Flow of Oldroyd-B Fluid between Two Inclined and Oscillating Plates

Transcription

1 J. Appl. Environ. Biol. Sci., 5()59-67, 05 05, TexRoad Publicaion ISSN: Journal of Applied Environmenal and Biological Sciences Flow of Oldroyd-B Fluid beween Two Inclined and Oscillaing Plaes Abdul Shakoor,Taza Gul,3, Zahir Shah, Muhammad Alaf khan, S.Islam, F. Khan Mahemaics Deparmen, ISPaR/ Bacha Khan Universiy Charsadda, KPK Pakisan Mahemaics Deparmen, Abdul Wali Khan Universiy Mardan, KPK Pakisan 3 Higher Educaion Deparmen Peshawar, KPK, Pakisan Received: June 9, 05 Acceped: Sepember, 05 ABSTRACT The sudies of his paper relaed o he unseady magneohydrodynamics (MHD) incompressible Oldroyd-B fluid beween wo periodically vibraed inclined plaes. The basic governing equaions of he above menioned fluid are modeled and simplified in he form of nonlinear PDEs. The soluion of he above non-linear PDEs has been obained by using wo analyical mehods, namely HPM and OHAM. The comparison of hese wo mehods are also analyzed and found in very good agreemen. Finally, he physical effecs of modelled parameers have been sudied graphically. KEYWORDS: Unseady moion, hin film, MHD, Oldroyd-B fluid, inclined bel, OHAM and HPM. I. INTRODUCTION In everyday life and in engineering flow of non-newonion fluid are frequenly occurred. I is ubiquious in naure and echnologies. Therefore, o undersand here mechanics is essenial in mos of applicaions. Mos of i problems appeared in several areas varying from blood o he engineering and echnologies problem. In non-newonian fluids Thin film flows have a large variey of pracical applicaions in nonlinear sciences and engineering indusries. For modelling of non-newonion fluid flow problem we used several model lik (second grade, hird grad, fourh grad oldoryd four, oldoyed-b) ec fluids model by which we go linear or nonlinear ode,s or pde,s which may be solved exacly, analyicaly or numerically. For soluion of hese problems differen varieies of mehods are used, in which (ADM,VAM,HAM,HPM,OHAM) are frequenly used. In his work we have modelled a non linear pde,s by using oldroyd-b fluid model. Soluion of he problem is obained by using OHAM and HPM. Anakira e all [] have used OHAM for he soluion of Delay differenial equaions. Aksele all [] have been examined he soluions of for some unseady of oldroyd-b fluids in unidirecional flows. Ayube all [3] discussed he oscillaing moions in plane wall for said fluid. Burdujan [4] ake i in paricular class and give a brief discussion on i. Feecau e al. [5-6] sudied oldroyd B fluid in form of consanly acceleraing flow over a fla plae and in second paper he discussed ransien oscillaing moion using same model in cylindrical domains. In boh case he obained he exac soluions. The relaed work can be seen in [3-6].Ghosh e all [7-9] examined Oldroyd-B fluid under he effec of MHD in differen case ha is pulsaing flow, Chanel flow and induced by recified sine pulses. All of deails is given briefly in heir works. Gul e al. [0, ] invesigaed he MHD hin film flow of non-newonian fluid on a verical and in inclined oscillaing bels. They obained analyical soluion of using he Asympoic perurbaion mehods (ADM, OHAM and HPM). The obained resuls of he lif and drainage problems for velociy and as well as for emperaure field are compared and given graphically. The properies of several physical parameers are also explained. Hameede all [3] discussed hin film fluids in verical bel. Haiao [4] e all sudied unidirecional flow of an oldoyd-b fluid. He JH [5-0] has been given a brief disrupion on HPM mehods. Khan e al [-3]sudied Oldroyed b fluid in roaing sysem wih effec of MHD and also sudied i in unseady sa and find some exac soluions for i. They examined i in porous space. Kashkari [4] examined he OHAM applicaion and approximae soluion of nonlinear Kawahara equaion. Lie e all [5-6] sudied Axial MHD flow of generalized Oldroyd-B fluid due o wo oscillaing cylinder. They gives basic inroducion of HPM wih deail examples. Marinca e al. [6-9] applied OHAM on non-linear seady flow problem modelled from fourh order fluid and find approximae soluion. They used OHAM for hin film flow problem and for nonlinear oscillaors wih disconinuiies. Mabood e al [30-3] gives applicaion of Opimal homoopy asympoic mehod for he approximae soluion and hey also examined he hea ransfer on MHD sagnaion poin flow in porous medium. Nofel [3] examined emperaure conducion and fracional vanderpol damped nonlinear oscillaor. Shahid e al [33] sudied unseady flow of Oldroyd-B fluid in seady sas. Transien soluion have been obained wih used of Laplace and Fourier series. Shah e al. [34] examined hin film flow of hird order fluid on moving inclined plane and found i soluion by using OHAM. Siddiqui e al. [35-37] examined he hin film *Corresponding Auhor: Abdul Shakoor, Mahemaics Deparmen, ISPaR/ Bacha Khan Universiy Charsadda, KPK Pakisan 59

2 Shakoor e al.,05 flow of a hird order fluid over an inclined plane as well hey sudied i in moving verical bel. For soluion of he non-linear problems hey used HPM in firs case and OHAM in oher case. Momenum and coninuiy equaion are wrien as II. Basic Equaion Where div v = 0, () Dv ρ = divτ+ J B + ρgsin θ. D () J = 0, σb0u,0, (3) = σ E+ U B, B = µ J ( ). J 0 D shows he maerial ime derivaiveρ is densiy, andg is body force andvis he velociy.where B is he D uniform inducion filed,σ is he elecrical conduciviy andj is curren densiy. DΨ Ψ T = + ( v. ) Ψ ( v) Ψ Ψ ( v). (5) D The above model can be reduced o differen ypes of fluid depend on λ r(relaxaion ime) and λ (reardaion ime).in equaion (5), if r λ = λ he fluid becomes viscous. Whenλ = 0, i becomes a Maxwell fluid and reduced o Oldroyd-B fluid when 0< λ < λ <. The cauchy sress ensor,τ is piis isoropic sress. T = Ρ I+ τ, (6) D Τ D Τ + λ = µ + λ, D D A (7) Τis used for exra ress ensorand µ is used for cofficien of viscosiy. Ais he Rivlin Ericksen sress ensor`. III. Formulaion of he Problem T A0 = I, A = v+ ( v ), = v, (8) Consider incompressible non-newonian Oldroyd-B fluid beween wo inclined and parallel plaes y= h and y= h.boh plaes are oscillaing.the configuraion of fluid flow is along he y-axis and perpendicular o x-axis. A ransvers uniformmagneic field is applied on inclined direcionl where he plae is non conduciong.the pressure Ρ is kep consan, ha is pressure gradien is zero.the flow is supposed o be unseady flow, laminar.graviaional force and magneics force causes he fluid moion. Fig.,Geomery of he problem The velociy field is of he form 60

3 J. Appl. Environ. Biol. Sci., 5()59-67, 05 subjec o he boundary condiions v= ( u( y, ),0,0), and = ( y,) ( ) ω ( ) Where ωis he frequency of he oscillaing bel. The momenum equaion () is reduced o I follows from (7) and (9) ha Equaion (6) reduces o Where ( y) u h, = Ucos, u h, = Ucosω,(0) τ (9) v p τxy ρ = + + ρgsinθ σb0 v, x y p τyy =, y y p =0. z τxx v v τxx + λ τ xy = µλ, y y τxy v v v τxy + λ τ yy = µ + λ µ, y y y yy τyy + λ τ = 0, ( ). () () (3) (4) (5) (6) τ yy =Ψ y e λ (7) Ψ any arbirary funcion. When << 0, hen τ = 0. Equaions () and (5) and in he presence of zero pressure gradien, we obain v v σb0 + λ = v + λ + λ v+ ρgsin θ. y ρ (8) Inroducing non-dimensional variables ( ( ( ( v y v λv λv ωδ ρ δ ρgsinθ σb0δ v=, y =, =, Κ =, Κ =, ω =, m=, M =. (9) V δ ρδ ρδ ρδ µ µ V µ where,m is he magneic parameer, ω is he oscillaing parameer, Κis he relaxaion paramer, Κ is he reardaion paramee and m is he graviaional parameer. Using (9) in (8) and (0) and dropping bars we obain yy u u M u m λ +Κ = +Κ + +. y (0) v, = cosω, v, = cosω, () ( ) ( ) IV. The HPM Soluion for he Problem By applying HPM o (0) and () componen form is ast v y 0 0 Ρ : = v y v0 v0 v 0 v0 Ρ : = k + k Mk + m Mv 0, y y y y m, (45) (46) 6

4 Shakoor e al.,05 v y v v v v Ρ : = Κ M +Κ Mv, (47) y y y Soluions of equaions (-4) using boundary condiion in equaion (0) is v0( y, ) = m( y ) + cos [ ω]. (48) 4 m M( 5-6 y + y ) MCos[ ω]( y ) ωsin[ ω]( y ) ω Cos[ ω] + Κ v ( y, ) = 4 ( y ) MωSin[ ω] Κ( y ) (49) Mm(300+ 6M 360y 75mMy + 60y + 5 My My ) + M cos[ ω] ( M 70y 80My + 30Mx ) + ω cos[ ω] ( y 30y ) + ωsin[ ω] c ( M + 70y + 360My 60My ) + M( m+ 60x 80x 80mx 60x mx ) + ω cos[ ω] Κ( M + 70y + 70My 0My ) + Mωsin[ ω] Κ( v( y, ) = 300M + 70y 60My ) + ω sin[ ω] Κ( 360y + 60y ) + M ω cos[ ω] Κ( y y ) + ω cos[ ω] Κ (50 80y + 30y ) + Mω sin[ ω] Κ ( y + 60y ) + 360ω (50) cos[ ω] Κ(y ) + 360Mωsin[ ω] Κ(y ) 360Mω cos[ ω] ΚΚ(y ) ω sin[ ω] ΚΚ( y ). The series soluions of velociy profile up o second componen is v(y,) m( y ) cos [m M 5-6 y + 4 ( ) ( ) ( ) ( ) v y, = v y, + v y, + v y,. (5) 0 [ ω] ( y ) MCos[ ω] c( y ) ωsin[ ω] 4 = ( y ) ω [ ω] Κ( ) ω [ ω] Κ( ) Cos y M Sin y ] + ( Mm(300+ 6M 360y 75mMy 4 [ ω] ω cos[ ω] y 5 My My ) M cos (700 50M 70y 8 My Mx ) ( y 30y ) + ωsin[ ω] ( M + 70y + 360My 60My ) + Mc (80+ 50m+ 60x ) [ ] ( [ ω] M y My ω [ ω] k y y M ω os[ ω] y y [ ] y + y M [ ] y + y [ ω] (y ω [ ω] ω [ ω] 60ω sin[ ω] x 80mx 60x + 30mx + ω cos ω Κ 0 600M + 70y + 70My 0My ) + Mω sin Κ ( ) + sin ( ) + c Κ ( ) + ω cos ω Κ ( ) + ω sin ω Κ ( ) + 360ω cos Κ ) + 360M sin Κ (y ) 360M cos Κ Κ (y ) + 3 Κ Κ ( y ). 3 V. The OHAM Soluion for he Problem (5) The componen problems of velociy profile are when OHAM mehod is applied Ρ v y = y 0 0 : m. v y v v v v Ρ = Κ + + +Κ + Κ + + y y y (54) : c ( c) c c( M ) m( c) Mcv0 (53) 6

5 J. Appl. Environ. Biol. Sci., 5()59-67, 05 v y v v0 v 0 v v p : = Κ c + c + kc + ( + c) + kc y y y y y (55) v v0 c( + MΚ ) + c( m Mv0) Mcv c( + MΚ). Soluions o equaions (30-3) using boundary condiion in equaion () are v0( y, ) = m( y ) + cos [ ω]. (56) 4 ( ) + MCos[ ω]( y ) ωsin[ ω]( y ) [ ] ( y ) M [ ] Κ( y ) m Mc 5-6 y + y c c v( y, ) = 4 cω Cos ω Κ c ωsin ω (57) The second componen soluion for velociy (in OHAM) is very larg, herefore, only graphical represenaions up o second order are given. The series soluions of velociy profile is obained as The values of for he velociy componens are v y = v y + v y + v y (58) 0 C = , C = Table Numarical Comparison of OHAM and HPM for he velociy profile, when M = 0.0, = 5, k= 0.0, k = 0.09, = 0, ω = 0.09, m = 4. x OHAM ADM Absolue Error Fig., Comparison of OHAM and HPMsoluions for velociy profileby when ω = 0., m = 0.4, M = 0.5, = 5, k= 0.6, k = 0.3, 63

6 Shakoor e al.,05 Fig. 3 Influence of differen ime level whenω = 0., m = 0.4, M = 0.5, = 5, k= 0.6, k = 0.3, Fig. 4,Influence of magnaic parameer on velociy profile whenω = 0., m = 0.5, = 5, k= 0.5, k = 0.7. Fig. 5,velociy disrubaion a diffrien ime level ω = 0., m = 0.4, =, k= 0.5, k =

7 J. Appl. Environ. Biol. Sci., 5()59-67, 05 Fig. 6,Velociy disribuion graphs whenω = 0., m = 0.4, M = 0.5, = 5, k= 0.6, k = 0.3, Fig.7, Effec of non-newonian parameerk on velociy profiles whenm = 0.4, M = 0.3, =, k= 0.7. Fig.8,Effec of non-newonian parameerkon velociy profiles when ω = 0., y = 0.4, M = 0.5, = 5, k=

8 Shakoor e al.,05 Fig.9,Effec of graviy parameer on he velociy profile when ω = 0., y = 0.5, M = 0.5, = 5, k= 0.7. VI. RESULTS AND DISCUSSION In his work we have sudied Oldroyd-B fluid in wo inclined oscillaing parallel plaes.the fluid is considered in unseady form under he effec of MHD.The obained resuls from boh OHAM and HPMmehods are compared.we have found ha hese resuls are in excellen agreemen.tables [] show he numerical comparisons of OHAM and HPM.The absolue errors beween boh mehods are calculaed and shown.fig.[] shows he geomery of he problem. Fig. analyise he comparison of OHAM and HPM soluions by using various values of parameers ha is( ω,m,, k, k).fig[3,5,6]show he influence of velociy disribuion using parameers (,M,, k, k) ω and show ha velociy field is increases by increasing hese parameers.all resuls for he Oldroyd-B fluid near he plaes are showed in he y-coordinae only for a specific domain y 0,.The influencem on velociy field is shownin Fig [4] which give a resul ha if we increase [ ] i,hemoionof fluidflowis also increased. The effecs of Κ (relaxaion ime parameer) and Κ (reardaion ime parameer) are shown Figs. [7, 8]. Increase in hese parameers increases he velociy profile.fig [9] show he graviy effec m. Increasing i fluid moion is clearly increased. VII. CONCLUSION We have concluded he excellen agreemen of HPM and OHAM for he menioned modelled problem. REFERENCES []Anakira NR, Alomari AK and Hashim I (03) Opimal Homoopy Asympoic Mehod for Solving Delay Differenial Equaions. Hindawi Pub Corp Mah Problems in Eng: -3. [] Aksel N, Feecau C and Scholl M (006) Saring soluion for some unseady unidirecional flows of oldroyd-b fluids. Z augew Mah Phy 57: [3]Anjum A, Ayub M, Khan M (0) Saring soluions for oscillaing moions of an Oldroyd-B fluid over a plane wall. Communicaions in Nonlinear Science and Numerical SimulaionVol 7(): [4]Burdujan L (0) The flow of a paricular class of Oldroyd-B fluids. j. Series on Mah and is Applicaions Vol 3: 3-45 [5]Feecau C, Shara C, Prasad SC and Rajagopal KR (007) A noe on he flow induced by a consanly acceleraing plae in an Oldroyd-B fluid. Applied Mahemaical Modeling 3: [6]Feecau C,Haya T and Feecau C (008) Saring soluions for oscillaing moions of Oldroyd-B fluids in cylindrical domains.journal of Non-Newonian Fluid Mechanics vol 53(): 9-0 [7]Ghosh AK and Sana P (009) On hydromagneic flow of an Oldroyd-B fluid near a pulsaing plae. Aca Asronauica 64: 7-80 [8]Ghosh AK and Sana P (009)on hydromagneic channel flow of an Oldroyd-B fluid induced by recified sine pulses. Compuaional and applied mah Vol 8(3):

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