# MHD-Conjugate Free Convection from  an Isothermal Horizontal Circular Cylinder with  Joule Heating and Heat Generation

Authors: NHM. A. Azim; M. K. Chowdhury
Journal: Journal of Computational Methods in Physics
 (2013)

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## Abstract

The present work is devoted to the numerical study of laminar magnetohydrodynamic (MHD) conjugate natural convection flow from a horizontal circular cylinder taking into account Joule heating and internal heat generation. The governing equations and the associated boundary conditions for this analysis are made nondimensional forms using a set of dimensionless variables. Thus, the nondimensional governing equations are solved numerically using finite difference method with Keller box scheme. Numerical outcomes are found for different values of the magnetic parameter, conjugate conduction parameter, Prandtl number, Joule heating parameter, and heat generation parameter for the velocity and the temperature within the boundary layer as well as the skin friction coefficients and the rate of heat transfer along the surface. It is found that the skin friction increases, and heat transfer rate decreases for escalating value of Joule heating parameter and heat generation parameter. Results are presented graphically with detailed discussion.


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## Body

## 1. Introduction

Two-dimensional laminar natural convection from a horizontal cylinder under various surface boundary conditions such as isothermal, uniform heat flux, and mixed boundary conditions was investigated by several researchers employing the different numerical techniques. For example, Merkin [1] analysed the free convection boundary layer on an isothermal horizontal cylinder. Kuehn and Goldstein [2] determined numerical solution for the Navier-Stokes equations for laminar natural convection about a horizontal isothermal circular cylinder. They obtained the solutions of the Navier-Stokes and energy equations for natural-convection heat transfer from a horizontal isothermal cylinder and found that boundary layer conditions reached at the lower portion of the cylinder when the Rayleigh number becomes very large. Wang et al. [3] investigated numerical computation of natural convection flow about a horizontal cylinder using splines. They reported some new computations at very high Rayleigh numbers which indicate the existence of attached “separation” vortices in the downstream plume region.The combined effect of conduction and free convection which is known as conjugate free convection has a significant importance in many practical applications. Gdalevich and Fertman [4] investigated conjugate problems of natural convection. Miyamoto et al. [5] investigated the effects of axial heat conduction in a vertical flat plate on free convection heat transfer. They discovered that axial heat conduction in the flat plate significantly affects the temperature distribution within the boundary layer. Pozzi and Lupo [6] investigated the entire thermofluid-dynamic (TFD) field resulting from the coupling of natural convection along the surface of the flat plate  and conduction inside a heated flat plate by two expansions: regular series and asymptotic expansions. Kimura and Pop [7] analysed conjugate natural convection from a horizontal circular cylinder. The results obtained by Kimura and Pop [7] showed that the ratio of thermal conductivities of the solid wall to that of fluid within the boundary layer substantially influenced the flow and heat transfer characteristics.On the other hand, the electromagnetic fields are used to control the heat transfer as in the convection flows and aerodynamic heating and heat is also produced by electromagnetic fields such as MHD generators and pumps. A substantial quantity of research has been done in the presence of electromagnetic field of the flow and heat transfer characteristics over a variety of geometries. For example, Wilks [8] studied MHD-free convection about a semi-infinite vertical plate in a strong cross field. Hossain [9] studied the effects of viscous dissipation and Joule heating on MHD-free convection flow with variable plate temperature. He concluded that dissipative heat reduces the velocity field more in the lower Prandtl number fluid than that of the higher Prandtl number. The author also revealed that the dissipative heat reduced temperature field faster in the higher Prandtl number than that in lower Prandtl number fluid. Aldoss et al. [10] analysed MHD mixed convection from a horizontal circular cylinder. They found that the increase in the magnetic field leads to a decrease in the flow field, a rise in the temperature values of the flow field, and a decrease in the local wall shear stress and local Nusselt number. The combined effect of viscous dissipation and Joule heating on MHD forced convection over a nonisothermal horizontal cylinder embedded in a fluid saturated porous medium was studied by El-Amin [11].Many practical heat transfer applications involve the conversion of some form of mechanical, electrical, nuclear, or chemical energy to thermal energy in the medium. Such mediums are said to involve internal heat generation. For example, a large amount of heat is generated in the fuel elements from atomic reactors as a result of atomic fission that serves as the heat source for the nuclear power plants. The heat generated in the sun as a result of fusion of hydrogen into helium makes the sun a large nuclear reactor that supplies heat to the earth. Possible heat generation effects may modify temperature distribution and, therefore, the particle deposition rate. The heat transfer in a laminar boundary layer flow of a viscous fluid over a linearly stretching continuous surface with viscous dissipation/frictional heating and internal heat generation was analysed by Vajravelu and Hadjinicolaou [12]. They considered the volumetric rate of heat generation, q′′′[W/m3], as q′′′=Q0(Tf-T∞), for Tf≥T∞ and q′′′=0, for Tf<T∞, where Q0 is the heat generation constant. The effects of heat generation/absorption and the thermophoresis on hydromagnetic flow with heat and mass transfer over a flat plate were investigated by Chamkha and Camille [13]. Méndez and Treviño [14] studied the conjugate-natural convection heat transfer along a thin vertical plate with nonuniform internal heat generation. Natural convection flow from an isothermal horizontal circular cylinder in the presence of heat generation was studied by Molla et al. [15]. They found that the skin-friction coefficient increases and the Nusselt number decreases for increasing values of the heat generation parameters. Mamun et al. [16] investigated MHD-conjugate heat transfer analysis for a vertical flat plate in presence of viscous dissipation and heat generation. Recently, Azim et al. [17] analysed viscous Joule heating MHD-conjugate heat transfer for a vertical flat plate in the presence of heat generation.In this study, MHD-conjugate natural convection flow from a horizontal circular cylinder with Joule heating in the presence of heat generation is considered. To the best of our knowledge, this problem has not been considered before. The governing equations are transformed into a nondimensional form, and the resulting nonlinear partial differential equations are solved numerically using the implicit finite difference method together with the Keller box technique [18, 19].


## 2. Mathematical Analysis

Let us consider a steady, laminar, two-dimensional, incompressible, electrically conducting natural convection flow from an isothermal horizontal circular cylinder of radiusa placed on a fluid of uniform temperature  T∞ (see Figure 1). The cylinder has a heated core region of temperature Tb and the normal distance from the inner surface to the outer surface is b with Tb>T∞. A uniform magnetic field having strength B0 is acting normal at the cylinder surface. It is assumed that the fluid properties are constant, and the induced magnetic field is ignored. In addition, it is also considered that the boundary layer thickness is very small compared with the external radius “a” of the cylinder. The  x--axis is taken along the circumference of the cylinder measured from the lower stagnation point, and the y--axis is taken normal to the surface. Under the balance laws of mass, momentum, and energy and with the help of  Boussinesq approximation to the body force term in the momentum equation, the equations governing this boundary layer natural convection flow can be written as follows:
(1)∂u¯∂x¯+∂v¯∂y¯=0,(2)u¯∂u¯∂x¯+v¯∂u¯∂y¯=ν∂2u¯∂y¯2+gβ(Tf-T∞)sin(x¯a)-σB02u¯ρ,(3)u¯∂Tf∂x¯+v¯∂Tf∂y¯=κρcp∂2Tf∂y¯2+σB02u¯2ρcp+Q0ρcp(Tf-T∞).
The appropriate boundary conditions [7, 20] of the system are
(4)u¯=v¯=0,Tf=T(x¯,0),∂Tf∂y¯=κsbκf(Tf-Tb)ony¯=0,x>0,u¯⟶0,Tf⟶T∞asy¯⟶∞,x¯>0.
To transform the governing equations (1)–(3) and the boundary conditions (4) from the (x-,y-) coordinates to the dimensionless coordinates (x,y) for the free convection dominated regime, the following nondimensional variables are introduced with Grashof number Gr=[gβa3(Tb-T∞)]/ν2:
(5)x=x-a,y=y-aGr1/4,u=u-aνGr-1/2,v=v¯aνGr-1/4,θ=Tf-T∞Tb-T∞,
where  θ is the dimensionless temperature. The nondimensional forms of the governing equations (1)–(3) are as follows:
(6)∂u∂x+∂v∂y=0,(7)u∂u∂x+v∂v∂y+Mu=∂2u∂y2+θsinx,(8)u∂θ∂x+v∂θ∂y=1Pr∂2θ∂y2+Ju2+Qθ,
where  M=σa2B02/νρGr1/2 is the magnetic parameter,  J=σνB02Gr1/2/ρcp(Tb-T∞) is the Joule heating parameter, Q=Q0a2/μcpGr1/2 is the heat generation parameter, and Pr=μcp/κ  is the Prandtl number.Figure 1
Physical model and coordinate system.The boundary condition (4) can be written as in the following dimensionless form:
(9)u=v=0,θ-1=χ∂θ∂y,ony=0,x>0,u⟶0,θ⟶0asy⟶∞,x>0,
where  χ=(bκf/aκs)Gr1/4 is the conjugate conduction parameter. The present problem is governed by the magnitude of magnetic parameter M,  Joule heating parameter J, heat generation parameter Q, and conjugate conduction parameter  χ. The values of  χ  depend upon the ratios of b/a and κf/κs and Grashof number Gr. The ratios  b/a and  κf/κs  are less than unity whereas Gr is large for free convection. Therefore, the value of  χ  is greater than zero. Present analysis will refer to the free convection problem without conduction for χ=0.To solve (6)–(8), subject to the boundary condition (9), we assume the following transformations:
(10)ψ=xf(x,y),θ=θ(x,y),
where θ is the dimensionless temperature and ψ is the stream function usually defined as follows:
(11)u=∂ψ∂y,v=-∂ψ∂x.
Substituting (11) into (6)–(9), new forms of the dimensionless equations (7) and (8) are
(12)∂3f∂y3+f∂2f∂y2-(∂f∂y)2-M∂f∂y+θsinxx=x(∂f∂y∂2f∂x∂y-∂2f∂y2∂f∂x),(13)1Pr∂2θ∂y2+f∂θ∂y+Jx2(∂f∂y)2+Qθ=x(∂f∂y∂θ∂x-∂θ∂y∂f∂x).
The corresponding boundary conditions as mentioned in (9) take the following form:
(14)f=∂f∂y=0,θ-1=χ∂θ∂yaty=0,x>0,∂f∂y⟶0,θ⟶0asy⟶∞,x>0.
Equations (12) and (13) are solved numerically using implicit finite difference method using Keller box scheme [18, 19] based on the boundary conditions described in (14). The shearing stress and the rate of heat transfer in terms of skin friction coefficient and Nusselt number, respectively, can be written as [15]
(15)Cf=τwρU∞2,Nu=aqwκ(Tw-T∞),
where  τw=μ(∂u-/∂y-)y-=0 and  qw=-κ(∂T/∂y-)y-=0.Using the variables (5) and the boundary conditions into (14), we have
(16)CfGr1/4=xf′′(x,0),NuGr-1/4=-θ′(x,0).
The results of the velocity profiles and temperature distributions can be calculated by the following relations:
(17)u=f′(x,y),θ=θ(x,y).
In (16) and (17) primes denote differentiation with respect to y only.


## 3. Method of Solution

In the present studies, we have employed implicit finite difference method, which was introduced by Keller [18] and elaborately described by Cebeci and Bradshaw [19]. A brief discussion of the development of an algorithm of implicit finite difference method together with the Keller box elimination scheme of the present analysis is given below.Equations (12) and (13) are written in terms of first-order equations in y, which are then expressed in finite difference form by approximating the functions and their derivatives in terms of the central differences in both coordinate directions. Denoting the mesh points in the (x,y) plane by xi and  yj, where  i=1,2,3,…,181 and j=1,2,3,…,305, central difference approximations are made such that the equations involving x explicitly are centered at  (xi-1/2,yj-1/2) and the remainder at  (xi,yj-1/2), where yj-1/2=(yj+yj-1)/2 and so forth. This results  in a set of nonlinear difference equations for the unknowns at xi in terms of their values at  xi-1. These equations are then linearised by the Newton’s method and are solved using a block-tridiagonal algorithm, taken as the initial iteration of the converged solution at  x=xi-1. Now to initiate the process at x=0, we first provide guess profiles for all five variables (arising the reduction to the first-order form) and use the Keller box method to solve the governing ordinary differential equations. Having obtained the lower stagnation point solution, it is possible to march step by step along the boundary layer. For a given value of x, the iterative procedure is stopped when the difference in computing the velocity and the temperature in the next iteration is less than 10-6, that is, when |δfi|≤10-6, where the superscript denotes the iteration number. The computations were not performed using a uniform grid in the y direction, but a nonuniform grid was used and defined by  yj=sinh((j-1)/p), with j=1,2,…,305 and p=100.


## 4. Results and Discussion

The dimensionless governing equations (12) and (13) and the boundary conditions (14) contain a set of physical parameters: Prandtl number Pr, magnetic parameter M, conjugate conduction parameter  χ, Joule heating parameter J, and heat generation parameter Q. The Prandtl numbers are considered to be 1.63, 1.44, 1.0, and 0.733 that correspond to Glycerin, water, steam, and hydrogen, respectively. The remaining parameters are taken as follows: magnetic parameter M=0.10–0.70; conjugate conduction parameter  χ=1.0–2.5; Joule heating parameter J=0.01–1.0; and heat generation parameter Q=0.01–0.12.A comparison of the local Nusselt number and the local skin friction factor obtained in the present work withM=0.0, χ=0.0,  J=0.0, Q=0.0, and Pr=1.0 and obtained by Merkin [1] and Molla et al. [15] has been shown in Table 1, and it has been observed that there is an excellent agreement among these three results.Table 1
Comparisons of the present numerical values with Merkin [1] and Molla et al. [15] for different values of x while with Pr=1.0, M=0.0, χ=0.0, J=0.0, and Q=0.0.
x
Nu G
r
-
1
/
4
=
-
θ
′
(
x
,
0
)
C
f
G
r
1
/
4
=
x
f
′
′
(
x
,
0
)
Merkin [1]
Molla et al.  [15]
Present
Merkin [1]
Molla et al. [15]
Present
0.0
0.4214
0.4214
0.4216
0.0000
0.0000
0.0000
π/6
0.4161
0.4161
0.4163
0.4151
0.4145
0.4139
π/3
0.4007
0.4005
0.4006
0.7558
0.7539
0.7528
π/2
0.3745
0.3740
0.3742
0.9579
0.9541
0.9526
2π/3
0.3364
0.3355
0.3356
0.9756
0.9696
0.9678
5π/6
0.2825
0.2812
0.2811
0.7822
0.7739
0.7718
π
0.1945
0.1917
0.1912
0.3391
0.3264
0.3239A grid independent test has been shown by taking three different grid configurations,181×305, 361×405, and 451×505, in Figures 2(a) and 2(b) as the skin friction coefficient and the rate of heat transfer while  Pr=1.0,  M=0.1,  χ=1.0,  J=0.01, and Q=0.01. From these figures, it can be concluded that the numerical solutions are completely independent of the grid orientations. In the present investigation 181×305 grid configuration has been chosen for the numerical computation.(a) The skin friction coefficient and (b) the rate of heat transfer againstx for different mesh configurations while  Pr=1.0, M=0.1, χ=1.0, J=0.01, and Q=0.01.
(a)
(b)Figures3, 5, 7, 9, and 11 illustrate the velocity and temperature distributions at x=π/2 against y, the direction along the normal to the surface of the cylinder, and Figures 4, 6, 8, 10, and 12 depict the skin friction coefficients and heat transfer rates against  x at y=0 (along the surface of the cylinder) for different values of the magnetic parameter, conjugate conduction parameter, Prandtl number, Joule heating parameter, and heat generation parameter, respectively.(a) Variation of velocity profiles and (b) variation of temperature distributions againsty for varying of magnetic parameter M with  Pr=1.0,  χ=1.0,  J=0.01, and Q=0.01.
(a)
(b)(a) Variation of skin friction coefficients and (b) variation of rate of heat transfer againstx for varying of  M with Pr=1.0,  χ=1.0,  J=0.01, and Q=0.01.
(a)
(b)(a) Variation of velocity profiles and (b) variation of temperature distributions againsty for varying of conjugate conduction parameter  χ with Pr=1.0, M=0.1,  J=0.01, and Q=0.01.
(a)
(b)(a) Variation of skin friction coefficients and (b) variation of rate of heat transfer againstx for varying of conjugate conduction parameter  χ with Pr=1.0, M=0.1,  J=0.01, and Q=0.01.
(a)
(b)(a) Variation of velocity profiles and (b) variation of temperature distributions againsty for varying of Pr with M=0.1,  χ=1.0,  J=0.01, and Q=0.01.
(a)
(b)(a) Variation of skin friction coefficients and (b) variation of rate of heat transfer againstx  for varying of  Pr with  M=0.1,  χ=1.0,  J=0.01, and Q=0.01.
(a)
(b)(a) Variation of velocity profiles and (b) variation of temperature distributions againsty for varying of  J with  Pr=1.0, M=0.1, χ=1.0, and Q=0.01.
(a)
(b)(a) Variation of skin friction coefficients and (b) variation of rate of heat transfer againstx for varying of  J with Pr=1.0, M=0.1, χ=1.0, and Q=0.01.
(a)
(b)(a) Variation of velocity profiles and (b) variation of temperature distributions againsty for varying of  Q with Pr=1.0, M=0.1, χ=1.0, and  J=0.01.
(a)
(b)(a) Variation of skin friction coefficients and (b) variation of rate of heat transfer againstx for varying of Q with Pr=1.0, M=0.1,  χ=1.0, and J=0.01.
(a)
(b)The magnetic parameter is the ratio of the magnetic force to the inertia force. Hence the magnetic force is important when it is the order of one, and the flow is considered as hydromagnetic flow. The flow is hydrodynamic forM≪1. For small value of M, the motion is hardly affected by the magnetic field and for large value of M, the motion is largely controlled by the magnetic field. The increasing values of the magnetic parameters increase magnetic-field strength, which is acting normal to the cylinder surface that reduces fluid motion as observed in Figure 3(a). As a result the skin friction at the surface to the cylinder is decreased, which is shown from Figure 4(a). Heat is produced due to the interaction between magnetic field and fluid motion; consequently, temperature within the thermal boundary layer increases for increasing value of the magnetic parameters as revealed from Figure 3(b). Increasing thermal energy within the boundary layer reduces the temperature difference between core region and boundary layer region, which ultimately decreases the heat transfer rate as illustrated in Figure 4(b).The velocity and temperature are illustrated in Figure5 and the variation of the local skin friction coefficient and local rate of heat transfer are depicted in Figure 6 for different values of conjugate conduction parameter  χ with Pr=1.0, M=0.1,  J=0.01, and Q=0.01. Increasing value of conjugate conduction parameter χ resists conduction from the core region to the boundary layer and consequently, decelerates convection within the boundary layer, as a result velocity and temperature decrease for increasing values of the conjugate conduction parameters at a particular value of y, which are presented in Figures 5(a) and 5(b), respectively. As the velocity decreases, the skin friction at the surface decreases for increasing value of conjugate conduction parameter χ, as observed in Figure 6(a). Since increasing value of the conjugate conduction parameters resists conduction from the core region to the boundary layer as mentioned earlier, it of course resists thermal energy transfer which is observed from Figure 6(b).The increasing value of Prandtl number increases viscosity and decreases the thermal action of the fluid. Therefore, the velocity and temperature of fluid are expected to decrease with the increasing Prandtl number which are observed in Figures7(a) and 7(b), respectively. Decreasing velocity of the fluid leads to decrease skin friction  and the decreasing temperature within in the boundary layer increases the temperature difference between core (inner) region and boundary layer region, which eventually increases heat transfer rate from the core region to the boundary layer region as depicted in Figures 8(a) and 8(b), respectively.The effects of the Joule heating parameters on the velocity and temperature are presented in Figure9, and on the skin friction and rate of heat transfer they are illustrated in Figure 10, respectively, with Pr=1.0, M=0.1, χ=1.0, and Q=0.01. The Joule heating parameter having the magnetic-field strength increases the temperature and eventually the fluid motion is accelerated as plotted in Figures 9(b) and 9(a), respectively. The increased temperature for increasing Joule heating parameter increases the temperature within the boundary layer which results in the heat transfer rate decrease as illustrated in Figure 10(b). The variation of local skin friction coefficient increases for the increasing  J as depicted in Figure 10(a). This is an expected behavior as fluid motion increases for increasing Joule heating parameter.Figure11 illustrates the effect of the heat generation parameters on the fluid velocity and temperature profiles, respectively. It is clear that as the heat generation parameter increases both the fluid velocity and temperature of the fluid increase. Figure 12 depicts the variation of the heat generation parameters on the skin friction coefficient and the heat transfer rate with  Pr=1.0, M=0.1, χ=1.0, and  J=0.01. It is observed that the local Nusselt number decreases, but the skin friction coefficient increases with increasing heat generation parameter.


## 5. Conclusion

MHD-conjugate natural convection flow from horizontal cylinder taking into account Joule heating in the presence of heat generation is studied. Numerical results were obtained for the physical parameters and discussed. From the results, it is established that the velocity of the fluid within the boundary layer decreases with increasing magnetic parameter, conjugate conduction parameter, and Prandtl number while it increases for increasing Joule heating parameter, and heat generation parameter. The temperature within the boundary layer increases for increasing magnetic parameter, Joule heating parameter and heat generation parameter whereas it decreases for increasing conjugate conduction parameter and Prandtl number. Moreover, the skin friction along the surface of the cylinder decreases for increasing magnetic parameter, conjugate conduction parameter, and Prandtl number, and it increases for increasing Joule heating parameter and heat generation parameter. Furthermore, the rate of heat transfer along the surface increases for increasing Prandtl number while it decreases for remaining parameters.



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