Finite Element Analysis of Heat and Mass Transfer of a MHD / Micropolar fluid over a Vertical Channel

Transcription

1 International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Volme 2, Isse 5, Ma 214, PP ISSN X (Print) & ISSN (Online) Finite Element Analsis of Heat and Mass Transfer of a MHD / Micropolar flid over a Vertical Channel V.V Redd 1, L. Anand Bab 2, Narsiml.Gitti 3 1 Department of Mathematics, Kamala Nehr Poltechnic College, Hderabad venkatreddiv@gmail.com 2 Department of Mathematics, Osmania Universit, Hderabad. 3 Department of Mathematics, CBIT, Hderabad. dr.narsiml@gmail.com Abstract: An analsis is presented for the problem of the fll developed natral convection nder the inflence of micropolar flid flow of heat and mass transfer in a vertical channel. Asmmetric temperatre and convection bondar conditions are applied to the walls of the channel. Soltions of the copled nonlinear governing eqations are obtained for different vales of the boanc ratio and varios material parameters of the micropolar flid and magnetic parameters. The reslting non dimensional bondar vale problem is solved b the Galerken Finite element method sing MATLAB Software. Inflence of the governing parameters on the flid flow as well as heat and solte transfers is demonstrated to be significant. Kewords: Finite Element Method, MHD, Natral convection, Micropolarflid, viscos dissipation 1. INTRODUCTION Problems dealing with free convection are often fond in man indstrial applications. Flldeveloped free convection heat transfer in a vertical channel or parallel plates was considered b Bodia et al. [3] for smmetric walls temperatre. Ang et al. [1, 2] and Miatake et al. [7] stdied the case of asmmetric bondar conditions which was later examined b Nelson et al. [8]. The earliest formlation of a general theor of flid micricontina was attribted to Eringen[5].His theor of micro flids has opened p new areas in research in the phsics of flid flow. B Eringen s definition, a simple microflid is a flent medim whose properties and behavior are affected b the local motions of the material particles contained in each of its volme elements sch a flid possesses local inertia. Phsicall, the represent flids consisting of randoml oriented particles sspended in a viscos medim Lkaszewicz[6]. The basic idea of micropolar flids has originated from the need to model the non-newtonian flow of flids containing rotating micro-constitents. Besides the sal eqations for Newtonian flow, this theor introdces some new material parameters, an additional independent vector field-the microrotation-and new constittive eqations that mst be solved simltaneosl with the sal Newtonian flow eqations. Sbseqent stdies showed that the model can be sccessfll applied to a wide range of applications inclding blood flow, lbricants, poros media, trblent shear flows, and flow in capillaries, and microchannels. The free micropolar convection for the case of asmmetricbondar conditions or asmmetric heating was investigated b Chamkha et al. [4], Thin film flow of non-newtonian flidsona moving beltstdied A.M. Siddiqiet al.[9], HabibisSaleh et al.[1] stdied the Simlation of Micropolar Flid Flow in a vertical Channel sing HPM.As pointed ot b Rawat and Bhargava [11], the std of heat and mass transfer in micropolar flids is of importance in the fields of chemical engineering, aerospace engineering and also indstrial manfactring effects processes. Snil et al. [12] stdied the Effect of Rotation on Doble-Diffsive Convection in a Magnetized Ferro flid with Internal Anglar Momentm. Recentl Bala Siddl Malga et al.[13] stdied the MHD Effects on Fll Developed Natral Convection Heat and Mass transfer of a micropolar Flid in a Vertical Channel. ARC Page 515

2 V.V Redd et al. 2. MATHEMATICAL MODEL Consider the free convection of a micropolar flid between two vertical plates, the space between the plates being h. The flow is assmed laminar, stead and fll-developed, i.e. the transverse velocit is zero. A niform magnetic field is applied to the flow. It is also assmed that the walls are heated niforml bt their temperatres ma be different reslting in an asmmetric heating sitation. The dimensionless governing eqations are: Sbject to the bondar conditions () =, θ() = R, N() =, (4) (1) =, θ(1) = R,N(1) =, (5) where is dimensionless velocit, N the dimensionless microrotation, θ dimensionless temperatre, K is the material parameter and temperatre of the left cooled wall and temperatre of the right wall. 3. METHOD OF SOLUTION The above model is a sstem of second-order BVP. Eqation (1) sbject to the bondar conditions (4) (5), possesses the following Finite Element soltion, obtained with the help of the MATLAB software. In order to redce the above sstem of differential eqations to a sstem of dimensionless form, we ma represent the velocit and microrotation, temperatre and concentration b appling the Galerkin finite element method for eqation (1) over a tpical twonoded linear element. In eqation (2)-(3), taking i=1(1) n and sing bondar conditions (4) and (5), the following sstem of eqation are obtained. Where s are matrices of order n and are colmn matrices having n-components. The soltion of above sstem of eqations are obtained sing Thomas algorithm for velocit, anglar velocit and temperatre. Also, nmerical soltions for these eqations are obtained b MATLAB program. In order to prove the convergence and stabilit of the method, the same MATLABprogram was rn with slightl changed vales of h and k, no significant change was observed in the vales of. 4. RESULTS AND DISCUSSION To get the phsical insight of the problem the reslts are discssed throgh graph. The comptations are carried ot for the governing flow of the problem with the Galerkin Finite Element Method. The obtained reslts are compared and these are a good agreement with the reslt of Habibis Saleh at al [1]. The nmerical reslts are obtained for the velocit and microrotation has been shown graphicall for different flow parameters. The effect of magnetic field parameter M on the velocit profiles and mocrorotation N for K=5, when R=.5 is shown in Fig.1 and 2. Here it is observed that the velocit profiles decreases with an increase of M, microrotation profiles increases p to centre of the channel the reverse phenomenon is observed in the other part of the channel. It can be seen that the velocit profiles increases with an increase of M, The microrotation N decreases with the increase of M, p to middle of the channel (flow direction is pward) and it is increases with increase of M, in the other part of the channel is observed. (1) (2) (3) (6) International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 516

3 Finite Element Analsis of Heat and Mass Transfer of a MHD / Micropolar flid over a Vertical Channel.12.8 M= M=5 M= Figre1. Effects of Magnetic parameter M on the velocit profiles for K=5and R= M= M=5 M=1.5 N Figre 2. Effects of Magnetic parameter M on the microrotation N for K=5and R= K= K=5 K= Figre 3. The velocit profiles for R=.5 International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 517

4 V.V Redd et al. The Fig.3 and 4 illstrates the inflence of vortex viscosit parameter K on the distribtion of velocit and microrotation N for R=.5. It is observed that with the increasing the vale of K the intensit of convective velocit is redced as compared to the Newtonian flid sitation (K=). In fact it is fond that as K,.The inflence of parameter K on the microrotation N it is noticed that the variation with K of the vale of N evalated at the position half of the channel also presented in the graphs it can be seen that the intensit of N first increases with increase of K, the reverse phenomenon is observed later. The velocit and microrotation profiles for the case R =.5 and K=, 1, 3 are depicted in Fig.3 Clearl; good agreement is observed when we compared with Habibis Saleh at al [1]..7.5 K=5 K=1.3 N K= Figre 4. The microrotation profiles for R= Ec=, 5, 1 & M= Figre 5. Effects of Viscos dissipation (Ec) on the Velocit Profiles International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 518

5 Finite Element Analsis of Heat and Mass Transfer of a MHD / Micropolar flid over a Vertical Channel Ec=.5, M=, 5, Figre 6. Effects of Viscos dissipation (Ec) and Manetic Parameter M on the Velocit Profiles It is seen from the Fig. 5 and 6 for the vales of magnetic parameter M =, 5, 1, the velocit decreasing at a point and then cross the side and increasing with magnetic parameter M. This is becase of the velocit profiles having lower peak vales for higher vales of magnetic parameter M tend to decreases comparativel slower along -direction than velocit profiles with higher peak vales for lower vales of magnetic parameter M. We ma conclde that for increasing vales in M; the Lorentz force, which opposes the flow, there is a fall in velocit maximm de to the retarding effect of the magnetic force in the region. As a reslt the momentm bondar laer thickness becomes larger and the separation of the bondar laer will occr earlier. Here, it is observed that the increase in the viscos dissipation (Ec) decreases the velocit θ.4 Ec=, 5, 1, 15,2, Figre 7. Effects of Viscos dissipation(ec) on the Temperatre Profiles θ International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 519

6 V.V Redd et al. It is also observed from Fig.7 that as the viscos dissipation parameter (Ec) increases, the temperatre profiles increases. The increase in the viscos dissipation cools the flid. The temperatre profile for varios vales of the viscos dissipation parameter (Ec) while the other parameters are kept constant. It is fond that increase in viscos dissipation parameter (Ec) leads to a corresponding increase in the temperatre profile. It is also seen that the temperatre decreases at certain portion of the channel and then increases this cold be de to the dissipation effect and the harmonic pressre term. It is known that the viscos dissipation prodces heat de to drag between the flid particles and this extra heat cases an increase of the initial flid temperatre (see Fig. 7). This increase of temperatre cases an increase of the boant force. The increase of the boant force cases an increase of the flid velocit. The bigger flid velocities case bigger drag between the flid particles and conseqentl bigger viscos heating of the flid. The new increase of flid temperatre inflences the boant force and this procedre goes on. There is a continos interaction between the viscos heating and the boant force. This mechanism prodces different reslts in the pward and downward flow. In the pward flow where the flid is warmer than the ambient the extra viscos heat is added to the initial heat (the warm flid becomes warmer) and the flid velocit increases. In the downward flow the flid is cooler than the ambient and the viscos heating cases an increase in the initial flid temperatre (the cold flid becomes warmer). REFERENCES [1] W. Ang, Fll developed laminar free convection between vertical plated heated Asmmetricall, International Commnications in Heat and Mass Transfer, Vol. 15,1972, pp [2] W. Ang, L.S. Fletcher, V. Sernas, Developed laminar free convection between vertical plateswith asmetric heating, International Commnications in Heat and Mass Transfer, Vol. 15, 1972, pp [3] J.R. Bodia, J.F. Osterle, The development of free convection between heated vertical plates,asme Jornal of Heat Transfer, Vol. 84, 1962,pp [4] A.J. Chamkha, T. Grosan, I. Pop, Fll developed free convection of a micropolar flid in avertical chanel, International Commnications in Heat and Mass Transfer, Vol. 29, 22, pp [5] A.C. Eringen, Theor of micropolar flids, Jornal of Mathematics and Mechanics, Vol.16,1966, pp [6] G. Lkaszewich, Micropolar flids: theor and applications, Birkhaser: Basel,1999. [7] O. Miatake, T. Fjii, Natral convection heat transfer between vertical parallel plates atneqal niform temperatres, Heat Transfer Japanese Research, Vol. 2, 1973, pp [8] D.J. Nelson, B.D. Wood, Combined heat and mass transfer natral convection between verticalparallel plates, International Jornal of Heat and MassTransfer, Vol. 32, 1973, pp [9] A.M. Siddiqi, M. Ahmed, Q.K. Ghori, Thin film flow of non-newtonian flids on a movingbelt, Chaos, Solitons and Fractals, Vol. 33, No. 3, 27, pp [1] HabibisSaleh,IshakHashim, Simlation of Micropolar Flid Flow in a Vertical Channel singhomotop-pertrbation Method,Applied Compting Conference (ACC '8), Istanbl, Trke, Ma 27-3, 28,pp [11] S. Rawat, R. Bhargava, Finite element std of natral convection heat and mass transferin a micropolar flid-satrated poros regime withsoret/dfor effect, Int. J.Appl.Math. Mech. 5 (29), pp [12] Snil, P. Chand, A. Mahajanand P. Sharma, Effect of Rotation on Doble-Diffsive Convection in amagnetized Ferroflid with Internal Anglar Momentm Jornal of Applied Flid Mechanics, 4(4), (211), pp [13] Bala Siddl Malga,N.Kishan, MHD Effects on Fll Developed Natral Convection Heat and Mass Transferof a Micropolar Flid in a Vertical Channel, Elixir Appl. Math. 63 (213) pp: International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 52