International Jornal of Engineering and Technical Research (IJETR) ISSN: 2321-0869 (O) 2454-4698 (P), Volme-3, Isse-8, Agst 2015 Elastico-Viscos MHD Free Convective Flow Past an Inclined Permeable Plate with Dfor Effects in Presence of Chemical Reaction Kamal Debnath Abstract An attempt has been made to std the nstead MHD two-dimensional free convective elastico-viscos flid past an inclined permeable plate with Dfor effects in presence of chemical reaction. The elastico-viscos flid flow is characterized b Walters liqid (Model B'). The pertrbation scheme is sed to solve the governing eqations of the flid motion. The approximate soltions have been derived for the velocit profile, temperatre field, concentration field, skin friction, rate of heat transfer and rate of mass transfer. It is noticed that the flow field is considerabl affected b the elastico-viscos parameter. The velocit profile and the approximate skin friction coefficient have been presented graphicall to observe the elastico-viscos effects for varios vales of the flow parameters involved in the soltion. Index Terms Chemical Reaction, Dfor effects, MHD, Skin friction I. INTRODUCTION The analsis of free convection flow adjacent to inclined srface bonded b an extensive bod of flid is of considerable importance in micrometeorolog and indstrial applications. Investigation of chemical reaction, heat and mass transfer in MHD elastico-viscos flid flow for an electricall condcting flid past a heated srface has attracted the interest of man researchers in view of its applications in man engineering problems sch as plasma stdies, petrolem indstries, MHD power generator, cooling of nclear reactors, MHD proplsion thermo flid dnamics, bondar laer control in aerodnamics, etc. Process involving heat and mass transfer in poros media are often encontered in the chemical indstr and in reservoir engineering in connection with thermal recover process. A comprehensive review of the stdies of heat transfer mechanism throgh poros media has been made b Neild and Bejan [1]. The std of natral convection flow of inclined plate has been presented b the athors Ganesan and Palani [2] and Sparrow and Hsar [3]. Free convection bondar laer flow over a horizontal and slightl inclined srface has been stdied b Pera and Gebhart [4]. Karks [5] has applied the pertrbation techniqe to std the natral convection flow adjacent to inclined isothermal and finite-length srfaces. Hossain et al. [6] have investigated the free convection flow from an isothermal plate inclined at a small angle to the horizontal. Raptis et al [7] have investigated the viscos flow over a non-linearl stretching sheet in presence of chemical reaction and magnetic field. Rajesh et al Kamal Debnath, Department of Mathematics, Roal School of Engineering and Technolog, Roal Grop of Instittions, Gwahati, Assam, India, PH:0-7035263679 [8] have stdied the chemical reaction and radiation effects on MHD flow past an infinite vertical plate with variable temperatre. Diffsion thermo and radiation effects on MHD free convective heat and mass transfer flow past an infinite plate in the presence of chemical reaction of first order has been stdied b Raveendra Bab et al [9]. The heat and Mass Transfer in MHD Bondar Laer Flow past an inclined plate with viscos dissipation in poros medim has been analzed b Singh P K [10]. Considering the model of elastico-viscos flid, man scientists have solved problems of engineering interests viz. Saxena and Dbe [11], Chodhr and Mahanta [12],Chodhr and De [13], Chodhr and Dhar [14]. The aim of the present work is to std the diffsion thermo (Dfor effect) and radiative effects on MHD nstead free convective flow of elastico-viscos flid past an inclined permeable plate characterized b Walters liqid (Model B'). The effects of the elastic-viscos flid on velocit component and skin friction coefficient have been shown graphicall with the combination of other flow parameters involved in the soltion. The constittive eqation for Walters liqid (Model B') is (1) where is the stress tensor, p is isotropic pressre, is the metric tensor of a fixed co-ordinate sstem, is the velocit vector, the contravariant form of is given b It is the convected derivative of the deformation rate tensor defined b (3) Here is the limiting viscosit at the small rate of shear which is given b and (4) being the relaxation spectrm as introdced b Walters [15]-[16]. This idealized model is a valid approximation of Walters liqid (Model B') taking ver short memories into accont so that terms involving (5) have been neglected. The mixtre of Polmethl Methacrlate and pridine at 250 C containing 30.5 gm of polmer per litre and having densit 0.98 gm/ml. fits ver nearl to this model. For this mixtre, the relaxation spectrm as given b Walters is where poises (gm/cm. sec), sec and is the Dirac s delta fnction so that k0=0.60557 gm/cm. (2) 186 www.erpblication.org
Elastico-Viscos MHD Free Convective Flow Past an Inclined Permeable Plate with Dfor Effects in Presence of Chemical Reaction II. MATHEMATICAL FORMULATION The region of nstead two-dimensional free convective hdromagnetic flow of an elastico-viscos flid past an infinite inclined permeable plate has been analzed in presence of chemical reaction. A niform magnetic field is applied normal to the plate. Here the mechanism of heat and mass transfer in connection with Dfor effect are stdied. Since the plate is of infinite length, all phsical variables are fnctions of and onl. Let -axis is taken along the inclined plate and -axis is taken normal to the plate. Let and be the velocit components in and direction respectivel. Let α be the small angle made b the plate with the vertical so that sinα=0 as shown in Figre1. where is the Grashof nmber for heat transfer, is the Grashof nmber for mass transfer, is the magnetic field, is the volmetric co-efficient of expansion for heat transfer, is the volmetric co-efficient of expansion for mass transfer, g is the acceleration de to gravit, and are the temperatre of the flid inside the thermal bondar laer and the flid temperatre in the free stream respectivel, and are the corresponding concentrations, is the co-efficient of mass diffsivit, is chemical reaction parameter, K is the permeabilit parameter. Introdcing the non-dimensional parameters in the eqations (7), (8), (9), we get g where is the visco-elastic parameter. The bondar conditions of the problem in the dimensionless form are Figre 1: Geometr of the problem Under the above assmptions the flow is governed b the following set of eqations: Eqation of continit: III. METHOD OF SOLUTION We assme the soltions of the non-linear partial differential eqations (12) to (14) of the following form Eqation of motion: Using (16) in the eqations (12), (13) and (14) and eqating the coefficient of like powers of and neglecting the highest powers of we get Energ eqation: Concentration eqation: With relative bondar conditions: The modified bondar conditions are Solving eqations (21) and (22 ) and sing bondar conditions (23), we get The non-dimensional parameters are: Solving eqations ( 19) and (20) and sing bondar conditions ( 23), we get Solving eqations (17) and (18) and sing bondar conditions (23), we get The constant are obtained bt not presented here for the sake of brevit. 187 www.erpblication.org
IV. RESULTS AND DISCUSSION The velocit profile for the flid flow is given b The skin friction coefficient at the plate in non-dimensional form is given b The non-dimensional form of the rate of heat transfer coefficient in terms of Nsselt nmber is given b The non-dimensional form of the rate of mass transfer coefficient in terms of Sherwood nmber is given b The prpose of this std is to bring ot the effects of elastic-viscos parameter on the nstead MHD free convective flow past an inclined permeable plate with Dfor effects in presence of chemical reaction. The elastico-viscos effects is exhibited throgh the non-dimensional parameter.the corresponding reslts for Newtonian flid are obtained b setting. To nderstand the phsics of the problem the flid velocit is depicted against in Fig. 1 to Fig. 6 for different vales of Prandtl nmber P r, Magnetic nmber M, Grashof nmber for heat transfer G r, Grashof nmber for mass nmber G m, Dfor nmber D, Chemical reaction K r, Schmidt nmber Sc and elastico-viscos parameter k 1. The nmerical calclations are carried ot for Permeabilit parameter K=2 and L=2. Fig. 1 and Fig. 2 reveals that an increase in the Grashof nmber does not bring an significant change in the flow pattern bt with the rising effect of visco-elasticit the flid velocit enhances in comparison with Newtonian case. It is also observed that the flid velocit diminishes de to the rise of magnetic parameter (Fig. 1 and Fig. 3) and Dfor nmber (Fig. 1 and Fig. 4) in both Newtonian and non-newtonian cases. The variation of chemical reaction ( Fig. 1 and Fig. 5) and Schmidt nmber (Fig. 1 and Fig. 6) shows that the flid velocit enhances in both Newtonian and non-newtonian cases. Fig. 7 to Fig. 12 depict the variation of the shearing stress for different vales of the flow parameters involved in the soltion. Fig. 7 reveals as Prandtl nmber increases viscos drag declines for both Newtonian and non-newtonian parameter. It is observed from Fig. 8 that the increasing vales of Magnetic parameter amplif magnitde of shearing stress. Fig. 9 illstrates that the rising vales of Dfor nmber increase the shearing stress for both Newtonian and non-newtonian flid flow phenomenon. The variation of shearing stress against chemical reaction K r (Fig. 10) shows that the growth of visco-elastic parameter sbdes the shearing stress. Fig. 11 shows that the magnitde of shearing stress increases with amplified vales of Grashof nmber for cooling plate (G r >0) bt it decreases with the increasing vales of elastico-viscos parameter in comparison with Newtonian flid.. The reverse pattern of flow is observed for the heated plate (G r <0) (Fig. 12). It is frther noticed that the rate of heat and mass transfer coefficients are not significantl affected b the elastico-viscos parameter. International Jornal of Engineering and Technical Research (IJETR) ISSN: 2321-0869 (O) 2454-4698 (P), Volme-3, Isse-8, Agst 2015 V. CONCLUSIONS The nstead two-dimensional free convective elastico-viscos flid past an inclined permeable plate with Dfor effects in presence of chemical reaction has been investigated for different vales of flow parameters. In the analsis, the following conclsions are made: The flid velocit diminishes de to the rise of magnetic parameter and Dfor nmber in both Newtonian and non-newtonian cases. The speed of both Newtonian and non-newtonian flids throgh the inclined permeable plate enhances with the increasing vales of chemical reaction and Schmidt nmber. The growth of Grashof nmber does not bring an significant change in the flow pattern of the elastic-viscos flid. Viscos drag declines for both Newtonian and non-newtonian flid with the increase of Prandtl nmber and chemical reaction parameter. The increasing vales of Magnetic parameter amplif magnitde of shearing stress. The rising vales of Dfor nmber increase the shearing stress for both Newtonian and non-newtonian flid. The shearing stress enhances with amplified vales of Grashof nmber for cooled and heated plate. Fig: 1. Variation of against for P r =5, G r =6, G m =7, M = 3, K = L= 2, D =0.03, K r = 0.6, S c =3 Fig: 2. Variation of against for P r =5, G r =10, G m =7, M = 3, K = L= 2, D =0.03, K r = 0.6, S c =3. 188 www.erpblication.org
Elastico-Viscos MHD Free Convective Flow Past an Inclined Permeable Plate with Dfor Effects in Presence of Chemical Reaction Fig: 3. Variation of against for P r =5, G r =6, G m =7, M = 5, K = L= 2, D =0.03, K r = 0.6, S c =3 Fig: 7. Variation of against P r for G r = 3, G m =8, M = 3, K =L=2, D =0.03, K r =0.6, S c =3. P r Fig: 4. Variation of against for P r =5, G r =6, G m =7, M = 3, K = L= 2, D =0.05, K r = 0.6, S c =3. M Fig: 8. Variation of against M for P r =5, G r = 3, G m =8, K=L=2, D =0.03, K r =0.6, S c =3. Fig: 5. Variation of against for P r =5, G r =6, G m =7, M = 3, K = L= 2, D =0.03, K r = 0.9, S c =3. D Fig: 9. Variation of against D for P r =5, G r = 3, G m =8, M=3, K=L=2, K r =0.6, S c =3. Fig: 6. Variation of against for P r =5, G r =6, G m =9, M = 3, K = L= 2, D =0.03, K r = 0.6, S c =4. K r Fig: 10. Variation of against K r for P r =5, G r = 3, G m =8, M = 3, K =L=2, D =0.03, S c =3. 189 www.erpblication.org
International Jornal of Engineering and Technical Research (IJETR) ISSN: 2321-0869 (O) 2454-4698 (P), Volme-3, Isse-8, Agst 2015 Fig: 11. Variation of against G r (> 0) for P r =5, G m =8, M = 3, K =L=2, D =0.03, K r =0.6, S c =3. G r [12] R. Chodhr and M. Mahanta, Mixed convective MHD flow of visco-elastic flid past a vertical infinite plate with mass transfer, Int, lornal of Scientific and Engineering Research, 3(2), 2012, 1-7. [13] R. Chodhr and D. De, Mixed convective MHD flow with heat and mass transfer of an elastic-viscos flid from a vertical srface with ohmic heating in presence of radiation, Jornal of flids and thermal Sciences, 1(2), 131-143, 2012. [14] R. Chodhr and P. Dhar, Diffsion thermo effects of visco-elastic flid past a vertical poros srface in presence of magnetic field and radiation. Int. J. of Innovative Research in Science, Engineering and Technolog, 2(3), 2013, 805-812. [15] K. Walters (1960), The motion of an elastico-viscos liqid contained between co-axial clinders, Qart. J.Mech. Appl. Math., 13: 44. [16] K. Walters (1962), The soltions of flow problems in the case of materials with memories, J. Mecaniqe, 1: 473. G r Fig: 12. Variation of against G r (< 0)for P r =5, G m =8, M = 3, K =L=2, D =0.03, K r =0.6, S c =3. Dr. Kamal Debnath, M.Sc. B.Ed. Ph.D. is an Assistant Professor of Mathematics in Roal School of Engineering and Technolog, RGI, Gwahati, Assam, India. He served for five ears as a lectrer in Don Bosco College and Don Bosco College of Teacher s Edcation, Tra, Meghalaa. His research interest lies in the field of Flid Mechanics. He has pblished nearl a dozen of research papers in national and international jornals of repte. He is the life member of man mathematical societies like AAM, ISTE, IMS, ISTAM of India. REFERENCES [1] D.A. Nield and A Bejan., convection in poros media 2 nd edition, Springer-verlag, Berlin, 1998. [2] P. Ganesan and G Palani (2004), Finite difference analsis of nstead natral convection MHD flow past an inclined plate with variable srface heat flx, Int J Heat Mass Transfer, 47:4449. [3] E. M. Sparrow and R B Hsar (1967), Longitdinal vortices in natral convection flow on inclined plate, J Flid Mech, 37: 251. [4] L. Pera and B. Gebhart (1972), Natral convection bondar laer flows over horizontal and slightl inclined plate, Int J Mass Transfer, 16: 1131. [5] W. T. Karks (1968), Natral convection flow adjacent to inclined isothermal and finite-length srfaces, Int J Mass Transfer, 11: 241. [6] Hossain et al (1996), MHD free convection flow from an isothermal plate, J Theo and Appl Mech, 1:94. [7] A. Raptis, C Perdikis, Viscos flow over a non-linearl stretching sheet in the presence of a chemical reaction and magnetic field, Int. J. Non-linear Mech, 41 (2000), 527-529. [8] V. Rajesh and S.V.K. Verma., chemical reaction and radiation effects on MHD flow past an infinite vertical plate with variable temperatre, Far East Jornal of Mathematical Sciences,32 (1), 2009, 87-106. [9] B. K. Raveendra, A.G Kmar.V Kmar and S.V.K Varma., Diffsion-thermo and radiation effects on MHD free convective heat and mass transfer flow past an infinite vertical plate in the presence of a chemical reactionof first order, Advances in Applied Science Research, 3 (4), 2012, 2446-2462. [10] P. K. Singh (2012), Heat and Mass Transfer in MHD bondar laer flow past an inclined platewith viscos dissipation in poros medim, Int J Sc Engg Research, 6: 213. [11] S.S. Saxena and G.K Dbe., Heat and mass transfer effects on MHD free convection flow of a visco-elastic flid embedded in poros medim with variable permeabilit in the presence of radiation and heat sorce in slip flow regime, Advances in Applied Science Research, 2(5), 2011, 115-126. 190 www.erpblication.org