TIME-SPACE DEPENDENT FRACTIONAL VISCOELASTIC MHD FLUID FLOW AND HEAT TRANSFER OVER ACCELERATING PLATE WITH SLIP BOUNDARY

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1 HERMAL SCIENCE: Year 7, Vol., No. A, pp. 7-7 IME-SPACE DEPENDEN FRACIONAL VISCOELASIC MHD FLUID FLOW AND HEA RANSFER OVER ACCELERAING PLAE WIH SLIP BOUNDARY b Shenging CHEN a, Liancn ZHENG a*, Chnri LI a,b, and Jize SUI a,b a School of Mechanical Engineering, Universi of Science and echnolog Beijing, Beijing, China b School of Mahemaics and Phsics, Universi of Science and echnolog Beijing, Beijing, China Original scienific paper hps://doi.org/.98/scic he MHD flow and hea ransfer of viscoelasic flid over an acceleraing plae wih slip bondar are invesigaed. Differen from mos classical wors, a modified ime-space dependen fracional Maxwell flid model is proposed in depicing he consiive relaionship of he flid. Nmerical solions are obained b explici finie difference approximaion and exac solions are also presened for he limiing cases in inegral and series forms. Frhermore, he effecs of parameers on he flow and hea ransfer behavior are analzed and discssed in deail. Ke words: MHD flow, Maxwell flid, ime-space fracional derivaives, slip bondar Inrodcion Non-Newonian flids do no saisf he linear relaionship beween sress ensor and he rae of deformaion ensor, i has received mch aenion de o he varios applicaions in engineering and indsr, inclding food sffs, molen plasics, plps, perolem drilling, and oher similar aciviies. As an imporan class of non-newonian flids, viscoelasic flids show he properies of boh elasici and viscosi. Plen of models have been proposed o describe he response characerisics of hese flids, among which he Maxwell model has been sdied mos widel [-]. Fracional calcls has been applied sccessfll in describing he complex viscoelasic flids []. Generall, hese governing eqaions are derived from classical eqaions which are modified b replacing he ime ordinar derivaives of sress wih he fracional calcls operaors. his ind of generalizaion allows s o define non-ineger order inegrals or derivaives precisel. Wih he developmen of research, he ineres in viscoelasic flids has considerabl increased. Nazar e al. [7] sdied he veloci field and he shear sresses of generalized Maxwell flid on oscillaing recanglar dc. Jamil e al. [8] and Feeca e al. [9] discssed he nsead flow of Maxwell flid wih fracional derivaive. Yang and Zh [] sdied he flow of a viscoelasic flid in a pipe. Cao e al. [] derived in ime domain he fndamenal solion and relevan properies of he fracional order weighed disribed parameer Maxwell model. Haa e al. [] sdied he hea and mass ransfer effecs in -D flow of Maxwell flid * Corresponding ahor, liancnzheng@sb.ed.cn

2 8 HERMAL SCIENCE: Year 7, Vol., No. A, pp. 7- over a sreching srface wih convecive bondar condiions. Vier e al. [] invesigaed he ime-fracional free convecion flow of an incompressible viscos flid near a verical plae wih Newonian heaing and mass diffsion. Msafa e al. [] addressed he flow of Maxwell flid de o consanl moving fla radiaive srface wih convecive bondar condiion. Some aemps concerning his field we refer o he recen papers [-8]. However, mos of he analical solions of he fracional flid model conaining complex series or special fncions, which is no condcive o approximae calclaion. he finie difference mehod, becase of is flexibili, conines o be an efficien and reliable mehod. he finie differen mehod is now fond large applicaions in solving fracional differenial eqaions [9-]. Moivaed from he afore menioned sdies, he aim of his paper is o exend he resls of Zheng e al. [] o consider he MHD flow and hea ransfer of viscoelasic flid over an acceleraing plae wih slip bondar. A modified ime-space dependen fracional Maxwell flid model is proposed in depicing he consiive relaionship of he flid. Moreover, for he limiing cases, he similar solions are obained b means of Laplace ransform, which are presened in erms of series. Finall, he effecs of differen parameers on veloci and emperare fields are invesigaed and analzed. he basic eqaions Consider he flow and hea ransfer of a modified Maxwell flid, which depics b he ime-space dependen fracional derivaive (in he consiive relaionship), ignoring he pressre gradien, he governing eqaion can be wrien: λ (,) + λ + D = D (,) + D (,) ν M () λ γ (,) + + D D (,) γ = () ρ cp In he relaionship, (,) is he veloci, (,) is he emperare, D and D are fracional calcls operaors based on Capo definiion and Riemann-Lioville definiion, respecivel, in he form: ( ) D () = d < Γ( ) f τ f τ () ( τ) d f ( ξ ) D f( ) = d < Γ( ) d ξ () ( ) ξ where Γ () is he gamma fncion, D γ he fracional calcls operaors based on Capo definiion as eq. (), ν = µρ / he inemaic viscosi, ρ he consan densi of he flid, and λ, λ he relaxaion imes. Parameer and characerize he fracional srcres [] and he are inrodced for dimensional balance, M = B / ρ where is he elecric condcivi, B he magneic inensi, he hermal condcivi, and c p he specific hea capaci of flid. his model is redced o he generalized Maxwell model [] when = and o he ordinar Maxwell model when =, =.

3 HERMAL SCIENCE: Year 7, Vol., No. A, pp. 7-9 Saemen of he problem and solions I is assmed ha he flids are saic on he plae a firs, and a he ime = +, he plae achieves an acceleraed veloci in he x-direcion wih slip bondar. he shear sress resls in he moion of he flid. he governing eqaion is given b eq. (). Accordingl, he iniial and bondar condiions are: λ λ + D (, ) = + D + > A θ () = (,) (,) = = > () (, ), (, ) as > (7) where A is he consan acceleraion and θ is he slip coefficien. If θ = hen he no-slip bondar condiion is obained. If θ is finie, flid slip occrs a he wall b is effec depends on he lengh scale of he flow. We assme is he emperare of he flid a he momen =, + ( w ) f() denoes he emperare of he plae for (wih f() be a nown fncion). he corresponding iniial and bondar condiions for energ eqaion are: (, ) = + ( ) f( ) (8) w (,) (,) =, = > (9) (,), (,) as () Emploing he non-dimensional qaniies: ( + ) + * * ( Aν) * ( Aν) = = = ν ν ( Aν) ( + ) + * ( Aν) * ν * = M = M ( ) ν + + ( Aν) ( + ) γ + * ( Aν) γ * λ = λ, a = ν µ cp,,, λ λ,, =, Dimensionless moion eqaions can be derived (for brevi he dimensionless mar * is omied here): (,) λd D (,) M λ D (,) () γ (,) + ( + λ D ) = a D (,) () + ( + ) = ( + ) w ()

4 HERMAL SCIENCE: Year 7, Vol., No. A, pp. 7- he corresponding iniial and bondar condiions become: + λ = A + λ + θ () ( D ) (, ) ( D ) (,) (,) = =, (,), (,) as () = (, ) = f( ) () (,) (,) = =, (,), (,) as (7) Disperse space and ime a grid poins and ime insans, define: = + jh, h > j =,,, (8) j n = nτ τ > n =,,, (9) j j where h and τ are he lengh of space and ime seps, respecivel. Define i and i as he nmerical approximaion o (, ) and (, ), we obain: i n i Γ n j n j D (,) = b i ( bj bj ) i bj i + O( τ ) < < i τ ( ) = Γ j j j n i i i i D (,) = c ( cj cj ) + O( τ ) < < i τ ( ) τ = τ scheme: Inrodce he coefficiens: i+ n j i + i h = () () D (,) = d + Oh () < < () d τ b = ( + ) τ c = ( + ) + =, d = = d,, he ime firs-order derivaive can be approximaed b he Eler bacward difference ( i, j) ( i, j) ( i, j ) = + O( τ ) τ he explici finie difference approximaion for eqs. () and () are: j j j i i λ j j + c( i i ) ( cj cj )( i i ) = τ τ Γ( ) = () () () ()

5 HERMAL SCIENCE: Year 7, Vol., No. A, pp. 7- i+ = ( ) + + j j j M λ j di Mi b i bj bj i bj i h = τ Γ( ) = (7) λ a Γ j j j i+ i i j j j + c( i i ) ( cj cj )( i i ) = + d i + τ τ ( γ) = h = Special cases we obain: Leing in eq. (), we aain he veloci field eqaion: Denoing b: ( + ) = M( + ) (8) (,) (,) D λ λ D (, ) (9) { } s ( s, ) = L (, ) = e (, )d () he image fncion of (,) and appling he Laplace ransform o eq. (9): = ( s+ M) ( + λ s ) () According o he bondar condiions: A (, s) = + s ( s, ), θ ( + λs ) ( s, ) as = () () { λ } A = exp ( s+ M)( + s ) s+ M s + θ + λs In order o avoid he complexi of calclaing he resides and conor inegrals, we appl discree inverse Laplace ransform o ge he veloci and express eq. () as series form: () l n l + l l m n λ Γ ( ) ( ) ( ) ( ) m Γ n θ M = A l= l! =! m= m! n= n! l + l Γ Γ s Appling he discree inverse Laplace ransform, we have: l+ l m + n + () l l+ l n m n l m n + + ( θ) ( ) ( M) ( ) λ = A l!! m! n! l= = m= n=

6 HERMAL SCIENCE: Year 7, Vol., No. A, pp. 7- l + l Γ m Γ n l + l l + l Γ Γ Γ m + n + which is similar o he solion, eq. () of [] when θ = M = and =. In he same wa, we obain he solion of eq. () (le λ =, see eq. () in []). n n λ Γ n ( ) ( ) ( γ ) + γn = f( ss ) ds = n= n! ( ) a! Γ Γ γ + + γn () (7) Resls and discssion In his paper, he flow and hea ransfer for modified Maxwell flid wih ime-space fracional derivaives are sdied, where he flow is de o an infinie consanl acceleraing plae wih slip bondar. he generalizaion here is a pe of new fracional operaors and defined in he Capo and Riemann-Lioville sense. For vales of he parameers of he flid, he veloci field and he emperare field disribions are shown as in figs. -. For he sae of he simplici, we ae A = in all figres. Figres and show he veloci field disribion wih he fracional parameers. I is seen ha he smaller he, he more slowl he veloci decas. However, one can see ha an increase in maerial parameer has qie he opposie effec o ha of. Meanwhile, he resls also indicae he inflence of he magneic parameer, which decreases he veloci. =., M = =., M =. =.9, M = =.9, M =. =., M = =., M =. =.7, M = =.7, M =..... Figre. Veloci profiles for differen vales of and M λ =., =., =, θ =.... Figre. Veloci profiles for differen vales of and M λ =., =., =, θ = Figre shows he veloci field disribion wih he change of relaxaion parameer, he resls indicae ha he greaer he vale of λ, he more rapidl he veloci declines. Figre shows he effecs of slip parameer on veloci field disribion, we can see ha he changes of veloci wih differen vales of slip coefficien. he resl indicaes ha he increasing in he slip parameer a he wall resl in he decreases in veloci profiles. Figre is he veloci profile (,) vs. he ime. Resls indicae ha wih he increasing he vale of, he veloci rapidl speeds p. As seen from figs.7-9, he bigger he

7 HERMAL SCIENCE: Year 7, Vol., No. A, pp. 7- vale of is, he more slowl he veloci decas. However, one can see ha an increase in maerial parameer γ ( or λ ) has qie he opposie effec o ha of. λ =. λ =. λ =. θ = θ =. θ =..... Figre. Veloci profiles for differen vales of λ =., =., =, θ =, M =.... Figre. Veloci profiles for differen vales of θ =., =., =, λ =., M = = =. = Analical solion Nmerical solion.... Figre. Veloci profiles for differen vales of =., =., λ =, θ =, M = ) Figre 7. emperare profiles for differen vales of γ =., λ =, =, a =., f() = γ =. γ =. γ =..... Figre. Veloci profiles for generalized Maxwell flids wih =., =, λ =., M =., θ =. 9 =. 8 =. =. 7 Figre 8. emperare profiles for differen vales of γ =., λ =, =, a =., f() =

8 HERMAL SCIENCE: Year 7, Vol., No. A, pp. 7- Figres and presen he comparisons of nmerical and exac analical solions for boh veloci and emperare fields. he reliabili and efficienc of he nmerical solions are verified b analical resls wih good agreemen. 9 λ = 8 λ =. 7 λ = Figre 9. emperare profiles for differen vales of λ =., γ =., =, a =., f( ) = 9 Analical solion 8 Nmerical solion Figre. emperare profiles for generalized Maxwell flids wih γ =., =, a =, f( ) = Acnowledgmen he wor of he ahors is sppored b he Naional Naral Science Fondaions of China (No. 7, 79). Nomenclare B magneic indcion, [] c p specific hea of flid a a consan pressre, [Jg K ] h space sep, [m] hermal condcivi, [Wm K ] M magneic field inensi, [Am ] w emperare a he wall, [K] emperare a infini, [K] ime, [s] space, [m] Gree smbols,, γ order of fracional derivaive, [ ] θ slip parameer,[ ] λ, λ relaxaion ime, [s] µ dnamic viscosi, [Nsm ] ν inemaic viscosi, [m s ] ρ consan densi of he flid, [gm ] parameer for dimensional balance, [s] parameer for dimensional - - balance, [ s m ] elecric condcivi, [Sm ] τ ime sep, [s] References [] Renard, M., Sabili of Creeping Flows of Maxwell Flids, Archive for Raional Mechenics and Analsis, 98 (),, pp. 7-7 [] Savelev, E., Renard, M., Conrol of Homogeneos Shear Flow of Mlimode Maxwell Flids, Jornal of Non-Newonian Flid Mechanics, (), -, pp. - [] Karra, S., e al., On Maxwell Flids wih Relaxaion ime and Viscosi Depending on he Pressre, Inernaional Jornal of Non-Linear Mechanics, (),, pp [] Haa,., e al., Radiaion Effecs on MHD Flow of Maxwell Flid in a Channel wih Poros Medim, Inernaional Jornal of Hea and Mass ransfer, (),, pp. 8-8 [] Salah, F., e al., New Exac Solion for Raleigh-Soes Problem of Maxwell Flid in a Poros Medim and Roaing Frame, Resls in Phsics, (),, pp. 9- [] Podlbn, I., Fracional Differenial Eqaions, Academic Press, New Yor, USA, 999 [7] Nazar, M., e al., Flow hrogh an Oscillaing Recanglar Dc for Generalized Maxwell Flid wih Fracional Derivaives, Commn Nonlinear Sci Nmer Simla., 7 (), 8, pp. 9-

9 HERMAL SCIENCE: Year 7, Vol., No. A, pp. 7- [8] Jamil, M., e al., New Exac Analical Solions for Soes Firs Problem of Maxwell Flid wih Fracional Derivaive Approach, Compers and Mahemaics wih Applicaions, (),, pp. - [9] Feeca, C., e al., Flow of Fracional Maxwell Flid beween Coaxial Clinders, Archive of Applied Mechanics, 8 (), 8, pp. - [] Yang, D., Zh, K. Q., Sar-p Flow of a Viscoelasic Flid in a Pipe wih a Fracional Maxwell s Model, Compers and Mahemaics wih Applicaions, (), 8, pp. -8 [] Cao, L. L., e al., ime Domain Analsis of he Fracional Order Weighed Disribed Parameer Maxwell Model, Compers and Mahemaics wih Applicaions, (),, pp. 8-8 [] Haa,., e al., Sore and Dfor Effecs in hree-dimensional Flow of Maxwell Flid wih Chemical Reacion and Convecive Condiion, Inernaional Jornal of Nmerical Mehod for Hea & Flid Flow, (),, pp. 98- [] Vier, D., e al., ime-fracional Free Convecion Flow near a Verical Plae wih Newonian Heaing and Mass Diffsion, hermal Science, 9 (), Sppl., pp. S8-S98 [] Msafa, M., e al., Saiadis Flow of Maxwell Flid Considering Magneic Field and Convecive Bondar Condiions, AIP Advances, (),, pp. 7 [] Feeca, C., e al., Unsead Flow of a Generalized Maxwell Flid wih Fracional Derivaive de o a Consanl Acceleraing Plae, Compers and Mahemaics wih Applicaions, 7 (9),, pp. 9- [] Jamil, M., Feeca, C., Helical Flows of Maxwell Flid beween Coaxial Clinders wih Given Shear Sresses on he Bondar, Nonlinear Analsis: Real World Applicaions, (),, pp. - [7] Laadj,., Renard, M., Iniial Vale Problems for Creeping Flow of Maxwell Flids, Nonlinear Analsis, 7 (),, pp. - [8] Li, Q. S., e al., ime Periodic Elecroosmoic Flow of he Generalized Maxwell Flids beween wo Micro-Parallel Plaes, Jornal of Non-Newonian Flid Mechanics, (), 9-, pp [9] Yse, S. B., Weighed Average Finie Difference Mehods for Fracional Diffsion Eqaions, Jornal of Compaional Phsics, (),, pp. -7 [] Jia, J. H., Wang, H., Fas Finie Difference Mehods for Space-Fracional Diffsion Eqaions wih Fracional Derivaive Bondar, Jornal of Compaional Phsics, 9 (), Jl, pp. 9-9 [] X, Y. F., e al., Nmerical and Analical Solions of New Generalized Fracional Diffsion Eqaion, Compers and Mahemaics wih Applicaions, (),, pp. 9-9 [] Zheng, L. C., e al., Exac Solions for Generalized Maxwell Flid Flow de o Oscillaor and Consanl Acceleraing Plae, Nonlinear Analsis: Real World Applicaions, (),, pp. 7-7 [] Gomez-Agilar, J. F., e al., A Phsical Inerpreaion of Fracional Calcls in Observables erms: Analsis of he Fracional ime Consan and he ransior Response, Revisa Mexicana de Física, (),, pp. -8 [] Feeca, C., e al., A Noe on he Flow Indced b a Consanl Acceleraing Plae in an Oldrod-B Flid, Applied Mahemaical Modelling, (7),, pp. 7- [] Feeca, C., e al., General Solions for Magneohdrodnamic Naral Convecion Flow wih Radiaive Hea ransfer and Slip Condiion over a Moving Plae, Zeischrif fer Narforschng A, 8a (), -, pp. 9-7 Paper sbmied: Jne, Paper revised: Sepember, Paper acceped: Sepember, 7 Socie of hermal Engineers of Serbia Pblished b he Vinča Insie of Nclear Sciences, Belgrade, Serbia. his is an open access aricle disribed nder he CC BY-NC-ND. erms and condiions