# Heat Transfer in MHD Flow due to a Linearly Stretching Sheet with Induced Magnetic Field

Authors: Tarek M. A. El-Mistikawy
Journal: Advances in Mathematical Physics
 (2018)

---

## Abstract

The traditionally ignored physical processes of viscous dissipation, Joule heating, streamwise heat diffusion, and work shear are assessed and their importance is established. The study is performed for the MHD flow due to a linearly stretching sheet with induced magnetic field. Cases of prescribed surface temperature, heat flux, surface feed (injection or suction), velocity slip, and thermal slip are considered. Sample numerical solutions are obtained for the chosen combinations of the flow parameters.


---

## Body

## 1. Introduction

The problem of the two-dimensional flow due to a linearly stretching sheet, first formulated by Crane [1], has a simple exact similarity solution. This invited several researchers to add to it new features allowing for self-similarity. As a boundary-layer problem, Pavlov [2] added uniform transverse magnetic field. P. S. Gupta and A. S. Gupta [3] added surface feed (suction or injection). These problems were recognized as being exact solutions of the corresponding Navier-Stokes problems by Crane [1], Andersson [4], and Wang [5], respectively. To the Navier-Stokes problem, Andersson [6] added velocity slip. Fang et al. [7] combined the effects of transverse magnetic field, surface feed, and velocity slip.Heat transfer was treated in several publications, mostly neglecting viscous dissipation and Joule heating (in MHD problems). This allowed self-similar formulation in cases of the surface having constant temperature [3, 8] or temperature or heat flux proportional to a power of the stretch-wise coordinate x [9–11]. Prasad and Vajravelu [12] treated the boundary-layer flow of a power law fluid retaining viscous dissipation and Joule heating, in case of the surface temperature being proportional to x2.The abovementioned MHD problems adopted the small magnetic Reynolds number assumption, thus neglecting the induced magnetic field. In [13], it was shown that the full MHD problem, that is, Navier-Stokes and Maxwell’s equations with adherence conditions and appropriate magnetic conditions, allowed for self-similarity.In this article, the work of [13] is extended to the heat transfer problem including viscous dissipation and Joule heating, in cases of prescribed surface temperature or heat flux. Surface feed, velocity and thermal slip, and shear work are also included.The problem is of both theoretical and practical value. Theoretically, it indicates the importance of the traditionally ignored physical processes of induced magnetic field, viscous dissipation, Joule heating, and shear work. Practically, the problem is encountered in several situations. For example, extrusion processes in polymer and glass industries involve stretching sheets extruded in an otherwise quiescent fluid. The quality of the product depends on the controlled heat transfer between the sheet and the fluid. Four control agents are in mind, in this study, the MHD effect of a magnetic field permeating a conducting fluid, surface feed (fluid injection or suction), fluid additives (possibly, nanoparticles) associated with velocity and thermal slip [14], and convective heating or cooling [15] which has the same effect as thermal slip.


## 2. Mathematical Model

An electrically conducting, incompressible, and Newtonian fluid is driven by a nonconducting porous sheet, which is stretching linearly in thex-direction. At the surface, we consider cases of prescribed temperature or heat flux and allow for velocity and thermal slip. In the far field, the fluid is essentially quiescent under pressure p∞ and temperature T∞ and is permeated by a stationary magnetic field of uniform strength B in the transverse y-direction.The equations governing this steady MHD incompressible flow are the continuity and Navier-Stokes equations with Lorentz force [16](1)∇·V=0(2)ρV·∇V+∇p=ρν∇2V+J×B,the energy equation with viscous dissipation and Joule heating(3)ρcV·∇T=k∇2T+Φ+σ-1J2,and Maxwell’s equations and Ohm’s law, in the absence of surplus charge and electric field,(4)∇×B=μJ(5)∇·B=0(6)J=σV×B.V is the velocity vector, p is the pressure, T is the temperature, J is the current density, B is the magnetic field, and Φ is the dissipation function. Constants are the fluid density ρ, kinematic viscosity ν, specific heat c, thermal conductivity k, the electric conductivity σ, and magnetic permeability μ.Use of Ohm’s law (6) to eliminate J from (2) to (4) and casting in Cartesian components lead to the following equations for two-dimensional flow.(7)ux+vy=0,ρuux+vuy+px=ρνuxx+uyy+σB+srv-B+s2u,ρuvx+vvy+py=ρνvxx+vyy+σB+sru-r2v,sx-ry=σμB+su-rv,rx+sy=0,ρcuTx+vTy=kTxx+Tyy+ρν2ux2+vy2+uy+vx2+σB+su-rv2.They are complemented with the surface conditions(8)y=0:u=ωx+λwuy,v=vw,T=Tw+γwTy,orqw=-kTy-ρνλwuy2,and the far field conditions (9)y~∞:u~0,p~p∞,r~0,s~0,T~T∞.(u,v) are the velocity components in the (x,y) directions, respectively, and (r,s) are the corresponding induced magnetic field components. The stretching rate ω and the velocity and thermal slip coefficients λw and γw are assumed constant. In the condition for qw, the last term represents the shear work [17]. In the far field, the condition for r translates the physical requirement of the absence of any current density, while that on s indicates that B stands for the far field total magnetic field imposed and induced [13].The problem admits the similarity transformations(10)y=vωη,v=-νωfη,u=ωxf′,s=Bσμνgη,r=-Bσμνωxg′,p=p∞-ρωνf′+12f2-12f2∞-12B2σ2μνωx2g′2,T=T∞+νωcθ0η+ωνxθ1η+ωνx2θ2η,where primes denote differentiation with respect to η. The fact that the temperature is quadratic in x allows its constituents θ0, θ1, and θ2 to be dependent on η only.The problem becomes(11)f′′′+ff′′-f′2-βf′=-Pmβg′2+1+Pmgfg′-2+Pmgf′g(12)g′′=f′+Pmgf′-fg′(13)Pr-1θ2′′+fθ2′-2f′θ2=-βf′+Pmgf′-g′f2-f′′2(14)Pr-1θ1′′+fθ1′-f′θ1=0(15)Pr-1θ0′′+fθ0′=-Pr-12θ2-4f′2,where Pm=σμν is the magnetic Prandtl number, β=σB2/ρω is the magnetic interaction number, and Pr=ρνc/k is the Prandtl number.Consistent with the similarity transformations, we take the surface values to be(16)vw=-νωfw,Tw=T∞+νωcΘ0+ωνΘ1x+ωνΘ2x2,qw=kωνωcQ0+ωνQ1x+ωνQ2x2,where fw, Θ0, Θ1, Θ2, Q0, Q1, and Q2 are prescribed values.Withλ=λwω/ν and γ=γwω/ν, we get the following conditions on the flow variables:(17)f0=fw,f′0=1+λf′′0,f′∞=0(18)g∞=0,g′∞=0(19)θ20=Θ2+γθ2′0orθ2′0=-Q2-Pr⁡λf′′02,θ2∞=0(20)θ10=Θ1+γθ1′0orθ1′0=-Q1,θ1∞=0(21)θ00=Θ0+γθ0′0orθ0′0=-Q0,θ0∞=0.


## 3. Numerical Method

We start by solving forf(η) and g(η), since their nonlinear problem is uncoupled from the problems for θ2(η), θ1(η), and θ0(η). A closed form solution is not possible, so we seek an iterative numerical solution. In the nth iteration, we solve, for fn(η), (11) with its right hand side evaluated using the previous iteration solutions fn-1(η) and gn-1(η), together with conditions (17). Then we solve, for gn(η), (12) with the known fn(η), together with conditions (18). The iterations continue until the maximum error in f(η∞), f′′(0), g(0), and g′(0) becomes less than 10−10. For the first iteration, we zero the right hand side of (11) which corresponds to g0(η)=0.The numerical solution of the problem forfn(η) and gn(η) utilizes Keller’s two-point, second-order accurate, finite-difference scheme [18]. A uniform step size Δη=0.01 is used on a finite domain 0≤η≤η∞. The value of η∞=60 is chosen sufficiently large in order to insure the asymptotic satisfaction of the far field conditions. The nonlinear terms in the problem for fn(η) are quasi-linearized, and an iterative procedure is implemented, terminating when the maximum error in fn(η∞) and fn′′(0) becomes less than 10−10.Having determinedf(η) and g(η), we solve the linear problems: (13) with conditions (19) for θ2(η), (14) with conditions (20) for θ1(η), and then (15) with conditions (21) for θ0(η), using Keller’s scheme on the same grid.


## 4. Sample Results and Discussion

The problem forf(η) and g(η) involves four parameters: Pm, β, λ, and fw. For Pm=0.1, β=1, Figure 1 depicts f′(η) at different values of λ, when fw=0, and at different values of fw, when λ=0. The corresponding results for g(η) together with g′(η) are depicted in Figures 2 and 3, respectively. The induced magnetic field is primarily affected by the streamwise velocity component represented by f′(η). As f′(η) decreases due to higher surface slip or suction rate, both g(η) and -g′(η) decrease.Figure 1
Streamwise velocity profile.Figure 2
Profiles of induced magnetic field components;fw=0.Figure 3
Profiles of induced magnetic field components;λ=0.Tables1 and 2 give values of the surface shear and the entrainment rate represented, respectively, by f′′(0) and f(∞), as well as the induced magnetic field components at the surface represented, respectively, by g′(0) and g(0). Refer to [13] for values of f′′(0), f(∞), g′(0), and g(0) at different values of Pm and β, when λ=fw=0.Table 1
Variation off′′(0), f(η∞), g(0), and g′(0) with λ; fw=0.
λ f ′ ′ ( 0 ) f ( η ∞ ) g ( 0 ) g ′ ( 0 ) 0 −1.43222 0.69822 0.51249 −0.73400 1 −0.54904 0.37039 0.31377 −0.38201 2 −0.34863 0.26290 0.23363 −0.26904 3 −0.25674 0.20564 0.18749 −0.20949 4 −0.20357 0.16944 0.15701 −0.17210 5 −0.16879 0.14431 0.13524 −0.14626Table 2
Variation off′′(0), f(η∞), g(0), and g′(0) with fw; λ=0.
f w f ′ ′ ( 0 ) f ( η ∞ ) g ( 0 ) g ′ ( 0 ) −2 −0.70972 −0.59099 1.83271 −1.30070 −1 −1.00000 0.00000 1.00000 −1.00000 0 −1.43222 0.69822 0.51249 −0.73400 1 −2.02629 1.49351 0.26293 −0.53278 2 −2.75888 2.36247 0.14369 −0.39641Presented in Figures4–9 are the results for the temperature constituents θ0(η), θ1(η), and θ2(η) obtained, in case of prescribed surface temperature Tw∝xa, and in case of prescribed surface heat flux qw∝xa; a=0, 1 or 2. Table 3 summarizes the given surface values and the constituents involved in Figures 4–9, noting that (13)–(15) indicate that θ1 and θ2 are independent of the other constituents, while θ0 depends on θ2.Table 3
Data for Figures4–9.
Case Prescribed surface temperature Prescribed surface heat flux Given Θ 0 withΘ1=Θ2=0 Θ 1 withΘ0=Θ2=0 Θ 2 withΘ0=Θ1=0 Q 0 withQ1=Q2=0 Q 1 withQ0=Q2=0 Q 2 withQ0=Q1=0  Figure Figure4for θ0 Figure6for θ1 Figure8(a) for θ2 Figure 8(b) for θ0 Figure5for Q0 Figure7for Q1 Figure9(a) for Q2 Figure 9(b) for Q0  Fixed parameters:Pr=0.72, Pm=0.1, β=1, and λ=fw=γ=0Figure 4
Temperature constituentθ0(η) at different values of Θ0.Figure 5
Temperature constituentθ0(η) at different values of Q0.Figure 6
Temperature constituentθ1(η) at different values of Θ1.Figure 7
Temperature constituentθ1(η) at different values of Q1.Figure 8 (a)
Temperature constituentθ2(η) at different values of Θ2 (b)
Temperature constituentθ0(η) at different values of Θ2Figure 9 (a)
Temperature constituentθ2(η) at different values of Q2 (b)
Temperature constituentθ0(η) at different values of Q2The following is noticed.(i)
Constant surface temperature: at Θ0=Θ0c≈4.04660, θ0′(0)=0. For Θ0>Θ0c, θ0′(0)<0 and θ0 decreases monotonically with η, while for Θ0<Θ0c, θ0′(0)>0 and θ0 has a peak that gets farther from the surface as Θ0 decreases.(ii)
Constant heat flux: when Q0=0, θ0(0)≈8.39634. As more heat is added to the fluid, that is, for increasing Q0>0, θ0(0) rises and θ0 decreases monotonically with η. Removing more heat from the fluid, that is, for decreasing Q0<0, θ0(0) decreases and θ0 has a peak that gets farther from the surface.(iii)
Linear surface temperature and heat flux: noting that θ1(-Θ1)=-θ1(Θ1) and θ1(-Q1)=-θ1(Q1), the presented results for nonnegative Θ1 and Q1 indicate that θ1 decreases monotonically with η. Higher Θ1 results in smaller θ1′(0)<0, while higher Q1 results in higher θ1(0).(iv)
Surface temperature∝x2: at Θ2=Θ2c1≈0.65967, θ2′(0)=0. For Θ2>Θ2c1, θ2′(0)<0 and θ2 decreases monotonically with η. For Θ2c1>Θ2>Θ2c2≈-0.33596, θ2′(0)>0 and θ2 has a peak that gets farther from the surface as Θ2 decreases. For Θ2<Θ2c2, θ2′(0)>0 and θ2 rises monotonically with η. At Θ2=Θ2c3≈-0.61370, θ0′(0)=0 and θ0 drops from its zero surface value to a local minimum and then rises to its zero far field value. Similar behavior of θ0 is observed for decreasing Θ2<Θ2c3, but with decreasing minimum and θ0′(0)<0. For increasing Θ2≥Θ2c4≈-0.33807, θ0′(0)>0 and θ0 rises to a higher peak. For Θ2c3<Θ2<Θ2c4, θ0′(0)>0 and θ0 rises to a peak, falls to zero and then to a bottom, and rises again to zero.(v)
Heat flux∝x2: when Q2=0, θ2(0)≈0.65967. For increasing Q2>0, θ2(0) rises and θ2 decreases monotonically with η. For decreasing 0>Q2>Q2c1≈-0.96228, θ2(0) decreases from positive to negative values with θ2 having a positive peak. For Q2≤Q2c1, θ2 rises monotonically from a negative surface value to zero. For Q2≥Q2c2≈-0.98114, θ0 is monotonically decreasing, while for Q2≤Q2c3≈-1.96520, θ0 is monotonically increasing from its surface value to zero. For Q2c3<Q2<Q2c2, θ0 decreases to a negative local minimum and then rises to zero.To complement Figures4–9, we give in Table 4 the numerical values of θ0′(0) at different values of Θ0, θ1′(0) at different values of Θ1, and θ2′(0) and θ0′(0) at different values of Θ2, and in Table 5 the numerical values of θ0(0) at different values of Q0, θ1(0) at different values of Q1, and θ2(0) and θ0(0) at different values of Q2.Table 4
Dependence of temperature gradients at surface on surface temperatures;γ=0.
Θ 0 θ 0 ′(0) Θ 1 θ 1 ′(0) Θ 2 θ 2 ′(0) θ 0 ′(0) −2 2.33284 −2 1.39994 −2 2.57058 −3.52671 −1 1.94703 −1 0.69997 −1 1.60408 −0.98274 0 1.56122 0 0.00000 0 0.63757 1.56122 1 1.17541 1 −0.69997 1 −0.32893 4.10518 2 0.78960 2 −1.39994 2 −1.29543 6.64915 3 0.40379 3 −2.09991 3 −2.26194 9.19311 4 0.017978 4 −2.79988 4 −3.22844 11.73707 5 −0.36783 5 −3.49985 5 −4.19495 14.28104Table 5
Dependence of temperatures at surface on surface heat fluxes;λ=0.
Q 0 θ 0(0) Q 1 θ 1(0) Q 2 θ 2(0) θ 0(0) 2 13.58023 2 2.85727 2 2.72898 22.04100 1 10.98828 1 1.42863 1 1.69433 15.21867 0 8.39634 0 0.00000 0 0.65967 8.39634 −1 5.80439 −1 −1.42863 −1 −0.37499 1.57401 −2 3.21245 −2 −2.85727 −2 −1.40964 −5.24832 −3 0.62051 −3 −4.28590 −3 −2.44430 −12.07065 −4 −1.97144 −4 −5.71454 −4 −3.47896 −18.89298 −5 −4.56338 −5 −7.14317 −5 −4.51361 −25.71531Table6 shows the effect of the thermal slip coefficient γ. As the first legs of conditions (19)–(21) indicate, the sign of the surface derivative of the temperature constituent determines whether the surface value of the constituent increases or decreases with γ. Thus, for example, θ2(0) increases when Θ2=0, for which θ2′(0)>0, and decreases when Θ2=1, for which θ2′(0)<0.Table 6
Effect of thermal slip coefficientγ; λ=0.
γ θ 0(0) θ 2(0) θ 0(0) θ 1(0) θ 2(0) 0 0.00000 0.00000 1.00000 1.00000 1.00000 1 1.72175 0.32422 2.44335 0.58825 0.83273 2 3.01106 0.43476 3.57551 0.41668 0.77570 3 3.90609 0.49050 4.36960 0.32259 0.74695 4 4.55249 0.52410 4.94569 0.26317 0.72961 5 5.03861 0.54657 5.38001 0.22223 0.71802  Case Θ 0 = Θ 1 = Θ 2 =0 Θ 0 = 1 Θ 1 = 1 Θ 2 = 1 Θ 1 = Θ 2 = 0 Θ 0 = Θ 2 = 0 Θ 0 = Θ 1 = 0The shear work is represented by the term involving the velocity slip coefficientλ in the second leg of conditions (19) for θ2′(0). Table 7 demonstrates its importance. Neglecting the shear work reduces the predicted surface temperature.Table 7
Effect of shear work;Q0=Q1=Q2=0, γ=0.
λ θ 0(0) θ 2(0) θ 2 ′(0) θ 0(0) θ 2(0) 0 8.39634 0.65967 0.00000 8.39634 0.65967 0.2 9.69402 0.66777 −0.16337 7.84848 0.47205 0.4 11.16024 0.65306 −0.21129 7.62154 0.36953 0.6 12.75638 0.63416 −0.22379 7.54814 0.30418 0.8 14.46870 0.61552 −0.22308 7.56234 0.25866 1 16.29111 0.59830 −0.21704 7.63203 0.22505  Case Retaining shear work Neglecting shear workOn the right-hand side of (13) and (15), the first terms represent Joule heating and streamwise heat diffusion, respectively, while the second terms represent heat dissipation. Table 8 demonstrates the effect of neglecting these three processes. The predicted heat flux to the surface, represented by θ0′(0) and θ2′(0), is reduced considerably by neglecting viscous dissipation, streamwise diffusion, and/or Joule heating. Neglecting one or more may even predict heat flux in the wrong direction.Table 8
Neglect of viscous dissipation, streamwise diffusion, or Joule heating;Θ1=Θ2=0.
Θ 0 θ 0 ′(0) θ 0 ′(0) θ 0 ′(0) θ 0 ′(0) 0 0.22395 0.92158 1.33727 1.56122 1 −0.16186 0.53577 0.95146 1.17541 2 −0.54767 0.14996 0.56564 0.78960 3 −0.93348 −0.23585 0.17983 0.40379 4 −1.31929 −0.62166 −0.20598 0.017978 5 −1.70510 −1.00748 −0.59179 −0.36783  Case Neglectingdissipation Neglectingdiffusion NeglectingJoule heating Retainingall  θ 2 ′(0) 0.22323 0.63757 0.41434 0.63757


## 5. Conclusion

The problem of the flow due to a linearly stretching sheet in the presence of a transverse magnetic field has been formulated to include surface feed, velocity slip, and thermal slip. The problem has been shown to admit self-similarity of the full MHD fluid flow equations. Included in the thermal equation and conditions are physical processes such as viscous dissipation, Joule heating, streamwise heat diffusion, and shear work which were traditionally ignored or approximated.The presented numerical results are sample results. They show that the self-similar model (with the several physical processes involved) and the method of solution are capable of producing meaningful and useful results. No attempt is made to present a detailed study of the individual or collective effect of the 12 parameters at hand. It is up to the interested reader to choose his/her set of processes and associated parameters to concentrate on. Moreover, other physical processes can be added to the model such as thermal radiation, heat generation, and/or flow through a porous medium, as long as self-similarity is preserved.Nonetheless, the presented results pinpoint some interesting observations described below.The velocity slip coefficientλ and the suction rate fw have opposite effects on the curvature of the streamwise velocity profile f′(η). While increasing the first flattens the profile causing the surface shear f′′(0) to decrease, increasing the second curls the profile causing f′′(0) to increase. The more the curling (flattening) of the f′(η) profile, the higher (the lower) the far field entrainment rate f(∞), to compensate for the faster (the slower) streamwise flow close to the surface.The induced magnetic field is primarily affected by the streamwise velocity componentf′(η). As f′(η) decreases due to higher surface slip or suction rate, both g(η) and -g′(η) decrease.The effect of the velocity slip coefficientλ and the suction rate fw on the temperature is through their effect on the velocity and magnetic field components, with their involvement in viscous dissipation and Joule heating. Moreover, the velocity slip coefficient λ has the added effect of shear work, neglect of which results in considerable reduction in the predicted surface temperature.Even when the surface is maintained at the ambient temperatureT∞, the fluid adjacent to the surface acquires a higher temperature that increases as the thermal slip coefficient γ increases. This is due to viscous dissipation and Joule heating, which are traditionally ignored. The same applies to the equivalent situation of convective heating when the temperature on the other side of the sheet is maintained at T∞, with γ being the heat convection coefficient.The streamwise heat diffusion, which is neglected in boundary-layer models, is as important as viscous dissipation and Joule heating. Neglecting one or more of these thermal processes may predict lower heat flux or even heat flux in the wrong direction.Depending on the surface thermal conditions, the profiles of the temperature constituents my decay monotonically toward the far field or may have an extremum at a finite distance from the surface. The critical condition corresponding to the extremum being at the surface separates cases of heat transfer to and from the surface.



---

*Source: 5686089-2018-02-25.xml*