Second Law Analysis of Ion Slip Effect on MHD Couple Stress Fluid

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1 ITERATIOAL JOURAL OF MECHAICS Volume, 8 Secod Law Aalysis of Io Slip Effect o MHD Couple Stress Fluid Abiodu A. Opauga, Hilary I. Okagbue, Olasumbo O. Agboola, Sheila A. Bishop Abstract This paper is cocered with the umerical ivestigatio of etropy geeratio i viscous icompressible MHD couple stress fluid i a rotatig frame of referece. A approximate solutio of the dimesioless velocity ad temperature profiles are obtaied ased to calculate the etropy geeratio rate ad Beja umber. The iflueces of the goverig parameters o velocity, temperature, etropy geeratio ad Beja umber are preseted with the aid of graphs. eywords Adomia decompositio method, Couple stress fluid, Etropy geeratio, Io slip, MHD, Beja umber C I. ITRODUCTIO OUPLE stress fluid theory was proposed by Stokes [] to demostrate its microrotatio effects which could ot be described by the classical avier-stokes theory. This type of fluid has attracted the attetio of may researchers due to its umerous applicatios i various egieerig ad idustrial processes such as extrusio of polymer fluids, solidificatio of liquid crystals, coolig of metallic plate i a bath, colloidal solutios, syovial joits (shoulder, hip, kee ad akle), geophysics, chemical egieerig ad astrophysics. Sice the extesio has bee doe, to iclude studies of hydrodyamic couple stress fluid [-5], blood flows [6-7], joural bearigs [8-9], squeeze films [-], couple irreversibility [-8] where various factors associated with couple stress irreversibility are ivestigated. Others are [9-]. The aim of this paper is to aalyse the ifluece of Io slip o the etropy geeratio of MHD couple stress fluisig the rapidly coverget Adomia decompositio method to solve the dimesioless velocity ad temperature profiles obtaied from the goverig equatios. This techique has bee applied to various flow problems [-5] ad other liear ad o-liear models [6-3]. The orgaizatio of the paper is as follows: sectio presets formatio of the goverig equatios, Adomia decompositio method of solutio is treated i sectio 3 ad sectio 4 is based o This work was supported by Coveat Uiversity Cetre for Research, Iovatio ad Developmet, Ota, igeria. A. A. Opauga, H.I. Okagbue, O.O. Agboola ad S.A. Bishop are with the Departmet of Mathematics, Coveat Uiversity, Ota, igeria.( abiodu.opauga@coveatuiversity.edu.g, hilary.okagbue@coveatuiversity.edu.g, ola.agboola@coveatuiversity.edu.g, sheila.bishop@coveatuiversity.edu.g). the discussio of results ad fial remarks are made i sectio 5. II. PROBLEM FORMULATIO Cosider the viscous icompressible ad electrically coductig couple stress fluid bouded by two ifiite horizotal parallel plates with covective heatig separated by a distace d. Let a cartesia co-ordiate system be chose such that the x -axis is alog the lower plate i the directio of the flow, the y -axis is perpedicular to the plates while the z -axis is ormal to xy - plae. Assumptio of relatively high electro-atom collisio frequecy is take so that the impact of io slip caot be eglected. The etire system is rotatig with a agular velocity about the ormal to the plate. The flow is fully hydrodyamically ad thermally developed, therefore the fluid velocity ad the temperature i the chael are fuctios of y oly. Followig the assumptios above, the goverig equatios for the flow ca be writte [3] as Figure : Geometry of the problem 4 d p w B u ; 4 d x u ( ) ( ) u ( d ) ( d ), () ISS:

2 ITERATIOAL JOURAL OF MECHAICS Volume, 8 4 d w d w u B w ; 4 d w d w w ( ) ( ) w ( d ) ( d ), d T d w k d w B u w ; T ( ) T ( d ). where u ad w are the velocity compoets i the x ad z directios respectively, is the dyamic viscosity, is the coefficiet of couple stresses, is the electrical coductivity, B is the magetic parameter, k is the thermal coductivity, T is fluid temperature ad d the chael width. The dimesioless variables itroduced are as follows y u w T T d y, u, w,, a d U U T T the equatios (-3) become 4 w G M u ; 4 a u ( ) u ( ) u () u (), 4 d w d w u M u ; 4 a w ( ) w ( ) w () w (), d d w Br d B r d w d w a B r M u v ; ( ), (). d () (3) (4) (5) (6) (7) umber, a d is the couple stress parameter. III. ADOMIA DECOMPOSITIO METHOD OF SOLUTIO Applicatio of Adomia decompositio method requires that the differetial equatios (5) (7) are writte i the itegral forms as follows 3 u ( y ) y y a w G M u d Y d Y d Y d Y, dy 3 w ( y ) y y d w a u M w d Y d Y d Y d Y, dy ( y) f y y d w Br d Y d Y B r d w a d Y d Y B rm u w + d Y d Y. Where,,, ad f are costats to be computed at y. Assumig followig solutio for equatios (7)-(9) u ( y ) u ( y ), w ( y ) w ( y ), ( y) ( y), Substitutig () i (7)-(9) yields the followig u y y y GdYdYdYdY 3 ( ) ; (7) (8) (9) () where G T d d P is the pressure gradiet parameter, U d x T d is the temperature differece parameter, T U Br k T T is the Brikma umber, d is u ( y) w a d Y d Y d Y d Y M u dy, () the rotatio parameter, M B d is the Hartma ISS:

3 ITERATIOAL JOURAL OF MECHAICS Volume, 8 3 w ( y ) y y ; w ( y) u a d w d Y d Y d Y d Y M w dy ( y ) f ; ( y ), () covective coolig. The rise i temperature is expected due to exchage of heat betwee the lower wall ad fluid particles where there is covective heatig, however a reverse pheomeo is experieced at the upper plate with ewtoia coolig as depicted i Fig 3c. d v Br d Y d Y y y B r d v + d Y d Y, a d Y d Y B rm A B (3) Where A ad B are the Adomia polyomials that ca be computed as follows A u, A u u, A u u u ; B w, B w w, B w w w (4) The fial stage of implemetatio of Adomia techique is the codig of equatios ()-(3) i a computer algebra package kow as Mathematica. The graphical results are preseted i Figures -5. Figurea: Primary velocity for differet IV. RESULTS AD DISCUSSIO To study the ifluece of rotatio ad covective heatig parameters o the dimesioless velocity, temperature, etropy geeratio expressio ad Beja umber, some plots are preseted i Figures -5. Figures a ad b display the rotatio effect o fluid velocity; the plots idicate a decrease i primary velocity while secodary velocity is ehaced. This is physically correct sice rotatio is kow to ehace secodary fluid velocity i the flow-field by retardig the primary fluid velocity. The reaso is that Coriolis force is domiat i the regio ear to the axis of rotatio. I Figures c ad d it is show that both primary ad secodary velocities decay with a rise i magetic parameter. The Loretz force effect is attributed to this pheomeo; the applied magetic field clutches fluid particles together ad thus impedes fluid flow. I Figures 3a, 3b ad 3c the temperature respose for several values of rotatio parameter, lower Biot umber Bi apper Biot umber Bi are displayed. It is see that temperature rises as rotatio parameter ad lower Biot umber icrease; while it decreases with a rise i ISS: Figureb: Secodary velocity for differet Figurec: Primary velocity for differet M

4 ITERATIOAL JOURAL OF MECHAICS Volume, 8 Figured: Secodary velocity for differet M Figure 3a: Temperature profile for differet Figure 3b: Temperature profile for differet Bi Figure 3c: Temperature profile for differet Bi ETROPY GEERATIO AALYSIS The etropy geeratio expressio (see Beja [3]) for the flow is writte as E G k d T d w T T (5) d w B w u, T T Etropy geeratio expressio i equatio (5) cosists of heat trasfer irreversibility, fluid frictio irreversibility, couple stress irreversibility ad magetic field irreversibility respectively. Substitutig (4) i equatio (5) yields the dimesioless etropy geeratio expressio as d w d B r d w s, d a M w u Where s T G k ( T T ) d h E (6) The ratio of heat trasfer irreversibility to fluid frictio irreversibility ca be represeted as f (7) where h h d d ; d w (8) B r d w f a M w u A alterative irreversibility distributio ratio parameter is the Beja umber, it represets the ratio of heat trasfer irreversibility to the total irreversibility due to heat trasfer ad fluid frictio. Be h (9) s ISS:

5 ITERATIOAL JOURAL OF MECHAICS Volume, 8 I equatio (9) Be is Be. Be gives the limit at which fluid frictio irreversibility domiates etropy geeratio while Be deotes the domiace of heat trasfer irreversibility over fluid frictio irreversibility ad Be.5 is the case whe heat trasfer ad fluid frictio etropy geeratio rates are equal. Figures 4 ad 5 preset the ifluece of rotatio parameter, lower Biot umber Bi apper Biot umber Bi o etropy geeratio ad Beja umber. It is observed from Figure 4a that etropy geeratio decreases at plate y while it rises at plate y with icrease i the rotatio parameter. Figures 4b ad 4b depict that etropy productio is ehaced with icrease i Biot umbers. Furthermore, Figure 5a depicts that Beja umber reduces at plate y while it rises at y. Figures 5b ad 5c show that Beja umber icreases sigificatly at the middle of the chael but the icrease is ot oticeable at the chael walls. Geerally, etropy geeratio is domiated by heat trasfer irreversibility betwee.3 y. Figure 4c: Etropy geeratio for differet Bi Figure 5a: Beja umber for differet Figure 4a: Etropy geeratio for differet Figure 5b: Beja umber for differet Bi Figure 4b: Etropy geeratio for differet Bi Figure 5c: Beja umber for differet Bi ISS:

6 ITERATIOAL JOURAL OF MECHAICS Volume, 8 V. COCLUSIOS A umerical ivestigatio ito the rotatio effect o the etropy geeratio of steady, viscous ad icompressible MHD couple stress fluid has bee coducted. The fidigs are; Rotatio parameter retards primary fluid flow while secodary velocity is accelerated Magetic parameter retards fluid motio, Rotatio parameter ad lower Biot umber icrease fluid speed while fluid temperature is reduced by upper Biot umber Rotatio parameter ad Biot umbers icrease temperature. Rotatio parameter reduces Beja umber at the lower wall but icreases it at the upper wall, The Biot umbers icrease Beja umber. REFERECES [] V.. Stokes, Couple-stresses i fluids, Phys. Fluids 9, 79 75, 966. [] S.O. Adesaya, O.D. Makide, Heat trasfer to magetohydrodyamic o-ewtoia couple stress pulsatile flow betwee two parallel porous plates. Z aturforsch, vol. 67a: pp ,. [3].T.M. Eldabe, A.A. Hassa, M.A.A. 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