mixed convective rarefied flows with symmetric and asymmetric heated walls

Computational Thermal Sciences, 5 (4): 261–272 (2013)

MIXED CONVECTIVE RAREFIED FLOWS WITHSYMMETRIC AND ASYMMETRIC HEATED WALLS

Hamid Niazmand∗ & Behnam Rahimi

Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad 9177948974,Iran

∗Address all correspondence to Hamid Niazmand E-mail: [email protected]

Mixed convection gaseous slip flows in an open-ended vertical parallel-plate channel with symmetric and asymmetricwall temperatures are numerically investigated. Buoyancy effects on developing and fully developed solutions are stud-ied by solving the Navier-Stokes and energy equations using a control volume technique and SIMPLE algorithm. Thevelocity and temperature fields are examined for different values of Knudsen number, mixed convection parameter, andtemperature ratio. It is found that decreasing temperature ratio leads to higher velocity slip along this wall. Moreover,the friction coefficient increases with increasing mixed convection parameter on the hot wall, whereas it decreases onthe cold wall. It is also noticed that with an increase in mixed convection parameter, the net heat absorbed by the fluidbecomes higher and the Nusselt number increases. Furthermore, the centerline pressure drops in the early sections of thechannel due to the dominance of the viscous effects; however, it builds up later along the channel, when the buoyancyforces gain weight and play the dominant role.

KEY WORDS: microchannel, mixed convection, numerical simulation, slip/jump, buoyancy force

1. INTRODUCTION

Gas flows in microchannels are associated with some de-grees of rarefaction effects, which are well characterizedby a dimensionless parameter called the Knudsen num-ber, defined as the ratio of the mean free path of thegas molecules to the characteristic length of the system(Gad-El-Hak, 2001). For Knudsen numbers in the rangeof 0.001 < Kn < 0.1, which is called the slip flowregime, slight rarefaction effects are present. It is well es-tablished that for this regime the continuum conservationequations can simulate the gaseous bulk flow. However,the boundary conditions should be modified with the ve-locity slip and temperature jump conditions (Karniadakisand Beskok, 2002; Beskok et al., 1996; Beskok and Kar-niadakis, 1999).

In contrast to the forced convection that has receivedproper attention in the literature, very limited informationis available with regard to the rarefaction effects on mixedconvection. One of the early studies on the mixed convec-tion in microchannels was carried out by Avci and Aydin(2007a) in which they analytically investigated the fully

developed mixed convection in a vertical parallel-platemicrochannel with asymmetrical wall temperatures. Thesame authors further extended their study (Avci and Ay-din, 2007b) to microchannels with walls at uniform heatfluxes. Recently, Avci and Aydin (2009) studied analyti-cally the fully developed mixed convective heat transferin a vertical microannulus between two concentric micro-tubes.

Regarding the rarefaction effects on the natural con-vection, there are a few studies available in the literature.Chen and Weng (2005) analytically studied fully devel-oped natural convection in a vertical parallel-plate mi-crochannel. Biswal et al. (2007) numerically investigatedflow and heat transfer characteristics in the developing re-gion of an isothermal microchannel. They showed that therarefaction effects result in heat transfer enhancements.Recently, Buonomo and Manca (2010) numerically inves-tigated the steady-state developing natural convection ina vertical parallel-plate channel for asymmetric uniformheat fluxes at reduced pressure environment. To the bestof our knowledge, no investigation has yet been made toanalyze the rarefaction effects on developing mixed con-

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262 Niazmand & Rahimi

NOMENCLATURE

cp specific heat at constant pressure Greek SymbolsD channel width β thermal expansion coefficientg gravitational acceleration γ specific heat ratio,Cp/CvGr Grashof number,ρ2βg (Th − T0)D

3/µ2 Γ dimensionless local frictionH channel height coefficient,k thermal conductivity (D/v0) [(∂v)/(∂x)]|gKn Knudsen number,λ/D θ dimensionless temperature,Nu local Nusselt number, (T − T0)/(Th − T0)

[−k(∂T/∂x|g)D]/[(Tw − Tm)k] λ molecular mean free pathp pressure µ dynamic viscosityP dimensionless pressure,(p− p0)/pv

20 ρ density

Pr Prandtl number,(µCp)/k σt thermal accommodation coefficientrT temperature ratio,(Tc − T0)/(Th − T0) σv tangential momentum accommodationRe Reynolds number,(ρv0D)/µ coefficientT temperature (K)u velocity component inx direction (m/s) SubscriptsU dimensionless velocity component 0 ambient values

in x direction,(uRe)/v0 c cold wallv velocity component iny direction (m/s) h hot wallV dimensionless velocity component g gas value near the wall surface

in y direction,v/v0 m mean temperaturex, y coordinate system s slip/jump valuesX, Y dimensionless coordinate system w wall values

vective gas flows. The aim of the present study is to carryout a computational study of mixed convective slip flowin the entrance and fully developed regions of the verti-cal open-ended parallel-plate channel with symmetric andasymmetric uniform wall temperatures.

2. PROBLEM FORMULATIONS

Consider a vertical parallel-plate channel and Cartesiancoordinatesx and y, as shown in Fig. 1. The channelheight,H, is chosen 50 times larger than its width,D,to ensure fully developed flow conditions at the channelexit. The ambient air at temperatureT0 with uniform ve-locity v0 enters the channel. The channel walls are kept atsymmetrical or asymmetrical uniform and constant tem-peratures. Considering the common Boussinesq approxi-mation, the governing equations described by continuity,

momentum, and energy equations for a two-dimensional,steady, laminar and incompressible flow are as follows:

∂u

∂x+

∂v

∂y= 0, (1)

ρ

(u∂u

∂x+v

∂u

∂y

)=−∂p

∂x+

∂

∂x

(µ∂u

∂x

)+

∂

∂y

(µ∂u

∂y

), (2)

ρ

(u∂v

∂x+v

∂v

∂y

)=−∂p

∂y+

∂

∂x

(µ∂v

∂x

)+

∂

∂y

(µ∂v

∂y

)+ ρgβ (T − T0) , (3)

u∂(ρcpT )

∂x+v

∂(ρcpT )

∂y=

∂

∂x

(k∂T

∂x

)+

∂

∂y

(k∂T

∂y

), (4)

whereβ is volumetric thermal expansion coefficient,µ isdynamic viscosity,cp is specific heat at constant pressure,

Computational Thermal Sciences

Mixed Convective Rarefied Flows 263

FIG. 1: Flow geometry and the coordinates system

andk is thermal conductivity. Based on the gas kinetictheory, the Maxwell slip model relates the velocity slip tothe local velocity gradient at the wall as (Maxwell, 1879)

vs =2− σv

σvKnD

(∂v

∂x

)g

. (5)

For molecules that are not thermally accommodatedwith the wall, there is a temperature discontinuity knownas temperature jump, which can be expressed accordingto gas kinetic theory as (Kennard, 1938)

Ts − Tw =2− σt

σt

2γ

γ+ 1KnD

(1

Pr

∂T

∂x

)g

, (6)

whereTs is the temperature of the gas layer adjacent tothe wall, Tw is the wall temperature, andγ is the spe-cific heat ratio. Also,σv andσt are tangential momentumand energy accommodation coefficients, which are deter-mined experimentally. However, their values for most en-gineering applications are approximately around 1, whichis also adopted in the present study (Chen and Weng,2006). For inlet boundary conditions,u = 0, v = v0, andT = T0 are assumed. Also, for all flow variables at the out-let, zero gradients are applied. The local Nusselt numbersat hot and cold walls are evaluated according to

Nu =−k[(∂T )/(∂x)|g

]D

(Tw − Tm) k, (7)

where the mean temperature is defined as

Tm =

∫D

0vTdx∫D

0vdx

. (8)

The local friction coefficients at hot and cold walls aredefined as

Γ =D

v0

(∂v

∂x

∣∣∣∣g

). (9)

3. NUMERICAL MODELING AND VALIDATIONS

The governing equations are solved with a homemadenumerical code written in FORTRAN language using afinite volume approach. The convective terms are dis-cretized using the power-law scheme, which is a vari-able scheme between the second-order central differenc-ing and first-order upwind schemes (Patankar, 1980),while for diffusive terms the central differencing is em-ployed. Coupling between the velocity and pressure ismade with the SIMPLE algorithm (Patankar, 1980). Also,the boundary conditions are introduced implicitly as lin-ear source terms. The resulting system of the discretizedlinear algebraic equations is solved with an alternatingdirection implicit (ADI) scheme. Extensive computationshave been performed to identify the number of grid pointsthat produces reasonably grid-independent results. Thequantities examined for this purpose are the maximumand minimum of both velocity components and also themaximum temperature within the solution domain. It wasfound that the cross-sectional velocity component is moresensitive to the grid resolution. An illustration of thisstudy is presented in Table 1, for Kn = 0.1, Gr/Re =50, channel aspect ratio = 50, and the temperature ratio,

TABLE 1: Grid resolution effects on the cross-sectionalmaximum and minimum velocities

Grid umax umin

30×300 0.01106 –0.012030×600 0.01204 –0.012960×300 0.01183 –0.012660×600 0.01253 –0.0134120×600 0.01264 –0.013260×1200 0.01246 –0.0131120×1200 0.01245 –0.0131

Volume 5, Number 4, 2013


rT = (Tc − T0)/(Th − T0), equal to 0.5. Consequently,a system of 60×600 grid point with the expansion ratiosof 1.03 and 1.005 is adopted for the cross-sectional andaxial directions, respectively.

As for convergence of the iteration procedure, therelative variation of a given unknown must satisfy the|(Ψ−Ψold)/Ψmax| ≤ 10−5 criterion, whereΨ repre-sents the variable for which the problem is solved at thecurrent iteration, “old” represents the corresponding valueat the previous iteration, andΨmax is the maximum valueof the variable in the entire domain. The convergence cri-terion, 10−5, was chosen as the result of numerical stud-ies for having a reasonable computational cost and accu-racy altogether. A more conservative convergence crite-rion leads to higher computational costs. The effects ofa more conservative criterion such as 10−6 on the localNu number value for the case of Kn = 0.05 andrT = 0.5as compared to the convergence criterion of 10−5 is lessthan 0.3%.

It is essential to note that in typical low Pe flows asso-ciated with mixed convection in mini- and microchannels,the effect of axial conduction is considerable, such thatthe inlet temperature can be affected. Therefore, care mustbe taken in imposing a uniform inlet ambient temperaturein numerical simulations. In fact it is found that when ax-ial conduction is noticeable, applying constant inlet tem-perature leads to such considerable temperature gradientsjust at the inlet that the overall energy balance along alldomain boundaries is affected. To reduce the relative im-portance of the axial conduction with respect to convec-tion and to ensure that the uniform inlet temperature isphysically relevant in the present study, the flow Pe num-ber must be high enough. The available parameters forincreasing the Pe number in the present problem are theinlet velocity and the channel width. However, variationsin the inlet velocities are rather limited due to the natureof mixed convective flows. Therefore, the channel widthis the only parameter left for adjusting the Pe number.Our numerical experimentation indicated that the overallenergy balance to a certain accuracy level cannot be metif the uniform inlet temperature is not physically correctwhen the axial conduction is comparable to the convec-tion. Therefore, the overall energy balance has been im-posed to all cases considered in the present study to en-sure that the applied uniform inlet boundary conditions isconsistent with the physics of the problem and the resultsare reliable.

It is also worth mentioning that in the case of consid-erable axial conduction, the extended domain should beconsidered in order to properly simulate the flow and ther-

mal behaviors in the outflow region (Bello-Ochende andBejan, 2004; Naylor et al., 1991; Andreozzi et al., 2002,2005, 2007, 2009; Da Silva and Bejan, 2005; Khanaferand Vafai, 2000, 2003; Boetcher and Sparrow, 2009; An-dreozzi and Manca, 2010). This is also true for low chan-nel aspect ratios at moderate Rayleigh or Reynolds num-bers flows (not the case of the present study).

The numerical scheme has been validated by compar-ing the fully developed axial velocity profiles with the an-alytical solution of Avci and Aydin (2007a) as shown inFig. 2. The flow parameters are the temperature ratio andmodified mixed convection parameter, Gr/Re, which areset equal to 0.5 and 50, respectively, for Knudsen numbersof 0.1 and 0.05. The axial velocity and length are nondi-mensionalized by the inlet velocity,v0, and the chan-nel width, D, respectively. Fully developed velocity pro-files obtained numerically are in agreement with the cor-responding analytical results of Avci and Aydin (2007a)within less than 3%.

4. RESULTS AND DISCUSSION

The mixed convection parameter, Gr/Re = [ρ2βg× (Th − T0)D

3]/µ2/(ρv0D)/µ, indicating the relative

importance of natural convection over forced convection,is a nondimensional parameter that governs the flow andwill be used in presenting the results. All thermophysi-cal properties are evaluated at the inlet air temperature of

FIG. 2: Comparison of the fully developed velocity pro-files with those of Avci and Aydin (2007a) atrT = 0.5,Gr/Re = 50 and Kn = 0.05 and 0.1



T0 = 273.15 K. The volumetric thermal expansion coeffi-cient,β, is obtained according to the ideal gas approxima-tion. The slip flow regime, 0.001≤ Kn ≤ 0.1, is consid-ered which is typical in microdevices and systems in rar-efied atmosphere. The applied hot wall temperatures are313.15◦ and 353.15◦, corresponding to the Gr/Re equalto 30 and 60. The cold wall temperatures are determinedbased on the hot wall and inlet air temperatures and thespecified temperature ratio,rT = (Tc − T0)/(Th − T0).The effects of the Knudsen number, the temperature ra-tio, 0 ≤ rT ≤ 1, and the mixed convection parameter,Gr/Re, on fluid flow and heat transfer through a verticalparallel-plate channel with asymmetric wall temperaturesare examined.

The developments of velocity profiles in axial andcross-sectional directions are presented in Figs. 3(a) and3(b), respectively. Both the axial and cross-sectional co-ordinates and axial velocity are normalized with respectto the inlet velocity and channel width, respectively. Theslip velocity is considerable close to the inlet, where theuniform inlet velocity profile is dragged by walls lead-ing to the formation of large normal velocity gradients.In the early sections of the channel, the maximum veloc-ity shifts toward the hot wall, due to the strong buoyancyeffects which impose an upward flow in close vicinity ofthe hot wall. However, as the buoyancy force extends tothe core section of the channel, the maximum velocityslightly moves toward the middle of the channel. A cross-flow field is formed close to the entrance region becauseof the uniform inlet velocity profile and the dragging ef-fects of the walls as shown in Fig. 3(b). The negativecross-flow field is stronger than the positive field, which isa direct consequence of mass conservation and the higheraxial velocities near the hot wall.

The effects of the Knudsen number and mixed convec-tion parameter on the axial variations of the slip velocitieson the hot and cold walls are presented in Figs. 4(a)–4(b).As indicated by Eq. (5), the velocity slip is directly re-lated to Kn, and therefore, increases at higher Knudsennumbers. For a constant temperature ratio ofrT = 0.5,increasing Gr/Re at a given Kn results in a stronger buoy-ancy force, and therefore, shifts the maximum velocitytoward the hot wall, which in turn leads to an increaseand a decrease in slip velocity on hot and cold walls, re-spectively. The effect of temperature ratio,rT , on the slipvelocity is illustrated in Figs. 5(a) and 5(b) for Kn = 0.1and Gr/Re = 60. DecreasingrT shifts the maximum axialvelocity toward the hot wall leading to higher and lowerslip velocities close to the hot and cold walls, respectively.The increases in slip velocities after their initial drops in

(a)

(b)

FIG. 3: (a) Streamwise velocity profiles at different axiallocations for Gr/Re = 60, Kn = 0.1 atrT = 0.5;(b) cross-flow velocity profiles at different axial locations for Gr/Re= 60, Kn = 0.1 atrT = 0.5

Figs. 4(a) and 5(a) can be explained as follows. In the en-try region of the channel due to asymmetrical temperaturedistributions in cases withrT ̸= 1, the buoyancy forceshifts the axial velocity profiles toward the hot wall. Thiseffect, which is stronger at lowerrT and higher mixedconvection parameter, results in a rise in slip velocity near



FIG. 4: Variations of slip velocities along(a) the hot wall;and(b) the cold wall forrT = 0.5 and different values ofGr/Re and Kn

the hot wall. However, it must be noted that the buoyancyforce is almost negligible close to the channel inlet, andgains weight along the channel as heat spreads across thechannel and reduces the fluid density. Clearly, this slip ve-locity rise cannot exist near the cold wall since the buoy-ancy effects decrease the velocity gradients and slip ve-locities accordingly.

The variations of the friction coefficient as a functionof the streamwise location forrT = 0.5 and different val-ues of Gr/Re and Kn are plotted in Figs. 6(a) and 6(b)

FIG. 5: Variations of slip velocities along(a) the hot wall;and (b) the cold wall for Gr/Re = 60 and Kn = 0.1 anddifferent values ofrT

for hot and cold walls, respectively. As expected, the fric-tion coefficient decreases with increasing Kn number forboth hot and cold walls. It is further noticed that with anincrease in mixed convection parameter the friction coef-ficients increase on the hot wall, while they decrease onthe cold wall. It is also apparent from Fig. 6(a) that thereis an increase in friction coefficient after its initial drop inearly sections of the channel, which is mainly because ofasymmetrical temperature distribution and the buoyancyeffects as discussed with respect to Fig. 4(a).



FIG. 6: Variations of the local friction coefficients along(a) the hot wall; and(b) the cold wall forrT = 0.5 anddifferent values of Gr/Re and Kn

Figure 7 exhibits the nondimensional cross-sectionaltemperature profiles,θ = (T − T0)/(Th − T0), at differ-ent axial locations for flow conditions the same as thosein Fig. 3(a). Clearly, the temperature jump is consider-able close to the inlet, where uniform temperature inflowis exposed to the heated walls leading to the formationof large normal temperature gradients at walls. It is alsonoticeable that fluid temperature gradients are positive onthe cold wall in early sections of the channel, where its ad-jacent fluid temperature is lower than the cold wall. How-

FIG. 7: Cross-sectional temperature profiles at differentaxial locations for Gr/Re = 60, Kn = 0.1 atrT = 0.5

ever, they become negative later on (Y > 10) in asymmet-ric cases, indicating the fluid temperature adjacent to thecold wall is higher than the cold wall temperature itself,θw,c = 0.5 for the case of temperature ratio ofrT = 0.5due to the temperature jump condition. In the fully devel-oped region the heat is basically transferred from the hotwall to the cold wall.

For the same flow conditions as those in Figs. 4 and 5,rarefaction and buoyancy effects on the axial variations ofthe nondimensional temperature of gas layers adjacent tothe hot and cold walls are presented in Figs. 8(a) and 8(b).The influence of Gr/Re and Kn is illustrated in Fig. 8(a)for a constant value of the temperature ratio ofrT = 0.5.According to Eq. (6), the temperature jump is directlyrelated to Kn, and therefore, increases at higher Knud-sen numbers. The points of zero temperature jump on thecold wall, as seen in Fig. 8(a), correspond to the adiabaticpoints, which will be discussed later in connection withFig. 11. The effects of temperature ratio on temperaturevariations along the walls for Kn = 0.1 and Gr/Re = 60 areillustrated in Fig. 8(b). Lower values ofrT correspond tothe stronger asymmetrical temperature fields, leading tohigher temperature gradients and temperature jumps.

In Figs. 9(a) and 9(b) the effects of the flow parameterson the axial variations of nondimensional mean tempera-ture are studied. Figure 9(a) shows the effects of Gr/Reand Kn for a givenrT = 0.5, where it is observed that themean temperature requires a longer distance to approach



(a)

(b)

FIG. 8: (a) Streamwise variations of the gas layer temper-ature adjacent to the hot and cold walls forrT = 0.5 anddifferent values of Gr/Re and Kn;(b) streamwise varia-tions of the gas layer temperature adjacent to the hot andcold walls for different values ofrT at Gr/Re = 60 andKn = 0.1

its fully developed asymptotic value at higher values ofKn number. It is further noticed that with an increase inmixed convection parameter, the net heat absorbed by thefluid becomes higher due to shifting of the axial velocitytoward the hot wall. The effects of the temperature ra-tio on the mean temperature for Gr/Re = 60 and Kn =0.1 are illustrated in Fig. 9(b). Clearly, for a symmetric

(a)

(b)

FIG. 9: (a) Axial variations of nondimensional meantemperatures for different values of Gr/Re and Kn atrT = 0.5; (b) axial variations of nondimensional meantemperatures for Gr/Re = 60 and Kn = 0.1 and differentvalues ofrT

heating condition, higher mean temperature is obtained ascompared to any asymmetrical cases. The rapid increasein mean temperature in the early sections of the channel,which is stronger at higherrT , is due to the higher heattransfer rates in this region as discussed next.



The axial variations of heat transfer rate, Nu, forrT = 0.5 and several values of the Kn and Gr/Re areplotted in Figs. 10(a) and 10(b) for hot and cold walls,respectively. It can be seen that the Nusselt number de-creases with an increase in Knudsen number, which re-quires some considerations. It is expected that slip veloc-ity increases the heat transfer rates at walls due to theincrease in convective effects, where they are relativelyweak. On the other hand, considering the nature of thetemperature jump that reduces the normal temperature

FIG. 10: Variations of the Nusselt number along(a) thehot wall; and(b) the cold wall for different values ofGr/Re and Kn atrT = 0.5

gradients, it basically acts as a thermal contact resistance,which reduces the heat transfer. Note that both effects areenhanced at higher Knudsen numbers. Apparently, the re-duction effect of the temperature jump is the prevailingfactor in the present case, and therefore heat transfer ratedecreases with an increase in Kn. It is further noticed thatthe Nu number increases with an increase in Gr/Re be-cause of the stronger buoyancy effects.

Despite the fact that the Nusselt number of the hot wallis always positive and continuously decreasing along thechannel, the Nusselt number of the cold wall exhibits to-tally different behavior. A distinct feature is that Nuc dis-plays a discontinuity with a vertical asymptote. This dis-continuity occurs when the mean temperature reaches thetemperature of the cold wall and the denominator in thedefinition of Nuc becomes zero. Therefore, there is an un-defined point in the heat transfer rate of the cold wall. Leftof this point, Nu is positive, whereas on the right, Nu isfirst negative, becomes zero, and finally obtains a positivevalue. The zero Nu value after the undefined point is basi-cally corresponding to an adiabatic point on the cold wallwith zero temperature gradient as shown in Fig. 11. Thevariations of the cold wall temperature gradients along thechannel are illustrated in this figure for more clarification.The temperature gradients approach zero from the highinitial positive values relatively close to the channel inlet.With the help of this figure the negative value of the Nus-selt number along the cold wall can be explained. Notethat there is a change in the temperature gradient sign at

FIG. 11: Variations of temperature gradients along thecold wall for different values of Gr/Re and Kn atrT = 0.5



the cold wall along the channel, and also the sign of thedifference between cross-sectional mean temperature andthe wall temperature changes. However, these changesare not occurring at the same location, therefore, thereis a portion of channel with the negative Nusselt numberalong the cold wall.

It must be emphasized that this behavior of Nusseltnumber along the cold wall has nothing to do with rarefac-tion effects. In fact our numerical calculations show thatchanges in the sign of Nusselt number close to the unde-fined point also occur for Kn = 0, although in a more lim-ited axial length. Apparently, rarefaction effects extendthis phenomenon to a somewhat larger channel length. Itis also clear that increasing Kn and Gr/Re increases theabsolute values of the temperature gradients leading tohigher Nu on the cold wall.

The variations of the pressure along the channel arealso interesting to examine. Figure 12 shows the axialvariations of nondimensional pressure along the center-line of the channel for different values of Gr/Re andrTat Kn = 0.05. Pressure is nondimensionalized accordingto P = (p− p0)/pv

20 , where the pressure in the center of

the channel inlet,p0, is taken as the reference pressure.The centerline pressure obtains negative values close tothe entrance region, and then becomes positive thereafter,which can be physically explained as follows. Close to theinlet, the dominant viscous forces decrease the pressure,while buoyancy force, which becomes more effective af-ter the entry region, increases the pressure especially at

higher mixed convection parameters and temperature ra-tios. It is also interesting to note that the buoyancy forcenot only overcomes the viscous effects but also results ina considerable pressure buildup along the channel.

To obtain a more detailed picture of the pressure vari-ations in the solution domain, Fig. 13 is plotted, whichshows the axial variations of the dimensionless pressuregradients for the same flow conditions and values ofGr/Re andrT as those in Fig. 12. It is observed thatthe pressure gradients develop from high negative val-ues at the channel inlet, which sharply become positive;and then gradually approach their asymptotic values inthe fully developed region. The maximum pressure gra-dients that occur in early sections of the channel can beattributed to the rapid increase in fluid temperature in thisregion, as evidenced by the sharp increase in the meantemperature shown in Figs. 9(a) and 9(b).

5. CONCLUSIONS

A numerical analysis on the developing and developedmixed convective rarefied flows through a symmetric andasymmetric heated vertical channel has been performed.The governing equations subject to the slip/jump bound-ary conditions are solved using a control volume tech-nique. The numerical scheme validation was establishedthrough comparison of the numerical velocity profileswith their analytical counterparts. The effects of rarefac-tion, buoyancy, and temperature ratio on both developing

FIG. 12: Axial variations of the nondimensional centerline pressure for Kn = 0.05 and(a) rT = 0.5 and differentvalues of Gr/Re; and(b) Gr/Re = 60 and different values ofrT



FIG. 13: Axial variations of the centerline pressure gradients for Kn = 0.05 and(a) rT = 0.5 and different values ofGr/Re; and(b) Gr/Re = 60 and different values ofrT

and fully developed velocity and temperature profiles areexamined in detail. The major findings from the presentstudy can be summarized as follows:

(i) Increasing Gr/Re and decreasingrT result in an in-crease and decrease in slip velocities on the hot andcold walls, respectively.

(ii) The friction coefficient increases and decreases withincreasing mixed convection parameter on the hotand cold walls, respectively.

(iii) In asymmetric cases, the temperature jumps in theearly sections of the channel are positive on the coldwall; afterwards, they become negative, causing thefluid temperatures to be higher than the wall temper-ature.

(iv) The net heat absorbed by the fluid becomes higherwith an increase in mixed convection parameter.

(v) The Nu on the hot wall increases at higher values ofGr/Re and lower values of Kn.

(vi) Increasing Kn and Gr/Re increases the absolute val-ues of the temperature gradients and Nusselt numberon the cold wall.

(vii) The centerline pressure drops in the early sections ofthe channel because of the viscous effects, and thenstarts to build up due to the buoyancy effects, which

is more enhanced at higher temperature ratios andmixed convection parameters.

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mixed convective rarefied flows with symmetric and asymmetric heated walls

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