Free Convection from an Elliptic Tube with its Major Axis Vertical and Placed in a Micropolar Fluid

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1 Proceedigs of the 5th IASME/WSEAS It. Coferece o Heat Trasfer, Thermal Egieerig ad Eviromet, Athes, Greece, August 5-7, 7 5 Free Covectio from a Elliptic Tube with its Major Axis Vertical ad Placed i a Micropolar Fluid F. M. MAHFOUZ Mechaical Departmet, UET, Taxila, Pakista, O leave from Meoufia Uiversity, Egypt fmahfouz64@hotmail.com Abstract: - I this paper free covectio from a horizotal isothermal elliptic tube placed i a micropolar fluid with its major axis vertical is ivestigated. The goverig equatios are based o the coservatio of mass, liear mometum, agular mometum ad eergy. The full goverig equatios writte i stream fuctio-vorticity formulatio are solved umerically usig the Spectral method based o Fourier series expasio. Beside the classical cotrollig parameters (Rayleigh umber, Pradtl umber ad ellipse axis ratio ) the effect of the material parameters of the micropolar fluid are also cosidered. These parameters are the vortex viscosity, micro-iertia desity ad spi gradiet viscosity. I compariso with ewtoia fluids, the study has show that micropolar fluids display a clear reductio i heat trasfer rate. The study shows also that the effect of vortex viscosity is the most sigificat material parameter o heat trasfer rate. Key-Words:- Micropolar free covectioo elliptic - tube vortex viscosity - microrotatio Itroductio The theory of micropolar fluids has bee proposed by Erige []. I this theory the local effects arisig from the microstructure ad itrisic motio of the fluid elemets are take ito accout. Such fluids ca support surface ad body couples which are ot preset i the theory of ewtoia fluids. Micropolar fluids are believed to be successful i describig the behavior of heterogeeous mixtures such as ferro liquids, colloidal fluids, aimal blood, most slurries ad some liquids with polymer additives. Erige [] developed the theory of thermo-micropolar fluids by extedig the theory of micropolar fluids. Previous studies of covective heat trasfer i micropolar fluids have focused maily o relatively simple geometry such as flat plates ad circular cyliders. Refs. [3-8] are oly examples. Most of these studies were maily based o the umerical solutio of simplified (boudary layer) goverig equatios. There were oly a few attempts to ivestigate the case of atural covectio from a elliptic cylider placed i a micropolar fluid. Amog these attempts were those made by Bhattacharyya ad Pop [9] ad Mahfouz []. Bhattacharyya ad Pop solved the boudary layer equatios to ivestigate the steady atural covectio from a isothermal elliptic tube with its major axis either horizotal or vertical. They preseted results for local usselt umber alog with velocity ad temperature fields. While Mahfouz solved the full goverig equatio without boudary layer simplificatios to ivestigate the trasiet atural covectio from a isothermal elliptic tube with its major axis horizotal. The mai objective of this work is to study the effect of Rayliegh umber ad material parameters o atural covectio from a isothermal elliptic tube placed i micropolar fluid with its major axis vertical. The buoyacy drive flow is assumed to be lamiar ad two dimesioal. omeclature a legth of semi-major axis Ar axis ratio (=b/a) b legth of semi-mior axis c dimesioless focal distace (= Ar ) F b buoyacy force K v vortex viscosity j micro-iertia desity u local usselt umber u average usselt umber x, y Cartesia coordiates Y * y b distace alog mior axis (= Ra.5 ) a Greek symbols α thermal diffusivity β coefficiet of thermal expasio γ spi gradiet viscosity η, ξ elliptical coordiates µ viscosity coefficiet. ν kiematics viscosity Subscripts

2 Proceedigs of the 5th IASME/WSEAS It. Coferece o Heat Trasfer, Thermal Egieerig ad Eviromet, Athes, Greece, August 5-7, 7 6 s,o at cylider surface at ifiite distace from the surface Problem Formulatio Fig. shows the physical model ad coordiates system, cosistig of a isothermal horizotal elliptic tube of ifiite legth placed with its major axis vertical i a quiescet micropolar fluid at temperature T. The effect of temperature variatio o fluid properties is cosidered egligible except for the buoyacy force term i the mometum equatio ( Boussiesqu approximatio). The coservatio equatios of mass, liear mometum, agular mometum ad eergy i terms of the vorticity, stream fuctio, microrotatio ad temperature read the followig: x ' y ' ξ F b ξ ο b Fig.. Physical model ad coordiate system ζ ζ ζ Kv + = ( ν + ) ζ τ ρ K F v y Fx - ω + () ρ ρ ζ + ψ = () ω ω ω γ + = ω τ ρj T T + τ a K + v ( ζ ω ) (3) ρ j T k - = T (4) ρc where = + x τ is the time, ρ is the desity, ν is the kiematic viscosity, k is the thermal coductivity ad c v is the specific heat. K v, j ad are the vortex viscosity, micro-iertia desity ad spi-gradiet viscosity. ζ is the vorticity, ψ is the stream fuctio, T is the temperature ad ω is the compoet of γ v g microrotatio vector whose directio of rotatio is i the x -y plae. F x = ρgβ(t T ) ad Fy = are the compoets of the buoyacy force, where β is the coefficiet of thermal expasio of the fluid. The boudary coditios are maily the o-slip, impermeability ad o-spi coditios o the tube surface ad the stagat ambiet coditios very far away from it. -o the tube surface ψ = =, =, T = Ts ad ω =. (5a) -far away from the tube surface,, T = T y ad ω (5b) ow it is more coveiet to use the followig dimesioless variables: x y τα ψ a x =, y =, t =, ψ =, ζ = ζ, a a a α α a ω K M =, v j γ =, J =, λ = α µ a jµ T T ad φ = Ts T Usig elliptic coordiates ξ, η such that x = c cosh( ξ ) cos( η), y = c si sh( ξ )si( η), Eqs. ()- (4) ca ow be writte i terms of the above dimesioless variables as :. ζ ζ ζ D = Pr( + ) ζ Pr M t φ φ + ( c / 8) Ra Pr coshξ siη + sihξ cosη (6) Dζ + ψ = (7) M M M D = - + Pr λ M t D Pr (ζ + M) / J (8) φ φ φ D = φ + (9) t where D = c ( cosh ξ cos η ) is the Jacobia of trasformatio matrix, Ra = gβ (a) ( Ts T ) / να is the Rayleigh umber ad Pr = ν / α is the Pradtl umber. The boudary coditios Eq. (5) ca ow be expressed as: -o the tube surface ( ξ = ξ o ), ψ = =, =, M = ad φ = (a) 3

3 Proceedigs of the 5th IASME/WSEAS It. Coferece o Heat Trasfer, Thermal Egieerig ad Eviromet, Athes, Greece, August 5-7, 7 7 ad very far away from the tube surface ( ξ ),,, M ad φ (b) where ξ o defies the ellipse surface (= tah - Ar ) The temperature of the stagat fluid aroud the tube at times t < is T ( φ = ) which is the same as that of the tube surface. At the start of computatios ( t = ) the tube surface assumes a sudde temperature icrease from T to T s ( φ = ), ad from that momet the time developmet of both flow ad thermal fields commeces. 3 Method of Solutio The method used for solvig the goverig equatios (6)-(9) to obtai the time developmet of both velocity ad temperature fields is based o approximatig the stream fuctio, vorticity, microrotatio ad temperature usig Fourier series expasio. The approach is similar to that used by Badr ad Deis []. The stream fuctio ψ, vorticity ζ, microrotatio Γ ad temperature φ are ow approximated as ψ = f si ( η) =,,. (a) = ζ = g si ( η) (b) = r = M = si (η ) (c) φ = H + H cos ( η) (d) o = where is the umber of terms i the Fourier series. The fuctios f, g, r, H o ad H are Fourier coefficiets ad all are depedet o ξ ad t. The rest of the details of the method of solutio is similar to that i Mahfouz [] ad Badr [] ad will ot be repeated for the sake of brevity. The local usselt umber is defied as u = ah/k () where h is the local heat trasfer coefficiet defied as h= q& /( To T ), q& = -k( T/ s ) ξ o q& is the rate of heat trasfer per uit area, S is the ormal directio to the tube surface. From the above defiitios the u ca be expressed i terms of Fourier coefficiets H ad H as = H H = + u cos η (3) D = o ξ = ξo The average usselt umber ca be expressed as u = P P u dp = πa H P ξ = ξo where P is the perimeter of the elliptic sectio. (4) 4 Results ad Discussio The goverig equatios alog with the boudary coditios were solved i order to get the details of both flow ad thermal fields. The simulatios were carried out oly after validatig the method of solutio by comparig the preset results for the case of ewtoia fluids ( = ) with the most relevat results i the literature. Some of these comparisos are give i Mahfouz [8,]. Fig Steady patters of streamlies (right) ad isotherms (left) The mai cotrollig parameters are Rayleigh umber Ra, Pradtl umber Pr, axis ratio Ar ad the material parameters λ, ad J. For the sake of brevity oly the effect of Ra, ad J are cosidered while Pr, λ ad Ar are fixed at 7,,.5 respectively. The Rayleigh umber Ra, is cosidered i its moderate rage up to 4. The material parameter, which characterizes vortex viscosity is cosidered i the rage from to while the material parameter J, which characterizes microiertia desity, is cosidered i the rage from. to. These values for material parameters satisfy the thermodyamics restrictios give by Erige []. The time developmet of both flow ad thermal fields is more or less similar to that of ewtoia fluids. That is immediately after the temperature of the tube surface is raised, a temperature gradiet is established withi the fluid layer adjacet to the tube surface, causig predomiatio of coductio mode of heat trasfer. At this early time stages the ewbor buoyacy force causes the commecemet of fluid motio. As the time goes, the buoyacy iduced motio itesifies with gradual trasitio to covectio mode domiatio heat trasfer. At late

4 Proceedigs of the 5th IASME/WSEAS It. Coferece o Heat Trasfer, Thermal Egieerig ad Eviromet, Athes, Greece, August 5-7, 7 8 times, the covectio mode domiates ad the flow ad thermal fields i the viciity of the tube surface gradually ted to be almost steady. The steadiess i the earby flow ad thermal fields at late time leads to steady rates of heat trasfer. A typical example for such flow ad thermal fields at late time is show i Fig.. The figure shows the flow field, i terms of streamlies, ad the thermal field, i terms of isotherms, for the case of Ra=, λ =, = ad J=. Sice these fields are symmetrical about the vertical axis, oly oe half of each field is cosidered. The figure shows that these distributios are geerally similar to those for ewtoia fluids. Table Effect of Ra ad material parameters ad J o steady state average usselt umber. Ar Ra J u refers to a ewtoia fluid Table shows the effect of Rayleigh umber Ra, ad the material parameters ad J of micropolar fluid o the steady state average usselt umber, u. It ca be see that the effect of Ra o steady state u is quite clear, that is at ay fixed value of fluid material parameters as Ra icreases the u icreases. This is expected sice icreasig of Ra leads to icreasig of covectio currets ad so icreasig the heat trasfer rate. Also, it ca be see that as the material parameter icreases at ay fixed value of Ra the u decreases. The table also shows that at fixed values of Ra, ad the material parameter J has almost egligible effects o the u i the rage cosidered for the parameters. Fig. 3 shows the time variatio of averaged usselt umber u, for the case of Ra=, ad at differet values of dimesioless vortex viscosity =,,, 5. The figure clearly shows that the geeral variatio of u is similar to that for ewtoia fluids ( = ). That is u evolves i a sequece of pure coductive, trasiet covective ad steady covective processes. The pure coductive process prevails immediately after the tube surface temperature is icreased. The high temperature gradiet established ear the tube surface results i high heat flux ad so high values of u. I this early time stages ad as a result of quick developig of thermal boudary layer a quick decrease i u occurs, reachig to a miimum value at a certai short time. Beyod this time, the buoyacy force becomes more effective, causig trasiet covective process. This trasitio as show i the figure takes a form of overshoot i u. At late times the covective process gradually prevails with steady rates of heat trasfer ad so steady u gradually approach. u Ra=, J=, λ= Pr=7. Ar=.5, = = = = t Fig. 3 The time developmet of u The figure also shows that the steady heat trasfer rates ( i.e steady u ) i the case of micropolar fluids ( =,, 5) is lower tha that for ewtoia fluid ( = ). This decrease may be attributed, as explaied i Hsu et al. [7], to the icrease of the flow viscosity as a result of vortex viscosity. Icreasig of flow viscosity weakes the flow covectio currets ad icreases the thickess of the thermal boudary layer which i tur decreases heat.

5 Proceedigs of the 5th IASME/WSEAS It. Coferece o Heat Trasfer, Thermal Egieerig ad Eviromet, Athes, Greece, August 5-7, 7 9 trasfer rates ( ad so u ). Sice the coductio mode of heat trasfer is predomiat i the iitial stages the vortex viscosity has o effect ad the heat trasfer rates for micropolar fluid ad that ewtoia fluid are idetical. As the covectio domiatio mode starts developig the vortex viscosity of micropolar fluid ehaces the flow viscosity ad so decreases the heat trasfer rate. The larger the value of the larger the flow viscosity ad the lower the value of steady state u. The steady state local usselt umber distributios at Ra = ad at differet values of dimesioless vortex viscosity are show i Fig. 4. Sice the thermal field is symmetrical about the vertical axis, oly oe-half of u distributio is show. It ca be see that at the topmost poit o the tube surface ( η = ) the u is miimum for all values of though it is slightly smaller for bigger values of. As η icreases from topmost poit toward the bottommost poit, the u slightly icreases up to almost poit ( η = 3 ), ad.keeps almost costat up to poit ( η = 9 ) the icreases rapidly, reachig maximum at the bottommost poit ( η = 8 ). The figure clearly shows that, at ay surface poit u decreases as icreases. Decreasig of u o the tube surface as icreases explais the decrease of steady state u as icreases as show i Table. ad Fig. 3 8 Ra=, Pr=7 λ= Ar=.5, J= = bottommost poit. The figure also shows that as icrease the surface vorticity decreases at all poits of the tube surface. Decreasig of surface vorticity reflects the decrease of velocity gradiets at the tube surface which reflects i tur the weakess i the flow covectio currets. The weaker the flow currets the smaller the heat trasfer rates. ζ ο Ra=, Pr=7 λ= Ar=.5, J= = = =5 = η Fig. 5 Steady state surface vorticity distibutio φ Ra=, Pr=7 λ= Ar=.5, J= η=7 = = =5 = = u 6 =5 = Y * Fig. 6 Temperature variatio alog mior axis of the tube η Fig. 4 Steady state local usselt umber distibutio. Fig. 5 shows the surface vorticity distributio for the same case. The surface vorticity at ay fixed value of icreases rapidly from zero at topmost poit ( η = ) to maximum at almost η = 3 the slightly decreases as η icreases till almost η = ad the sharply decreases to zero at Fig. 6 shows the temperature variatio alog the extesio of the ellipse mior axis ( η = 7) for the case of Ra =, J= ad at differet values of dimesioless vortex viscosity,. It ca be see that the fluid temperature decays with distace from tube surface till it reaches evetually the stagat fluid temperature (i.e φ = ). Also. the figure clearly shows that as icreases the temperature gradiet at the tube surface decreases ad accordigly local heat trasfer decreases which i tur meas a decrease i u as ca be see i Fig.4.

6 Proceedigs of the 5th IASME/WSEAS It. Coferece o Heat Trasfer, Thermal Egieerig ad Eviromet, Athes, Greece, August 5-7, 7 Fig. 7 shows the variatio of both vorticity ad microrotatio alog the mior axis of the tube for the case of Ra=, J= ad =. The figure shows that the values of both vorticity ad microrotatio are sigificat i the earby regio of the tube ( i the boudary layer regio ) ad almost egligible elsewhere alog the axis extesio. It ca also be iferred that the directio of rotatio of both mea flow ad fluid elemets alog the mior axis is the same. The mea flow rotatio is represeted by vorticity while fluid elemets rotatio is represeted by the microrotatio. Moreover, the poit of zero vorticity ad zero microrotatio or the poit at which the mea flow ad fluid elemets chage directio of rotaio is the same for both of them. The cosisitecy of this result with the logical exepectatio from oe side supports the valdity of the micropolar fluid model developed by Erigi [] ad from the other side supports the validaty ad accuracy of the preset umerical techique. ζ, Μ Ra=, Pr=7. = η=7 vorticity, ζ microrotatio, Μ 3 4 Y * Fig.7. The vorticity ad microrotatio variatio alog mior axis 5 Coclusio The effect of material parameters of micropolar fluid ad Rayleigh umber o atural heat covectio from a isothermal elliptic tube ad placed horizotally with its major axis vertical i a micropolar fluid was ivestigated. The study cosidered a rage for Ra up to 4, a rage for material parameter, from to ad a rage for material parameter, J from. to. While the material parameter, λ Pradtl umber ad axis ratio are kept uchaged at, 7 ad.5 respectively. The study showed that at certai values for material parameters as Rayleigh umber icreases the local heat trasfer rate icreases at all poits o the cylider surface which i tur icreases the average heat trasfer rate. The study has also show that the vortex viscosity is the most importat material parameter. A oticeable reductio i local ad average heat trasfer rates is observed as vortex viscosity icreases. Geerally, the study showed that the covective heat trasfer rate decreases i the micropolar fluids i compariso with the ewtoia fluids. Refereces [] A. C Erige, Simple micropolars, It. J. Eg. Sci.,, 964, pp.5-7 [] A. C. Erige, Theory of thermomicrofluids, J. Math. Aal. Appl., 38, 97 pp [3] G. Ahmadi, Self-Similar solutio of icompressible micropolar boudary layer flow over a semi-ifiite plate, It. J. Eg. Sci., 4, 976, pp [4] I. A. Hassaie, Mixed covectio i micropolar boudary layer flow over a horizotal semiifiite plate, ASME, J. of Fluids Egieerig, 8, 996, pp [5] R. S. R. Gorla, Axisymmetric thermal boudary layer of a micropolar fluid o a cylider, It. J. Eg. Sci., 3, 995, pp [6] A. A. Mohammedie, R. S. R. Gorla, ad I. A. Hassaie, Mixed covectio i a axisymmetric stagatio flow of micropolar fluid o a vertical cylider, Acta Mecaica, 4, 996, pp [7] T. Hsu, P. Hsu ad S. Tsai, atural covectio of micropolar fluids i a eclosure with heat sources, It. J. Heat Mass Trasfer, 4(7), 997, pp [8] F. M. Mahfouz, Trasiet free covectio from a horizotal cylider placed i a micropolar fluid, Heat ad Mass Trasfer, 39, 3, pp [9] S. Bhattacharyya ad I. Pop, Free covectio from cyliders of elliptic cross sectio i micropolar fluids, It. J. Eg. Sci., 34, 996, pp. 3-3 [] F. M. Mahfouz, atural covectio from a elliptic tube with major axis horizotal ad placed i a micropolar fluid, It. J. Heat ad Mass Trasfer, 47(6-7), 4, pp [] H. M. Badr, ad S. C. R. Deis, Timedepedet viscous flow past a impulsively started rotatig ad traslatig circular cylider, J. Fluid Mechaics, 58, 985, pp [] H. M. Badr, Lamiar atural covectio from a elliptic tube with differet orietatios, ASME J. of Heat Trasfer, 9, 997, pp

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