An investigation of MHD Flows of Nanofluid over. an oscillating surface with Joule Heating

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1 Jornal of Alied ahemaics & Bioiormaics, vol.7, no., 7, 7-48 ISSN: (rin, (online Scienress Ld, 7 An invesigaion of HD Flows of Nanoflid over an oscillaing srface wih Jole Heaing B.. Namai, J.K. Bio, A.W. anonge 3 and J.K. Kandie 4 Absrac he resen sd aims a invesigaing nsead hdromagneic flow of viscos incomressible, condcing nanoflid over an iinie oscillaing srface wih Jole heaing. he governing eqaions involved are ransformed ino dimensionless form sing aroriae non-dimensional variables. he resling dimensionless momenm eqaion is solved analicall b sing he mehod of searaion of variables. he Jole heaing erms in he energ eqaion are exressed exlicil sing his solion. Wih he hel of he serosiion rincile, a comlemenar fncion is creaivel snhesized and a ariclar inegral is obained b he mehod of ndeermined coefficiens. he comlee solion is obained b smming he wo. he effecs of magneic field srengh, oscillaion freqenc, nanoaricle concenraion, Prandl nmber and he Ecer nmber are illsraed grahicall. oi Universi Kena. namaib@mail.com Universi of Eldore, Kena. jacobbio@ahoo.com 3 aseno Universi. drmanonge@ahoo.com 4 Universi of Eldore, Kena. andieichirchir@ahoo.com Aricle Io: eceived : Ags 3, 6. evised : Ocober, 6. Pblished online : Aril 3, 7.

2 8 An Invesigaion of HD Flows of Nanoflid ahemaics Sbjec Classificaion: 76W5 Kewords: Jole heaing; oscillaing srface; nanoflid; (HD flow Nomenclare V veloci Pr Prandl nmber veloci comonen in he x-direcion Ec Ecer nmber HD magneohdrodnamics emerare B magneic field θ dimensionless emerare ρ densi υ inemaic viscosi E elecrical field φ nanoaricle concenraion c secific hea caaci a consan ressre hermal condcivi J elecrical crren ( dimensionless variables oscillaion freqenc iini ( w srface/wall condiions ( free sream condiions Inrodcion he movemen of elecricall condcing hdromagneic flow resls o flow of elecrical crren, J. his crren resls o a force F J B called he Lorenz force and Jole heaing exressed mahemaicall as J E. he heaing effec is a manifesaion of energ dissiaion iorm of hea de o oosiion of crren. his sd aims a analzing he amon of Jole heaing which occrs as a resl of nanoflid movemen riggered b he oscillaion of srface and he role laed b nanoflid in removing he hea generaed de o Jole heaing. Sdies on HD flows have received a lo of aenion b researchers de o heir man alicaions in areas lie asrohsics, geohsics, Plasma coinemen, HD ms ec.. Sch flows over oscillaing srfaces have been done. However, he Jole heaing which occrs de o he oscillaion of he srface has no been given he seriosness i deserves. Osrach [] analzed aerodnamic heaing of an oscillaing srface. I was demonsraed ha aerodnamic heaing increases wih

3 B.. Namai, J.K. Bio, A.W. anonge and J.K. Kandie 9 an increase on he amlide of oscillaion and ha oscillaion considerabl increases mechanical dissiaion. Sondalgelar [] sdied free convecion effecs on HD flow as an iinie verical oscillaing lae wih consan hea flx. I was fond ha veloci decreases wih increasing freqenc of oscillaion b emerare increases wih an increase in freqenc of oscillaion. hcmaraswam and eena [3] heoreicall sdied chemical reacion effecs on verical oscillaing lae wih variable emerare. I was fond ha chemical reacion has significan effec on hea ransfer. An invesigaion on combined hea and mass ransfer on HD free convecion flow as an oscillaing lae embedded in oros medim was done b Chadhanr and Jain [4]. Here, i was discovered ha cooling he oscillaing lae leads o a decrease in veloci and vice versa. Kishore e al. [5] sed finie difference mehod o invesigae he effec of hermal radiaion and viscos dissiaion on HD hea on mass diffsion flow as an oscillaing verical lae embedded in a oros medim wih variable srface condiions. his sd showed ha lae oscillaion, variable mass diffsion, radiaion, viscos dissiaion and oros medim affec he flow aern significanl. Ahmed and Kalia [6] ver recenl, analicall and nmericall sdied HD flow over an iinie oscillaing verical srface bonded b oros medim in resence of chemical reacion. In his sd, i was illsraed ha resls b Lalace echniqe comare ver well wih hose b Cran-Nicolson nmerical echniqe. I was also coirmed ha an increase in chemical reacion arameer leads o a decrease in flid veloci and concenraion rofiles. Ver recenl Abid e al. [7] invesigaed HD flow of a casson flid as an oscillaing verical lae wih Newonian heaing. I was roved ha flow searaion can be conrolled b increasing he vale of Casson flid arameer as well as increasing he Prandl nmber. I was also shown ha veloci decreases as Casson arameer increases and hermal bondar laer hicness increases wih increasing Newonian heaing arameer. Anoher ver recen sd on HD as a verical oscillaing lae wih radiaion and chemical reacion in a oros medim was

4 3 An Invesigaion of HD Flows of Nanoflid done b dra and Nhaben [8]. I was discovered ha emerare decreases wih chemical arameer. his sd also showed ha concenraion decreases as Schmid and radiaion arameers increases. I was fond ha veloci increases wih decreasing magneic arameer and chemical reacion. he science of nanoflid and echnolog was inrodced b Nobel rizing winning hsicis ichard in 959. Since hen mch research wor has been done o his new e of maerial becase of is high raed roeries and behavior associaed wih hea ransfer as reored b Choi [9]. Xan and oefel [] invesigaed conceions for hea ransfer correlaion of nanoflids. I was roved ha hermal disersions las a e role in increasing hea ransfer in he flid and in he wall. Li and Xan [] examined he characerisics of C-waer nanoflid. Here i was demonsraed ha nanoflid hermal condcivi is serior o ha of re flid. aisinmai e al. [] invesigaion showed ha he imoran nanoaricles seles slowl han he larger aricles and his revens channel clogging and an oher desrcive effec. Wen and Ding [3] exerimenall invesigaed convecive hea ransfer of nanoflids a he enrance region nder laminar flow condiions. B measring, i was fond ha hea ransfer coefficien is greaes a he enr and ha an increase in aricle concenraion enhances i. hermall develoing elecroosmoic (wih Jole heaing ransor of nanoflids in a micro channels was invesigaed b Charabor and o [4]. I was demonsraed ha ressre gradien as well as Jole heaing affecs veloci rofile and emerare rofile of he flid significanl.ver recenl, Ghasemi and Aminossadai [5] invesigaed mixed convecion in a lid driven rianglar enclosre filled wih nanoflids. I was shown ha addiion of AL O 3 nano aricles enhances hea ransfer rae for ever vale of ichardson nmber and for each sliding wall moion. Chad [6] invesigaed he hermal insabili of roaing nanoflid. he effecs of he alor nmber, concenraion aleigh nmber, Prandl nmber and Lewis nmber were invesigaed for sabili roses. A Sd on hea exchangers was done b eha [7]. his sd gave a deailed

5 B.. Namai, J.K. Bio, A.W. anonge and J.K. Kandie 3 descriion of hea exchangers, roeries of nanoflids ogeher wih heir rearaion. A ver imoran bacgrond of nanolids is also given. Aninda and Asho [8], nmericall analzed hea ransfer characerisics of combined elecroosmoic (wih Jole heaing and ressre-driven fll develoed flow of ower law nano-flid in microchannels. hrogh his sd i was observed ha Nsse nmber decreases wih decreasing viscosi raio and increasing ermiivi raio b i increases wih increasing resisivi raio. Had [9] sdied he flow and hea ransfer characerisics of a viscos nanoflid over a non linearl sreching shee in he resence of hermal radiaion. In his sd i was fond ha an increase in solid volme and Ecer nmber ields an incremen in he nanoflids emerare. In addiion, an increase in he hermal radiaion arameer and he non linear sreching shee arameer ields a decrease in he nanoflids emerare. Ver recenl ohammed [] sdied magneohdrodnamic free convenion of nanoflid over a verical fla lae aing ino accon Newonian heaing bondar condiion. I was fond ha dimensionless veloci and emerare disribions increase wih he increase of Newonian heaing arameer. Ferdows and Khan [] sdied HD bondar laer flow of a nanoflid over an exoneniall sreching shee. I was fond ha momenm, hermal and concenraion bondar laer hicness increase as he viscos raio arameer increases. On he oher hand sin-fricion coefficien, srface hea and mass ransfer rae decreases. I was also observed ha an increase in Ecer nmber resls o increase in momenm and hermal bondar laer. Bar and aizah [] invesigaed analicall HD mixed convecion flow of a viscos dissiaing microolar nanoflids in a bonding laer sli flow wih Jole heaing. he obained resls showed ha hsical arameers sch as hermal Grashof nmber ilence veloci, emerare and he concenraion rofiles. his lierare review shows ha, sdies on HD flow of nanoflid over oscillaing srface have no been done e. In his sd we shall ae he nanoflid as Coer waer nanoflid. he roeries his e of nanoflid are

6 3 An Invesigaion of HD Flows of Nanoflid fond in Anjali and Jlie []. U nanoaric les momenm laer hermal laer U cos x Figre : Phsical coigraion. ahemaical formlaion of he roblem We shall consider nsead, forced convecive HD flow of incomressible and elecricall condcing viscos nanoflid aing lace over a dobl iinie srface oscillaing in he x direcion a a veloci U cos as shown in figre. A he srface and he flid are a he same emerare everwhere. Also, a magneic field of niform srengh is alied erendiclarl o he srface. A combinaion of he Navier-Soe s (N-S eqaions of flid dnamics and axwell s eqaions of elecromagneism gives he general governing eqaions of his flow. Since he srface is dobl iinie, he veloci and he emerare of he flid can be visalized in he and geomeries resecivel. Since here is no moion in he direcion v. 3 D Now J σe σ ( V B

7 B.. Namai, J.K. Bio, A.W. anonge and J.K. Kandie 33 i σ j B σb. he Lorenz force J B σb σb i. B And he Jole heaing erm is exressed as; J E ( σ B,, ( B,, σ B. Since, he srface is doble iini, all gradiens wih resec o x and z of veloci comonens ms varnish. In view of he above discssion he governing eqaions which describe he flow become omenm i j σbz υ ρ. ( Energ ρc z σb ρc. ( Sbjec o:, v,, for all, U cos, w a, > (3,, as, > Here, σb ρ is he HD erm and σb ρ c reresens Jole heaing.

8 34 An Invesigaion of HD Flows of Nanoflid 3 Solion of he roblem o see he solion he following dimensionless variables are inrodced:,, L, U, Pr µc, U B ρ υ σ w θ, w,, ( U υ, L υ, (4 (,. c U Ec In view of his variables eqaion ( becomes. ρ σ υ U B L U U (5 lil boh sides of eqaion (5 b 3 U υ o ge U B L U U ρ υ σ υ υ (6 B maing se of he dimesionless variables as in (4, (6 becomes (7 Eqaion (7 is he momenm eqaion in dimensionless form. In view of eqaion(4 eqaion ( becomes w w c U B L c ρ σ θ ρ θ (8 Dividing boh sides b w ields

9 B.. Namai, J.K. Bio, A.W. anonge and J.K. Kandie 35 θ ρc θ σb U (9 L ρc ( w Ping he relevan dimensionless variables in (9 gives θ θ Ec Pr Ec accons for Jole heaing.. ( Eqaion ( is he energ eqaion in dimensionless form. he ICS ogeher wih he BCS in dimensionless form are:, θ, for all cos,, θ a (, θ, as, > Le he solion of he momenm eqaion be ( Y ( (, or siml Y d d Y cons an. ( d Y d. Since he LHS is a fncion of indeenden variable and HS is a fncion of indeenden variable, and he wo canno be eqal o each oher nless boh redce o a consan vale. Le he consan be so ha d d and d Y Y d. he solions are A ( ex and Y A ex( A3 ex( (, ( Aex( B ex( ex(.. I follows ha (3 Now since (, (, B ex( ex(, we ms have A. he solion redces o. (4

10 36 An Invesigaion of HD Flows of Nanoflid Aling he bondar condiion ( cos, we have ( ( ( B i ex ex( ex e meaning ha B and i or i. he niqe solion becomes, ( ( ex ex(, or ( ( [ ] i i e ex, (5 or an cos 4 ex e, i an sin 4 or ( ( ( 4 4, e ex cos an ex sin an i or (6 an sin 4 cos an cos 4 ex,

11 B.. Namai, J.K. Bio, A.W. anonge and J.K. Kandie 37 elacing and in eqaion ( in view of (6 we have, θ Pr ( θ ' Ec ex 4 cos an cos eqaion (7 ma be wrien in comac form as 4 sin an (7 θ θ Ec ex( a ( ( ex a cos b ( Pr ( a sin( b abe c ex. Ec (8 Where a ( cos an 4, b Le ( 4 sin an Ec ( a ex( a cos( Ec ( b f, ex So ha ( a sin( b abec ex θ θ ( Pr f (,. (9 Now, ( Pr υ α Where ν and α are he inemaic viscosi and hermal diffsivi of he

12 38 An Invesigaion of HD Flows of Nanoflid nanoflid resecivel. he inemaic viscosi is given b, µ υ ( ρ And he hermal diffsivi is given b, α ( ρ ( c hs he Prandl nmber ma be exressed as, ( ( c µ Pr ( or Pr ( ( c µ f µ r ( φ( c φ( c ( ( P φ 5 f s (3 Eqaion (3 ma be wrien as: θ µ ( φ( c φ( c θ f f s 5 ( Pr ( φ f (, or siml, θ θ λ Whereb, f (, µ f λ. ( φ( c φ( c ( Pr ( φ f 5 Assme a ariclar inegral of he form θ (, Aex( a B ex( a cos( b ( a sin( b C ex. s. (4 (5 I follows ha θ B C ex b. (6 ex( a sin( b ( a cos(

13 B.. Namai, J.K. Bio, A.W. anonge and J.K. Kandie 39 θ 4a Aex( a [( 4a 4b B 8abC] ex( a cos( b [ 8abB ( 4a 4b C] ex( a sin( b. (7 Sbsiing eqaion (6 and (7 ino eqaion (4 and eqaing coefficiens we ge Ec A, 8λa ( ( 8λab B λ 4a 4b ( 8 ( λab C λ 4a 4b B C Ec Afer solving for B and C in eqaion (8 we ge B C Ec λ ( 4a 4b ( 8λab ( 4λa 4λb Ec λ( 8λab ( 8 ( 4 4. λab λa λb he ariclar inegral becomes θ ( ex( az Ec 8λa Ec λ( 4a 4b ( 8λab ( 4λa 4λb (8 ( b, cos Ec λ( 8λab ( 8λab ( 4λa 4λb sin ( b We shall creaivel snhesize he comlemenar fncion of eqaion (4 from some solion of he homogeneos ar of he eqaion so ha he bondar condiions are saified. Some of hese solions inclde: (9

14 4 An Invesigaion of HD Flows of Nanoflid ex λ cos λ saionar srface is secified o be harmonic of ime. ex λ sin λ saionar srface is secified o be harmonic of ime., is a solion if he emerare of a, is a solion if he emerare of a erf λ is a solion o or from a saionar srface de o a emerare Ec difference or oenial of. w 8λa B he rincile of serosiion he comlemenar fncion ma be wrien as, θ c (, Ec Ec ex cos 8λa 8 ab 4 a 4 b λ λ 8 a λ ( 4a 4b ( λ ( λ λ Ec λ( 8λab sin ( b ex sin ( 8λab ( 4λa 4λb λ λ Ec erf θ λ w θ θw erf λ λ (. (3 he sm of eqaion (9 and (3 gives s he comlee solion Ec λ ( 4a 4b Ec θ(, ex( a cos b 8λa ( 8λab ( 4λa 4λb λ( 8λab ( 8λab ( 4λa 4λb ( Ec sin ( b

15 B.. Namai, J.K. Bio, A.W. anonge and J.K. Kandie 4 Ec Ec ex cos 8λa 8 ab 4 a 4 b λ λ λ ( 4a 4b ( λ ( λ λ Ec λ( 8λab sin ( b ex z sin ( 8λab ( 4λa 4λb λ λ Ec erf θ 8λa w θ θw erf λ λ (. (3 Where he error fncion erf ( is defined as erf ( γ ex( r dr has he roeries erf ( and ( erf. π γ and 4 esls and Discssions Below are grahs of veloci and emerare rofiles in connecion o he arameers of ineres..9.8 wi/ wi/3 wi/4 wi/.7 Veloci decreasing Figre : Veloci rofiles for differen vales of when and.8

16 4 An Invesigaion of HD Flows of Nanoflid Veloci..5 decreasing Figre 3: Veloci rofiles for differen vales of when i / and saionar srface w4i w8i wi w6i emerare.8.6 increasing Figre 4: emerare rofiles for differen vales of Pr7.. when,, E.3,

17 B.. Namai, J.K. Bio, A.W. anonge and J.K. Kandie saionar srface emerare.8.6 decreasing Figre 5: emerare rofiles for differen vales of when 4 i,, E.4, Pr saionar srface Ec. Ec. Ec.3 Ec.4 emerare.8.6 increasing Figre 6: emerare rofiles for differen vales of Ec when i /,, Pr7. and 5.

18 44 An Invesigaion of HD Flows of Nanoflid saionar srface v.9 v.7 v.4 v. emerare increasing Figre 7: emerare rofiles for differen vales of v (nanoaricle concenraion when i /,, Ec. and saionar srface Pr7. Pr5.3 Pr3.39 Pr.6 emerare decreasing Figre 8: emerare rofiles for differen vales of Pr when i /,, Ec. and 5.

19 B.. Namai, J.K. Bio, A.W. anonge and J.K. Kandie 45 Figre shows he variaion of emerare wih. Here, i can be seen ha veloci decreases wih an increase in. Figre 3 deics he effec of on veloci. Evidenl, veloci decreases wih an increase in. increasing leads o an increase of he Lorenz force which ooses he flow of he nanoflid. Figre 4 illsraes he relaionshi beween emerare and oscillaions freqenc. I is observed ha emerare increases wih he increase of oscillaion freqenc. Figre 5 shows ha emerare increases wih a decrease in which is aribed o he fac ha smaller corresonds o higher veloci which in rn leads o high elecrical crren hence he increase in Jole heaing. Figre 6 indicaes ha emerare increases wih an increase in Ec. Figre 7 demonsraes he effecs of nanoaricle concenraion on emerare. Clearl, he higher he concenraion he higher he emerare. his is de o he fac ha nanoaricles imroves he hermal condcivi of he flid. From figre 8 i is observed ha emerare decreases wih an increase in Pr. his is becase Pr redces he hermal condcivi of he nanoflid. From Figres 4-8 i can be observed ha he emerare of he nanoflid over a saionar srface is he lowes as comared o ha of over oscillaing srface. his means ha oscillaion of he srface leads movemen of he nanoflid which in rn leads o flow of elecrical crren which resls o Jole heaing hence he increase in emerare. 5 Conclsion Sdies on HD flow of nanoflid over an oscillaing srface have been carried o. I was fond o ha veloci decreases wih he freqenc of oscillaion as well as wih magneic arameer. I was also observed ha emerare increases wih an increase in Erer nmber (Ec, nanoaricle concenraion and he freqenc of oscillaion b decreases wih an increase in magneic arameer ( and he Prandl nmber (Pr. In smmar, oscillaion of

20 46 An Invesigaion of HD Flows of Nanoflid he srface over which a nanoflid flows increases Jole heaing deending on he alied magneic field srengh, oscillaion freqenc of he srface. Increasing nanoaricle concenraion increases he hermal condcivi of he nanoflid and his imroves he oenial o remove hea added ino he flid as a resl of Jole heaing. eferences [] S. Osrach, Noe on he Aerodnamic heaing of an Oscillaing srface, NACA, Washingon, 954. [] V.. Sondalgear, U.N. Das and E.K. Ea, Free convecion effecs on HD flow as an Iinie verical oscillaing lae wih consrain hea flx, Indian Jornal of ahemaics, 39(3, (997, 95-. [3]. hcmaraswam and S. eena Kshisndaam, heoreical sd of chemical eacion effecs of verical Oscillaing lae wih variable emerare, heoreical Alied echanics, 33(3, (6, [4].C. Chandhanr and A. Jain, Combined Hea and ass ransfer Effecs on HD Free or Convecion Flow Pas an Oscillaing Plae Embedded in Poros edim, om. Jornal of Phs., 5(5-7, (7, 55-54, Bchares. [5] P.. Kishore, V. ajesh and S. Vijaamar, he effecs of hermal radiaion and viscos dissiaion and HD hea and ass diffsion flow as an Oscillaing verical lae embedded in oros medim wih variable srface condiions, heoreical. A. ech., 39(, (, 99-5, Belgrade. [6] S. Ahmed and K. Kalia, Analical and Nmerical sd for HD adiaion Flow over an iinie Oscillaing verical Srface Bonded b a oros medim in resence of Chemical eacion, Jornal of Alied Flid echanics, 6(4, (3, , [7] H. Abid, S ohd,. afman and K. Ilas, Unsead Bondar laer flow and hea ransfer of a Casson Flid as an oscillaing verical lae wih

21 B.. Namai, J.K. Bio, A.W. anonge and J.K. Kandie 47 Newonian Heaing, (4, available a Dol:.37/jnral.one.8763,h.//jornals.los.org/-lasone/aricle?d.37/jornal.one.8763 PLOS (4. [8] D. dra and C.Nhaben, HD flow as a verical Oscillaing lae wih adiaion and chemical eacion in oros medim, Jornal of mahemaics (5-J, II, (, (5, 46-5, IOS. [9] S. Choi, Enhancing hermal condcivi of flids wih nano aricles, ASEefed, ( 3, (995, [] Y. Xan and W. oefe, Conceions for Hea ransfer Correlaion of Nanoflids, Inernaional Jornal of Hea and ass ransfers, 43, (, [] Q. Li and Y.. Xan, Convecive Hea ransfer and Flow Characerisics of C-waer nano-flid, Sciences in China Series E, echnological Sciences, 45(4, (, [] P. asinmai, J.A. Asrom,. Kaaja,. Lava-Koo, A. Koonen, A. Jasberg,. Shab and J. imonen, Clsering and viscosi in a Shear flow of a ariclar Ssension, Phsical eviews E, 68, (3, 643. [3] D.Wen and Y.Ding, Exerimenal Invesigaion ino Convecive Hea ransfer of Nanoflids a he Enrance egion nder Laminar flow Condiions, In. J. of Hea and ass ransfer, 47, (4, [4] S.Charabor and S.o, hermall Develoing elecroosmosic ransor of nanoflids in microchannels, Jornal of microflids and Nanoflid, 4, (8, 5-5. [5] B. Ghasemi and S.. Aminossadai, ixed Convecion in a lid driven rianglar enclosre filled wih nano flids, Inernaional Jornal of Alied ahemaics and echanics, 7(6, (, 5-7. [6]. Chad, hermal insabili of roaing nano flid, J. of Alied flid mechanics, (.

22 48 An Invesigaion of HD Flows of Nanoflid [7] A. eha, Hea Exchanger in nano flid, In. Jornal of Advanced Engineering echnolog, 3(4, (, [8] A. Aninda and K. Asho, Nmerical Analsis and hea ransfer ower Law NanoFlids in icrochannels, Proceedings of he 3 rd Eroean Coerence of microflids icro flids, Heidelberg, (December 3-5,. [9] F.. Had, Sringer oen access Jornal, eceived (December,, h:// /7//9 (. [] Ferdows and Khan, HD mixed convecive Bondar laer flow of a Nanoflid hrogh a oros medim de o exoneniall sreching shee, ahemaical roblems in Engineering,, (, -. [] A. Bar and Z. aizah, Unsead HD ixed convenion flow of a viscos dissiaing micro olar flids in a bondar laer sli flow regime wih Jole heaing, Inernaional Jornal of Scienificall & Engineering esearch, 3(8, (, [] S.P. Anjali, and A. Jlie, Laminar Bondar Laer Flow of Nanoflid over a Fla Plae, Inernaional Jornal of Alied ah and echanics, 7(6, (, 5-7.