Finite Element Solution for Heat Transfer Flow of. a Third Order Fluid between Parallel Plates

Transcription

1 Adv. Studes Theor. Phs., Vol. 5,, no. 3, 7 - Fnte Element Soluton for Heat Transfer Flow of a Thrd Order Flud between Parallel Plates R. Mahmood a, M. Sajd a, and A. Nadeem b a Theoretcal Plasma Phscs Dvson, PINSTECH, P.O. Nlore Islamabad 44, Pakstan b Federal Urdu Unverst of Art, Scence and Technolog, Islamabad 44, Pakstan Abstract. Ths paper deals wth the numercal soluton of the heat transfer flow of a thrd order flud between parallel plates. Three dfferent flow problems namel, () plane Couette flow, () plane Poseulle flow and () plane Couette-Poseulle flow have been analzed usng the fnte element method. The same problems have alread been studed b Sddqu et al. [ ] usng the tradtonal perturbaton and homotop perturbaton methods. The solutons obtaned n [ ] are onl vald for the weak nonlneart. However the numercal solutons presented n ths paper are vald for all values of the parameters. Fnall the results are presented graphcall n all three cases and the effects of the thrd grade parameter, pressure gradent and Brnkman number are dscussed. Kewords: Thrd order flud, Couette flow, Poseulle flow, Fnte element method.. Introducton In a recent attempt Sddqu et al. [ ] analzed the heat transfer flow of a thrd order flud between two parallel plates. The have found a two term soluton of the same problem usng homotop perturbaton method (HPM). Lke the tradtonal perturbaton technque, whch s strongl based on the presence of a small parameter n the equaton, the HPM soluton s also vald onl for weakl nonlneart. It s explctl proved b Sajd et al. [,3] that HPM does not over come the dsadvantages of tradtonal perturbaton Correspondng author. e-mal address: sajdqau@ahoo.com (M. Sajd)

2 8 R. Mahmood, M. Sajd and A. Nadeem methods. It s also proved n [,3] b consderng examples of the thn flm flows that the HPM results are dvergent for strong nonlneart. Keepng ths fact n mnd t s requred to use such a technque whch can handle strongl nonlnear problems. Fnte element method s an effcent method and s wdel used for fndng the solutons of lnear and nonlnear dfferental equatons n dfferent felds rangng from sold mechancs [ 4 7] to flud mechancs [ 8 3]. It has been successfull appled to all engneerng dscplnes, but cvl, mechancal and aerospace engneers are the most frequent users of the method. The applcaton of fnte element method n the flud mechancs are developng ver fast and wll reman the feld of nterest n the future. However, n the lterature there are ver few studes relatng the applcaton of fnte element method to strongl nonlnear flud mechancs problems. Some of them can be seen n [ 8 3] and references theren. Organzaton of the paper s as follows. Secton contans the basc equatons, the dervaton s omtted here because t s alread gven n Ref. [ ]. In secton 3 the procedure of fnte element method adopted here s gven n detal. The numercal results and dscusson s ncluded n secton 4. Fnall, the conclusons are presented n secton 5.. Governng equatons The detaled dervaton s omtted here and we have drectl taken the dmensonless form of the governng flow and heat transfer equatons n a thrd order flud from Sddqu et al. [ ]. Whch are gven b d u du d u + 6β + B =, () d d d d θ du + λ d d u ( ) =, u( ) = ( ) =, θ ( ) =, 4 du + βλ d A, =, θ where u s the veloct, θ s the temperature, β s dmensonless materal constant of thrd order flud, λ s the Brnkman number, B the non-dmensonal form of constant appled pressure gradent and A s an constant ntroduced to dfferentate for Couette, Poseulle and Couette-Poseulle flows. In Ref. [ ] all these three problems are treated separatel but here we have combned them nto a sngle problem gven b Eqs. () ( 3). One has for B = and A = the problem of plane Couette flow, for B and A = we have plane Poseulle flow and for B and A = we have plane Couette-Poseulle () (3)

3 Fnte element soluton for heat transfer flow 9 flow. The values of these dmensonless parameters β, λ and B are gven n [. ] 3. Fnte element method The fnte element method has been emploed for the soluton of non-lnear sstem of dfferental equatons gven b Eqs. ( ) (3). A fnte element model of a problem gves a pecewse approxmatons to the governng equatons. In ths method, a gven boundar value problem s frst transformed nto a weak form or varatonal form. A weak form s a weghted ntegral statement of a dfferental equaton n whch the dfferentaton s dstrbuted among the dependent varable and the weght functon or test functon, and ncludes the natural boundar condtons of the problem. The weak formulaton has two desrable characterstcs. Frst, t requres weaker contnut of the dependent varable b dstrbutng the dfferentaton between the soluton and the weght functon w (due to ts weaker requrement of contnut, t has been gven the name weak form). Second, the natural boundar condtons of the problem are ncluded n the weak form, and the soluton s requred to satsf onl the essental boundar condtons of the problem. Whenever the classcal soluton exsts t concdes wth the weak soluton of the problem. In ths method the contnuous phscal model or doman s dvded nto fnte number of smaller elements/sub-domans whch s called dscretzaton. The doman for the boundar value problem s vewed as an assemblage of these sub-domans usuall known as fnte element mesh/grd. The ponts at whch these elements are connected are called nodes or nodal ponts. An approxmate soluton s then computed on these node ponts. Instead of solvng the problem for the entre doman n one step, attenton s manl devoted to the formulaton of the propertes of the consttuent elements. A standard element s selected from the mesh and then fnte element formulaton s constructed for ths element. Results are recombned to represent the whole doman/mesh. Snce these elements can be put together n a varet of was, the can be used to represent ver complex doman shapes. The mesh conssts of lne segments n one dmenson, n two dmensons t ma consst of trangles or quadrlaterals and n three dmensons t ma consst of tetrahedra or hexahedra. All these are known as fnte elements or smpl elements. A varet of element shapes ma be used, and, wth care dfferent element shapes ma be emploed n the same soluton regon. If we partton the doman Ω nto a fnte number of elements Ω, Ω,..., Ω E, then these elements should be non overlappng and cover the doman Ω n the sense that, Ω e Ω f E U = φ for e f and Ω = Ω. (4) e= e The number and the tpe of elements to be used n a gven problem are matters of mathematcal or engneerng judgement. The fnte element method works b expressng the unknown feld varable n terms of

4 R. Mahmood, M. Sajd and A. Nadeem assumed approxmatng functons wthn each element. The approxmatng functons (sometmes called nterpolaton functons) are defned n terms of lnear combnatons of algebrac polnomals called bass functons and the values of the feld varables at node ponts. The nodal values of the feld varable and the bass functons for the elements completel defne the behavor of the feld varable wthn the elements. For the fnte element representaton of a problem the nodal values of the feld varable become unknowns to be determned. A fnte element approxmate soluton s of the tpe, where U h N = uψ, (5) = u are soluton values at the node ponts to be determned and ψ are chosen approxmatng functons. The choce of algebrac polnomals as a bass functon has two reasons. Frst the nterpolaton theor of numercal analss can be used to develop the approxmate functons sstematcall over an element. Second, numercal evaluaton of ntegrals of algebrac polnomals s eas. The degree of the polnomal chosen depends on the number of nodes assgned to the element, the nature and number of unknowns at each node and certan contnut requrements mposed at the nodes and along the element boundares. For example, n two dmensons on trangles the feld varables ma be approxmated b lnear polnomals p = α + α x + α 3, wth three nodes at the vertces of the trangle or b quadratc polnomals p = α + α x + α 3 + α 4x + α 5x + α 6, wth sx nodes, three at the vertces and three at the md ponts of the trangle edges. Bass functons ψ have the followng propertes. ) The functons ψ are bounded and contnuous, that s, ψ C(Ω). ) The total number of bass functons s equal to the number of nodes present n the mesh and each functon ψ s nonzero onl on those elements that are connected to node : ψ ( x) f / Ωe. Ω e 3) ψ s equal to at node, and equal to zero at the other nodes, f = j ψ ( x j ) = (6) otherwse In Galerkn approach the approxmate soluton of orgnal problem for an element s sought b choosng test/weght functon equvalent to the bass functon for that element. On substtutng the approxmate fnte element soluton n the weak form we get the algebrac element equatons. Ths elds a large set of smultaneous algebrac equatons. After mposng the essental and natural boundar condtons the problem s thus reduced to one of solvng the set of smultaneous equatons where the number of equatons s equal to the number of nodes at whch the soluton s requred. In matrx form ths set of equatons can be wrtten as K u = f, (7)

5 Fnte element soluton for heat transfer flow where the matrx K s known as the stffness matrx and f s known as the load vector. Snce ψ = for all elements that do not have node as a node, t follows that ths propert of bass functons wll result n the matrx K havng a sparse structure or, wth an approprate orderng, a banded structure n whch all nonzero entres are clustered around the man dagonal. 3. Implementaton of fnte element method The varatonal form assocated wth equatons ( ) and ( ) over a tpcal two node lne element, ) s gven b ( + + d u du d u φ + 6β + B d =, (8) d d d 4 d θ du du + φ + λ + βλ d =, (9) d d d where φ and φ are arbtrar test functons whch can be consdered as the varatons of u and θ, respectvel. We assume the fnte element soluton over ths element s of the form, u = uψ, () = θ = uψ, () = For Galerkn fnte element soluton approxmaton we use φ = φ = ψ ( =, ) () Where ψ are the bass functons for a tpcal element, ) and the are defned as ( + + ψ =, ψ = +. (3) + + Equaton ( 8) s non lnear n veloct varable however t does not nvolve the temperature varable. It can be solved ndependentl to get the veloct values at requred node postons. Fnte element formulaton of Eq. ( 8) wll also generate a nonlnear set of algebrac equatons. Equaton ( 8) can be wrtten n matrx form as, f( u) ( u) f F ( u) =. =,. f n ( u) (4)

6 R. Mahmood, M. Sajd and A. Nadeem where f (u) can be found b puttng n the value of u from Eq. ( ) n Eq. ( 8). The doman of the problem s dvded nto a set of elements of equal length. As we know the veloct values at the boundar of the doman therefore after ncorporatng these boundar condtons we obtan a set of 99 smultaneous non lnear algebrac equatons havng 99 unknowns. For the soluton of ths set of equatons we used Newton's teratve method wth an ntal guess provded to t as Zero vector. Newton's teratve method s generall mplemented n a two step procedure. Frst a vector z s found whch wll satsf ( k ) ( k ) J ( u ) z = F( u ), (5) (k ) (k ) where J ( u ) s the Jacoban of F ( u ). After ths has been carred out the new ( k+) approxmaton u, can be obtaned b ( ) ( ) u k+ = u k + z. (6) Equaton ( 9) n the sstem s lnear n temperature varable and non-lnear n the veloct varable. The fnte element mesh used for the soluton of ths equaton s kept the same whch s used for the soluton of Eq. ( 8). Soluton values for the veloct varable at the requred node ponts from the frst equaton have alread been computed, therefore the stffness matrx for the second equaton s assembled b takng all the veloct varable terms to the rght hand sde of the equaton. Temperature values at the boundar of the doman are gven therefore after ncorporatng these boundar condtons we obtan a set of 99 smultaneous lnear algebrac equatons havng 99 unknowns whch are solved teratvel. Effectvel we have transformed ths sstem of two non-lnear equatons nto two separate equatons thus reducng the computatonal efforts b calculatng the soluton for the two equatons separatel rather than for the whole sstem. 4. Results and dscusson To see the nfluence of pertnent parameters lke thrd order parameter, pressure gradent and Brnkman number on the veloct and temperature profles, Fgs. have been dsplaed for the plane Couette, plane Poseulle and plane Couette-Poseulle flows. 4. Plane Couette flow The behavor of the temperature under the nfluence of thrd grade parameter β and the Brnkman number λ s shown n Fgs. and. It s evdent from Fg. that an ncrease n the thrd order parameter ncreases the temperature. The same effect on the temperature s seen n the case when we ncrease Brnkman number b keepng thrd order parameter fxed. It s further noted here that the results of Ref. [ ] are matchng wth the numercal results obtaned here.

7 Fnte element soluton for heat transfer flow Fg.. Influence of thrd order parameter β on the temperature Fg.. Influence of the Brnkman number λ on the temperature. 4. Plane Poseulle flow To see the effects of parameters β, λ and B on the veloct and temperature Fgs. 3 7 have been plotted. Fg. 3 elucdate that the veloct decreases wth an ncrease n parameter β. Moreover these results are qute opposte to that presented n Ref. [ ]. Hence the results n Ref. [ ] are not predctng the real phscal stuaton. If one checks the presented two term approxmate soluton n Ref. [ ] for the values of parameter β <.3 the results show that veloct decreases wth an ncrease n β but for larger

8 4 R. Mahmood, M. Sajd and A. Nadeem values of β there soluton s showng opposte behavor hence not correct. Ths shows that there results are onl vald from ver small values of β.e. for β <. 3 as t s alread known from the lterature that perturbaton results are vald onl for small value of the perturbaton parameter. The nfluence of thrd order parameter on the temperature s shown n Fg. 4. It depcts that temperature decreases b an ncrease n the thrd order parameter. Fgs. 5 and 6 shows that the veloct and temperature both ncreases b ncreasng the pressure gradent. One can see that the magntude of ncrease s ver small as compared to that presented n [ ]. The onl reason s that the two-order soluton presented n [ ] s not vald for all the range of the parameter values. Fg. 7 shows an ncrease n temperature b ncreasng the Brnkman number. However, agan ths fgure shows that the results presented n [ ] are not correct..5 B u ;.4 ;.7 ;..5.5 Fg. 3. Influence of thrd order parameter β on the veloct., B ;.4 ;.7 ;..5.5 Fg. 4. Influence of thrd order parameter β on the temperature.

9 Fnte element soluton for heat transfer flow 5.4 B.5 ; B.5 ; B.75 ; B. u Fg. 5. Influence of constant pressure gradent on the veloct..8, B.5 ; B.5 ; B.75 ; B Fg. 6. Influence of constant pressure gradent on the temperature.

10 6 R. Mahmood, M. Sajd and A. Nadeem B,.5 ;. ;.5 ;..5.5 Fg. 7. Influence of Brnkman number λ on temperature. 4.3 Plane Couette-Poseulle flow The effects of dfferent parameters n the case of plane Couette-Poseulle flow are shown n Fgs. 8. Fg. 8 shows that veloct decreases b ncreasng thrd order parameter. The effect of thrd grade parameter on the temperature s opposte to that of veloct and s shown n Fg. 9. Moreover, both veloct and temperature ncreases b an ncrease n the pressure gradent and ths fact s evdent from Fgs. and. The effect of Brnkman number on the temperature s depcted n Fg.. It elucdate that temperature ncreases b an ncrease n the Brnkman number.. B u ;. ;.4 ; Fg. 8. Influence of thrd order parameter β on the veloct.

11 Fnte element soluton for heat transfer flow , B. ;. ;.4 ; Fg. 9. Influence of thrd order parameter β on the temperature. u , B. ; B.4 ; B.6 ; B Fg.. Influence of the pressure gradent on the veloct.

12 8 R. Mahmood, M. Sajd and A. Nadeem.8.6.4, B. ; B.4 ; B.6 ; B Fg.. Influence of the pressure gradent on the temperature B,.3 ;.5 ;.7 ; Fg.. Influence of the Brnkman number λ on the temperature. 5. Concludng remarks In ths paper the numercal soluton usng fnte element method of the heat transfer flow of a thrd order flud between parallel plates s studed. The obtaned numercal solutons are vald for all range of the values of the pertnent parameters of the consdered problem. The results for plane Couette flow, plane Poseulle flow and plane Couette-Poseulle

13 Fnte element soluton for heat transfer flow 9 flow are graphcall presented and the nfluence of the thrd grade parameter, pressure gradent and Brnkman number s dscussed through graphs. It s worthmentonng here that the results presented n [ ] are not vald for the range of values used n [ ] because the perturbaton and homotop perturbaton solutons are vald onl for weak nonlneart. References [] A. M. Sddqu, A. Zeb, Q. K. Ghor and A. M. Benharbt, Homotop perturbaton method for heat transfer flow of a thrd grade flud between parallel plates. Chaos, Soltons and Fractals 36 ( 8) 8 9. [] M. Sajd, T. Haat and S. Asghar, Comparson of the HAM and HPM solutons of thn flm flows of non-newtonan fluds on a movng belt, Nonlnear Dnamcs 5 ( 7) [3] M. Sajd and T. Haat, The applcaton of homotop analss method to thn flm flows of a thrd order flud, Chaos, Soltons and Fractals 38 ( 8) [4] T. Horbe and K. Takahash, Crack dentfcaton n beam usng genetc algorthm and three dmensonal p-fem, J. Sold Mech. Mater. Eng. ( 7) [5] R. Mahmood and P.K. Jmack, Locall optmal unstructured fnte element meshes n 3-dmensons, Comput. Struct. 8 ( 4) 5 6. [6] M. Tur, J. Fuenmaor, A. Mugadu and D. A. Hlls, On the analss of sngular stress felds: part : fnte element formulaton and applcaton to notches, J. Stran Anal. 37 ( ) [7] L. T. Tenek and J. Argrs, Fnte element analss for composte structures (Sold Mechancs and ts Applcatons), Sprnger, 997. [8] M. Aouad, Numercal stud of mcropolar flow over a stretchng sheet, Comput. Mater. Sc. 38 ( 7) [9] R. Bhargava, L. Kumar and H. S. Takhar, Fnte element soluton of mxed convecton mcropolar flow drven b a porous stretchng sheet, Int. J. Eng. Sc. 4 ( 3) [] W. J. Yan and Y. C. Ma, Fnte element method for the vscous ncompressble flud, Appl. Math. Comput. 85 ( 7)

14 R. Mahmood, M. Sajd and A. Nadeem [] M. A. Seddeek, Fnte element method for the effect of varous njecton parameter on heat transfer for a power law non-newtonan flud over a contnuous stretched surface wth thermal radaton, Comput. Mater. Sc. 37 ( 6) [] M. Amoura, N. Zerab, A. Smat and M. Gareche, Fnte element stud of mxed convecton for non-newtonan flud between two coaxal rotatng clnders, Int. Comm. Heat Mass Transf. 33 ( 6) [3] D. V. Krshna, D. R. V. P. Rao and V. Sugunamma, Fnte element analss of convecton flow through a porous medum n a horzontal channel, Trans. Porous Meda 36 ( 999) Receved: September,