# Numerical Investigation of Heat and Mass Transport in the Flow over a Magnetized Wedge by Incorporating the Effects of Cross-Diffusion Gradients: Applications in Multiple Engineering Systems

Authors: Aisha M. Alqahtani; Adnan; Umar Khan; Naveed Ahmed; Syed Tauseef Mohyud-Din; Ilyas Khan
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The flow over a wedge is significant and frequently occurs in civil engineering. It is significant to investigate the heat and mass transport characteristics in the wedge flow. Therefore, the analysis is presented to examine the effects of preeminent parameters by incorporating the cross-diffusion gradients in the energy and mass constitutive relations. From the analysis, it is perceived that the temperature drops against a higher Prandtl number. Due to concentration gradients in the energy equation, the temperature rises slowly. Moreover, it is examined that the mass transfer significantly reduces due to Schmidt effects and more mass transfer is pointed against the Soret number. The shear stresses increase due to stronger magnetic field effects. The local thermal performance of the fluid enhances against more dissipative fluid, and DuFour effects reduced it. Furthermore, the mass transport rate drops due to higher Soret effects and increases against multiple Schmidt number values.


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## Body

## 1. Introduction

Importance of boundary layer flow cannot be ignored because of its diverse class of applications in daily life and industries as well. These comprised in civil engineering, aerodynamics, and many more.The Newtonian flow over a stagnant wedge was firstly developed by Falkner and Skan [1]. They transformed a dimensional model to a third-order nonlinear self-similar differential equation by applying similarity variables. Afterwards, in 1937, Hartree [2] explored the boundary layer model approximately. Later, researchers turned toward the study of wedge flow under different flow conditions. In 1961, Koh et al. [3] explored the shear stresses and quantities of practical interest such as nusselt and Sherwood numbers over a wedge in the existence of a porosity parameter. They discussed the model by encountering the impacts of inconstant wall temperature and suction property. In 1987, Lin et al. [4] discussed the approximate solutions for wedge flow of any Prandtl number. They analyzed the temperature regimes in the existence of forced convection.Lately, Hussanan et al. [5] examined the boundary layer model in the existence of porous media by considering constant concentration gradients and resistive heating. Impacts of various physical quantities such as radiative heat flux and suction/injection on the chemically heating boundary layer model are reported in [6]. Chamber et al. [7] reported the boundary layer in the existence of a chemical reaction and diffusion gradients. The Falkner Skan flow model for a static or moving wedge saturated with nanofluid was discussed by Yacob et al. [8] in 2011. Flow in a wavering sheet was examined in [9]. They encountered the effects of natural convection and Ohmic heating. Su et al. [10] inspected the magnetohydrodynamic flow of Newtonian fluid composed by nanoparticles. Influences of applied Lorentz forces, slip flow condition, and ohmic heating were discussed in their study.In 2016, Khan et al. [11] investigated the bioconvection model in the existence of a porosity parameter over a wedge. They explored the influence of viscous dissipation, resistive heating, and imposed Lorentz forces on the flow of a gyrotactic microorganism. They discussed the impacts of different self-similar physical parameter on the momentum, thermal, density motile, and concentration profiles of the microorganism. Graphical analysis for shear stresses, local mass, and heat transfer comprised in their study. They treated a particular model numerically and, for the accuracy of the results, made comparison with the prevailing literature. They investigated that velocity field increases for stronger magnetic field. Furthermore, significant analysis regarding to the characteristics of the flow behavior in various geometries under multiple flow conditions is perceived in [12–22].In 2007, Ishak et al. [23] studied Falkner Skan flow through an accelerating wedge with the addition of suction/injection properties. Forced convection magnetohydrodynamic flow over a nonisothermal wedge by prevailing time-dependent viscosity was discussed by Pal et al. [24] in 2009. A magnetonanofluid model by considering the heat generation/absorption and the influence of a convective flow condition was inspected by Rahman et al. [25] in 2012.Lately, Ullah et al. [26] investigated the non-Newtonian model past a wedge in the existence of imposed Lorentz forces. Ullah et al. [27] inspected the non-Newtonian model past a nonlinearly stretchable sheet by encountering the influences of various physical parameters. Aman et al. [28] studied a nanofluid model composed of gold nanoparticles. They also explored the influences of radiative heat flux and crisscross diffusion on the flow characteristics. Analysis of a ferrofluid composed of cylindrical-shaped nanoparticles was conducted in [29]. Various flow models under certain boundary flow conditions in different channels are investigated in [30–33]. Brinkman sort of a nanofluid model was examined in [34]. A natural convection flow model stretchable sheet was studied by Ullah et al. [35]. A nanofluid model bounded by Riga plates and a second-grade flow model between an oblique channel were studied in [36, 37], respectively. Impacts of the effective Prandtl model and thermal radiation on the Newtonian model between a converging/diverging channel were explored in [38, 39], respectively. For further study regarding nanofluids and regular fluid from different aspects, we can analyze [40–46].From a careful science literature review, it is pointed that, to date, no one analyzed the energy and mass transportation in MHD wedge flow by prevailing crisscross diffusion gradients. Initially, formulation of the model is carried out, and then, mathematical analysis of the model is performed by means of the RK technique. Section3 is dedicated to highlight the impacts of varying a nondimensional parameter in the flow characteristics. A fruitful comparison has been made for reliability of the study. In the end, major results of the work are highlighted in the last section.


## 2. Self-Similar Analysis

Consider the time-independent Newtonian flow past a wedge positioned in the main stream. The fluid is electrically conducting, and the effects of cross diffusion are also under consideration. The main stream velocity of the fluid isU∗x. The flow is considered in the Cartesian coordinate system, and x and y axis are chosen in such a way that the surface of the wedge is along the x−axis, and the y−axis makes a right angle with the x−axis and perpendicular to the wedge surface. Furthermore, the wedge surface is kept at variable temperature Tw∗x and concentration Cw∗x. The ambient temperature and concentration of the fluid are denoted by T∗ and C∗, respectively. Inconstant magnetic field B∗x imposed perpendicularly to the wedge with the assumptions of a smaller magnetic Reynolds number and inconsequential induced magnetic field. The schematic theme of the MHD flow model is demonstrated in Figure 1:Figure 1
Physical configuration.In the light of the aforementioned restrictions, a particular flow model can be described in the existence of various physical quantities in the following manner [47]:(1)∂u∗∂x∗+∂v∗∂y∗=0,(2)u∗∂u∗∂x∗+v∗∂u∗∂y∗=U∗xdU∗xdx∗+μρ∂2u∗∂y∗2+σρB∗2U∗x−u∗,(3)u∗∂T∗∂x∗+v∗∂T∗∂y∗=kρcp∂2T∗∂y∗2+DKTρcpCs∂2C∗∂y∗2,(4)u∗∂C∗∂x∗+v∗∂C∗∂y∗=D∂2C∗∂y∗2+DKTTm∂2T∗∂y∗2.Equation (1) is the dimensional mass conservation law. Momentum, energy, and concentration equations by prevailing magnetic field and thermal and concentration gradients are embedded in equations (2)–(4), respectively. Furthermore, u∗,v∗ represent the velocities along xandy coordinates. Moreover, U∗x is the main stream velocity, dynamic μ, density ρ, imposed magnetic field B∗, temperature T∗, concentration C∗, heat capacity cp, mass diffusivity D, mean temperature of the fluid Tm, thermal diffusion KT, and concentration susceptibility Cs∗.The feasible set of auxiliary conditions for the current wedge flow is as follows [47]:(5)u∗↑y=0=0,v∗↑y=0=0,T∗↑y=0=Tw∗x,C∗↑y=0=Cw∗,(6)u∗↑y⟶∞⟶U∗x=u0xm,T∗↑y⟶∞⟶T∞∗,C∗↑y⟶∞⟶C∞∗.The governing equations (1)–(4) can be reduced into the dimensionless form by utilizing the following defined nondimensional transformations and stream functions [47]:(7)u=∂ψ∂y,v=−∂ψ∂x,ψ=νu0xm+1Fη,η=u0xm+1νyx−1,βη=T∗−T∞∗Tw∗−T∞∗,ϕη=C∗−C∞∗Cw∗−C∞∗.Here,B∗ shows the applied magnetic field which is a function of x and is in the form B=B0xm−1/2, where B0denotes the uniform magnetic field (for precedence, we can see [48, 49]), U∗x denotes the free stream velocity and described in function of x, and is U∗x=u0/x−m, in which u0 is an invariable quantity and m (0≤m≤1) describes the Falkner Skan power law parameter. The expression β=β2−β−1, in which β=Ω/π describes the Hartree pressure gradient. Furthermore, β=0 and β=1 describe the cases along the x and y coordinates, respectively.The following self-similar model is attained from the dimensional system after incorporating the feasible similarity variables which comprised the influences of cross diffusion and magnetic:(8)F‴−mF'2−m+12FF'′−m+M21−F′=0,(9)β″+Prm+12Fβ′+PrDfϕ″=0,(10)ϕ″+Scm+12Fϕ′+SrScβ″=0.The corresponding feasible boundary conditions are in the following manner:(11)Fη↑η=0=0,F′η↑η=0=0,βη↑η=0=1,ϕη↑η=0=1,(12)F′η↑η⟶∞⟶1,βη↑η⟶∞⟶0,ϕη↑η⟶∞⟶0.In equations (8)–(10), with nonhomogeneous auxiliary conditions described in equations (11) and (12), self-similar quantities are the Hartmann number M=σB02/ρu0, Prandtl number (Pr=νρCp/k), Dufour number (Df=DkTCw∗−C∞∗/CsCpνTw∗−T∞∗), Soret number (Sr=DmkTTw∗−T∞∗/νTmCw∗−C∞∗), and Schmidt number (Sc=ν/D).The following are the formulas to estimate the shear stresses, local Nusselt (Nux), and Sherwood numbers Shx[47]:(13)CF=μρU2x∂u∗∂y∗↑y=0,(14)Nux=kxkTw−T∞∂T∗∂y∗↑y=0,(15)Shx=xCw−C∞∂C∗∂y∗↑y=0.Hence, the nondimensional form for shear stress, local heat, and mass transfer is defined as follows:(16)CFRex=F'′0,NuxRex−1/2=−β′0,ShRex−1/2=−ϕ′0,where Rex=xU∗x/ν describes the local Reynolds number.


## 3. Mathematical Treatment

Usually, closed form solutions are very rare or not even exist for those models which are coupled and attain high nonlinearity. It is better to tackle such sort of models either numerically of asymptotically. A particular model is coupled and nonlinear in nature. Therefore, we adopted a numerical method called the Runge–Kutta method [50]. To initiate the technique, feasible substitutions are as follows:(17)y1=F,y2=F′,y3=F′',y4=β,y5=β′,y6=ϕ,y7=ϕ′.Then, the system of equations (8)–(10) is transformed into the following pattern:(18)F‴=mF'2−m+12FF″−m+M21−F′,(19)β'′=−Prm+12Fβ′−PrDfϕ'′,(20)ϕ'′=−Scm+12Fϕ′−SrScβ'′.By entreating the substitutions in equations (18)–(20), the following system is obtained:(21)y1′y2′y3′y4′y5′y6′y7'=y2y3my22−m+12y1y3−m+M21−y2y5−Prm+12y1y5−PrDfy7′y7−Scm+12y1y7−SrScy5′.Consequently, supporting initial conditions are(22)y1y2y3y4y5y6y7=00n11n21n3.After this, we performed the numerical calculation and obtained the tabulated results for the model. Table1 presents the solutions for velocity, temperature, and concentration fields over the domain of interest.Table 1
Solutions for the velocity, temperature, and concentration.
ηF′ηβηϕη0.08.13152×10−201.0000001.0000000.50.4145450.8551800.8551801.00.6808840.7119650.7119651.50.8404080.5737530.5737532.00.9276900.4448140.4448142.50.9706310.3291710.3291713.00.9894060.2297470.2297473.50.9966410.1479180.1479184.00.9990850.0834860.0834864.50.9998090.0349580.0349585.01.0000002.306680×10−92.306680×10−9


## 4. Graphical Results and Discussion

The impacts of ingrained self-similar quantities on the momentum, temperature, and mass of the fluid are incorporated in this section. These physical quantities areM, Pr, Df, Sr, and Sc. The impacts of the magnetic field and thermal and concentration gradients are also under consideration. It is very important to mention that, in the current nonlinear flow model, if m=1/2, then it represents the wedge flow. If m=0 and m=1, it represents the flow of the horizontal plate and stagnation point flow, respectively. For an example, we can refer [47].The flow behavior against an imposed magnetic field is given in Figure2. It is explored that, against a stronger magnetic field, fluid motion increases. For horizontal plate flow, the velocity F′η rises promptly comparative to the stagnation and wedge flow case. Physically, over a horizontal surface, the fluid particles move freely due to which the velocity upturns. Near the surface, these alterations are almost inconsequential. The physical reason behind this behavior is the force of friction between the surface and fluid layer adjacent to the surface. Due to the force of friction, the motion of fluid particles near the surface declines and rest of the fluid layer flow abruptly. Furthermore, for wedge and stagnation cases, increment in the velocity is observed quite slowly. Figure 2(b) presents a 3D image of the velocity against multiple values of M.Figure 2
Influence of the magnetic numberM on the velocity distribution F′η. (a) Two-dimensional; (b) 3D view.
(a)(b)The behavior of thermal performanceβη against Prandtl and Dufour parameters are given in Figures 3(a) and 3(b), respectively. The declines in βη are noticed against stringer Prandtl values. For the stagnation point, these alterations are prompt, and maximum declines are perceived in the region 2≤η≤6. The temperature βη is against the horizontal plate flow case. Physically, the fluid flows abruptly over the horizontal surface due to which the collision between the particles increases; consequently, the temperature drops slowly. The significant increasing variations in βη are pointed against Dufour effects. Due to the Dufour number, the temperature βη upturns. However, maximum increment in the temperature is pointed against horizontal plate flow. Figures 4(a) and 4(b) show the 3D image of βη against Prandtl and Dufour parameters, respectively.Figure 3
Stimulus of the (a) Prandtl numberPr and (b) Dufour number Df on βη.
(a)(b)Figure 4
3D Plot for temperature distributionβη against the (a) Prandtl number Pr and (b) Dufour number Df.
(a)(b)The mass transfer for multiple values of Schmidt and Soret parameters is given in Figures5(a) and 5(b), respectively. It is perceived that the mass transfer ϕη is in inverse proportion to the Schmidt number. Due to higher Schmidt effects, less mass transfer at the surface is noted. However, maximum decrement is pointed against the stagnation point case. Near the surface, an almost inconsequential behavior of ϕη is noted for wedge, stagnation, and horizontal flat plate cases. The Soret number which appears due to cross-diffusion gradients favor the mass transfer ϕη. The mass transfer profile rises against stronger Soret effects. For a horizontal plate, maximum alterations in the mass transfer trends are detected comparative to wedge and stagnation cases. The 3D behavior of the fluid concentration is shown against Sc and Sr in Figures 6(a) and 6(b), respectively.Figure 5
Stimulus of the (a) Schmidt numberSc and (b) Soret number Sr on βη.
(a)(b)Figure 6
3D Plot for mass distributionϕη against the (a) Schmidt number Sc and (b) Soret number Sr.
(a)(b)The behavior of shear stress against the magnetic field andm is shown in Figures 7(a) and 7(b), respectively. It is pointed out that the shear stresses upturn against both the parameters. Due to stronger Hartmann effects, the fluid motion reduces; consequently, the shear stresses increase.Figure 7
Stimulus of (a)m and (b) the Hartmann number M on F″0.
(a)(b)The local thermal performance and mass transfer rate against multiple parameters are shown in Figures8 and 9, respectively. From inspection of the plotted results, it is noticed that the local thermal performance declines against the Dufour number. For the stagnation case, a drop in the local heat transfer rate is slow comparative to wedge and horizontal plate cases. On the other hand, viscous dissipative effects favor the local thermal performance rate. It is pointed out that, against more dissipative fluid, the local heat transfer rate increases. The mass transfer rate against Soret and Schmidt parameters is accessed in Figures 9(a) and 9(b), respectively. It is examined that the mass transfer rate enhances due to a stronger Schmidt number, and inverse variations are perceived against the Soret number.Figure 8
Stimulus of the (a) Dufour numberDf and (b) Prandtl number Pr on −β′0.
(a)(b)Figure 9
Stimulus of the (a) Soret numberSr and (b) Schmidt number Sc on ϕ′η.
(a)(b)Table2 shows the comparative study against restricted flow parameters. The value of m is fixed at one and the computation is carried out. From the inspection of Table 2, it is perceived that the presented results and adopted technique are reliable and acceptable.Table 2
Comparison table forF′'0.
MPresent[51][47]01.23281567651277561.2325881.23259651960.21.2487828222228801.248572—0.41.29553569769488731.295368—0.61.37000076394500161.369884—0.81.46804821270334431.467974—1.01.58537103193050121.5853311.5852800424


## 5. Conclusions

The presented work encountered MHD flow over a wedge. The stimuli of cross diffusion are taken under consideration. The model is effectively treated by using a numeric technique called the RK technique. The impacts of ingrained quantities on the flow behavior are presented, and lastly, the following key findings are noted:(i)
The fluid velocityF′η is accelerating for increasing M, and in the case of stagnation point flow, these variations are rapid(ii)
The Dufour parameter favors the temperatureβη, whereas the Prandtl number is against the fluid temperature(iii)
increasing Schmidt number leads to a decrease in the concentration of the fluid, and for a varying Soret number, these variations are reverse(iv)
Less heat and mass transfer are investigated for Dufour and Soret parameters(v)
More heat and mass transfer are noted for Prandtl and Schmidt parameters in the wedge, stagnation point, and horizontal plate flows



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