International Journal of Mathematical Archive-3(7), 2012, Available online through ISSN

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1 International Jornal of Mathematical Archive-3(7) 49- Available online throgh ISSN 9 46 CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A PARALLEL FLAT WALL DIVIDED BY A PERFECTLY CONDUCTIVE BAFFLE J. C. Umavathi Professor in Department of Mathematics Glbarga Universit Glbarga-8 6 Karnataka India (Received on: -6-; Accepted on: -7-) ABSTRACT The stead two-dimensional free convection flow of a viscos flid in a vertical doble passage wav channel has been investigated analticall. The channel is divided into two passages b means of a thin perfectl condctive plane baffle and the velocit and temperatre will be individal in each streams. The governing eqations of the flid and the heat transfer have been solved sbject to the relevant bondar conditions b assming that the soltion consists of two parts; a mean part and distrbance or pertrbed part. To obtain the pertrbed part of the soltion the long wave approximation has been sed and to solve the mean part well known approximation sed b Ostrach [] has been tilized. Reslts are presented graphicall for the distribtion of velocit and temperatre fields for varing phsical parameters sch as baffle position Grashof nmber wall temperatre ratio and prodct of non-dimensional wave nmber and space-coordinate at different positions of the baffle. The relevant flow and heat transfer characteristics namel skin friction and the rate of heat transfer at both the walls has been discssed in detail. Kewords: Wav vertical channel baffle viscos flid. INTRODUCTION Natral convection in vertical channels and tbes has been stdied extensivel becase of its interest in man practical sstems inclding cooling of electronic eqipment [ 3] chimnes and frnaces heat exchangers and solar energ collectors nclear engineering and geo-phsical flows. Earl experiments were carried ot b Elenbaas [4] and Ostromove [] and later b Sparrow and Bahrami [6] for isothermal tbes and plates heated at temperatres above the ambient temperatre. Bodoia and Osterle [7] analzed the flow in a vertical channel with niform wall temperatre; Engel and Mller [8] applied an integral method to channels of infinite height with niform wall temperatre or niform srface heat flx. When the channel is divided into several passages b means of plane baffles as sall occrs in heat exchangers or electronic eqipment it is qite possible to enhance the heat transfer performance between the walls and flid b the adjstments of each baffle position and strengths of the separate flow streams. In sch configrations perfectl condctive and thin baffles ma be sed to ovoid significant increase of the transverse thermal resistance. Stronger streams ma be arranged to occr within the passages near the channel wall srfaces in order to cool or heat the walls more effectivel. Even thogh the sbject of channel flow has been investigated extensivel few stdies have so far evalated for these effects. It is necessar to std the heat and mass transfer from an irreglar srface becase irreglar srfaces are often present in man applications. It is often encontered in heat transfer devices to enhance the heat transfer. Mixed convection from irreglar srfaces can be sed for transferring heat in several heat transfer devices. For example flat-plate solar collectors and flat-plate condensers in refrigerators. The natral convection heat transfer from an isothermal vertical wav srface was first stdied b Yao [9 ] and sing an extended Prandtl s transposition theorem and a finite difference scheme. He proposed a simple transformation to std the natral convection heat transfer from isothermal vertical wav srfaces sch as sinsoidal srface. Molic and Yao [] solved for mixed convection with thermal diffsion. Along a vertical wav srface Chi and Cho [] stdied the natral convection heat transfer in micropolar flids. Chen and Wang [3] analzed transient free convection along a wav srface in microflids. Malashett et al. [4] stdied on magnetoconvective flow and heat transfer between vertical wav wall and a parallel flat wall. Umavathi et al. [-7] also stdied two flid flow in a vertical wav channel. Corresponding athor: J. C. Umavathi Professor in Department of Mathematics Glbarga Universit Glbarga-8 6 Karnataka India International Jornal of Mathematical Archive- 3 (7) Jl 49

2 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- Das and Mahmd [8] analzed the free convection inside both the bottom and the ceiling wav and isothermal enclosre. The indicated that onl at the lower Grashof nmber the heat transfer rate rises when the amplitde wavelength ratio changes from zero to other vales. Recentl Dalal and Das [9] made a nmerical soltion to investigate the inclined right wall wav enclosre with spatiall variable temperatre bondar conditions. Oztop [] applied the electric grid generation and obtained sinsoidal dct geometr to enhance the forced convection heat transfer. Varol and Oztop [] investigated the effects of inclination angle on the laminar natral convection heat transfer and flid flow in a wav solar collector in stead state regime. The observed that the inclination angle is the most important and effective parameter on heat transfer which can be sed to control the heat transfer inside the collector. Most of the previos stdies abot vertical wav srfaces are concerned with micro flids or poros media. Recentl Jang et al. [] has stdied nmericall on natral convection heat and mass transfer along a vertical wav srface. Yet the preceding literatre srve shows that mixed convection heat and mass transfer in Newtonian flid along a vertical wav srface has not been well investigated. The flow and thermal field in an isothermal vertical wav enclosre was stdied b Mahmd et al. [3] for Grashof nmber and orientations. Khalil et al. [4] analzed the natral convection heat transfer in a wav poros enclosre sing non-darcian model. The fond that the amplitde of the wav srface and the nmber of ndlation affected the heat transfer characteristics. Hasnaoi et al. [] Ben-Nakhi and Camkha [6] Dagtekin and Oztop [7] investigated also the natral convection in enclosres with a partition. The presence of a partition was the effective parameter on heat transfer. Tansmim and Collins [8] did a nmerical std on heat transfer in a sqare cavit with a baffle located on the hot wall. The std showed that the baffle has a significant effect on increasing the rate of heat transfer compared with a wall withot baffle. Recentl Prathap Kmar et al. [9 3] analzed free convection in a doble passage wav channel sing Walters Flid. Cheng and Shia [3] stdied the effects of a horizontal baffle on the heat transfer characteristics of plsating opposing mixed convection in a parallel vertical open channel. The inflences of the dimensionless plsating freqenc Strohal nmber and magnitde Prandtl nmber and baffle position on the velocit and temperatre distribtion and long time average Nsselt nmber variation for the sstem at varios Renolds nmber and sqare of mixed convection parameter were explored in detail. The reslt showed that the channel with both flow plsation and a baffle gives the best heat transfer. Heat transfer enhancement in a heat exchanger tbe b installing a baffle was reported b Nasirddin and Siddiqi [3]. The effect of baffle size and orientation on the heat transfer enhancement was stdied in detail. Three different baffle arrangements were considered. The reslts showed that for the vertical baffle an increase in the baffle height case a sbstantial increase in the Nsselt nmber bt the pressre loss is also ver significant. For the inclined baffles the reslt show that the Nsselt nmber enhancement is almost independent of baffle inclination angle with the maximm and average Nsselt nmber was % and 7% higher than that for the case of no baffle respectivel. For a given baffle geometr the Nsselt nmber enhancement was increased b more than a factor of two as the Renolds nmber decreased from to. Simlations were condcted b introdcing another baffle to enhance heat transfer. The reslt show that the average Nsselt nmber for the two baffles case is % higher than the one baffle case and 8% higher than the no baffle case. The above reslts sggest that a significant heat transfer enhancement in a heat exchanger tbe can be achieved b introdcing a baffle inclined towards downstream side with the minimm pressre loss. Mixed convection heat transfer in an inclined parallel-plate channel with a transverse fin located at lower channel wall was investigated nmericall b Yang et al. [33]. Keeping the interest of heat transfer problems owing to their practical applications an attempt is made to nderstand the flow and heat transfer in a vertical wav channel b introdcing a perfectl thin baffle.. MATHEMATICAL FORMULATION Consider a stead two-dimensional laminar fll developed free convection flow in an open ended vertical channel filled with prel viscos flid. The geometr nder consideration illstrated in Figre consists of wav wall sitated at Y = ε cos ( kx ) and flat wall at Y = d. The X -axis is taken verticall pward and parallel to the direction of boanc and the Y -axis is normal to it. The wav and flat walls are maintained at a constant temperatre T w and T respectivel. The flid properties are assmed to be constant and the Bossinesq approximation will be sed so that the densit variation is retained onl in the boanc term. Frther it is also assmed that the wavelength of the wav wall is large compared with the breadth of the channel. The channel is divided into two passages b means of a perfectl condcting thin baffle for which the transverse thermal resistance can be neglected (Cheng et al. [34] and Salah El Din [3 36]). IJMA. All Rights Reserved 496

3 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- Fig. : Phsical configration of the doble-passage channel Introdcing the following non-dimensional variables in the governing eqations for velocit and temperatre as X Y Ud i Vd i T Ts p x = = i = vi = θ = p = d d ν ν Tw Ts ρνd () Doing this one obtains the eqation of continit as i x vi + = () and the momentm eqation becomes i i Pi i i i + i = θ i v G x x x i i i i i + vi = + + v v P v v i x x (3) (4) The energ eqation becomes P i i i i i + vi = + θ θ θ θ x x () IJMA. All Rights Reserved 497

4 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- Sbject to the bondar conditions = = = on = εcos( λx) v θ = = v = v = on θ θ θ θ θ = θ + = + on = x x = v = θ = m on = = (6) where P = µ c k the Prandtl nmber ε = d the dimensionless amplitde parameter λ = kd the dimensionless p ε freqenc parameter m = ( T T ) ( T T ) the wall temperatre ratio and G d g β( T T ) s w s nmber. The sbscript s denotes qantities in the static flid condition. 3. SOLUTIONS 3 x w s ν = the Grashof Eqations (3) to () are copled non-linear partial differential eqations and hence finding exact soltions is ot of scope. However for small vales of the amplitde parameter ε approximate soltions can be extracted throgh the pertrbation method. The amplitde parameter ε is sall small and hence reglar pertrbation method can be strongl jstified. Adopting this techniqe soltions for velocit and temperatre are assmed in the form i( x ) = i + ε i( x ) vi( x ) = ε vi( x ) pi = pi( x) + ε pi( x ) θi( x ) = θi + εθi( x ) (7) where the pertrbations i vi pi and θ i are small compared with the mean or zeroth order qantities. Eqations () to (6) ield the following eqations. Zeroth order eqations i d d + Gθ = ; i d θi d = (8) First order eqations i vi + = (9) x i i Pi i i i + i = θ i x v x x G () vi Pi vi vi i = + + x x () θi dθi θi θi P i + vi = + x d x () In deriving the eqation (8) the constant pressre gradient term ( p ps ) x has been taken eqal to zero following Ostrach []. In view of eqation (7) the bondar condition in eqation (6) can be split p into the following two parts. Zeroth order bondar conditions = θ = on = dθ dθ = = θ = θ = on = (3) d d = θ = m on = IJMA. All Rights Reserved 498

5 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- First order bondar conditions iλxd iλxdθ θ = rp e v = = rp e on = d d = = v = v= θ = θ on = (4) = v = θ = on = where rp represents the real part The soltions for zeroth order velocit i and zeroth order temperatre θ i satisfing the eqation (8) and the bondar conditions (3) are given b Stream : θ = c+ c () 3 = l + l + d+ d (6) Stream : θ = c + c (7) = l + l + d + d (8) In order to solve eqations (9) to () for the first order qantit it is convenient to introdce stream fnction ψ in the following form i ψ i ψ i = v i = for i = (9) x The stream fnction approach redces the nmber of dependent variables to be solved and also eliminates pressre from the list of variables. Differentiate eqation () with respect to and differentiate eqation () with respect to x and then sbtract eqation () with eqation () which will reslt in the elimination of pressre p i. We assme stream fnction and temperatre in the following form ψ iλx iλx i( x ) = e ψi θ i = i ( x ) e t From the above eqations (9) to () after elimination of p i can be expressed in terms of the stream fnction ψ and t in the form ( ) ( ) i ( ) iv 4 3 i i i i i i i ii Gti ψ ψ λ + λ + ψ λ + λ + λ = () i λ i λ i ψθ i i t t = P t + () where i is the coefficient of imaginar part and sffix i = denotes stream- and stream- respectivel. Bondar conditions as defined in eqation (4) can be written in terms of ψ and t as = = on = ψ t ψ = ψ = t = θ on = ψ = ψ = ψ = ψ = on = () t t t t ψ = = = on = IJMA. All Rights Reserved 499

6 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- We restrict or attention to the real parts of the soltions for the pertrbed qantitiesψ t i and v i. Consider onl small vales of λ. On sbstitting r ( ) = λψ ( λ ) ψ λ r = r t = λ t (3) r = r r into eqations () to () we obtain to the order of λ the following set of ordinar differential eqations Zeroth order i iv i Gti t = (4) ψ = () First order i i i i i t = P t + ψ θ (6) iv iv iv i ii i ii i i K i i i Gti ψ = ψ ψ + ψ ψ + (7) Zeroth order bondar conditions in terms of stream fnction and temperatre are ψ = ψ = t = θ on = = = on = ψ t ψ = ψ = ψ = ψ = on = (8) t t t t ψ = = = on = First order bondar conditions in terms of stream fnction and temperatre are ψ = ψ = t = on = = = on = ψ t ψ = ψ = ψ = ψ = on = (9) t t t t ψ = = = on = The set of eqations (4) to (7) sbject to bondar conditions as given in eqations (8) and (9) have been solved exactl for ψ and t. From these soltions the first order qantities can be pt in the form = ( rp + i ip ) = i + i ( i ) ψ ψ ψ ψ λψ t = t + t = t + λt (3) rp ip j i i where sffix rp denotes the real part and i p denotes the imaginar part. Considering onl the real part the expression for first order velocit and temperatre become ( λ ) ψ λψ sin ( λ ) cos sin ( x) t t ( x) = cos x + x (3) i i i i i i v = λ λx ψ λψ λx (3) θ = cos λ λ sin λ (33) i i i The first order and total soltions are given in Appendix. 3.. Skin friction and Nsselt nmber The shearing stress σ x at an point in the flid in non-dimensional form is given b d σ x iλx iλx σ x = = + ε e + i ελe v (34) ρν IJMA. All Rights Reserved

7 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- At the wav wall εcos( λx) = skin friction takes the form cos sin cos σ w = σ + ε λx + λψ λx ψ λx (3) and at the flat wall = skin friction takes the form ( ) cos ( ) sin σ f = σ + ε ψ λx + λψ λx (36) where σ d = d = and σ ( ) d = d = are the zeroth order skin-friction at the walls. The non-dimensional heat transfer coefficient known as Nsselt nmber ( N ) is given b θ N = = θ + ε e θ iλx Re (37) At the wav wall εcos( λx) = + Nsselt nmber N w takes the form ε θ ( λ ) ( λ ) λ ( λ ) N = N + cos x + t cos x t sin x (38) w and at the flat wall = f ( ) ε ( ) cos ( λ ) λ ( ) sin ( λ ) N = N + t x t x (39) where N dθ = d = and N ( ) dθ = d = are zeroth order Nsselt nmber at the walls. Velocit and temperatre soltions are nmericall evalated for several sets of vales of the governing parameters. Also the wall skin friction σ σ and the wall Nsselt nmber N N are calclated nmericall and some of the qalitative interesting featres are presented. w f w f 3. Comparison of the Soltions with Salah El Din [36] in the presence of baffle To validate the reslts of the present model the problem is solved in the absence of the prodct of non-dimensional wave nmber and space co-ordinate and pressre gradient.. The dimensionless basic eqations () to () become d θi d = (4) d d j + Gθ = (4) j To compare the reslts the bondar conditions on temperatre are taken as in Salah El Din [36] i.e. θ = on = ; θ = on = dθ dθ on θ = θ = = (4) d d IJMA. All Rights Reserved

8 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- The bondar conditions on velocit are no-slip conditions on the bondar and vanishing of velocit at the baffle. That is = on = ; = on = = = on = (43) The soltions of eqations (4) and (4) sing bondar conditions (4) and (43) become θ = θ = 3 G = + d+ d 3 3 G = + d + d (44) (4) The above soltions agree ver well with the soltions of Salah El Din [36]. 3.3 Comparison of the Soltions with Rita and Alok [37] in the absence of baffle The above case validates the reslts for flat wall. To validate the reslts for wav wall the problem is solved in the absence of the baffle and visco-elastic parameter and compared the reslts with Rita and Alok [37]. The comparison of the present model is carried ot in two was. Shifting the baffle to the right wall and comparing the soltions of stream- with Rita and Alok [37]. The bondar conditions for this case become θ = = t = θ ψ θ = m = t = ψ t = t = ψ ψ ψ ψ ψ = = on = = ψ = on = (46) = = on = = = on = (47) The soltion of stream- with above bondar conditions become 3 θ = c + c ; = l + l + d+ d (48) 4 d3 3 d4 t = c 3 + c4 ; ψ = l d + d6 (49) t = Pi m + m + m + m + m + c + c ψ n n n n n n n d d = i d9 + d With the above soltions the velocit and temperatre field become ( cos sin ) cos sin cos ( x) ( t ) t sin ( x) = + ε λx ψ + λψ λx () v = ε λ λx ψ λψ λx () θ = θ + ε λ λ λ (3) Eqations () to (3) are compted and are tablated in Table. The above soltions agree ver well with Rita and Alok [37]. Shifting the baffle to the left wall and comparing the soltions of stream- with Rita and Alok [37]. IJMA. All Rights Reserved ()

9 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- The bondar conditions for this case become θ = = θ = m = t = t θ ψ ψ = ψ = ψ = on = = = on = (4) t = t = ψ ψ ψ ψ = = on = = = on = () The soltions of stream- with above bondar conditions become θ = c7 + c8 ; = l + l + d + d (6) 4 d3 3 d4 t = c9 + c ; ψ = l d + d6 (7) t = Pi m + m + m + m + m + c + c ψ n n n n n n n d d = i d9 + d With the above soltions the velocit and temperatre filed become ( cos sin ) cos sin cos ( x) ( t ) t sin ( x) = + ε λx ψ + λψ λx (9) v = ε λ λx ψ λψ λx (6) θ = θ + ε λ λ λ (6) Eqations (9) to (6) are compted and are tablated in Table 3b. These soltions agree ver well with Rita and Alok [37]. All he constants appeared in the above eqations are defined in the Appendix section 4. RESULTS AND DISCUSSION Free convective flow and heat transfer of viscos flid in a vertical channel one of whose walls is wav containing a thin condcting baffle is stdied analticall. The non-linear partial differential eqations governing the motion have been solved b a linearization techniqe wherein the flow is assmed to be of two parts; a mean part and a pertrbed part. Exact soltions are obtained for the mean part and the pertrbed pert is solved sing long wave approximation. The Prandtl nmber wave nmber and amplitde are fixed as.7. and. respectivel for the comptation whereas the Grashof nmber wall temperatre ratio and prodct of non-dimensional wave nmber and space-coordinate are fixed as - and respectivel for all the graphs except the varing one. The effect of the Grashof nmber G on the main velocit is shown in figre abc at three different baffle positions ( =.. and.8). As the Grashof nmber increases the main velocit increases near the left (hot) wall and decreases at the right (cold) wall. The larger the vale of G the stronger the pward velocit. Especiall for sfficient large vales of G a flow reversal phenomenon is predicted near the right (cold) wall as shown in figre a which can also be observed for Newtonian single passage flow examined b Ang and Wark [38]. For each graph the maximm point of the main velocit moves to the left (hotter) wall for increasing vales of G and ths the velocit decreases near the right (colder) wall. The effect of Grashof nmber G on the cross velocit is exactl opposite to the effect of G on the main velocit as seen in figre 3abc. That is as G increases the cross velocit v decreases near the left (hot) wall and increases at the right (cold) wall. Also as G increases the maximm point of the cross velocit moves to the right (colder) wall and ths the cross velocit decreases at the left (hot) wall. The effect of the Grashof nmber G on the temperatre is shown in Table-. It is noticed that the temperatre remains almost invariant at all baffle positions for G = and G =. Figre 4abc show the effect of the wall temperatre ratio m on the main velocit ( m = means that the average of the temperatres of the two walls is eqal to that of the static temperatre m = corresponds to the case that the temperatre of the flat wall is eqal to the static temperatre m = means that the wav and flat wall are maintained at eqal IJMA. All Rights Reserved 3 (8)

10 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- temperatre and m > implies that the wall temperatres are neqal). As m increases the main velocit increases in both streams at the baffle positions.. and.8 and the flow reversal is observed in stream for m =. It is also observed that the main velocit profile for m = lies in between m = (below) and (above). When the baffle position is near the left wall the variations of m are not ver effective on the main velocit bt its effect is dominant when the baffle position moves to the center and near to the right wall as seen in figre 4. The effect of m on the cross velocit is opposite to its effect on the main velocit. That is as m increases the cross velocit decreases at all baffle positions as seen in figre abc. The velocit profile for m = lies in between m = (below) and - (above). The effect of m on the cross velocit is not effective for the baffle position at the right wall when compared to the baffle position at the left and at the center of the channel. The effect of wall temperatre ratio m on the temperatre field is shown in figre 6abc. As m increases temperatre increases at all the baffle positions and magnitde of promotion also remains the same for =.. and.8. The effect of the prodct of non-dimensional wave nmber and space coordinate λ x on the main velocit is shown in figre 7abc. As λ x increases the main velocit increases near the wav wall and reverses its direction near the baffle position and remains constant at the flat wall at all baffle positions. The effect of λ x on the cross velocit is shown in figre 8abc. As λ x increases the cross velocit decreases in stream and increases in stream for the baffle positions at the left and at the center of the walls. However there is no effect of λ x at the right wall when =.8. The effect of λ x on the temperatre is presented in figre 9abc. It is seen that increasing λ x cases the temperatre to decrease at the wav wall while it remains nchanged at the flat wall for the baffle positions near the left center and right walls. The magnitde of sppression remains the same at an position of the baffle. The effect of the Grashof nmber is seen to increase the skin friction at both the wav wall and at the flat wall at all baffle positions. The effect of wall temperatre ratio m is to increase the skin friction at the wav wall and decrease at the flat wall. The effects wave nmber and the amplitde parameter do not show mch variation in the vales of skin friction at both the walls as observed in Table-a. The effect of the Grashof nmber is fond to increase the rate of heat transfer at the wav wall and decrease at the flat wall in magnitde at all baffle positions. The effect of the wall temperatre ratio is seen to decrease in magnitde the heat transfer at both walls at all baffle positions. The effect of the wave nmber λ and amplitde parameter ε is fond to increase in magnitde the rate of heat transfer at the wav wall and decrease at the flat wall. However their effect on the rate of heat transfer is not ver significant at both the walls as seen Table-b. The reslts obtained in 3. and the reslts obtained b Salah El Din [36] are displaed in Table-3a. This table shows that both the models show same vales at all baffle positions which will jstif the present model. The soltions obtained in section 3.3 and the soltions obtained b Rita and Alok [37] in the absence of viscoelastic parameter K are shown in Table- 3b. When the baffle is shifted to the right wall (stream-) the problem redced to single passage whose soltions agree with Rita and Alok [37] which jstifies the soltions of stream-. To jstif the soltions obtained in stream- the baffle is shifted to the right wall which again redces to single passage. The soltions obtained in stream- agree ver well with Rita and Alok [37].. CONCLUSION The flow and heat transfer of viscos flid in a vertical wav doble-passage channel inserting a perfectl condcting baffle is investigated. According to the reslts the following conclsions can be drawn:. The maxima of the main velocit profiles are obtained for increasing vales of the Grashof nmber and the wall temperatre ratio especiall when the baffle position is in the middle of the channel. As λ x increases the main velocit decreases at the wav wall and remains constant at the flat wall. The effects of the Grashof nmber wall temperatre ratio and λ x on the cross velocit are exactl opposite to their effect on the main velocit.. The temperatre profiles remain invariant with changes in the Grashof nmber. The effect of the wall temperatre ratio promotes the temperatre whereas λ x redces the temperatre at the wav wall while it remains constant at the flat wall. Again these effects are more prononced in the wider passage than in the narrower passage. 3. The skin friction increases at both the wav and flat wall for increasing vales of the Grashof nmber. The increase in the wall temperatre ratio increases the skin friction at the wav wall and decreases at the flat wall. Wave nmber and the amplitde parameter do not affect the skin friction at the wav wall and at the flat wall significantl. IJMA. All Rights Reserved 4

11 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl The effect of the Grashof nmber is fond to increase the rate of heat transfer at the wav wall and decrease at the flat wall in magnitde at all baffle positions. The effect of the wall temperatre ratio is seen to decrease in magnitde the heat transfer at both walls at all baffle positions. The effect of the wave nmber λ and amplitde parameter ε do not significantl affect the rate of heat transfer at both the walls.. The reslts of the present model agreed ver well with the reslts obtained b Salah El Din [36] for both the walls to be flat. The reslts of the present model were also in good agreement with the reslts of Rita and Alok [37] for single passage with wav wall G= λx = π / P =.7 λ =. ε = Fig.a G= λx = π / P =.7 λ =. ε = Fig.b G= λx = π / P =.7 λ =. ε = Fig.c Fig. : Main velocit profiles for different vales of Grashof nmber G..4 G=. - - v x - v x - - v x λx = π / P =.7 λ =. ε = Fig.3a - λx = π / P =.7 λ =. - ε = Fig.3b λx = π / -4 P =.7 λ =. ε = Fig.3c Fig. 3: Cross velocit profiles for different vales of Grashof nmber G. IJMA. All Rights Reserved

12 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl m = Fig 4a: λx = π / G = P =.7 λ =. ε = m = - m = -. λx = π / G = -.3 P =.7 - λ =. ε = Fig 4b: m = λx = π / G = P =.7 λ =. ε = Fig 4a: Fig. 4: Main velocit profiles for different vales of wall temperatre ratio m v x - v x v x λx = π / G = λ =. ε = Fig a: -3-3 m = Fig b: λx = π / G = λ =. ε = λx = π / G = λ =. ε = Fig c: Fig. : Cross velocit profiles for different vales of wall temperatre ratio m. IJMA. All Rights Reserved 6

13 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl m =. m =. m =... θ. - θ. - θ λx = π / G = λ =. ε =. -. λx = π / G = λ =. ε =. -. λx = π / G = λ =. ε = Fig.6a Fig.6a Fig.6a Fig. 6: Temperatre profiles for different vales of wall temperatre ratio m λx = π /4 π / λx = π / π / λx = π / π / G = λ =. ε = Fig.7a G = -.3 λ =. ε = Fig.7b G = -.3 λ =. ε = Fig.7c Fig. 7: Main velocit profiles for different vales of λ x. IJMA. All Rights Reserved 7

14 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl λx = π / λx = λx = π /4 - - v x - - π /4 λx = v x π /4 v x π /4-4 π / G = λ =. ε =. - - π / G = λ =. ε =. -4 π / G = λ =. ε = Fig.8a Fig.8b Fig.8c Fig. 8: Cross velocit profiles for different vales of λ x..... λx = π /4 π /. λx = π /4 π /. λx = π /4 π /... θ θ θ G = λ =. ε = Fig.9a G = λ =. ε = Fig.9b G = λ =. ε = Fig.9c Fig. 9: Temperatre profiles for different vales of λ x. IJMA. All Rights Reserved 8

15 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- Table-: Temperatre vales for different Grashof nmber and baffle positions with m = P =.7 and λ =. =. =. =.8 G = G = G = G = G = G = Table-a: skin friction vales at the wav and flat wall. G =. σ w =. =.8 G =. σ f =. = m m λ λ ε ε G Table-b: Nsselt nmber vales at the wav and flat wall. =. N w =. =.8 N f =. IJMA. All Rights Reserved 9 G =. = m m λ λ

16 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl ε ε Table-3a: Comparison of velocit at different baffle positions with Salah El Din [36] for G = m = P =.7 and λ =. Present Model Salah El Din [36] Present Model Salah El Din [36] Present Model Salah El Din [36] =. =. =. =. =.8 = Table-3b: Validit of the present model with stream- (shifting the baffle to the right wall) and stream- (shifting the baffle to the left wall) with Rita and Alok [37] for G = m= P =.7 and λ =.. Stream- Present Model Stream- v θ v Rita and Alok [37] for viscoelastic parameter K = θ v θ e e e E E E E E E E E E E-..9-9E-..9-9E NOMENCLATURE d channel width d width of passage dimensionless specific heat at constant pressre c p g x acceleration de to gravit ( 3 x w s ν ) G Grashof nmber d g β( T T ) k N p p wavelength Nsselt nmber pressre dimensionless pressre P Prandtl nmber ( cpη k) p s static pressre IJMA. All Rights Reserved

17 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- rp real part i p T T s UV v XY x imaginar part temperatre static temperatre velocities along X and Y directions dimensionless velocities space co-ordinates dimensionless space co-ordinates GREEK SYMBOLS β dimensionless co-efficient of thermal expansion ε non-dimensional amplitde parameter ( ε d ) ε amplitde λ non-dimensional wave nmber ( k/ d ) µ viscosit ν kinematic viscosit θ dimensionless temperatre ρ densit ρ static densit σ x skin friction ψ stream fnction SUBSCRIPTS i refer qantities for the flids in stream and stream respectivel. Appendix 3 3 θ = c + c ; = l + l + d + d ; θ = c 7 + c8 ; = l 6 + l 7 + d+ d ; d3 n n n3 n4 n n6 n7 = cos ( λx) 4l3 + + d4 + d + λsin ( λx) d7 + + d 8 + d d n n n n n n n ( λx) l d d λ ( λx) d = cos sin d8 + d9 d d n n n n n n n v x l d d x d d = λsin( λ ) λ cos( λ ) d9 + d 6 d d n n n n n n n v λ λx l d d λ λx d d = sin( ) cos d9 + d θ = λx c 3 + c4 λ λx Pm + m + m 3 + m 4 + m + c + c6 cos sin( ) θ = cos( λx)( c + c ) λsin λx P m + m + m + m + m + c + c d n n n n l l d d x l d d x = ε cos( λ ) λsin( λ ) n 4 n6 3 n7 3 d d8+ d d n n n l l l d d ( x) l d d ( x) = ε cos λ λsin λ + + n4 n 4 n6 3 n7 3 d d8 + d d d n n n n n v x l d d x = ε λsin( λ ) λ cos( λ ) n6 n7 4 d7 3 d d9 + d IJMA. All Rights Reserved

18 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- d d n n n n n v x l d d x = ε λsin( λ )( ) λ cos( λ ) n6 4 n7 4 d7 3 d d9 + d ( cos sin( ( ) )) ( cos sin c = m c = ( m ) c9 = m c = ( m ) c7 = m 8 θ = c+ c + ε λx c+ c λ λx Pm + m + m + m + m + c+ c ) θ = c + c + ε λx c + c λ λx P m + m + m + m + m + c + c c m = c = 3 Gc d = l = 6 l 4 c = d = d 6 = d 9 = Gc Gc3 6d = l 3 = d = ( l + l) d = ( l + l) d3 = l d3 d4 = d 4l3 d l 4 = 3 d4 Gc7 Gc8 l6(- ) + l7(- ) l = l6 = l7 = d = d Gc9 l8 (( ) 4( ) + ( )( )) 3 4l8 d = l6 l7 d l 8 = d3 = d = ( ) ( )( ) ( ) + ( ) d3 d3 d4 d3 d4 cl 3+ cl 3 cl 3+ c4 l + cl 4 d = 4l 8 d4 d6 = l8 d l 9 = l = m = m = cd 3 + c4 l + cl cd 3 + c4d+ cd cd 4 cl 96+ cl 78 cl 97+ c6 l + cl 79 m3 = m4 = m = m6 = m7 = 6 3 cd 9 + cl7 + cl 7 cd 9 + cd + cd 7 cd + cd 7 6 m8 = m9 = m = (( ) ) c = ip m + m + m + m + m m + m + m + m + m c = Pi m + m + m + m + m c c ip m m m m m ip m m m m m c = n = 6l3 l + 6GPm n = l3 l + GPm n3 = d3 l + 4l4 l 4l l + 4GPm3 n4 = l3d + 6l4d 6ld + 3GPm4 n = 6ld 4 + ld 6ld 6 ld + GPm n6 = ld ld 6 n7 = GC n3 4 3n n6 n7 n n n d 7 = n3 4 n n6 n7 d7 n n n d 8 = n = 6l6l8 + 6GPm6 n = l7l8 + GPm n3 = dl8 + 4l9l7 4l6l + 4GPm8 n4 = l8d + 6l9d 6l6d + 3GPm9 n = 6l9d + ld 6l6d6 l7d + GPm n6 = ld ld 7 6 n7 = GC ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n n n n n = Z ( ) ( ) ( ) 3 ( n n n ) n4 ( ) ( ) n ( ) n n n3 n4 n Z = ( ) 6 ( ) ( n ) n7 ( ) n6 n IJMA. All Rights Reserved

19 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl ( ) ( ) ( ) ( ) d7 = Z Z + d 6 4 n n n3 n n6 n7 n8 d7 d9 = d d7 ( ) 8 = Z ( ) n n n3 n4 n n6 n7 d7 d8 9 d = d Case 3. Comparison of the Soltions with Salah El Din [36] in the presence of baffle θ = θ = 3 G 3 G = + d + d 3 = + d3 + d4 3 G 3 G G d = d 3 = d3 = c8 d4 = + ( ) 3 6 Case 3.3 Comparison of the Soltions with Rita and Alok [37] in the absence of baffle. Shifting the baffle to the left wall and comparing the soltions of stream- with Rita and Alok [37]. Gc Gc Gc3 c = m c = l = l = d = l l d = c3 = m c4 = ( m ) l 3 = d3 = l3 + 6d 6 4 d3 cl 3+ cl 3 cl 3+ c4 l + cl 4 cd 3 + c4 l + cl d4 = d 4l3 d = l l d 6 = m = m = m3 = 3 cd 3 + c4d+ cd cd 4 m4 = m = c = ip( 6m6 + m7 + 4m8 + 3m9 + m ) ip( 6m+ m + 4m3 + 3m4 + m) + c 6 c 6 = n = 6l3 l + 6GPm n = l3 l + GPm n3 = d3 l + 4l4 l 4l l + 4GPm3 n4 = l3d + 6l4d 6ld + 3GPm4 n = 6ld 4 + ld 6ld 6 ld + GPm n6 = ld ld 6 n7 = GC n 3n n3 n4 3n n6 n7 n n n3 n4 n n6 n7 d7 d 7 = d 8 = d 9 = d = Shifting the baffle to the right wall and comparing the soltions of stream- with Rita and Alok [37]. d3 d4 Gc7 Gc8 c9 = m c = ( m ) c7 = m c 8 = l 4 = l = l6 = l7 = d = l6 l7 6 6 Gc9 d3 d3 d3 d4 d3 d = l6 l7 d l 8 = d3 = 6l8 d4 = 4l8 d = 4l8 d4 d6 = l8 d l 9 = d4 cl 96+ cl 78 cl 97+ c6 l + cl 79 cd 9 + cl7 + cl 7 cd 9 + cd + cd 7 l = m6 = m7 = m8 = m9 = 3 6 cd + cd 7 6 m = c = ip( ( m6 + 4m7 + 3m8 + m9 + m ) ( m + 4m + 3m3 + m4 + m )) c = Pi ( m6 + m7 + m8 + m9 + m ) c n = 6l6l8 + 6GPm6 n = l7l8 + GPm7 n3 = dl8 + 4l9l7 4l6l + 4GPm8 n4 = l8d + 6l9d 6l6d + 3GPm9 n = 6l9d + ld 6l6d6 l7d + GPm n6 = ld ld 7 6 n7 = GC n n n3 n4 n n6 n7 8n 7n 6n3 n4 4n 3n6 n7 Z = Z = Z d n d7 = Z + 7 n n3 n n6 n7 n8 d7 d8 = Z d9 = d n n n3 n4 n n6 n7 d7 d8 d = d IJMA. All Rights Reserved 3

20 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl REFERENCES [] Ostrach S. An analsis of laminar free-convection flow and heat transfer abot a flat plat parallel to the direction of the generating bod force NACA TN-63 Accession nmber-93r [] Jaloria. Y. Natral convective cooling of electronic eqipment in: S. Kakac W. Ang R. Viskant (Eds) Natral Convection Fndamentals and Applications Hemisphere Washington DC 98. [3] Peterson G.P. and Ortega A. Thermal control of electronic eqipment and devices Adv. Heat Transfer vol. pp [4] Elenbaas W. Heat Dissipation of parallel plates b free convection Phsica vol. 9 pp [] Ostromov G.A. Free convection nder the condition of the internal problem NACA TM [6] Sparrow E.M. and Bahrami P.A. Experiments on natral convection from vertical parallel plates with either open or closed edges ASME. J. Heat Transfer vol. pp [7] Bodoia J.R. and Osterne J.F. The development of free convection between heated vertical plates ASME. J. Heat Transfer vol. 84 pp [8] Engel R.K. and Mller W.K ASME paper 67-HT [9] Yao L.S. Natral Convection along a wav srface ASME. J Heat Transfer vol. pp [] Yao L.S. A note on Prandtl s transposition theorem ASME. J Heat Transfer vol. pp [] Molic S.G. and Yao L.S. Mixed convection along a wav srface ASME. J Heat Transfer vol. pp [] Chi C.P. and Cho H.M. Transient analsis of natral convection along a vertical wav srface in micropolar flids Int. J. Eng. Sci. vol. 3 pp [3] Chen C.K. and Wang C.C. Transient analsis of force convection along a wav srface in miropolar flids J. Thermophs. Heat Transfer vol. 4 pp [4] Malashett M.S. Umavathi J.C. and Leela V. Magnetoconvective flow and heat transfer between vertical wav wall and a parallel flat wall Int. J. Applied Mechanics and Engng. vol. 6 No. pp [] Umavathi J.C. Prathap-Kmar J. and Shekar M. Mixed convective flow of immiscible viscos flids confined between a long vertical wav wall and a parallel flat wall Int. J. Eng. Sci. Tech. vol. No. 6 pp [6] Umavathi J.C. and Shekar M. Mixed convection flow and heat transfer in a vertical wav channel containing poros and flid laer with traveling thermal waves Int. J. Eng. Sci. Techno. Vol. 3 No. 6 pp [7] Umavathi J.C. and Shekar M. Mixed convective flow of two immiscible viscos flids in a vertical wav channel with traveling thermal waves Heat Transfer Asian Research vol. 4(8) pp [8] Das P.K. and Mahmd S. Nmerical investigation of natral convection inside a wav enclosre Int. J. Therm. Sci. vol. 4 pp [9] Dalal A. and Das M.K. Laminar Natral convection in an inclined complicated cavit with spatiall variable wall temperatre Int. J. Heat Mass Transfer vol. 48 pp [] Oztop H.F. Nmerical std of flow and heat transfer in crvilinear dcts: applications of electric grid generation Appl. Math. Compt. vol.68 pp [] Varon Y. and Oztop H.F. Effect of inclination angle on natral convection in wav solar air collectors: A comptational modeling the nd International Green Energ Conference UOIT Oshawa Canada -9 Jne 6. IJMA. All Rights Reserved 4

21 J. C. Umavathi/ CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN WAVY WALL AND A. / IJMA- 3(7) Jl- 49- [] Jang J.H. Yan W.M. and Li H.P. Natral convection heat and mass transfer along a vertical wav srface Int. J. Heat and Mass Transfer vol. 46 pp [3] Mahmd S. Das P.K. Hder N. and Islam A.K.M. Free convection in an enclosre with vertical wav wall Int. J Thermal and Science vol. 4 pp [4] Khalil Khanafer Baber Al-Azim Alia Marafree Ioan Pop Non-Darcian effects on natral convection heat transfer in a wav poros enclosre Int. J. Heat Mass Transfer vol. pp [] Hasnaoi M. Bilgen E. and Vasser P. Natral convection above an arra of open cavities heated from below Nmerical Heat Transfer vol.8 pp [6] Ben-Nakhi A. and Chamkha A.J. Natral convection in an inclined partitioned enclosre Int. J. Heat Mass Transfer vol. 4 pp [7] Dagtekin I. and Oztop H.F. Natral convection heat transfer b heated partitions within enclosres Int. J. Heat and Mass Transfer vol. 8 pp [8] Tansmim S.T. and Collins M.R. Nmerical analsis of heat transfer in a sqare cavit with a baffle on the hot wall Int. Com. Heat Mass Transfer vol. 3 pp [9] Prathap Kmar J. Umavathi J.C. Ali J. Chamkha and Prema H. Free convection in a vertical doble passage wav channel filled with a Walters flid (model B ) Int. J. Energ & Technolog vol. 3 No. pp. 3. [3] Prathap Kmar J. Umavathi J.C. and Prema H. Free convection of Walter s flid flow in a vertical doblepassage wav channel with heat sorce Int. J. Eng. Sci. Techno. vol. 3 No. pp [3] Chang T.S. and Shia Y.H. Flow plsation and baffle s effects on the opposing mixed convection in a vertical channel Int. J. Heat and Mass Transfer vol. 48 pp [3] Nasirddin and Siddiqi M.H.K. Heat Transfer agmentation in a heat exchanger tbe sing a baffle Int. J. Heat and Flid Flow vol. 8() pp [33] Yang M.H. Yeh R.H and Hwang J.J. Mixed convective cooling of a fin in a channel Int. J Heat and Mass Transfer vol. 3(4) pp [34] Cheng C.H. Ko H.S. and Hang W.H. Laminar fll developed forced-convection flow within an asmmetric heated horizontal doble-passage channel Appl. Energ vol. 33 pp [3] Salah El-Din M.M. Fll developed laminar convection in a vertical doble-passage channel Appl. Energ vol. 47 pp [36] Salah El-Din M.M. Effect of viscos dissipation on fll developed laminar mixed convection in a vertical doblepassage channel Int. J. Therm. Sci. vol. 4 pp [37] Rita Chodhr and Alok Das Free convection flow of a non-newtonian flid in a vertical channel Defense Science Jornal vol. pp Sorce of spport: Nil Conflict of interest: None Declared IJMA. All Rights Reserved