Scientia Iranica B ( ) ( ), –

Sharif University of Technology

Scientia IranicaTransactions B: Mechanical Engineering

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Mixed convection in cylindrical annulus with rotating outer cylinderand constant magnetic field with an effect in the radial directionH.R. Mozayyeni, A.B. Rahimi ∗Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, P.O. Box 91775-1111, Iran

Received 20 January 2011; revised 24 July 2011; accepted 3 December 2011

KEYWORDSCombined convection;Steady;Transient;Cylindrical annulus;Viscous dissipation;Magnetic field.

Abstract In the present study, mixed convection of a fluid in the fully developed region in a horizontalconcentric cylindrical annulus with different uniform wall temperatures, is numerically investigated inboth steady and unsteady states in the presence of radial MHD force, as well as in consideration of heatgeneration due to viscous dissipation. Also, cylinder length is assumed to be infinite. Moreover, radiationheat transfer from the hot surface is assumed to be negligible. Buoyancy effects are also considered,along with Boussinesq approximation. The forced flow is induced by the cold rotating outer cylinderat slow constant angular velocity, with its axis at the center of the annulus. Investigations are madefor various combinations of non-dimensional group numbers; Reynolds number (Re), Rayleigh number(Ra), Hartmann number (Ha), Eckert number (Eck) and annulus gap width ratio (σ0). These dimensionlessparameters used in the present study will be investigated over a wide range to present the basic flowpatterns and isotherms in a concentric cylindrical annulus. A finite volume scheme, consisting of theTri-Diagonal Matrix Algorithm (TDMA), is used to solve governing equations, which are continuity, two-dimensional momentum and energy, by the SIMPLE algorithm. The numerical results reveal that the flowand heat transfer are suppressed more effectively by imposing an external magnetic field. Furthermore, itis found that the external magnetic field causes the fluid velocity and temperature to be suppressed moreeffectively. Moreover, it will be shown that viscous dissipation terms have significant effects in situationswith high values of Eckert and Prandtl number and low values of Reynolds number.

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1. Introduction

The study of themagnetohydrodynamic (MHD) flow of elec-trically conducting fluids, such as liquid metals, has gainedincreased attention due to its wide range of industrial appli-cations, in engineering and geophysics, to control unwantedconvective flow in cases such as the design of MHD powergenerators, modification of the solidification processes of met-als, optimization of crystal growth processes, plasma industries

∗ Corresponding author.E-mail address: rahimiab@yahoo.com (A.B. Rahimi).

and the cooling of nuclear reactors. Although the analysis offlow and heat transfer in cylindrical annuli has been investi-gated in several pieces of literature, there is not any significantresearch conducted using a magnetic field. Recently, laminarconvection in either vertical or horizontal concentric annuli hasreceived attention by various investigators because of its widerange of applications, such as in heat exchangers, cooling sys-tems in electrical devices, solar collectors, the cooling of tur-bine rotors and high speed gas bearings. In one primary study,Rokerya and Iqbal [1] investigated the effect of viscous dissi-pation in the thermal energy balance for viscous flow in verti-cal concentric annuli. So far, most work on mixed-convectionproblems has been performed for flows in vertical rotating sys-tems [2–4]. Relatively few studies, however, have been madefor flows in horizontal annuli. Fusegi et al. [5] and Lee [6,7] areinvestigators that performed simulations about annuli with aheated rotating inner cylinder. Busse and Finocchi [8] and Petryet al. [9] carried out numerical investigations about the problemof the onset of convection in a cylindrical annulus with coni-cal end boundaries in the presence of magnetic fields. Exact so-lutions for fully developed natural convection in open-endedvertical concentric annuli under a radial magnetic field have

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doi:10.1016/j.scient.2011.12.006

2 H.R. Mozayyeni, A.B. Rahimi / Scientia Iranica, Transactions B: Mechanical Engineering ( ) ( ), –

been presented by Singh et al. [10]. The flow patterns in rotat-ing cylinders are categorized into three basic types, according tothe number of eddies by Yoo [11]. Abu-Hijleh and Heilen [12]have proposed a correlation for the average Nusselt number,as a function of Reynolds number and buoyancy parameter,from a rotating cylinder. Also, mixed heat and fluid flow in in-ner cylinder rotating annuli [13], and heat convection from anisothermally heated horizontal rotating cylinder [14] have beenperformed. Barletta [15–17] conducted an analytical methodof perturbation to solve momentum balance and energy bal-ance equations for investigating flow and heat transfer in ver-tical ducts. Effects of viscous dissipation on mixed convectionin an inclined channel and in a vertical tube are also reported,using the perturbation method [18,19]. Non-uniform circum-ferential heating in horizontal concentric annuli [20] andanalysis of entropy generation in cylindrical annuli, due toconduction and viscous dissipation [21], are other pieces ofresearch in this field. Chen and Zhang [22] simulated nonlin-ear thermal convection in a fluid-filled gap between two co-rotating and concentric cylindrical annuli. Categorization of theflow regimes, according to the number of eddies, are estab-lished on the Ra–Re plane for various numbers presented byKhanafer and Chamkha [23]. Also, it is shown in [24] that heattransfer and fluid flow over a non-isothermal horizontal cylin-der in a porous medium can be controlled using electromag-netic fields. In the article of Mahmud and Fraser [25], the firstand second law (of thermodynamics) characteristics of fluidflow and heat transfer inside a cylindrical annulus have beeninvestigated analytically. The study of magnetic field effects onan electrically conducting fluid has also been presented in [26].In the paper by Sankar et al. [27], it is shown that the flow andheat transfer can be suppressed by imposing an external mag-netic field. A survey of the flow characteristics and the heattransfer of a non-Newtonian fluid in rotating concentric an-nuli were reported numerically by Amura et al. [28]. Some sur-veyed studies in the literature were concerned primarily withthe double-diffusive mixed convection of a binary fluid withina two-dimensional horizontal annulus [29,30]. Moreover, theeffect of viscous dissipation in a porous medium was investi-gated in [31,32]. For the first time, Dyko and Vafai [33] pre-sented the problem of convection in the annulus between twohorizontal coaxial cylinders, resulting from gravity modulationin a microgravity environment. The study of flow and heattransfer characteristics in the vertical porous systems receivedattention in [34,35]. Besides, an investigation to provide an in-verse approach for estimating the viscosity of the fluid and ther-mal behavior of concentric cylinders has been presented in [36].Recently, Zanchini [37] conducted an analytical study oflaminar mixed convection with a temperature-dependentviscosity in a vertical annular duct with uniform wall tempera-tures. Some attention has also been paid to laminar magneto-hydrodynamics convection heat transfer about a vertical flatplate [38] and in a rectangular cavity [39]. Discussion aboutthe stability of the circular Couette flow under imposition of ahomogeneous magnetic field has been analyzed by Leschhornet al. [40]. Moreover, Barletta et al. [41], in a recent paper, havestudied the combined forced and free flowof an electrically con-ducting fluid in a vertical annular porous medium surround-ing a straight cylindrical electric cable with a radially varyingmagnetic field.

The objective of the present study is to investigate numer-ically the problem of mixed convection of a fluid in the fullydeveloped region between two horizontally concentric cylin-ders with infinite lengths, in the presence of a constant mag-netic field with a radial MHD force direction, considering the

effects of viscous heat dissipation in the fluid in both steady andunsteady states. The forced flow is induced by the cold rotatingouter cylinder, in slow constant angular velocity, with its axisat the center of the annulus. The buoyancy effect is also con-sidered through Boussinesq approximation. Investigations areundertaken for various combinations of non-dimensional groupnumber, such as Reynolds number, Rayleigh number, Prandtlnumber, Hartmann number, Eckert number and the annulusgap width ratio. The dimensionless parameters used in thepresent study will be investigated over a wide range to presentthe basic flow patterns and temperature contours in a concen-tric cylindrical annulus.

2. Problem formulation

The schematic diagram of the annular duct under consid-eration and coordinate system are shown in Figure 1. At first,two infinitely long horizontal concentric circular cylinders filledwith incompressible fluid are fixed and held at the same tem-perature of T0. Suddenly, the outer cylinder starts to rotate ina counter-clockwise direction with constant angular velocity,ω, with its axis at the center of the annulus. Also, the temper-ature of the inner cylinder is increased to a higher value of Ti,concurrently. The cylindrical annulus is filled with viscous,Newtonian and incompressible fluid. The fluid is permeatedby a uniform external magnetic field, Bo, along an axial direc-tion. We assume that the magnetic field induced by the mo-tion of fluid and the heating produced by electromagnetic fields,are taken to be small enough. Further, the viscous dissipationterms in the energy equation are considered completely due tothe viscosity of the fluid. It is assumed that all other thermo-physical properties of the fluid are constant, except for the den-sity change that is taken in a Boussinesq approximation. Densityvariation can be described by the following equation:

ρ = ρ0(1 − β(T − T0)), (1)

where β is the thermal expansion coefficient, such that:

β = −1ρ0

∂ρ

∂T

P. (2)

Dimensionless forms of variables are put into consideration bythe following definitions:

r =r∗

D, Vr =

V ∗r

RoΩ, Vφ =

V ∗

φ

RoΩ,

θ=T − ToTi − To

, P =P∗

ρR2oΩ

2 , t=

t∗

D/ (RoΩ). (3)

Taking into account the above assumptions and dimensionlessvariables, the governing equations describing a set of conserva-tion of laws can be written as follows:

∇.V = 0, (4)

∂ V∂t

+

V · ∇

V

= −∇P +1Re

∇2V +

RaPr Re2

θ(cosφ) er − (sinφ) eφ

− Vr

Ha2

Reer , (5)

∂θ

∂t+

V · ∇

θ

H.R. Mozayyeni, A.B. Rahimi / Scientia Iranica, Transactions B: Mechanical Engineering ( ) ( ), – 3

Figure 1: Configuration of the problem.

=1

Re Pr∇

2θ +EckRe

2

∂Vr

∂r

2

+

1r∂Vφ∂φ

+Vr

r

2

+

∂Vφ∂r

−Vφr

+1r∂Vr

∂φ

2. (6)

Note that the MHD body force appeared on the right hand sideof r-momentum equation is evaluated due to an imposed axialconstant magnetic field according to the Lorentz force [24]. InEqs. (4)–(6), V is the dimensionless velocity vector and P is thedimensionless acting hydrodynamic pressure. Let us define thedimensionless group numbers:

Re =RoΩDν

, Ra =gβ (Ti − To)D3

να,

Pr =ν

α, Ha = B0D

σ

ρν,

Eck =R2oΩ

2

cp (Ti − To), σo = 2Ri/(Ro − Ri),

D = R0 − Ri. (7)

The dimensionless form of the initial conditions for the presentproblem is defined as:

Vr = Vφ = θ = 0 at t = 0. (8)

Moreover, the dimensionless forms of the boundary conditionsare given by:

Vr = 0, Vφ = 0, θ = 1 at r = ri, (9)

Vr = 0, Vφ = 1, θ = 0 at r = r0. (10)

If a pure conduction heat, the transfer rate in the dimensionlessform and in the absence of any fluid motion is defined as:

Nucond = 1/ ln(ro/ri). (11)

Then, the local Nusselt number, along the inner and outercylinders, can be expressed as the actual heat transfer dividedby this quantity as:

Nui(φ) = −

r∂θ

∂r

Nucond at r = ri, (12)

Nu0(φ) = −

r∂θ

∂r

Nucond at r = r0. (13)

Also, the average Nusselt number at both inner and outer cylin-ders are calculated, respectively, by:

Nui =12π

2π

0Nui(φ)dφ, (14)

Nu0 =12π

2π

0Nu0(φ)dφ. (15)

The important note is predicting the same result by both ex-pressions in Eqs. (12) and (13) for the steady state solution andin the absence of heat generation. The net circulation of fluid inthe direction of cylinder rotation is defined by:

Γ = |Ψ2 − Ψ1|. (16)

3. Numerical procedure

A finite volume procedure is used to numerically solvethe governing equations describing the sets of laws by thenumerical code accomplished completely by the authors. Theimportant problem on the overall solution procedure is thepressure gradients in both momentum equations, in r and φdirections. The remedy to solve this problem is to use staggeredgrids for the velocity components through the iterative SIMPLEalgorithm in a specified domain, with a prescribed boundaryand initial conditions. In the iterative SIMPLE algorithm, acentral differencing method with accuracy of the second orderwas introduced to obtain the discretized equations for thediffusion terms on the right hand side of Eqs. (5) and (6). In orderto calculate the value of each of the velocity components andtemperature of the convective terms at control volume faces,the power-law differencing scheme is used with accuracy ofthe second-order at low aspect ratios of the convective massflux per unit area to the diffusion conductance at cell faces,and with accuracy of the first-order at high aspect ratios. Also,a fully implicit transient scheme is employed for calculationsin an unsteady state. To assess the grid independence of thenumerical scheme, the distributions of Nusselt number on theinner cylinder were initially tested with different (r∗φ) meshsizes of 18∗5, 32∗9, 56∗17, 100∗31 and 180∗55 in Figure 2a.In this set of mesh sizes, as can be seen, the coefficient of1.8 was used in each test to increase the number of meshgrids in both directions of r and φ. It was found that thevariations of Nu number distributions on the inner cylinderwere not significant, between (r∗φ) andmesh sizes of (100∗31)and (180∗55). Hence, a 100∗31 grid in r − φ directionswas applied for the computational domain in the cylindricalannulus. Fine, non-uniform grid spacing is used in ther-direction to capture the rapid changes, such as grid lines,being closer packed near the inner and outerwalls. On the otherhand, a uniform mesh was implemented in the φ-direction.Figure 2b illustrates a sample of computational meshes usedin this investigation. The value of stream function, ψ , on thestagnant boundary of the inner cylinder is considered as zero.The other values of ψ are calculated using distributions ofvelocity component computed from the numerical solution byway of the integration method. After imposing the suddenchanges, an iterative SIMPLE algorithm is then repeated, such

4 H.R. Mozayyeni, A.B. Rahimi / Scientia Iranica, Transactions B: Mechanical Engineering ( ) ( ), –

Figure 2a: Nusselt number distributions on the inner cylinder for variousmeshsizes at Re = 100, Pr = 0.7, Eck = 0.0002, Ra = 10000.0, Ha = 20.0.

Figure 2b: Sample of grid system.

that the desired convergence of governing equations is attainedfor each time step until the steady state conditions are obtained.The convergence criterion for the numerical solution in atransient state is obtained by comparing the residue of tworepeated sequences of each of the momentum and energyequations in a specific time domain, characterized by thedefinition of a time step as follows:

|Re sidualnm+1 − Re sidualnm| ≺ 10−8.

In the above expression, n refers to time and m refers tothe number of iterations in that time domain. Moreover,the condition needed for passing the transient situation andattaining the steady state is defined by comparing the residueof each governing equation at two successive time domains as:

|Re sidual(n + 1)− Re sidual(n)| ≺ 10−4.

It must be noticed that the time step used in this proceduredepends strongly on values specified for Pr and Re numbers. Thecriterion employed for steady state conditions in the numerical

Figure 2c: Convergence history of φ-momentum equation for Re = 200,Pr = 2, Eck = 0.0007, Ra = 10, 000 and Ha = 20.

Figure 3: Comparison of the streamlines and isotherms between the presentwork and Yoo [11] for Ra = 104 , Pr = 0.7, σ0 = 2, Ha = 0.0, Eck = 0.0 andRe = 100.

technique is defined by |Re sidual(m + 1) − Re sidual(m)| ≺

10−9. In Figure 2c, the convergence history of theφ-momentumequation can be seen for specified values of non-dimensionalgroup numbers, such as Re = 200, Pr = 2, Eck = 0.0007,Ra = 10, 000 and Ha = 20.

In the next section, the results of solving the governingequations along with discussions are presented.

4. Presentation of results

In order to validate the numerical code, the steady-stateresults of the present numerical code have been comparedin Figure 3 against the results obtained by Yoo [11], for theconsidered non-dimensional numbers.

H.R. Mozayyeni, A.B. Rahimi / Scientia Iranica, Transactions B: Mechanical Engineering ( ) ( ), – 5

Figure 4: Effect of Hartman number on temperature contour for Re = 100, Eck = 0.0, Ra = 10, 000 and Pr = 1.

Aswe know, the dimensionless Hartman number representsthe magnitude of the applied magnetic field. Here, the effectof Hartman number on streamlines and isotherms at Ra =

10, 000, Eck = 0.0, Pr = 1.0 and Re = 100.0 is presented.In the absence of a magnetic field, the existence of buoyancyforces and forced convection brings about the solution regionin the fluid flow, but application of an outer magnetic fieldchanges the balance of these forces in the fluid flow. The effectsof application of a weak magnetic field, Ha = 20.0, on thestreamlines and temperature contours are shown in Figures 4and 5. These changes are more pronounced for Ha = 50.0,so that, in this situation, the existing eddies at the right handside of the annulus fade away and the intensity of the effectof buoyancy forces on temperature contours is reduced greatly.By an increase in Hartman number after around 200, not muchchange in the natural convection of the fluid is seen. In thissituation, the rotation of the outer cylinder is the sole cause offluid flow. Also, it can be seen from these contours that for largemagnetic fields, such as in the case of Ha = 100, the dominantheat transfer mechanism is conduction. Figure 6 shows theeffects of imposing a constant magnetic field on the flow andtemperature distributions for Re = 100, Pr = 1, Eck = 0.0 andRa = 10, 000 at φ = 0. In the absence of amagnetic field (Ha =

0.0), the circumferential velocity and temperature distributionsare with high-gradient slopes, especially near the cold cylinder.It is interesting to note that as the strength of themagnetic fieldincreases (Ha = 50.0), variations in temperature and flow fieldbecome more uniform than before. As the Hartmann numberis further increased (Ha = 100), these distributions are nearlyconduction-like, and this is due to suppression of convectionby the magnetic field. Also, it can be seen that the flow fieldbecomes completely stable and conduction-like at Ha = 200.

In Figure 7, the distribution of Nusselt number on the outerand inner surfaces is presented for different Hartman numbers.An increase in Hartman number, which causes a reductionin buoyancy forces, will bring about a gradual decrease inNusselt number on both surfaces of the cylinder, in such a waythat at Ha = 200, not many changes in the region between0° to 360° can be sensed. It is interesting to point out that inthis situation where forced convection is dominant on the fluidflow, the Nusselt number in the above region, on both cylindersurfaces, is equal to one. As we know, imposing amagnetic fieldcauses a change in the streamlines structure and the amountof stream function on them, in such a way that an increasein Hartman number not only causes existing vortices in thefluid flow to diminish, but causes the streamlines to becomeconcentric circles. Net circulation is a quantity related to therate of volume flow in unit depth between two concentriccylinders. Figure 8 shows a variation of net circulation with theHartman number and for different values of Rayleigh number.As can be seen in this region, net circulation and therefore theamount of flow passing through the circular section betweentwo cylinders increases up to 0.54, as the Hartman numberincreases at relatively low Rayleigh numbers (Ra = 1000, Ra =

10, 000). The explanation of this phenomenon can be the quickreduction of buoyancy forces in lower Rayleigh numbers andthe domination of forced convection on fluid flow. But, at highervalues of Rayleigh number, around Ra = 30, 000, this increaseis not as abrupt. Although, as Hartman number increases, forcedconvection will eventually dominate the fluid flow, and the netcirculation absolute value will tend to 0.54.

In Figure 9, variations of average Nusselt number on innerand outer surfaces versus Hartman number are shown fordifferent values of Eckert number. As can be seen from this

6 H.R. Mozayyeni, A.B. Rahimi / Scientia Iranica, Transactions B: Mechanical Engineering ( ) ( ), –

Figure 5: Effect of Hartman number on temperature contour for Re = 100, Eck = 0.0, Ra = 10, 000 and Pr = 1.

Figure 6: Variation of distribution (a) circumferential component of velocity, and (b) temperature φ = 0, and for different values of Hartman number at Re = 100,Eck = 0.0, Ra = 10, 000 and Pr = 1.

figure, for certain values of Hartman number, and for increasingEckert number, the average Nusselt number on the innercylinder decreases, and on the outer cylinder increases. In fact,as the Eckert number increases heat production inside thefluid increases because of viscous dissipation, and thereforefluid temperature at the cross-section increases. This causesthe non-dimensional temperature gradient to decrease in the

vicinity of the inner cylinder, and increase in the vicinity ofthe outer cylinder. So, if the intensity of the magnetic field isnot considerable, then the increase of Eckert number causesconsiderable decrease in average Nusselt number on the innercylinder and its increase on the outer cylinder. As Hartmannumber increases and thus the strength of buoyancy forcesdecreases, the average Nusselt number on both surfaces tends

H.R. Mozayyeni, A.B. Rahimi / Scientia Iranica, Transactions B: Mechanical Engineering ( ) ( ), – 7

Figure 7: Effect of Hartman number on local Nusslet number distribution on (a) inner and (b) outer cylinders, and for Re = 100, Ra = 10, 000, Eck = 0.0, andPr = 1.

Figure 8: Variation of net circulation with Hartman number in terms ofRayleigh numbers, and for Re = 100, Pr = 1 and Eck = 0.0.

to unity. This phenomenon ismore visible on the outer cylinder.One interesting point in both of these figures is the suddenchange of average Nusselt number on both inner and outersurfaces in the range of Hartman numbers from 10 to 100.Another interesting point is the unexpected behavior of averageNusselt number with an increase in Hartman number, and forEck = 0.5, in Figure 9(a). Aswe know, heat production in fluid isconsiderable for Eck = 0.5. This produced heat causes the fluidtemperature of different points of the cross-section, especiallynear the inner cylinder, to be more than in the past. In thissituation, and also because of the heat gradient being low, theaverageNusselt number on this cylinder is lower than expected.So, contrary to other cases, this number on the inner cylinderincreases as the strength of the magnetic field increases in therange of 10–100, and forced convection becomes dominant influid flow.

The effects of viscous dissipation terms on the rate ofheat transfer are presented in Figure 10. By increasing thevalue of the Eckert number, the influence of heat generationdue to viscous dissipations in the concentric annulus isenhanced, and therefore the average temperature of the fluidat the circumferential area sections is increased. Since the

dimensionless temperature of the inner and outer cylinders areheld at 1.0 and 0.0, respectively, it is obvious that by increasingthe amount of average temperature at every circumferentialarea section, the rate of heat transfer between the fluid andthe inner cylinder is decreased because of the reduction in thetemperature differences between them. On the other hand, theheat transfer rate between the fluid and the outer cylinderis increased because of the enhancement of temperaturedifferences between them. These figures, which are obtainedwith variations of Eckert number from 0.001 to 0.1, confirm theabove explanations. Also, it can be easily seen from Figure 10(b)that the fluctuations of local Nu distributions are suppressedby increasing the value of the Eckert number gradually. Theeffect of Eckert number on the averaged Nusselt number on theinner and outer cylinders against different values of Reynoldsnumber are shown in Figure 10(c) and (d). From these figures,one can see that for lower values of Reynolds number, thechange of Eckert number, which is representative of the viscousdissipation terms, becomes much more considerable.

The effects of Prandtl number on the streamlines andtemperature contours for Re = 100.0, Eck = 0.0007, Ra =

10000.0 and Ha = 20.0 are shown in Figure 11. By use of theabove non-dimensional values and for the fixed value of theRa number, the ratio of Grashof number to Re2 for lower Prnumbers is noticeable. Therefore, buoyancy-induced flow playsa more important role in the flow patterns, which causes theexistence of a two-eddy flow pattern for Pr = 0.5. As thevalue of Pr number increases, the ratio of Grashof number toRe2 decreases, with the Ra number remaining at a fixed value.This tends to enhance the influence of the force-induced flow.The above reasons, along with forced flow being opposed bybuoyancy-induced flow at the right hand side of the annulus,cause a reduction in the strength of the eddy in that regionand eventually the disappearance of the eddy at Pr = 2.The eddy remained at the left hand side within the annulusdisappears with an increase in the value of the Pr number, too.In a similar way, the effects of high values of Pr number onstratifying the temperature field in the radial direction can besimply observed in Figure 12. It is interesting to note that thethermal plumes near the upper portion of the inner cylinderare tilted in the direction of the cylinder rotation at Pr =

1.0. In Figure 13, the behavior of the average Nusselt numberagainst the Eckert number is shown for two different fluidswith different Prandtl numbers. The interesting point in this

8 H.R. Mozayyeni, A.B. Rahimi / Scientia Iranica, Transactions B: Mechanical Engineering ( ) ( ), –

Figure 9: Effect of Hartman number on average Nusslet number for different Eckert numbers for Re = 50, Ra = 10, 000 and Pr = 2. (a) On inner Cylinde; and (b)on outer cylinder.

Figure 10: Effect of Eckert number on distribution of local Nusslet number on (a) inner and (b) outer cylinders for Re = 100, Pr = 20, Ra = 10, 000, Ha = 0.0 andon averaged Nusselt number against Reynolds number on (c) inner and (d) outer cylinders for Ha = 0, Ra = 20, 000 and Pr = 2.

figure is the linear relation between these two variables for bothinner and outer surfaces. Of course, the slopes of these linesare relatively dependent on the Prandtl number of the fluid, insuch a way that the average Nusselt number for the fluid withlarger Prandtl number is more influenced by Eckert number

variations, which is because of the considerable heat generatedwithin the fluid with the larger Prandtl number, due to viscousdissipation. If the Eckert number is large enough (Eck = 0.2)the heat generated causes the fluid temperature in the vicinityof the inner cylinder surface to be larger than the cylinder

H.R. Mozayyeni, A.B. Rahimi / Scientia Iranica, Transactions B: Mechanical Engineering ( ) ( ), – 9

Figure 11: Effect of Prandtl number on the streamlines for Re = 100.0, Eck = 0.0007, Ra = 10, 000, and Ha = 20.0. (a) Pr = 0.5, (b) Pr = 1.0, (c) Pr = 2.0 and (d)Pr = 5.0.

Figure 12: Effect of Prandtl number on temperature contour for Re = 100.0, Eck = 0.0007, Ra = 10, 000, and Ha = 20.

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temperature. In this unexpected situation, the non-dimensionaltemperature gradient in the vicinity of the inner cylinder ispositive, and therefore the direction of the heat transfer wouldbe from the fluid to the inner cylinder. It is necessary tomentionthat in this situation (Eck = 0.2, Pr = 20), the average Nusseltnumber on the inner surface, in general, should be reported asan absolute value, in order to avoid negative values.

Next is the investigation of the effect of σ0, the ratio of innercylinder diameter to the outer and inner radius difference onflow and heat transfer. As was mentioned above, the effect ofapplication of a magnetic field is in the direction of suppressingthe flow and heat transfer. As can be seen in Figure 14, as theparameter σ0 decreases, there is need for a magnetic field oflesser intensity to suppress the heat transfer mechanism, or inother words, remove the buoyancy effects. For Ha = 50 andσ0 = 0.5, not much buoyancy force is observed, but for thesame magnetic field intensity, and σ0 = 5, many eddies areseen, which is an indication of the existence of these forces inthe annulus.

In this part, for completeness, some transient results arepresented in different non-dimensional time values for flowand heat transfer for selected values of Hartman number(Figures 15 and16), aswell as localNusselt number on inner andouter cylinders (Figures 17 and 18) and distribution of velocityand temperature in the annulus (Figures 19 and 20).

Figure 13: Average Nusselt number on the inner and outer surfaces of thecylinder against Eckert number for two different fluids with Prandtl numbersPr = 1 and Pr = 20 for Re = 100, Ra = 10, 000 and Ha = 0.0.

Aswe know, for larger fluid Prandtl number, themomentumflow transfers faster than heat transfer. This can be seen clearlyin Figure 21, which is for a fluid with Pr = 5, and shows

Figure 14: Comparison of streamlines in different σ0 situations for Re = 100, Pr = 1, Eck = 0.0, and Ra = 10, 000.

H.R. Mozayyeni, A.B. Rahimi / Scientia Iranica, Transactions B: Mechanical Engineering ( ) ( ), – 11

Figure 15: Streamlines at different times for Re = 100, Pr = 1, Eck = 0.0, Ra = 10, 000, and di erent values of Hartman number.

the distribution of the circumferential component of velocityreaching a steady-state quicker than the temperature at alocation of zero degrees. There is notmuchdifference in velocityat t = 50 compared to t = 210, but comparing temperaturedistribution at t = 50 with later times, it indicates that muchmore time is still needed to reach steady-state.

5. Conclusions

In the present work, the finite volume method was usedto solve the set of governing equations using the SIMPLEalgorithm. The investigations were carried out in a wide rangeof 0 < Ha < 200, 1000 < Ra < 30, 000, 0 < Eck < 0.5,0.5 < Pr < 20 and 0.5 < σ0 < 5 of the flow, with differentRe numbers, to explore the effects of the above dimensionlessgroup numbers on overall flow patterns and heat transfer

rates. The results show that the application of a magnetic fieldchanges the balance of the buoyancy and forced convectionforces, and is in the direction of suppressing the patterns of thestreamlines, as well as temperature contours and, in turn, theheat transfer. An increase in magnetic field strength, therefore,reduces the Nusselt number on both surfaces of the cylinders.As we know, an increase in the value of Eckert number causesthe influence of viscous dissipation terms to enhance, which,in turn, suppresses the fluctuations of local Nusselt number onthe cylinder surfaces. Also, it was noticed that by increasingthe Eckert number of the high-Prandtl flow, the heat generateddue to viscous dissipation causes the fluid temperature in thevicinity of the inner cylinder to be higher than the cylindertemperature. Furthermore, the obtained results depict that the

12 H.R. Mozayyeni, A.B. Rahimi / Scientia Iranica, Transactions B: Mechanical Engineering ( ) ( ), –

Figure 16: Temperature contours at different times for Re = 100, Pr = 1, Eck = 0.0, Ra = 10, 000 and different values of Hartman number.

Figure 17: Nusselt number distribution on inner cylinder surface at different times for (a) Ha = 20, and (b) Ha = 200 and for Re = 100, Pr = 1, Eck = 0.0 andRa = 10, 000.

H.R. Mozayyeni, A.B. Rahimi / Scientia Iranica, Transactions B: Mechanical Engineering ( ) ( ), – 13

Figure 18: Nusselt number distribution on outer cylinder surface at different times for (a) Ha = 20, (b) Ha = 200 and for Re = 100, Pr = 1, Eck = 0.0, andRa = 10, 000.

Figure 19: Distribution of circumferential velocity components at different times at φ = 0 for (a) Ha = 0.0, (b) Ha = 100 and for Re = 100, Pr = 1, Eck = 0.0, andRa = 10, 000.

Figure 20: Temperature distribution at different times at φ = 0 for (a) Ha = 0.0 and (b) Ha = 100 for Re = 100, Pr = 1, Eck = 0 and Ra = 10, 000.

effects of an increase in Prandtl number and a decrease in theratio of the inner cylinder diameter to the outer and innerradius difference, like the effect of the magnetic field, are in the

direction of suppressing the flow and heat transfer. Moreover,the results of the unsteady state were presented in differentcases to complete the results studied in the present work.

14 H.R. Mozayyeni, A.B. Rahimi / Scientia Iranica, Transactions B: Mechanical Engineering ( ) ( ), –

Figure 21: Distribution of (a) circumferential component of velocity and (b) temperature at φ = 0 and at different times for Pr = 5, Re = 100, Eck = 0.0007, Ra =

10, 000 and Ha = 20.

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Hamid Reza Mozayyeni was born in Torbat, Iran, in 1986. He received his B.S.degree in Mechanical Engineering from Bahonar University of Kerman in 2008and his M.S. degree in Mechanical Engineering from Ferdowsi University ofMashhad in 2010, where he is currently a Ph.D. degree student.

Asghar Baradaran Rahimiwas born in Mashhad, Iran, in 1951. He received hisB.S. degree in Mechanical Engineering from Tehran Polytechnic, in 1974, and aPh.D. degree inMechanical Engineering from theUniversity of Akron, Ohio, USA,in 1986. He has been Professor in the Department of Mechanical Engineering atFerdowsi University ofMashhad since 2001. His research and teaching interestsinclude heat transfer and fluid dynamics, gas dynamics, continuummechanics,applied mathematics and singular perturbation.