steady mhd flow of micropolar fluid between two rotating cylinders ...

1.G.NAGARAJU, 2. J.V.RAMANA MURTHY, 3. K.S.SAISTEADY MHD FLOW OF MICROPOLAR FLUID BETWEEN TWOROTATING CYLINDERS WITH POROUS LINING1.DEPARTMENT OF MATHEMATICS, GITAM UNIVERSITY, HYDERABAD CAMPUS,RUDRARAM, MEDAK DIST (A.P) - 506004, INDIA2.DEPARTMENT OF MATHEMATICS, NIT, WARANGAL (A.P) -506004, INDIA3. DEPARTMENT OF MATHEMATICS, MACHILIPATNAM ENGINEERING COLLEGE, MACHILIPATNAM, KRISHNA DISTRICT, AP, INDIAABSTRACT: The steady flow of an electrically conducting, incompressible micropolarfluid in a narrow gap between two concentric rotating vertical cylinders, with porouslining on inside of outer cylinder, under an imposed axial magnetic field is studied.Beavers and Joseph slip condition is taken at the porous lining boundary. The velocityprofiles, coefficient of skin friction on the cylinders are calculated. The effects ofHartmann number, the porous lining thickness parameter, coupling number, couplestress parameters and Reynolds number on azimuthal velocity, micro-rotationcomponent and coefficient of skin friction on cylinders are depicted through graphs.KEYWORDS: Micropolar fluid; Magnetic field; porous lining; Skin friction; Angularvelocity; micro-rotation vectorINTRODUCTIONMicropolar fluids are fluids with microstructures.They belong to a class of fluids with a non-symmetricstress tensor. Micropolar fluids consist of rigid,randomly oriented spherical particles with their ownspins and micro-rotations, suspended in a viscousmedium. The concept of micro-rotation was proposedby Cosserat and Cosserat in the theory of elasticity[1]. In the middle of the 1960s, Condiff and Dahler[2] and Eringen [3] applied the concept to describefluids with microstructures. Recently, acomprehensive text book on micropolar fluids hasbeen published [4]. Physical examples of micropolarfluids can be seen in ferrofluids [5], blood flows [6,7], bubbly liquids [8], liquid crystals [9], and so on,all of them containing intrinsic polarities. Thus,micropolar fluid mechanics is not a uselessgeneralization of the Navier-Stokes model, but is aphysically relevant model that has many applications.Flows through and past porous media with finitethickness are of relevance in many industrialapplications like lubrication and tapping of solarenergy. The control of shearing stress is important inthe design of rotating machinery like totallyenclosed fan cooled motors and lubrication industryin which centrifugal force plays a major role.Channabasappa et al [10] have examined the effectof porous lining thickness on velocity vector andshear stresses at the wall of inner and outer cylindersfor the flow between two rotating cylinders. Sai [11]has studied the steady motion of an electricallyconducting, incompressible viscous liquid in a narrowgap between two concentric rotating verticalcylinders in presence of an imposed magnetic fieldand he has presented the velocity profilesgraphically. Bathaiah et al [12] have studied theviscous incompressible, slightly conducting fluid flowbetween two concentric rotating cylinders with nonerodableand non conducting porous lining on theinner wall of the outer cylinder under the influenceof radial magnetic field of the form given in Hughesand Young and they have shown the effect ofmagnetic parameter, porous lining thickness, theratio of the velocities of the cylinders, the slipparameter on velocity and temperature distributionsgraphically. Beavers and Joseph [13] studied theflow of a viscous fluid in a channel bounded below bya naturally permeable wall. Ramamurthy [14] haveobtained the velocity distribution and magnetic fieldfor a viscous incompressible conducting fluid betweentwo coaxial rotating cylinders under the influence ofradial magnetic field. Singh et al [15] haveinvestigated the impulsive motion of a viscous liquidcontained between two concentric circular cylindersin the presence of radial magnetic field. Mahapatra[16] studied the unsteady motion of anincompressible viscous conducting liquid between twoporous concentric circular cylinders in presence of aradial magnetic field. Subotic et al [17] haveobtained the analytical solutions for the flow andtemperature fields in an annulus with a porous sleevebetween two rotating cylinders and they studied theeffects of Darcy number, Brinkman number andporous sleeve thickness on the velocity profile andtemperature distribution. Ranganna et al [18] havestudied the stability analysis of laminar flowbetween two long concentric circular rotatingcylinders with non-erodible porous lining on theouter wall of the inner cylinder and he has shown theeffect of porous lining thickness on critical Taylor© copyright FACULTY of ENGINEERING ‐ HUNEDOARA, ROMANIA 115

number graphically. Kamel [19] has studied thecreeping motion of a polar fluid in the annular regionbetween the two eccentric rotating cylinders. Hedepicted the dependence of the velocity componentsand the spin on the coupling number and the lengthratio graphically. Meena et al [20] have studied theflow of a viscous incompressible fluid between twoeccentric rotating porous cylinders with smallsuction/injection at both the cylinders and they havepresented stream lines and pressure plotsgraphically. Borkakati et al [21] have examined thesteady flow of an incompressible electricallyconducting fluid between two coaxial cylinders inpresence of radial magnetic field and they plottedgraphically the heat transfer rate from the cylindersagainst the Hartmann number.Srinivasacharya et al [22] have studied the steadyflow of incompressible and electrically conductingmicropolar fluid flow between two concentric porouscylinders and they have presented the profiles ofvelocity and micro-rotation components for differentmicropolar fluid parameters and magneticparameter. Pontrelli et al [23] studied the steadyflow of an Oldroyd-B fluid between two porousconcentric circular cylinders. They found velocity bynumerically solving a system of nonlinear ODEsobtained from the equation of motion andconstitutive equations and the effects of non-Newtonian parameters on velocity and on shear stressare shown through graphs.Fetecau et al [24] studied the velocity fieldscorresponding to the motion of an incompressiblesecond grade fluid due to longitudinal and torsionaloscillations of an infinite circular cylinder.In this paper, we study the steady incompressibleelectrically conducting micropolar fluid flow betweentwo concentric rotating cylinders with porous liningin the presence of an axial magnetic field.FORMULATION AND SOLUTION OF THE PROBLEMConsider an incompressible micropolar fluid betweentwo concentric circular cylinders of radii a and b(a

ACTA TECHNICA CORVINIENSIS – Bulletin of EngineeringThe velocity and micro-rotation are choosen in theformq = v() r e θ and υ = C() r e z(9)Substituting (9) in (6) and comparing the componentsalong e , directions, we getr eθdp v= (10)dr rdC 2 2− c + D v − M v = 0(11)dr22 d 1 d 1where D = + −2r dr2dr rSimilarly using (9) in the equation (7), the axialdirection component yields the following equationfor micro-rotation C.2⎛ dv v⎞⎛ d C 1 dC⎞− 2sC + s⎜+ ⎟+ ⎜ + ⎟ = 0 (12)2⎝ dr r ⎠dr r dr⎝ ⎠dCEliminating value from (11) and (12) we get thedrfollowing equation for v as4D v −This can be expressed aswhere22222( M + ( 2 − c)s) D v + 2sM v = 02 2 2 2( D λ ) ( D − λ ) v = 012− (13)2( 2 c)λ1 + λ2= M + s − and λ 1 λ2= 2sMAs q is finite between the interval r 0

where T118rθTrθ= is the non dimensional stress.Ω2( μ + k )Hence the coefficient of skin friction on the innerand outer cylinders is given by2TrθC f = at R=a and R=b–h (22)2ρUwhere U = Characteristic velocity = bΩ 2This can be written in the following non-dimensionalform as2TrθC f = at r=r 0 and r=1–e (23)ReTORQUEThe torque acting on the cylinders about the commonaxis of the cylinders is given byτ = ( Trθ × 2π R) × R . (24)Hence the non dimensional torque on inner and outercylinders are calculated asτin20= 2π rand τ 2π ( 1- e)outTrθr = r 02Trθr = 1-e= (25)RESULTS AND DISCUSSIONSThe velocity filed is considered along tangential(azimuthal) direction only. In fact we can start withnonzero radial velocity and come to conclude that itshould vanish by considering continuity equation andthe condition that there is no suction on the walls.(See Channabasappa, Umapathy and Nayak [10].)We have investigated the effects of the Hartmannnumber M, the porous lining thickness parameter e,Reynolds number Re, coupling number c and couplestress parameter s on azimuthal velocity v, microrotationvelocity C and the coefficient of skin frictionat the inner and outer cylinders are numericallyobtained and are depicted through Figures 2 to 19.The azimuthal velocity v and Micro-rotation velocityC are computed for different values of M and e.Figures 2–7 shows the azimuthal velocity v and microrotationvelocity C against distance r for differentvalues of M at a fixed value of porous lining thicknessparameter e = 0.1, 0.3, 0.4. We have observed that vdecreases as e increases and C increases as eincreases. Both v and C decrease as M increases. It isobserved that the nature of velocity profiles v isincreasing with distance where as the nature ofrotation C is increasing and decreasing as Mincreases. But for small values of M, C is decreasingwith r. When M takes large values, the values of Care near to zero. i.e rotation of the particles can beneglected.The Figures 2, 4 and 6 of velocity v are in goodagreement with the results obtained by Bathaih andVenugopal [12] and Subotic and Lai [17], when thereis no applied magnetic field (the curve with M=0.0001 indicates almost no magnetic field). Figures8 and 9 give the velocity profiles v and C fordifferent values of c. From this it is clear that anincrease in coupling parameter c increases the valuesof both the velocities v and C. i.e. if micro-polarityof the fluid increases, the velocity v and microrotationC will also increase. In Figures 10 and 11 thevelocity profiles v and C plotted against Re. FromACTA TECHNICA CORVINIENSIS – Bulletin of Engineeringthis it is clear that an increase in Re increases boththe velocities.Figure 2. Variation of v with r at e = 0.1Figure 3. Variation of C with r at e = 0.1Figure 4. Variation of v with r at e = 0.3Figure 5. Variation of C with r at e = 0.32013. Fascicule 3 [July–September]

ACTA TECHNICA CORVINIENSIS – Bulletin of EngineeringFigure 6. Variation of v with r at e = 0.4Figure 10. Variation of v with rFigure 7. Variation of C with r at e = 0.4Figure 11. Variation of C with rFigure 8. Variation of v with rFigure 12. Variation of v with rFigure 9. Variation of C with rFigure 13. Variation of C with r2013. Fascicule 3 [July–September] 119

Figures 12-13 we have seen that as the couple stressparameter s increases both the velocities v and C areincreasing. But the effect of s on the values of v isnot very significant. i.e., the variation in the valuesof s does not result in much variation in the values v.Figures 14–16 show the variation the coefficient ofskin friction C f at the inner cylinder against c forindifferent values of M, Re, s.We observe that C f decreases with increasinginvalues of M, Re, s whereas in Figures 17–19, the skinfrictionC f at the outer cylinder increases withoutincreasing values of M, Re, s.ACTA TECHNICA CORVINIENSIS – Bulletin of EngineeringFigure 17. Variation of C with cfoutFigure 14. Variation ofC with cfinFigure 18. Variation ofC with cfoutFigure 15. Variation ofC with cfinFigure 19. Variation ofC with cfoutFigure 16. Variation ofC with cfinFigure 20. Variation of C with efin1202013. Fascicule 3 [July–September]

ACTA TECHNICA CORVINIENSIS – Bulletin of EngineeringFigure 21. Variation of C with eFigures 20–21 show the variation of the coefficient ofskin friction at inner and outer cylinders against Mfor different values of e.We observe thatfinfoutC increases, whereasC fdecreases with increasing values of e. These resultsare in correlation with the results of Bathiah andVenugopal [12].CONCLUSIONSIn this paper, the effect of axial magnetic field onmicropolar fluid flow due to steady rotation ofconcentric cylinders with inner porous lining isexamined. It is observed that1. Micro-polarity of the fluid affects the velocitybut couple stress parameter can not affect thevelocity profiles2. As magnetic field strength increases velocitydecreases and micro-rotation of the particlesdecreases and there by skin friction decreases atinner cylinder and increases at outer cylinder3. As porous lining thickness increases, the skinfriction at the inner cylinder increases and atthe outer cylinder decreases.REFERENCES[1.] E. Cossera and F. Cosserat, Theorie des CorpsDeformables. (A. Hermann, Paris) (1909).[2.] D.W. Condiff and J.S. Dahler, Fluid mechanicalaspects of anti-symmetric stress, Phys. Fluids, 7,842–854, (1964).[3.] A.C. Eringen, Theory of micropolar fluids, J.Math. Mech, 16, 1–18, (1966).[4.] G.Lukaszewicz, “Micropolar fluids theory andapplications”, Birkhauser, Boston, (1999).[5.] R.E. Rosensweig, Ferrohydro-dynamics,Cambridge University Press, Cambridge, (1985).[6.] M.A. Turk, N.D. Syvester and T. Ariman,Amicrocontinuum model of blood withdeformable cells, ASME Biomechanics symposium(Edited by Y.C. Fung and J.A.Brighton), (1973).[7.] Bugliarello G., Sevilla J., Velocity distributionand other characteristics of steady and pulsatileblood flow in glass tubes, Biorheology, 7, 85–107,(1970).[8.] A.C. Eringen, Theory of thermo-microstretchfluids and bubbly liquids, Int. J. Eng. Sci. 28, 133–143, (1990).[9.] P.G. de Gennes and J. Prost, The Physics of LiquidCrystals, 2nd ed. Oxford University Press, Oxford,(1993).out[10.] M.N. Channabasappa, K.G.Umapathy and I.V.Nayak, Effect of porous lining on the flowbetween two concentric rotating cylinders, Proc.Indian Acad. Sci., 88 A III, 163–167, (1979).[11.] K.S. Sai, MHD flow between two rotatingcylinders with porous lining, Rev. Roum. Phys, 30,119–125, (1985).[12.] D. Bathaiah and R. Venugopal, Effect of porouslining on the MHD flow between two concentricrotating cylinders under the influence of a radialmagnetic field, Acta. Mech, 44, 141–158, (1982).[13.] G.S. Beavers and D.D. Joseph, Boundaryconditions at a naturally permeable wall, J.Fluid. Mec, 30,197–207, (1967).[14.] P. Ramamurthy, Flow between two concentricrotating cylinders with a radial magnetic field,Phys. Fluids 4, 1444, (1961).[15.] D. Singh and Syed Ali Tahir Rizvi Unsteady motionof a conducting liquid between two infinitecoaxial cylinders, Phys.Fluids, 7,760–761, (1964).[16.] J.R. Mahapatra, A note on the unsteady motion ofa viscous conducting liquid between two porousconcentric circular cylinders acted on by a radialmagnetic field, Appl. sci. res, 27, 274–282,(1973).[17.] M. Subotic and F.C. Lai, Flows between rotatingcylinders with a porous lining, J. of HeatTransfer, 130, 102601(1–6), (2008).[18.] M.N. Chinnabasappa, G. Ranganna and B.Rajappa, Stability of couette flow betweenrotating cylinders lined with porous material-I,Indian J.Pure Appl.Math, 14 (6), 741–756, (1983).[19.] M.T. Kamel, Flow of a polar fluid between twoeccentric rotating cylinders, J. of Rheol., 29 (1),37–48, (1985).[20.] S. Meena, P. Kandaswamy and Lokenath Debnath,Hydrodynamic flow between two eccentriccylinders with suction at the porous walls, Int. Jmath. maths sci,25 (2),93–113, (2001).[21.] A.K. Borkakati and I. Pop, MHD Heat transfer inthe flow between two coaxial cylinders, ActaMech., 51, 97–102, (1984).[22.] D. Srinivasacharya and M. Shiferaw Numericalsolution to the MHD flow of micropolar fluidbetween two concentric porous cylinders, Int. J.of Appl. Math. Mech, 4(2), 77–86, (2008).[23.] G. Pontrelli and R.K. Bhatnagar, Flow of aviscoelastic fluid between two rotating circularcylinders subject to suction or injection, Int. J forNumer. Methods in Fluids, 24, 337–349, (1997).[24.] C. Fetecau and Corina Fetecau, Starting solutionsfor the motion of a second grade fluid due tolongitudinal and torsional oscillations of acircular cylinder, Int. J of Engg. sci, 44, 788–796,(2006).ACTA TECHNICA CORVINIENSIS – Bulletin of EngineeringISSN: 2067-3809 [CD-Rom, online]copyright © UNIVERSITY POLITEHNICA TIMISOARA,FACULTY OF ENGINEERING HUNEDOARA,5, REVOLUTIEI, 331128, HUNEDOARA, ROMANIAhttp://acta.fih.upt.ro2013. Fascicule 3 [July–September] 121