# Effect of Joule Heating and Thermal Radiation of MHD Boundary Layer Oldroyd-B Nanofluid Flow with Heat Transfer over a Porous Stretching Sheet by Finite Element Method

Authors: Hakeem Ullah; Mehreen Fiza; Kamal Khan; Shamaila Batool; S. M. Ghufran; Seham M. Al-Mekhlafi
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

In this study, an incompressible two-dimensional Oldroyd-B nanofluid steady flow past a stretching sheet considering the outcomes of magneto-hydrodynamics (MHD) and porous medium with magnetic, electrical, and thermal radiation effects is investigated. Using a similarity transformation, the governing equations in the form of partial differential equations (PDEs) are converted into a nonlinear ordinary differential equations (ODEs) system. The acquired system is numerically solved by the finite element method (FEM). The effects of physical parameters like Deborah numbers “β1” and “β2”, Brownian motion “Nb”, thermophoresis parameter “Nt”, Prandtl parameter “Pr”, Lewis number “Le”, thermal conductivity “k”, dynamic viscosity “μ”, magnetic and electric effects as “M” and “E1”, and thermal radiation effect “Rd” on the flow are studied in detail. For higher Nb values, regional Nusselt numbers are increasing in magnitude. The local Sherwood number’s size rises for high Nb numbers.


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## Body

## 1. Introduction

The fluid which obeys the Newton’s law of viscosity is known as the Newtonian fluid, whereas the non-Newtonian fluid is recognized as to be the satisfaction of the Newton’s law of viscosity [1, 2]. The non-Newtonian fluid dragged the attraction of the researchers due to its significant applications in industrial and engineering such as mud drilling, plastic polymers, optical fiber, metal cooling and wire of polymer plates, damping agent in braking devices, and protective devices [3–5]. Sakiadis introduced the concept of flow due to a stretching sheet [6]. Scientists used this concept to develop the new results to various fluids [7–19]. Magyar and Keller extended this concept by using it as the exponential stretching sheet [20]. The boundary layer flow (BLF) of an incompressible fluid over a stretching sheet and the viscoelastic fluids is used commonly in engineering and industrial developments. The field has attracted researchers in the last few decades. In industries, BLF is used like wrapping thermal, cooling plates, condensation of thin film, fiber glass, heat exchangers, plastic processing, cosmetics, geology composites, paint flow, adhesives, tower generators, accelerators, electrostatic filters, and droplet filters [3, 4, 5]. By immersing them in quiescent liquids, many metal processes need to cool continuously such as fibers. The BLF is used by many scientists. The BLF based on the exponential stretching is studied by Bidin and Nazar who study the BLF due to exponential stretching sheet with thermal radiations [21]. This experiment is further extended with the partial slip effect by Mukhopadhyay and Gorla [22]. Singh and Agarwal [23] study the thermal radiation effect of the boundary layer flow with exponential stretching. The BLF and heat transfer HT over a moving surface are studied by Tsou et al. and Elbashbeshy [24, 25]. Choi proposed the concept of nanofluid in 1995 [26]. The nanofluid has many industrial and engineering applications like heat exchangers, engine radiators, and cooling processes [27, 28]. Khan et al. [29, 30] investigated the BLF of a nanofluid in a porous material. The thermal effect of the nanofluid flow of the boundary layer on the moving surface in various conditions was presented by Olanrewaju et al. [31], Crane [7], Koo and Kleinstreuer [32], and Khan et al. [29, 33]. The porous material containing the pores and the skeletal portion of the material is known as a matrix. The pores are filled by a fluid under consideration. Fiza et al. studied the nanofluid flows in a porous medium with viscoelastic properties [34]. The recent development in the study of nanofluid by considering various physical effects can be seen in [29, 33–59]. From the literature survey, it is clear that the effect of Joule heating and thermal radiation of MHD boundary layer Oldroyd-B nanofluid flow with heat transfer over a porous stretching sheet is not studied. This article is aimed at studying the effect of Joule heating and thermal radiation of MHD boundary layer Oldroyd-B nanofluid flow with heat transfer over a porous stretching sheet by a numerical computation method known as finite element method (FEM). The basic fundamental equations Navier-Stokes equations and continuity are used for the mathematical formulations. After using the similarity transformation, the formulation in the form of PDEs is converted into ODEs. The FEM is used for the solution of the modeled problem [60–63]. The physical paramaters are discussed with the help of graphs and tables. Organisation of the paper is as follows: Section 1 is dedicated to introduction, Section 2 to problem formulation, Section 3 to solution techniques, Section 4 to results and discussions, and Section 5 to conclusions.


## 2. Problem Formulation [57]

Consider the nanofluid motion of an incompressible two-dimensional Oldroyd-B across a stretching sheet. Nanoparticles are saturated when the sheet is stretched aty=0, and the flow is originated at y>0. The fluid is electrically conducted in the existence of magnetic field B⟶=0,B0,0 and electric field E⟶=0,0,−E0 which follow the Ohm’s law J⟶=σE⟶+V⟶×B⟶. The sheet is stretched linearly ux=ax, where “a>0” and the sheet is considered as porous. The sheet velocity is taken parallel to the flow. The inducted magnetic field and Hall current effects are disregarded due to the minute magnetic field. The governing equations are as follows:
(1)ux+vy=0,uux+vvy+A1u2uxx+v2uyy+2uvuxy=vuyy+A2uuxyy+vuyyy−uxuyy−uyvyy+σρE0B0−B02u−vku,uFx+vFy=αFxx+Fyy+τDBCxFx+CyFy+DtF∞Fx2+Fy2+σρuB0−E02−∂qr∂y,uCx+vCy=DBCxx+Cyy+DFF∞Fxx+Fyy⋅DBCxx+Cyy+DFF∞Fxx+Fyy.Hereu,v represent the velocity components, fluid density ρ, kinematic viscosity ν, electrical conductivity σ, A1/A2 the relaxation/retardation parameters, the thermal diffusivity α, the temperature T, the concentration C, DB the Brownian diffusions, DT the thermophoretic diffusion coefficient, and τ=ρcp/ρcf the nanoparticle to fluid heat capacity. ρp represents ithe idensity iof iparticle, J⟶ is the Joule current, σ is the electrical conductivity, and V⟶ is the velocity field of the flow. iIf Y⟶∞, then the values of F and C are, respectively, F∞ and i C∞ as shown in Figure 1.Figure 1
Schematic diagram of the flow.qr represents the radioactive heat fluctuation that is proposed by Rosseland approximation like as follows:
(2)qr=−16φ3Κ∂Τ4∂y,where K represents the mean absorption coefficient and φ denotes the Stefan Boltzmann constant. By Taylor series equation, we obtain the following:
(3)Τ4=Τ04+4Τ03Τ−Τ02+⋯On ignoring higher-order terms, we have the following:(4)Τ4=4ΤΤ03−3Τ04.Inserting Equation (10) in Equation (8), it reduced to the form of the following:
(5)∂qr∂y=−16TC3φ3Κ∂2T∂y2.The boundary conditions are as follows:(6)ux=ax,v=0,C=C0,F=F0,y⟶0,u∞=0,v∞=0,C=C∞,F=F∞,y⟶∞.Using the similarity transformation,(7)ψ=av1/2xFη,θη=F−F∞Fw−F∞,ϕη=C−C∞Cw−C∞,η=avy.Using the stream function asu=∂ψ/∂y and v=−∂ψ/∂x, we get the following:
(8)f′′′−f′2+ff′′+β1f2f′′′−2ff′f′′+β2ff′′′′−f″2+M2E1−f′−κf′=0,(9)1+45Rdθ′′+Prθθ′+Nbθ′ϕ′+Ntθ′2+M2Prf′−E12=0,(10)ϕ′′+LePrfϕ′+NtNbθ′′=0,with
(11)f0=0,f′0=1,f′∞=0,f′′∞=0,(12)θ0=1,θ∞=0,(13)ϕ0=1,ϕ∞=0,where β1=aA1,β2=aA2 are the Deborah numbers, Pr=ν/α, is the Prandtl number, Nb=ρcρDBCw−C∞/vρcρ and Nt=ρcρDTTw−T∞/vρcρ are the Brownian and thermophoresis parameters, respectively, and Le=α/DB is the Lewis number.Also,(14)Nux=xqwαTw−T∞,(15)Shx=xqmDBCw−C∞,where qw=−α∂T/∂yy=0,qm=−DB∂C/∂yy=0 are the heat and mass flux, respectively.The dimensionless form of Equation (14) is given as follows:
(16)Rex−1/2Nux=−θ′0,Rex−1/2Shx=−ϕ′0,where Rex=uwxx/viis the Reynolds local, κ=v/k is the porosity parameter, M2=σB02/ρa is the magnetic variable, and the electric parameter is E1=E0/B0ax.


## 3. Finite Element Method

The FEM is a powerful method to evaluate the nonlinear differential equations and can be used to engineering problems such as fluid mechanics, biomathematics, physics, and channel process [49–51].The important steps of FEM are as follows:(1)
Discretization into finite elements of the infinite domain(2)
Generation of component equations(3)
Gathering component equations(4)
Imposing boundary conditions(5)
Evaluation of gathered equationsIterative approach can be applied in the last step.A grid sophistication experiment is performed via dividing the domain in consecutive grids sized81×81,101×101 and 121×121 in z direction. Four functions are to be assessed at each node, and assembly of element equations, we get 404 nonlinear equations. An iterative scheme is acquired for solving the system introduced by BCs. If the relative difference among the sequential iteration is fewer than 10−6, so the solution is considered as convergent. The code is run for various grid sizes and observed that the solution is free of the grid. The effect of the step size for step h=0.01 is verified by achieving an excellent agreement for different profiles.


## 4. Results and Discussion

### 4.1. Figures Discussions

The nonlinear system of ODEs (Equations (8)–(13)) constrained by boundary conditions (Equations (11)–(13)) is assessed numerically by FEM. Figures 2–22 describe the actions of emerging parameters like Prandtl parameter Pr, relaxation time constant β1, retardation time constant β2, Brownian parameter Nb, Lewis number Le, thermophoresis parameter Nt, and porosity parameter κ on velocity profile f′η, mass fraction function ϕη, and temperature profile θη. Figures 2–4 represent the effect of β1 on f′η, θη, and ϕη. β1which is a function of relaxation timeA1assists the flow due to the viscoelastic characteristics of fluids. The BL thickness and the velocity profile f′η increased by increasing the values of β1. Also by the increase of β1, both θη and ϕη of the mass fraction increase. The effects of β2 on f′η, θη, and ϕη are given in Figures 5–7. Since the retardation time of the fluid increases the fluid movement, so as a result, f′η, θη, and ϕη increases by increasing β2. The effect of the porosity number for the velocity profile is given in Figure 8. It is noticed that by increasing the porosity parameter, the velocity profile decreases. Since in the porous medium, there exist the porous holes and these holes reduce the velocity profile. The Lewis numbers are plotted for the temperature and concentrations profiles in Figures 9 and 10. An increase in the Lewis value caused to increase θη and ϕη. The impacts on mass fraction function ϕη of Brownian motion and temperature profile θη and thermophoresis parameters are displayed in Figures 11, 12, 13, and 14, respectively. By increasing the values of Nb, the temperature profile increases while the concentration profile first decreases near the boundary and then increases as can be seen in Figures 11 and 12; for the effect of Nt, it is noticed that by uplifting Nt, the temperature profile increases while the concentration profiles decrease can be seen in Figures 13 and 14. The effects of the Prandtl number on the temperature and concentration profiles are presented in Figures 15 and 16. Higher values of Prandtl numbers caused to decrease the temperature and concentration profiles. It is due to the fact that the Pr has an inverse relation with α, increasing Pr is basically to reduce the value of α which turn to reduce the elastic collision of the nanoparticles caused to reduce the profiles of concentration and temperature. The effects of the electric field E1 on the velocity and temperature profiles are given in Figures 17 and 18, respectively. Increase in E1 causes to increase the profiles of temperature and velocity. Also, the effects of the magnetic field on velocity and temperature profiles are given in Figures 19 and 20. Increase in magnetic field causes to decrease the velocity profile, whereas by increasing the magnetic field decreases the temperature profile. The magnetic field is put in perpendicular to the flow and it resisted the flow; thus, the velocity profile reduced. The effect of the thermal radiations on the velocity and temperature field is given in Figures 21 and 22. Increase in thermal radiation caused to increase/decrease the velocity and temperature profiles. Results on the Sherwood number of the Brownian parameter, the Nusselt number, and thermophoresis parameter are given in Figures 23–26. It is noticed that the Nusselt number falls by rising Nt, whereas the Nusselt number rises by rising Nb as given in Figures 23 and 24. The effects of Nt and Nb on the Sherwood number are given in Figures 25 and 26.Figure 2
f′η versus β1.Figure 3
θη versus β1.Figure 4
ϕη versus β1.Figure 5
f′η versus β2.Figure 6
θη versus β2.Figure 7
ϕη versus β2.Figure 8
f′η versus k.Figure 9
θη versus Le.Figure 10
ϕη versus Le.Figure 11
f′η versus Nb.Figure 12
ϕη versus Nb.Figure 13
θη versus Nt.Figure 14
ϕη versusNt.Figure 15
θη versus Pr.Figure 16
ϕη versus Pr.Figure 17
f′η versus E1.Figure 18
θη versus E1.Figure 19
f′η versus M.Figure 20
θη versus M.Figure 21
f′η versus Rd.Figure 22
θη versus Rd.Figure 23
Nusselt number versusNt.Figure 24
Nusselt number versusNb.Figure 25
Sherwood number versusNt.Figure 26
Sherwood number versusNb.


### 4.2. Tables Discussions

In Tables1–4, an increase inNtcaused to reduce the Sherwood numbers and an increase in Nbcaused to rise the Sherwood number. The Nusselt number decreases whenever the Pr is less than Le and increases when the Pr is greater than Le for NbandNt. However, the Sherwood number decreases with the rise of NbandNt for both the cases when the Pr is greater or less than Le. Lastly, low thermal conductivity is caused by a high Prandtl fluid which decreases conductivity, resulting in an increase in sheet surface heat transfer. In the absence of non-Newtonian parameters β1 and β2, it is observed that the effect of both temperature and concentration profiles of nanoparticles brings the falling action. Consequently, as Pr grows, the thickness of the boundary layer shrinks. Table 5 shows a comparison of our results to the results presented by Jawad and Saeed [58]. In the absence of the nanoparticles and non-Newtonian paramaters, our results are identical to the Jawad and Saeed results. This comparison shows the accuracy and validity of our method.Table 1
Numerical values of the local Nusselt numberRe−1/2Nux.
Nb/Nt0.20.40.60.80.10.3453320.3370050.3289320.3211090.20.3405330.3324160.3245280.3168850.30.3359030.3279490.3202410.3127760.40.3313720.3236360.3160710.3087790.50.3269620.3193680.3120130.304892Table 2
Numerical values of the local Nusselt numberRe−1/2Nux.
Nt/Nb0.20.40.60.80.1-0.344720-0.335351-0.326466-0.3180490.2-0.340555-0.331372-0.322668-0.3144240.3-0.336454-0.327456-0.318931-0.3108570.4-0.332416-0.323620-0.315252-0.3073470.5-0.328441-0.319805-0.311629-0.303894Table 3
Numerical values of the Sherwood numberRe−1/2Shx.
Nb/Nt0.20.40.60.80.10.5551000.7595190.9700361.1861000.20.4578040.5602220.6657740.7739860.30.4252210.4937630.5643150.6365640.40.4089190.4605130.5135550.5678150.50.3991300.4405470.4830770.526535Table 4
Numerical values of the Sherwood numberRe−1/2Shx.
Nt/Nb0.20.40.60.80.10.4079240.3837410.3756660.3590170.20.4578040.4089190.3925970.3716180.30.50855910.4345200.4097890.3844160.40.5602220.4605130.4272230.4105390.50.6126370.4868680.4448790.423837Table 5
Comparison of local Nusselt results in the absence of non-Newtonian and nanoparticle paramaters.
ReWang [64]Present0.70.45390.453920.91140.911471.89541.8954203.35393.3539706.46226.4622


## 4.1. Figures Discussions

The nonlinear system of ODEs (Equations (8)–(13)) constrained by boundary conditions (Equations (11)–(13)) is assessed numerically by FEM. Figures 2–22 describe the actions of emerging parameters like Prandtl parameter Pr, relaxation time constant β1, retardation time constant β2, Brownian parameter Nb, Lewis number Le, thermophoresis parameter Nt, and porosity parameter κ on velocity profile f′η, mass fraction function ϕη, and temperature profile θη. Figures 2–4 represent the effect of β1 on f′η, θη, and ϕη. β1which is a function of relaxation timeA1assists the flow due to the viscoelastic characteristics of fluids. The BL thickness and the velocity profile f′η increased by increasing the values of β1. Also by the increase of β1, both θη and ϕη of the mass fraction increase. The effects of β2 on f′η, θη, and ϕη are given in Figures 5–7. Since the retardation time of the fluid increases the fluid movement, so as a result, f′η, θη, and ϕη increases by increasing β2. The effect of the porosity number for the velocity profile is given in Figure 8. It is noticed that by increasing the porosity parameter, the velocity profile decreases. Since in the porous medium, there exist the porous holes and these holes reduce the velocity profile. The Lewis numbers are plotted for the temperature and concentrations profiles in Figures 9 and 10. An increase in the Lewis value caused to increase θη and ϕη. The impacts on mass fraction function ϕη of Brownian motion and temperature profile θη and thermophoresis parameters are displayed in Figures 11, 12, 13, and 14, respectively. By increasing the values of Nb, the temperature profile increases while the concentration profile first decreases near the boundary and then increases as can be seen in Figures 11 and 12; for the effect of Nt, it is noticed that by uplifting Nt, the temperature profile increases while the concentration profiles decrease can be seen in Figures 13 and 14. The effects of the Prandtl number on the temperature and concentration profiles are presented in Figures 15 and 16. Higher values of Prandtl numbers caused to decrease the temperature and concentration profiles. It is due to the fact that the Pr has an inverse relation with α, increasing Pr is basically to reduce the value of α which turn to reduce the elastic collision of the nanoparticles caused to reduce the profiles of concentration and temperature. The effects of the electric field E1 on the velocity and temperature profiles are given in Figures 17 and 18, respectively. Increase in E1 causes to increase the profiles of temperature and velocity. Also, the effects of the magnetic field on velocity and temperature profiles are given in Figures 19 and 20. Increase in magnetic field causes to decrease the velocity profile, whereas by increasing the magnetic field decreases the temperature profile. The magnetic field is put in perpendicular to the flow and it resisted the flow; thus, the velocity profile reduced. The effect of the thermal radiations on the velocity and temperature field is given in Figures 21 and 22. Increase in thermal radiation caused to increase/decrease the velocity and temperature profiles. Results on the Sherwood number of the Brownian parameter, the Nusselt number, and thermophoresis parameter are given in Figures 23–26. It is noticed that the Nusselt number falls by rising Nt, whereas the Nusselt number rises by rising Nb as given in Figures 23 and 24. The effects of Nt and Nb on the Sherwood number are given in Figures 25 and 26.Figure 2
f′η versus β1.Figure 3
θη versus β1.Figure 4
ϕη versus β1.Figure 5
f′η versus β2.Figure 6
θη versus β2.Figure 7
ϕη versus β2.Figure 8
f′η versus k.Figure 9
θη versus Le.Figure 10
ϕη versus Le.Figure 11
f′η versus Nb.Figure 12
ϕη versus Nb.Figure 13
θη versus Nt.Figure 14
ϕη versusNt.Figure 15
θη versus Pr.Figure 16
ϕη versus Pr.Figure 17
f′η versus E1.Figure 18
θη versus E1.Figure 19
f′η versus M.Figure 20
θη versus M.Figure 21
f′η versus Rd.Figure 22
θη versus Rd.Figure 23
Nusselt number versusNt.Figure 24
Nusselt number versusNb.Figure 25
Sherwood number versusNt.Figure 26
Sherwood number versusNb.


## 4.2. Tables Discussions

In Tables1–4, an increase inNtcaused to reduce the Sherwood numbers and an increase in Nbcaused to rise the Sherwood number. The Nusselt number decreases whenever the Pr is less than Le and increases when the Pr is greater than Le for NbandNt. However, the Sherwood number decreases with the rise of NbandNt for both the cases when the Pr is greater or less than Le. Lastly, low thermal conductivity is caused by a high Prandtl fluid which decreases conductivity, resulting in an increase in sheet surface heat transfer. In the absence of non-Newtonian parameters β1 and β2, it is observed that the effect of both temperature and concentration profiles of nanoparticles brings the falling action. Consequently, as Pr grows, the thickness of the boundary layer shrinks. Table 5 shows a comparison of our results to the results presented by Jawad and Saeed [58]. In the absence of the nanoparticles and non-Newtonian paramaters, our results are identical to the Jawad and Saeed results. This comparison shows the accuracy and validity of our method.Table 1
Numerical values of the local Nusselt numberRe−1/2Nux.
Nb/Nt0.20.40.60.80.10.3453320.3370050.3289320.3211090.20.3405330.3324160.3245280.3168850.30.3359030.3279490.3202410.3127760.40.3313720.3236360.3160710.3087790.50.3269620.3193680.3120130.304892Table 2
Numerical values of the local Nusselt numberRe−1/2Nux.
Nt/Nb0.20.40.60.80.1-0.344720-0.335351-0.326466-0.3180490.2-0.340555-0.331372-0.322668-0.3144240.3-0.336454-0.327456-0.318931-0.3108570.4-0.332416-0.323620-0.315252-0.3073470.5-0.328441-0.319805-0.311629-0.303894Table 3
Numerical values of the Sherwood numberRe−1/2Shx.
Nb/Nt0.20.40.60.80.10.5551000.7595190.9700361.1861000.20.4578040.5602220.6657740.7739860.30.4252210.4937630.5643150.6365640.40.4089190.4605130.5135550.5678150.50.3991300.4405470.4830770.526535Table 4
Numerical values of the Sherwood numberRe−1/2Shx.
Nt/Nb0.20.40.60.80.10.4079240.3837410.3756660.3590170.20.4578040.4089190.3925970.3716180.30.50855910.4345200.4097890.3844160.40.5602220.4605130.4272230.4105390.50.6126370.4868680.4448790.423837Table 5
Comparison of local Nusselt results in the absence of non-Newtonian and nanoparticle paramaters.
ReWang [64]Present0.70.45390.453920.91140.911471.89541.8954203.35393.3539706.46226.4622


## 5. Conclusion

The Oldroyd-B nanofluid model was presented across a stretched sheet for this investigation. The quantitative analysis of the impacts of thermophoresis parameter, elastic parameter, and Brownian motion on heat and flow transfer is studied here. Below are the key features of the study.(1)
For the mass fraction, temperature, and speed functions,β1 and β2 impacts have the opposite behavior. Such anomalies arise only because of the influence of β1 and β2 viscoelastic parameters(2)
Prandtl’s actions are the same for both the temperature and mass fraction functions. As Pr is the link among visual and dynamic viscosity, at greater Pr levels, the temperature profile remains under control(3)
Similar effects ofNb and Nt are seen on the temperature profile as both Nb and Nt increase the temperature(4)
For higherNb values, regional Nusselt numbers are increasing in magnitude(5)
The local Sherwood number’s size rises for highNb numbers(6)
The electric field increasement increases the velocity and temperature profile, whereas the thermal radiation has reverse results(7)
The increases in magnetic field resist the flow, and so the velocity profile get decreases while it assists the temperature profile(8)
On increasingβ and porosity parameters, the velocity distribution decreases(9)
Increase in the Pr thermal boundary-layer thickness and contraction profiles is noticed to decrease(10)
Increase in the Lewis number causes to decrease the profiles of velocity, temperature, and concentration



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