasing grain size. This was proved through the use of measurements. In Chengara et al. [30], they explore how the disjoining pressure exerted by nanoparticles in a liquid film affects its spreading. Critical heat fluxes in boiling conditions increase (You et al. [31]). Convection of heat over a permissible elastic wall in a porous medium was investigated by Sheikholeslami et al. [32].Extrusion, wire drawing, glass fiber manufacture, continuous casting, crystal growth, and paper formation are among the engineering applications that have drawn researchers' attention to boundary layer flow and heat transmission over a constantly stretched surface. Crane [33] was the one who pioneered the study of boundary layer flow over an elastic sheet with variable velocity. In contrast, the convection aspect of the former problem was resolved by Carragher and Crane [34]. Temperature differences on surfaces and fluids were directly related to distances from fixed points [34].The introduction of nanofluids to fluid dynamics, which owned the property of effective thermal conductivity, created a chance to study non-Newtonian fluids under the roof of nanofluids for effective conductivity. Non-Newtonian nanofluid transport in various geometries and boundary conditions in porous and non-porous media has only been studied by a few researchers. The importance of non-Newtonian nanofluids in physical and biological systems, specifically in polymer melts and paints, is evident from [35–37], which significantly lifted this area for research purposes. These phenomena are found in physics, geophysics, astronomy, and chemical engineering applications, including crystal magnetic damping control, hydromagnetic chromatography, and conducting flow in trickle-bed reactors [38].Various authors studied the effect of dimensionless parameters on the flows over plates. Asjad et al. [39] worked on MHD nanofluid flow over an exponentially stretching sheet by studying microorganisms and the effects of thermal radiations, chemical reactions, and heat source dissipation. The differential equations obtained from the flow problems were solved using the shooting method based on the Runge–Kutta scheme. The study of microorganisms for mixed convection of Casson nanofluid flow in the stagnation region of the rotating sphere was given in [40]. Gyrotactic microorganisms have been considered to enhance heat transportation, and nano-sized particles were brought into the study to improve stability. The dimensionless partial differential equations were solved by the Galerkin finite element method. The study of microorganisms was also presented for MHD Williamson nanofluid flow [41] over the sheet with irregular thickness. The impacts of temperature-dependent thermal conductivity and nonuniform viscosity were also studied with the aforementioned flow phenomenon. The reduced set of ordinary differential equations was solved by the shooting method. The study of Reiner–Rivlin fluid over a disk has been presented in [42] with Brownian motion and thermophoresis effects.For simulating MHD Casson flow over a linearly shrinking/stretching sheet, the experimental relations for approximating the thermophysical properties of hybrid nanofluids have been explored in [43]. A fourth-order accurate solution was found using Matlab. The dual solution and stability analysis were also given. The results show that the Casson fluid parameter range enhances with second nanoparticle mass, suction parameter, and radiation parameter. The study of MHD stagnation-point flow over a wavy circular cylinder has been given in [44], considering aluminum-copper/water hybrid nanofluid as the working fluid with temperature jump and velocity slip boundary conditions. The obtained results revealed that a developed mass-based model can be employed further to study heat transfer of hybrid nanofluid flow in similar problems. A semianalytical study for two-dimensional boundary layer flow over a shrinking/stretching wedge has been given in [45]. The Tiwari–Das model with a combination of mass-based hybrid nanofluid procedures was applied to find governing equations of the considered flow problem. It was observed that the boundary layer thickness for the first solution was thinner than the second one. The steady laminar, incompressible, and two-dimensional hybrid nanofluid flow has been studied numerically in [46]. The effects of the convectively-warmed moving wedge with radiative transition have also been considered. The three primary geometries namely the flat plate, the wedge, and the vertical plate were considered. The mass-based method has been combined with the entropy generation analysis for the first time in [46].The finite element method can adequately predict boundary layer flow’s heat and mass transmission profiles. It is the most effective methodology that provides a great understanding of the time history of constitutive variables and is an energetic tool in computational fluid dynamics. In this work, a mathematical model of boundary layer flow is modified using nonlinear mixed convection and studied using the modified finite element method. A summary of each section is given in the next paragraph.The problem is defined in section2 with governing equations and boundary conditions, and also in this section, the partial differential equations are transformed into ordinary differential equations using transformations. The procedure for solving equations by the modified finite element method is given in section 3. The validation of the computed results is provided in section 4. The discussion related to graphs is given in section 5.


## 2. Problem Formulation

Consider incompressible, Newtonian, two-dimensional, and mixed convective unsteady flow over the stretching sheet. Let the sheet be placed horizontally. Thex-axis is taken along the sheet, while the y-axis is taken perpendicular to the sheet. The flow is generated by the sudden movement of the plate towards the positive x-axis. Let T be the temperature of the fluid, C be the concentration, Tw&Cw are respectively temperature and concentration at the wall, and T∞&C∞ are respectively temperature and concentration at the free stream. The geometry of this problem is given in Figure 1, which shows the direction of flow, coordinate axis and momentum, and thermal and concentration boundary layers. Considering thermal conductivity and mass diffusivity as temperature and concentration-dependent quantities, and following [47–49] the governing equations for this phenomenon can be stated as(1)∂u∂x+∂v∂y=0,(2)∂u∂t+u∂u∂x+v∂u∂y=ν∂2u∂y2+gΛ1T−T∞+Λ2T−T∞2+Λ3C−C∞+Λ4C−C∞2,(3)∂T∂t+u∂T∂x++v∂T∂y=1ρCp∂∂ykT∂T∂y−1ρCp∂qr∂y,(4)∂C∂t+u∂C∂x+v∂C∂y=∂∂yDC∂C∂y−k1C−C∞.Figure 1
The geometry of the problem.Subject to the following boundary conditions:(5)ut,y=Uw,vt,y=vw,Tt,y=Tw,CT,y=Cwwheny=0,t>0ut,y⟶0,Tt,y⟶T∞,CT,y⟶C∞wheny⟶∞,t>0,where kT=k∞1+ε1T−T∞/Tw−T∞,DC=D∞1+ε2C−C∞/Cw−C∞ temperatures, concentration-dependent thermal conductivity, and mass diffusivity have been considered in [50] and equations (1)–(4), and terms having derivatives with respect to x will be ignored, as it is assumed in [47]. Uw=ax denotes the velocity of the stretching sheet, kT and DC are, respectively, temperatures and concentration-dependent thermal conductivity and mass diffusivity, ν is the kinematic viscosity, g is the gravity, Λ1 and Λ2 are the coefficients of linear and nonlinear thermal expansion, respectively, Λ3 and Λ4 are coefficients of linear and nonlinear solutal expansions, respectively, ρ is the density of the fluid, Cp denotes the specific heat capacity, k1 represents the reaction rate, and qr denotes the radiative flux. In this study, a Rosseland radiative flux [51] is considered, which is expressed as(6)qr=−4σ∗3k∗∂T4∂y≈−16σ∗T∞33k∗∂T∂y.Consider the following transformations [47].(7)η=yδt,u=Uwfη,θ=T−T∞Tw−T∞,ϕ=C−C∞Cw−C∞.The continuity equation (1) is solved, and its solution is given by [47].(8)v=−voνδ,where vo=−vwδ/ν.Under transformations (7), equations (2)–(5) are reduced to(9)−f′ηδδ′ν+vo=f″+Gr0θ+β1θ2+Grmϕ+β2ϕ2=0,(10)−θ′ηδδ′ν+v∘=1Prε1θ′2+1+ε1θθ″+43RdPrθ″,(11)−ϕ′ηδδ′ν+v∘=1Scε2ϕ′2+1+ε2ϕϕ″−γϕ,subject to the dimensionless boundary conditions(12)fη=1,θη=1,ϕη=1whenη=0fη⟶0,θη⟶0,ϕη⟶0whenη⟶∞,where Gr0=gΛ1Tw−T∞δ2/Uwν,β1=Λ2/Λ1Tw−T∞,Grm=gΛ3Cw−C∞δ2/Uwν,β2=Λ4/Λ3Cw−C∞,Pr=ρcpν/k0,Rd=4σ∗T∞3/3k0k∗,Sc=ν/D0,γ=k1δ2/ν are thermal Grashof number, nonlinear thermal convection variables, solutal Grashof number, nonlinear solutal convection variables, Prandtl number, radiation parameter, Schmidt number, and dimensionless reaction rate parameter.To make equations (9)–(11) dimensionless, the following assumption is made [48]:(13)A=δδ′νis a constant.Under assumption (13), equations (9)–(11) can be expressed as(14)−f′ηA+v0=f″+Gr0θ+β1θ2+Grmϕ+β2ϕ2,(15)−θ′ηA+v0=1Prε1θ′2+1+ε1θθ″+43RdPrθ″,(16)−ϕ′ηA+v0=1Scε2ϕ′2+1+ε2ϕϕ″−γϕ.The skin friction coefficient is defined as(17)Cf=τ1/2ρUw2,where τ=−μ∂u/∂yy=0=−μUw/δf′0Therefore,(18)12ReCf=−f′0,where Re=Uwδ/ν represents the Reynolds number.


## 3. Finite Element Method

The problem is more complex because the governing equations for the velocity field are nonlinear. Mathematical equations describe generalized diffusion thermoelasticity concerning changing mass diffusivity and thermal conductivity, including dynamic and nonlinear variables. To solve equations (14)–(16) using boundary conditions (12), a finite element method is employed. The entire domain is partitioned into a finite number of subdomains for this method’s implementation. The solution is found at each subdomain. Let N denote the number of subdomains. The first step of the method is to approximate the solution by interpolation polynomials. For the present study, a linear interpolation polynomial is employed. So, let the solution of each equation (14)–(16) be approximated as(19)f=f0+f1η,θ=θ0+θ1η,ϕ=ϕ0+ϕ1η,where f0,f1,θ0,θ1,ϕ0, and ϕ1 are unknown to be found. Let each subdomain be consisting of two nodes located at each end. Let the ith subdomain be denoted by ηi,ηi+1. When the three interpolated polynomials for each dependent variable are evaluated at ith subdomain, then the resulting solution evaluated at ith subdomain can be expressed as(20)f=ξ1ηfi+ξ2ηfi+1,(21)θ=ξ1ηθi+ξ2ηθi+1,(22)ϕ=ξ1ηϕi+ξ2ηϕi+1,where ξ1η=ηi+1−η/ηi+1−ηi,ξ2η=η−ηi/ηi+1−ηi for ηi≤η≤ηi+1.The defined functionsξ1η and ξ2η are called shape functions, while the dependent variables f,θ, and ϕ are called trial functions. By using the Galerkin approach, weighted residuals for equations (14) to (16) can be expressed as(23)∫0η∞ξd2fdη2+Aη+v0dfdη+Gr0θ+β1θ¯θ+Grmϕ+β2ϕ¯ϕdη=0,(24)∫0η∞ξ1Pr1+43Rdd2θdη2+1Prε1θ¯d2θdη2+1Prε1dθdη¯dθdη+ηA+v0dθdηdη=0,(25)∫0η∞ξ1Scd2ϕdη2+1Scε2ϕ¯d2ϕdη2+1Scε2dϕdη¯dϕdη+ηA+v0dϕdη−γϕdη=0,where 0,η∞ is the domain of equations (14)–(16). Equations (23)–(25) are strong formulations and weak formulations. The second-order derivative terms are in equations (23)–(25) will be integrated once. This leads to the following weak formulations:(26)∫0η∞−dξdηdfdη+ξAη+v∘dfdη+Gr0θ+β1θ¯θ+Grmϕ+β2ϕ¯ϕdη=−ξdfdη0η∞,(27)∫0η∞−1Pr1+43Rddξdηdθdη−∈1θ¯Prdξdηdθdη+ξ1Pr∈1dθdη¯dθdη+ηA+v0dθdη=−ξdθdη0η∞1Pr1+43Rd−∈1θ¯Prξdθdη0η∞,(28)∫0η∞−1Scdξdηdϕdη−1Sc∈2ϕ¯dξdηdϕdη+ξ1Sc∈2dϕdη¯dϕdη+ηA+v0dθdη−γϕ=−ξdϕdη0η∞1Sc−∈2ϕ¯Scξdϕdη0η∞.These weak formulations (26)–(28) are constructed over the whole domain 0,η∞, and since the finite element method finds the solution on each subdomain, therefore weak formulations on ith subdomain are constructed as(29)∫ηiηi+1−dξdηdfdη+ξAη+v∘dfdη+Gr0θ+β1θ¯θ+Grmϕ+β2ϕ¯ϕdη=−ξdfdηηiηi+1,(30)∫ηiηi+1−1Pr1+43Rddξdηdθdη−∈1θ¯Prdξdηdθdη+ξ1Pr∈1dθdη¯dθdη+ηA+v0dθdη=−ξdθdηηiηi+11Pr1+43Rd−∈1θ¯Prξdθdηηiηi+1,(31)∫ηiηi+1−1Scdξdηdϕdη−1Sc∈2ϕ¯dξdηdϕdη+ξ1Sc∈2dθdη¯dϕdη+ηA+v0dϕdη−γϕ=−ξdϕdηηiηi+11Sc−∈2ϕ¯Scξdϕdηηiηi+1.The stiffness matrix for this problem can be expressed as(32)Pi=P11P12P13P21P22P23P31P32P33,where(33)Pij11=∫ηiηi+1−dξidηdfjdη+ξiAη+v0dfjdηdη,Pij12=∫ηiηi+1ξiGr0θj+β1θ¯θjdη,Pij13=∫ηiηi+1ξiGrmϕj+β2ϕ¯ϕjdη,Pij22=∫ηiηi+1−1Pr1+43Rddξidηdθjdη−∈1θ¯Prdξidηdθjdη+ξi1Pr∈1dθdη¯dθjdη+ηA+v0dθjdηdη,Pij33=∫ηiηi+1−1Scdξidηdϕjdη−1Sc∈2ϕ¯dξidηdϕjdη+ξi1Sc∈2dθdη¯dϕjdη+ηA+v0dϕjdη−γϕjdη.and the remaining Pijmn,1≤m,n≤3 are zero. The terms in bar ″−″ notation are kept fixed to linearize the equations.


## 4. Validation

An iterative procedure is also considered for the convergence of the solution obtained by employing the finite element method. The iterative solver is employed due to the nonlinearity of the differential equations. This iterative technique begins, finds the solution using the finite element method, and stops if the specified stopping criteria are satisfied. The stopping criteria for the solver are(34)Fi,jc−Fi,jp<tol,where Fi,jc denotes the solution for one of three dependent variables f,θ,ϕ where j=1,2,3 and each grid point i. Suppose the solution obtained at current and previous iterations is not close enough. In that case, the iterative procedure will continue, and it will continue until the obtained values at two consecutive iterations and each grid point are small enough. So, if the iterative method stops, a converged solution may expect.Also, the results computed by the finite element method are compared with those obtained by Matlab solver bvp4c. This Matlab solver is used to find the solution of linear and nonlinear ordinary differential equations. Using a collocation formula and collocation polynomial, the solver generates a fourth-order accurate continuous solutionC1 using the three-stage Lobatto-IIIa formula. The results for −f′0 are given in Table 1 using different values of involved parameters. The comparison between results can be seen in Table 1. The numerical values for −f′0 are calculated by employing two different numerical methods. Also, the comparison of consumed time calculated by employing two different approaches is made in Table 1. The Matlab solver bvp4c consumed less time than the considered finite element method.Table 1
Comparison of the finite element method and Matlab solverbvp4c in finding numerical values of −f′0 using Grm=1,β1=0.1,A=1,v0=1,ε1=0.1,ε2=0.1,γ=0.1.
Gr0β2RdPrSc−f′0Time (s)F.E.Mbvp4cF.E.Mbvp4c10.50.10.70.70.18210.20760.4690.32410−5.4696−5.34560.4510.3330.15−0.7610−0.70580.4750.3430.150.82830.83960.4320.3150.140.92300.93450.4360.3290.141.21461.22000.4410.348


## 5. Results and Discussions

The finite element method is employed for solving equations (14)–(16) using boundary conditions (12). The Galerkin finite element method is applied in which weighted residuals of a specific type are constructed, and linear interpolation polynomials are used. Interpolated polynomials are evaluated at two nodes of the subdomain, and shape functions are found. The weak formulations have an advantage over strong formulations, for the approach uses linear interpolating polynomials. For the standard finite element method, the numerical derivatives of solutions are not high-order accurate when interpolation is performed using linear polynomials. However, in the modified approach proposed in [41], numerical derivatives are found using finite difference formulas. So, in this study, a standard approach of the finite element method is employed for finding the solutions of equations (14)–(16) using boundary conditions (12), and the numerical values of −f′0 are found using the modified approach.Figure2 shows the velocity profile by varying thermal Grashof number Gr0. The velocity profile escalates by enhancing the thermal Grashof number. This increase is the increment in the buoyancy force that augments the velocity profile. Figure 3 shows the impact of the solutal Grashof number on the velocity profile. The velocity profile increases by rising solutal (modified) Grashof number values. Mathematically, the rise in the solutal Grashof number increases the flow's acceleration, and consequently, the velocity profile escalates. Figure 4 shows the velocity profile by increasing the parameter ε1. The velocity profile escalates by the rising values of the parameter ε1. The increase in the parameter ε1 escalates the coefficient of dimensionless temperature and yields an increase in thermal conductivity. For mixed convective flows, the temperature is one of the applied forces to initiate or escalate the flow's velocity; therefore, velocity increases. Figure 5 deliberates the velocity profile with the variation of nonlinear thermal convection parameters. The velocity profile increases with the escalation of the nonlinear thermal convection parameter. This is due to an increase in the temperature of the flow by enhancing the thermal convection parameter that leads to augments in the velocity profile. The temperature profile with the variation of the radiation parameter is deliberated in Figure 6. The temperature profile grows with the rise in the radiation parameter. This is the consequence of incoming radiations, which escalates in surface heat flux, and therefore, the temperature of the flow rises. Figure 7 deliberates the temperature profile by the change in the Prandtl number. The temperature profile decays by raising the Prandtl number. The growth in the temperature profile is the consequence of thermal conductivity that enhances by the escalation in thermal diffusivity due to increasing values of the Prandtl number. Figure 8 illustrates the effect of the Schmidt number on the concentration distribution. By increasing the Schmidt number, the concentration profile decreases. This increase is the growth of mass diffusivity by escalating the Schmidt number, leading to deescalation in the concentration profile. Figure 9 shows the effect of a chemical reaction parameter on the concentration profile. The concentration profile deescalates by the growth of the reaction rate parameter. This decay in the concentration profile is the consequence of either increasing the level of impurity or escalation of creating substances in the flow.Figure 2
Impact of the thermal Grashof number on the velocity profile usingA=1,v0=0.1,Grm=5,ε1=0.1,β1=0.4,Gr0=5,Pr=4,Rd=0.1,Sc=1.5,γ1=0.4.Figure 3
Impact of the solutal Grashof number on the velocity profile usingA=1,v0=0.1,ε1=0.1,ε2=0.1,β1=0.4,β2=0.4,Gr0=5,Pr=4,Rd=0.1,Sc=1.5,γ1=0.4.Figure 4
Impact of the parameter in variable thermal conductivity on the velocity profile usingA=1,v0=0.1,Gr0=5,ε2=0.1,Grm=5,β1=0.4,β2=0.4,Pr=4Rd=0.1,Sc=1.5,γ1=0.4.Figure 5
Impact of the nonlinear solutal convection parameter on the velocity profile usingA=1,v0=0.1,Gr0=5,ε1=0.1,ε2=0.1,β1=0.4,β2=0.4,Grm=5,Pr=4,Rd=0.1,Sc=1.5,γ1=0.4.Figure 6
Impact of the radiation parameter on the temperature profile usingA=1,v0=0.1,Gr0=5,ε1=0.1,ε2=0.1,β1=0.4,β2=0.4,Grm=5,Pr=4,Rd=0.1,Sc=1.5,γ1=0.4.Figure 7
Impact of the Prandtl number on the temperature profile usingA=1,v0=0.1,Gr0=5,ε1=0.1,ε2=0.1,β1=0.4,β2=0.4,Grm=5,Rd=0.1,Sc=1.5,γ1=0.4.Figure 8
Impact of the Schmidt number on the concentration profile usingA=1,v0=0.1,Gr0=5,ε1=0.1,ε2=0.1,β1=0.4,β2=0.4,Grm=5,Rd=0.1,Pr=4,γ1=0.4.Figure 9
Impact of the reaction rate parameter on the concentration profile usingA=1,v0=0.1,Gr0=5,ε1=0.1,ε2=0.1,β1=0.4,β2=0.4,Grm=5,Rd=0.1,Sc=1.5,Pr=4.Figures10–12 are drawn using the partial differential equations as governing heat transfer equations in mixed convection flow under the effect of radiations. The software uses the specified geometry and finds the solution using the finite element method. The considered flow problem’s geometry is a rectangle with one inlet and one outlet. The plate is considered to be the lower boundary of the rectangular region. The bottom wall of the rectangular region has a fixed temperature. The water is used as the base fluid. Instead of using the fixed velocity of the sheet, different velocities of the sheet are chosen, and corresponding to each velocity of the sheet, one of the Figures 10–12 is obtained. The variation in temperature surface plots can be seen in these Figures 10–12 with growing values of the velocity of the wall.Figure 10
(a) Surface plot for velocity, (b) streamlines, (c) surface plot for temperature, and (d) isothermal contours usingGr0=100,β1=0.7,Gtm=0,Uw=0.001.
(a)(b)(c)(d)Figure 11
(a) Surface plot for velocity, (b) streamlines, (c) surface plot temperature, and (d) isothermal contours usingGr0=100,β1=0.7,Grm=0,Uw=0.005.
(a)(b)(c)(d)Figure 12
(a) Surface plot for velocity, (b) streamlines, (c) surface plot temperature, and (d) isothermal contours usingGr0=100,β1=0.7,Grm=0,Uw=0.01.
(a)(b)(c)(d)


## 6. Conclusion

Many experimental and theoretical studies show that thermal conductivity and mass diffusivity are not constants. Inspired by these findings, an existing mathematical model for heat and mass transfer flow has been modified. A numerical approach has been employed for solving ordinary differential equations. The nonlinear equations have been linearized, and an iterative procedure has been considered to handle nonlinearities in the equations. Numerical integration of three point formula has been carried out using the Legendre polynomial. The variations of thermal and solutal Grashof numbers, a nonlinear solutal convection parameter, a small parameter in variable thermal conductivity, the Prandtl number, a radiation parameter, the Schmidt number, and a reaction rate parameter on velocity, temperature, and concentration profiles have been presented in the form of graphs. The concluded points can be expressed as(i)
Velocity profile escalated by enhancing thermal and solutal Grashof numbers(ii)
Velocity profile has grown by rising values of nonlinear solutal convection parameters(iii)
Temperature profile escalated by growing values of radiation parameters(iv)
Concentration profile deescalated by augmenting the reaction rate parameterFurthermore, the modified finite element method considered in this work can be employed to solve nonlinear problems of a similar type that arise in computational fluid dynamics with some extra effects. Following the completion of this work, it will be possible to propose other applications for the currently employed methodology, if desired [52–56]. In addition, the developed method is easy to use and can solve a broader range of differential equations in both practice and theory.



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