Ibrhim, R. W., e l.: Soluion for Sysem of Frcionl He Equions of THERMA SCIENCE, Yer 015, Vol. 19, Suppl. 1, pp. S51-S57 S51 SOUTION FOR A SYSTEM OF FRACTIONA HEAT EQUATIONS OF NANOFUID AONG A WEDGE by Rbh W. IBRAHIM b nd Hmid A. JAAB Insiue of Mhemicl Sciences, Universiy Mly, Kul umpur, Mlysi b Fculy of Compuer Science nd Informion Technology, Universiy Mly, Kul umpur, Mlysi Originl scienific pper DOI: 10.98/TSCI15S1S51I In his ricle, uhors se new sysem of frcionl he equions of nnofluid long wedge nd esblish he exisence nd uniqueness of soluion bsed on he Riemnn-iouville differenil operors. Sufficien condiions on he prmeers of he sysem re imposed. A numericl soluion of he sysem is discussed, nd pplicions re illusred. The echnique is bsed on he biliy of Podlubny s mrix in Mlb o formule he operion of frcionl clculus. Key words: frcionl clculus, frcionl diffusion equion, frcionl he equion Inroducion In he ls few decdes, frcionl differenil equions hve been considered successful models of rel-life phenomenon. One of he min pplicions of frcionl clculus is modeling of inermedie physicl processes. Some of he imporn models include frcionl diffusion nd wve equions. Vrious universl elecromgneic, mechnicl, nd cousic responses cn be ccurely represened by uilizing frcionl diffusionwve equions [1, 5]. Differen sudies on frcionl diffusion equions hve inroduced vrious frcionl operors, such s Riemnn-iouville, Cpuo, nd Rize [6, 7]. Moreover, he uhors imposed mximl soluion o he ime-spce frcionl he equion in complex domin. Frcionl ime is considered in Riemnn-iouville operor, wheres frcionl spce is inroduced in he Srivsv-Ow operor for complex vribles [8]. The problem of boundry lyer flow nd he rnsfer ppernces of nnofluids hs drwn much enion in recen yers in response o requess for soluion from he indusries of medicl equipmen, uomoive, nd power pln nd compuer cooling sysems. Since he pioneering work of Choi nd Esmn [9], numerous specs of he problem hve been sudied. Resuls of previous sudies climed h nnofluid velociy is lower hn he velociy of he bse fluid, nd he exisence of he nnofluid indices hinning of he hydrodynmic boundry lyer. Khn nd Pop [10] considered numericlly he boundry lyer flow ps moving wedge in nnofluid nd esblished h emperure increses wih boh Brownin nd hermophoresis prmeers. Furhermore, herml rdiion chnges he emperure disribuion by plying role in conrolling he rnsfer process such s in polymer processing Corresponding uhor; e-mil: rbhibrhim@yhoo.com
S5 Ibrhim, R. W., e l.: Soluion for Sysem of Frcionl He Equions of THERMA SCIENCE, Yer 015, Vol. 19, Suppl. 1, pp. S51-S57 nd nucler recor cooling sysem. Thus, much reserch on he effecs of herml rdiion hs been conduced. The sudy of inernl he generion or bsorpion is imporn in problems involving herml rdiion where he my be genered or bsorbed in he course of such rdiion. We will show he effeciveness of hese prmeers in our sudy. In his sudy, we se new sysem of frcionl he equions of nnofluid long wedge nd esblish he exisence nd uniqueness of soluion bsed on he Riemnn-iouville differenil operors. Sufficien condiions on he prmeers of he sysem re imposed. A numericl soluion of he sysem is discussed, nd pplicions re illusred. The mehod is bsed on he biliy of Podlubny s mrix in Mlb o formule he operion of frcionl clculus. Noe h numericl soluions re imposed by differen mehods [11]. Mhemicl seing We consider he -D, sedy, nd lminr boundry lyer flow over wedge immersed in nnofluid, where he sysem chnges in ime wih respec o velociy U(), emperure T(), nd volume V(). The nnofluid is dilued solid-liquid mixure wih uniform volume frcion of nnopricle dispersed wihin he bse fluid. The bse fluid nd nnopricles re in hermlly equilibrium. The effecs of Brownin moion nd hermophoresis re included for he nnofluid. Figure 1 shows he flow model nd physicl configurion. The free srem velociy of he poenil flow ouside he boundry lyer is denoed s G(x) = cx m, where c is posiive consn, m wedge ngle prmeer, m = β/( β) nd β is he Hrree pressure grdien prmeer h corresponds o β = Ω /π for ol ngle of he wedge. We hve he following ssumpions in formuling he sysem: U is he wo-dimensionl velociy, T he emperure of nnofluid, nd V he volume frcion of he nnopricles V = ( V V )/( Vw V ), where V w denoes he volume he wedge, nd V denoes he volume fr from he wedge Figure 1. Flow configurion long wedge nd he co-ordine sysem 1 R wih emperure T w nd T, respecively. η, η, nd η 3 re he viscosiy of nnofluid, he nnofluid herml diffusiviy, nd he Brownin diffusion coefficien, respecively. 1 λ =, λ, nd λ 3 re relxion consns such s rdiion prmeer, Prndl number nd Schmied number. None of hese prmeers is negive. Tking he bove ssumpions ino considerion, he boundry lyer form of he governing equions cn be wrien (fig. 1): U T D U = η 1 + λ 1, T U DT = η + λ + Gx ( ), V T D V = η 3 + λ 3, (1,b,c) x
Ibrhim, R. W., e l.: Soluion for Sysem of Frcionl He Equions of THERMA SCIENCE, Yer 015, Vol. 19, Suppl. 1, pp. S51-S57 S53 Frcionl clculus (including frcionl order indefinie inegrl nd differeniion) is used o nlyze phenomen wih ype. We pply he Riemnn-iouville operors: The frcionl (rbirry) order differenil of he funcion f of order >0 is given by: 1 ( ) I υ( ) = υ ( )d Γ ( ) d ( ) D υ( ) = υ ( )d d Γ(1 ) We use he noion D when = 0. We pply he following resul. e one of he ssumpions be chieved [1]: β µ ([0, Τ]) nd n ([0, Τ]), < β 1, n ([0, Τ]) nd µ ([0, Τ]), < β 1 γ β d µ ([0, Τ]) nd n ([0, Τ]), < β β + d, β, d (0,1) where ([0, Τ]) = { :[0, Τ] R/ ( s) ( s j) = O( j ) uniformly for 0 < τ j < s Τ}. Then we hve: s s s D ( )()= s () sd () s + () sd () s s [() τ ()][() s τ ()] s ()() s s dτ Γ Γ + 1 (1 ) 0 ( s τ ) (1 ) s poin-wise. We conclude h if µ nd ν hve he sme sign nd re boh incresing or boh decresing, hen: γ β nd for =, we obin: s s D ( )() s () sd () s + () sd () s Exisence oucome s s D ( )() s () sd () s () In his secion, we esblish globl exisence nd uniqueness resul for sysem (1,b,c) subjeced wih he iniil condiions ( U 0, T 0, V 0 ) nd he boundry condiion U (,0) = 0, nd U (,1) =, 0. Vrious soluions of hermodynmic sysems re inroduced in [13, 14]. Theorem 1. Assume h Ω is bounded domin in R wih smooh boundry Ω. Consider: where 1 1 1 0 0 0 0 0 0 ( U, T, V ) H ( Ω ) H ( Ω ) H ( Ω), U > 0, T 0, V 0 in Ω 1 H ( Ω)={ µ ( Ω): µ ( Ω )}, nd:
S54 Ibrhim, R. W., e l.: Soluion for Sysem of Frcionl He Equions of THERMA SCIENCE, Yer 015, Vol. 19, Suppl. 1, pp. S51-S57 If: λ1 λ + λ3 + 1 λ3 η1 + >0, η >0, η3 >0 + + η Τ λ1 λ + λ3 + 1 λ3 < 1, η: = mx η1 +, η +, η3 + Γ( + 1) hen here exiss one bounded soluion ( UTV,, ) for sysem (1). Proof. The proof is delivered in four seps. The firs hree seps describe he priori esimes, wheres sep 4 ddresses uniqueness. Sep 1. Firs esime. Our objecive is o show h (U, T, V) [ (Ω), (Ω), (Ω)]. We firs expnd eq. (1) by U, uilize () nd inegre over Ω. This pproch gives he following firs esime: 1 U 1 1 η λ Ω D U + U(,.) (3) Similrly, we expnd eq. (1b) nd inegre over Ω o receive: 1 η λ U D T + T (,.) Gx ( ) Ω + Ω (4) Finlly, we muliply eq. (1c) by V nd inegre over Ω, we obin: 1 V T 3 3 η λ Ω D V + V (,.) (5) Collecing (3)-(5) nd using he fc h U nd T vnish on he boundry, we find: 1 U V D ( ) 1 3 U T V η η η + + + + + U T + λ1 U (,.) λ T (,.) λ3 V (,.) Gx ( ) Ω + x Ω + + Ω Ω (6) Employing he Cuchy-Schwrz inequliy yields: + + + + + + 1 U V D ( ) 3 U T V η λ V + + + + + + + U V Gx ( ) λ3 ( U T V ) η G Ω (7) where η: = mx{ η1, η, η 3}. The frcionl Gronwll lemm implies h: sup [0, T] ( U + T + V ) Cε ( λ T ) + C, (8) 3
Ibrhim, R. W., e l.: Soluion for Sysem of Frcionl He Equions of THERMA SCIENCE, Yer 015, Vol. 19, Suppl. 1, pp. S51-S57 S55 where ε refers o he Mig-effler funcion, λ3 := λ3 / x, C, nd C re posiive consns depending on G, η, λ 3, nd. Sep. Second bound. We im o show h ( UTV,, ) lies in he spce [ ( Ω), ( Ω), ( Ω)]. We muliply he firs equion in U, where U is he plce operor nd inegre over Ω, king ccoun h U demerilizes on he boundry of Ω, hen he Cuchy-Schwrz inequliy nd he Young inequliy yield: D ( UDU)= D ( U U) UD U = η1 ( U) λ1 U Ω D Ω D Ω D + Ω D Ω x η1 ( D U) + λ1 U η1 ( U) λ1 U η1 U Ω D D + D D + Ω Ω Ω Ω + λ D η D + D + η + D + λ1 λ1 λ1 1 U 1 U U = 1 U Thus, we in o: λ1 λ1 D DU η1 + U D + In he sme mnner, muliplying eq. (1b) by T, we obin: λ + 1 λ U 1 D η + D + + D T T G Moreover, from eq. (1c), we conclude h: λ λ D V V T Joining eqs. (9)-(11) leds o: 3 3 D η3 + D + D U D ( D U + D T + DV ) η( D U + D T + D V ) + λ + + U V ( ) 1 1 + G η U + T + V + λ + + + G = mx{( + /), [ + ( + + 1)/], ( + /)}, λ = mx{ λ1/, λ/, λ3/}. By employing he frcionl Gronwll lemm nd he fc h (U 0, T 0, V 0 ) H 1 (Ω) H 1 (Ω) H 1 (Ω), we obin ( UTV,, ) lying in he spce [ ( Ω), ( Ω), ( Ω )]. Sep 3. Third bound. We emp o clcule he hird bound of he soluion (U, T, V). Se: where η η1 λ1 η λ λ3 η3 λ3 U V 1 φ Φ( ): = U + T + V, Ψ ( ): = + +, ( ): = G Thus inequliy (1) becomes: (1) (9) (10) (11)
S56 Ibrhim, R. W., e l.: Soluion for Sysem of Frcionl He Equions of THERMA SCIENCE, Yer 015, Vol. 19, Suppl. 1, pp. S51-S57 Opering (13) by I, D Φ() ηφ () + λψ () + φ() (13) we obin: 1 1 1 ( ) ( ) ( ) Φ( ) Φ 0 + η Φ ( )d + λ Ψ ( )d + φ( )d Γ ( ) Γ ( ) Γ ( ) 0 0 0 η Τ λτ Τ Φ 0 + sup Φ () + sup Ψ () + sup φ() Γ( + 1) [0, Τ] Γ( + 1) [0, Τ] Γ( + 1) [0, Τ] (14) which implies h: λτ Τ Φ Γ( + 1) Γ( + 1) () () φ() ητ ητ ητ 1 1 1 Γ( + 1) Γ( + 1) Γ( + 1) 0 sup Φ + sup Ψ + sup [0, Τ] [0, Τ] [0, Τ] Sep 4. Uniqueness. Assume h ( U1, T1, V 1) nd ( U, T, V ) re wo soluions for he sysem (1,b,c) subjeced o he iniil condiion: 0 0 0 1 1 1 1 1 1 ( U, T, V ) H ( Ω ) H ( Ω ) H ( Ω ) Se: ( UTV,, )=( U1 U, T1 T, V1 V) hen we receive he sysem: = η 1 λ 1, = η λ, = η 3 λ + + + 3 D U U T D T T U D V V T x Anlogous o Sep 1, we hve: sup [0, T] (15) ( U + T + V ) ρ (16) where ρ is n rbirry consn bsed on he prmeers T, nd he iniil condiion. Hence (1,b,c) commi only one bounded globl soluion ( UTV,, ). This complees he proof. Figure. Temperure when m = 0, c = 10, η = 1, nd λ = 1 Numericl resuls We use mrix mehod given in [15] o numericlly solve sysem (1,b,c). This echnique is bsed on he biliy of Podlubny s mrix in Mlb o formule he operion of frcionl clculus, which leds o he lef-sided Riemnn- iouville or Cpuo frcionl derivive simulneously pproximed in ll poins of he equidisn discriminion ( j = 0,1,..., n ) wih he help of he upper ringulr srip mrix. The Prndl number for he bse fluid is fixed in ll numericl compuions. In he flow over wedge, he effec of wedge ngle prmeer on
Ibrhim, R. W., e l.: Soluion for Sysem of Frcionl He Equions of THERMA SCIENCE, Yer 015, Vol. 19, Suppl. 1, pp. S51-S57 S57 velociy, emperure, nd nnopricle volume frcion profiles re imporn. Figure shows he dimensionless emperure, of nnopricle disribuions for vrious vlues of wedge ngle. Indeed, decrese in he wedge ngle prmeer cuses n increse in he he rnsfer coefficien nd he re of he rnsfer. The emperure profiles s he effec of sucion he wll of he wedge re demonsred. Noe h ll condiions of Theorem 1 re chieved. All soluions (U, T, V) of sysem (1,b,c) converge o he ineger cse (α = 1). Compred wih he ineger sysem, he soluions re pproximed o he sedy cse. Conclusions We conclude h he nnopricle volume frcion profiles decrese in ime s well s wih velociy, wheres he emperure increses in he boundry lyer flow over wedge becuse of he effec of sucion he wll of he wedge. Acknowledgmens This reserch is suppored by Projec No.: RG31-14AFR from he Universiy of Mly. References [1] Podlubny, I., Frcionl Differenil Equions, Mhemics in Science nd Engineering, Acdemic Press, Sn Diego, Cl., USA, 1999 [] Hilfer, R., Applicions of Frcionl Clculus in Physics, World Scienific, Singpore, 000 [3] Kilbs, A. A., e l., Theory nd Applicions of Frcionl Differenil Equions, Elsevier, Amserdm, 006 [4] Sbier, J., e l., Advnces in Frcionl Clculus: Theoreicl Developmens nd Applicions in Physics nd Engineering, Springer, The Neherlnds, 007 [5] kshmiknhm, V., e l., Theory of Frcionl Dynmic Sysems, Cmbridge Scienific Publisher, Cmbridge, UK, 009 [6] He, J.-H., e l., Convering Frcionl Differenil Equions ino Pril Differenil Equion, Therml Science, 16 (01),, pp. 331-334 [7] Guo, P., e l., Numericl Simulion of he Frcionl ngevin Equion, Therml Science, 16 (01),, pp. 357-363 [8] Ibrhim, R. W., Jlb, H. A., Time-Spce Frcionl He Equion in he Uni Disk, Absrc nd Applied Anlysis, 013 (013), ID 36404 [9] Choi, S. U., Esmn, J. A., Enhncing Therml Conduciviy of Fluids wih Nnopricles, ASME In. Mech. Eng. Congress Exposiion, Sn Fncisco, Cl. USA, 1995 [10] Khn, W. A., Pop, I., Boundry yer Flow ps Wedge Moving in Nnofluid, Mh. Probl. Eng. 013 (013), ID 63785 [11] Yng, X.-J., Blenu, D., Frcl He Conducion Problem Solved by ocl Frcionl Vriionl Ierion Mehod, Therml Science, 17 (013),, pp. 65-68 [1] Alsedi, A., e l., Mximum Principle for Cerin Generlized Time nd Spce Frcionl Diffusion Equions, J. Qurerly Appl. Mh., 73 (015), 1, pp. 155-4485 [13] Rn, H., Chong, C., The Thermodynmic Trnsiions of Aniferromgneic Ising Model on he Frcionl Muli-Brnched Husimi Recursive ice, Communicions in Theoreicl Physics, 6 (014), 5, pp. 1-16 [14] Toure, O., Audonne, F., Developmen of Thermodynmic Model of Aqueous Soluion Suied for Foods nd Biologicl Medi, Pr A: Predicion of Aciviy Coefficiens in Aqueous Mixures Conining Elecrolyes, The Cndin Journl Chemicl Engineering, 93 (015),, pp. 443-450 [15] Podlubny, I., e l., Mrix Approch o Discree Frcionl Clculus II: Pril Frcionl Differenil Equions, Journl of Compuionl Physics, 8 (009), 8, pp. 3137-3153 Pper submied: Ocober 10, 014 Pper revised: Februry, 015 Pper cceped: Februry 8, 015