MHD free convection heat and mass transfer flow through a porous medium bounded by a vertical surface in presence of hall current

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srface in presence of hall crrent Tavva. Sdhakar Redd, O. Siva Prasad Redd, M.. Raj and S. V. K. Varma 4 Department of Mathematics, Global ollege of Engineering and Technolog, kadapa, AP.india.

1 Available online at Pelagia Research Librar Advances in Applied Science Research,, (6):48-49 ISSN: ODEN (USA): AASRF MHD free convection heat and mass transfer flow throgh a poros medim bonded b a vertical srface in presence of hall crrent Tavva. Sdhakar Redd, O. Siva Prasad Redd, M.. Raj and S. V. K. Varma 4 Department of Mathematics, Global ollege of Engineering and Technolog, kadapa, AP.india. Departments of Mathematics, SRES, Tirpati, hitoor District, Andhra Pradesh, India. Department of Mathematics, AITS (Atonomos), Rajampet, India. Andhra Pradesh, India. 4 Department of Mathematics, S.V Universit, Tirpati, Andhra Pradesh,india. ABSTRAT In this paper the effects of hall crrent, chemical reaction and radiation on a free convection flow bonded b a vertical srface embedded in poros medim nder the inflence of niform magnetic field which is applied normal to the srface is stdied. The problem is solved analticall and the expressions for velocit, temperatre, concentration, skin friction and rate of heat and mass transfer are derived and the effects of varios phsical parameters like Magnetic parameter M, radiation parameter F, Grashof nmber Gr, modified Grashof nmber Gm, Prandtl nmber Pr, permeabilit parameter k and the chemical reaction parameter k are stdied thogh graphs and tables. It is observed that the velocit and concentration increase dring a generative reaction and decrease in a destrctive reaction. The same is tre for the behavior of the flid temperatre. The presence of magnetic field and radiation diminishes the velocit and also the temperatre. Ke words: Hall crrent, chemical reaction, MHD, Radiation, Poros medim. INTRODUTION In natre, there exist flows which are cased not onl b the temperatre differences bt also b concentration differences. These mass transfer differences do affect the rate of heat transfer. In indstries, man transport process exist in which heat and mass transfer takes place simltaneosl as a reslt of combined boanc effect of thermal diffsion and diffsion thermo chemical species. The phenomenon of heat and mass transfer freqentl exists in chemicall processed indstries sch as food processing and polmer prodction. Free convection flows are of great interest in a nmber of indstrial applications sch as fiber and granlar inslation, geothermal sstems etc. convection in poros media has applications in geothermal energ recover, oil extraction, thermal energ storage and flow throgh filtering devices. Magetohdrodnamics is attracting the attention of the man athors de to its applications in geophsics; it is applied to std the stellar and solar strctres, interstellar matter, radio propagation throgh the ionosphere etc. In engineering in MHD pmps, MHD bearings etc. at high temperatres attained in some engineering devices, gas, for example, can be ionized and so becomes an electrical condctor. The ionized gas or plasma can be made to interact with the magnetic and alter heat transfer and friction characteristic. Since some flids can also emit and absorb thermal radiation, it is of interest to std the effect of magnetic field on the temperatre distribtion and heat transfer when the flid is not onl an electrical condctor bt also when it is capable of emitting and absorbing thermal radiation. This is of interest becase heat transfer b thermal radiation is becoming of greater importance when we are concerned with space applications and higher operating temperatres. Pelagia Research Librar 48

2 Tavva. Sdhakar Redd et al Adv. Appl. Sci. Res.,, (6):48-49 Sondalgekar and Takhar [] first, stdied the effect of radiation on the natral convection flow of a gas past a semiinfinite plate sing the ogl-vincentine-gilles eqilibrim model. For the same gas Takhar et al. [] investigated the effects of radiation on the MHD free convection flow past a semi-infinite vertical plate. Later, Hossain et al. [] stdied the effect of radiation on free convection from a poros vertical plate. Mthcmarswam and Kmar [4] stdied the thermal radiation effects on moving infinite vertical plate in presence of variable temperatre and Mass diffsion. An analtical soltion for nstead free convection in poros media has been stdied b Magari et al. [5]. hamkha et al. [6] stdied the effects of Hdro magnetic combined heat and mass transfer b natral convection from a permeable srface embedded in a flid satrated poros medim. Mazmdar and Deka [7] stdied MHD flow past an implsivel started infinite vertical plate in presence of thermal radiation. The growing need for chemical reactions in chemical and hdrometallrgical indstries reqire the std of heat and mass transfer with chemical reaction. The presence of a foreign mass in water or air cases some kind of chemical reaction. This ma be present either b itself or as mixtres with air or water. In man chemical engineering processes, a chemical reaction occrs between a foreign mass and the flid in which the plate is moving. These processes take place in nmeros indstrial applications, for example, polmer prodction, manfactring of ceramics or glassware and food processing. A chemical reaction can be codified as either a homogenos or heterogeneos process. This depends on whether it occrs on an interface or a single phase volme reaction. A reaction is said to be of first order if its rate is directl proportional to the concentration itself [8]. The effect of chemical reaction on heat and mass transfer in a laminar bondar laer flow has been stdied nder different conditions b several athors [9-8]. The effect of a chemical reaction on a moving isothermal vertical srface with sction has been stdied b Mthcmarswam [9]. Recentl, Manivannan et al. [] investigated radiation and chemical reaction effects on isothermal vertical oscillating plate with variable mass diffsion. Inflence of chemical reaction and radiation on nstead MHD free convection flow and mass transfer throgh viscos incompressible flid past a heated vertical plate immersed in poros medim in the presence of heat sorce was investigated b Sharma et al. []. Mahapatra et al. [] stdied the effects of chemical reaction on free convection flow throgh a poros medim bonded b a vertical srface. Rajasekhar et al. [], Kishan and Srinivas [4], Anjalidevi and David [5], Kishan and Deepa [6] and Gaikwad and Rahldev [7] stdied the effects of varios parameters on flid flow qantities. In all the above stdies the combined effect of radiation and chemical reaction on MHD free convective flow in addition to Hall crrents have not been considered simltaneosl. Here we have made an attempt to std the Hall crrent effects on a stead flow of viscos flid throgh a poros medim bonded b a poros srface sbjected to sction with a constant viscosit in the presence of radiation and homogenos chemical reaction of first order. : We consider an electricall condcting, radiating, viscos incompressible flid throgh a poros medim occping a semi-infinite region of the space bonded b a vertical infinite srface. The x axis is taken along the srface in an pward direction and the axis is normal to it. A niform magnetic field B is assmed to be applied in a direction perpendiclar to the srface. The flid properties are assmed to be constant except for the densit in the bod force term. A chemicall reactive species is emitted from the vertical srface into a hdrodnamic flow field. It diffses into the flid, where it nder goes a homogenos chemical reaction. The reaction is assmed to take place entirel in the stream. Then the fll developed flow nder the above assmptions throgh a highl poros medim is governed b the following set of eqations: v B v gb T T gb v v ( ) ( ) T ( m ) k p T qr P P P T k D k c () () () (4) Pelagia Research Librar 48

3 Tavva. Sdhakar Redd et al Adv. Appl. Sci. Res.,, (6):48-49 It is assmed that the level of species concentration is ver low; hence the heat generated de to chemical reaction is neglected. The relevant bondar conditions are given as follows, T T, at = W W, T T, as (5) Eqation () gives that v constant v (6) In the opticall thick limit, the flid does not absorb its own emitted radiation in which there is no self absorption, bt it does absorb radiation emitted b the bonadries. ogle et al. [ ] showed that in the opticall thick limit for a non gra gas near eqilibrim as given below. r deb T T w K ww d I T Tw dt? q 4( ) ( ) / 4 ( ) (7) On introdcing the following non dimensional qantities, T T c,, TW T,Pr p W k 4I gt TW T gc W c D k K B kp,, P( T Tw) ( ) ( ) S, F, Gr, Gm, K E M k (8) The non-dimensional form of the governing eqations () to (4) redce to '' ' Gr Gm M (9) Where M M m k '' Pr ' PrE ' F () '' Sc ' kcsc () The corresponding bondar conditions are given b,, at = ;,, as () : In order to solve the copled nonlinear sstem of eqations (9) to () with the bondar conditions (), the following simple pertrbation is sed. The governing eqations (9) to () are expanded in powers of Eckert nmber E( ). E O( E ), E O( E ), E O( E ) () Sbstitting eqations () into eqations (9) to () and eqating the coefficients at the terms with the same powers of E, and neglecting the terms of higher order, the following eqations are obtained. Zero order terms: '' ' Gr Gm M (4) Pelagia Research Librar 484

4 Tavva. Sdhakar Redd et al Adv. Appl. Sci. Res.,, (6):48-49 '' Pr ' F (5) '' S ' S k (6) First order terms: '' ' Gr Gm M (7) '' Pr ' F Pr ' (8) '' S ' S k (9) The corresponding bondar conditions are, o,,,, at =,,,,, as () Solving eqations (4) to (9) nder the bondar conditions (), the following soltions are obtained. e k () k e () k k e k e k e l k k ( ) () 4 4 k k l l k e k e k e k e k e 8 9 k e k e k e k e l l k l k k l l l l4 l6 k e k e k e k e k e k e k e (5) l The expressions for the constants involved in eqations () to (5) are given in the appendix. RESULTS AND DISUSSION In order to point ot the effects of varios parameters on flow characteristic, the following discssion is set ot. The vales of Prandtl nmber are chosen Pr=7 (water) and Pr=.7 (air). The vale of the Schmidt nmber is chosen to represent the presence of species b hdrogen (.). Velocit profiles are presented in figres to 8. Figre depicts the velocit profile at the absence of radiation and MHD. From this figre it is observed that flid velocit increases and reaches its maximm over a ver short distance from the plate and then gradall to zero for varios vales of k. These reslts are in good agreement with the reslts of those Mahapatra et al. []. Figre shows the effect of magnetic parameter M on the velocit. From this figre it is observed that velocit decreases, in both the cases of air and water, as the vale of M is increased. It is tre as the magnetic force retards the flow, velocit decreases. Velocit increases as the Hall parameter increase as shown in Figre. Figre 4 depicts the velocit profiles for different vales of chemical reaction parameter k. A generative reaction (k <) increases the flid flow velocit, whereas a destrctive reaction (k >) redces it. In the case of water its magnitde is less than that of air. From figre 5 it is clear that an increase in the permeabilit parameter reslts in an increase in the velocit for both the cases of generative reaction and destrctive reaction. A radiation effect on velocit is shown in figre 6, from this figre it is observed that velocit decreases as the radiation parameter F increases. Velocit profiles for different vales of Gr and Gm, shown in figres 7 and 8. From these figres it is noticed that velocit increases with the increase Gr and Gm. Temperatre profiles are displaed throgh figres -. Effect of magnetic parameter M in the case of water and air is observed on the temperatre, it is observed that temperatre decreases with the increase in M; in the case of water the magnitde of the decrease of temperatre is ver low. From figres,and it is clear that temperatre decreases with the increase in radiation parameter F, the Schmidt nmber Sc and it shows the reverse effect in case of permeabilit parameter k. figre depicts the concentration profiles with the variations in (4) Pelagia Research Librar 485

5 Tavva. Sdhakar Redd et al Adv. Appl. Sci. Res.,, (6):48-49 chemical reaction parameter k, from this figre it is observed that a destrctive reaction redces the concentration. This is de to the fact that for k > the last term in the mass diffsion eqations (6) and (9) becomes positive and it contribtes to the concentration redction. At the same time, the same term mentioned an eqation becomes negative for a generative reactionk <and, as a reslt, it leads to a concentration increase. oncentration profiles for different vales of Schmidt nmber Scare shown in Figre 4, it is noticed that concentration decreases with an increase in Sc. The rate of heat transfer in terms of the Nsselt nmber is given b N k E(k k k k l k l k l k l k l k The Non-dimensional skin friction at the srface is given b ( k k ) l k k k k E( k k k k l k l k l k k l k l k k k e l k (7) Another important phsical qantit of interest is the Sherwood nmber which is in non-dimensional form is given S k ( E) b h (8) A variation in the heat transfer rate expressed in terms of the Nsselt nmber is shown table, from this table it is observed that N increases with the increase in magnetic parameter M and radiation parameter F, where as it shows reverse effect within the case of permeabilit parameter k,hall parameter m and chemical reaction parameter k.a variation in skin friction τ is shown in table, from this table it is noticed that skin friction decreases with an increase in M and F, where as it shows reverse effect in the case of k, m and k. Similarl a variation in the Sherwood nmber is shown in table, it is observed that Sherwood nmber (Sh) increases with an increase in Eckert nmber E and Schmidt nmber Sc and shows the reverse effect in the case of k. Table. Rate of heat transfer N for different vales of M, m, F, k, k With fixed vales for Pr=.7, Gr=5, Gm=5. M m F K K N (6) Pelagia Research Librar 486

6 Tavva. Sdhakar Redd et al Adv. Appl. Sci. Res.,, (6):48-49 Table. Skin friction for different vales of M, m, F, k, k with fixed vales for Pr=.7, Gr=5, Gm=5. E Sc K Sh Table. Variations of Sherwood nmber Sh for Varios vales of E, Sc, k with the fixed vales of M=, Gr=5, Gm=5, Pr=.7, k=. M m F k k τ k=. k=.7 Present reslts Reslts of Mahapatra et al. M=,Gr=5,Gm=5,F=,k=,Sc=.,Pr= M=,Pr=7 M=,Pr=7 M=,Pr=7 M=,Pr=.7 M=,Pr=.7 M=,Pr=.7 k=.4 k=. k=,k=,gr=5,gm=5,sc=.,f=.5,m= Fig.. Velocit profiles for different vales of k when M= and F= Pr=.7,m= Pr=.7,m= Pr=.7,m= Pr=.7,m=5 Pr=7,m= Pr=7,m= Pr=7,m= Pr=7,m=5 M=5,Gr=5,Gm=5,k==.,k=,F=.5, Fig.. Velocit profiles for different vales of M Pr=.7,k=-.4 Pr=.7,k=-. Pr=.7,k= Pr=.7,k=. Pr=.7,k=.4 Pr=7,k=-.4 Pr=7,k=-. Pr=7,k= Pr=7,k=. Pr=7,k= M=,k=,K=,Gr=5,Gm=5,Sc=.,F=.5,m= Fig.. Velocit profiles for different vales of m Fig.4. Velocit profiles for different vales of k Pelagia Research Librar 487

7 Tavva. Sdhakar Redd et al Adv. Appl. Sci. Res.,, (6): M=,Gr=5,Gm=5,Sc=., F=.5,m=5,pr=7 k=.,k=.4 k=.,k= k=.,k=.4 k=.5,k=-.4 k=.,k= k=.,k=.4 k=,k=-.4 k=,k= k=,k= M=,Gr=5,Gm=5,Pr=.7,k=, k=.,m=5,sc=. F=.5 F=. F=5. F= F= Fig.5. Velocit profiles for different vales of k Fig.6. Velocit profiles for different vales of F Gm=5 Gm= Gm=5 Gm= Gr=5 Gr= Gr=5 Gr= M=,Gr=5,F=.5,Pr=.7, k=,k=.,m=5,sc=. 4 M=,Gm=5,F=.5,Pr=.7,k=, k=.,m=5,sc= Fig.7. Velocit profiles for different vales of Gm Fig.8. Velocit profiles for different vales of Gr Pr=o.7,M= Pr=o.7,M= Pr=o.7,M= Pr=o.7,M= Pr=7,M= Pr=7,M= Pr=7,M= Pr=7,M= F=.5 F=5. F= F=.4. Gr=5,Gm=5,k=,k=,m=5,Sc=.,F=.5.4. M=,Gr=5,Gm=5,k==.,k=,F=.5,Sc=.,Pr= Fig.9. Temperatre profiles for different vales of M Fig.. Temperatre profiles for different vales of F. Pelagia Research Librar 488

8 Tavva. Sdhakar Redd et al Adv. Appl. Sci. Res.,, (6): k=. k=.4 k=.7 k= sc=. Sc=.6 Sc=. Sc= M=5,Pr=.7,k=,Gr=5,Gm=5,,k=,Sc=.,F= M=5,Pr=.7,k=,k=,F=.5,Gr=5,Gm= Fig.. Temperatre profiles for different vales of k Fig.. Temperatre profiles for different vales of Sc k=-.4 k=-. k=. k=. k= Sc=. Sc=. Sc=.6 Sc=.78 Sc=. Sc=..6.5 k=,gr=5,gm=5,f=.5,sc=.,e=..6.5 M=,Gm=5,Gr=5,k=,k=,F=.5, Fig.. oncentration profiles for different vales of k Fig.4. oncentration profiles for different vales of Sc ONLUSION In this paper we have stdied the effects of hall crrent, chemical reaction and radiation on MHD free convection flow throgh a poros medim bonded b a vertical srface. In the analsis of the flow the following conclsions are made i. The velocit of a flid increases with the permeabilit parameter k, Hall parameter m and decreases with the increase in Magnetic parameter M, Radiation parameter F. ii. A generative reaction (k <) increases the flid flow velocit, whereas a destrctive reaction (k >) redces it. iii. In most cases the velocit attains a maximm near the srface and there after decreases. iv. Temperatre decreases with the increase Magnetic parameter, radiation parameter bt it shows the reverse effect in the case of chemical reaction parameter. v. A destrctive reaction redces the concentration whereas the generative reaction increases it. vi. Nsselt nmber increases with the increase in magnetic parameter M and radiation parameter F, where as it shows reverse effect in the case of permeabilit parameter k and concentration parameter k. vii. Skin friction decreases with an increase in M and F, where as it shows reverse effect in the case of k,mand k viii. Sherwood nmber (Sh) increases with an increase in Eckert nmber E and Schmidt nmber Sc and shows the reverse effect in the case of k. REFERENES [] Sondalgekar V M, Takhar H S., Modelling Measre and ont.,99, 5, -4. [] Tahkar H S, Gorla S R, Sondalgekar V M., Int. J. Nmerical Methods heat flid flow,996, 6, [] Hossain A M, Alim M A and Rees D A S, Int.J.Heat Mass Transfer,999,4, 8-9. [4] Mthcmarswam R and Kmar G S, Theoret. Appl. Mach., 4,(), Pelagia Research Librar 489

9 Tavva. Sdhakar Redd et al Adv. Appl. Sci. Res.,, (6):48-49 [5] Magari E, Pop I and Keller B., 4, Analtical soltions for nstead free convection flow throgh a poros media., J.Eng. math., 48, 9-4. [6] hamaka A.J,,Int. J. Nm.Methods for heat and flid flow,, (5), [7] Mzmdar M K and Deka R K, Rom. Jorn. Phs., 7, Vol. 5, Nos P [8] ssler E L., (Ed) DiffsionMass transfer in Flid sstems, ambridge Universit press, London, 988. [9] Takhar H S, hamkha A J and Nath G., Int.J.Eng.Sci.,, 8, No., -4. [] Mthcmarswam R,Ganesan P, J.Appl.mech.Tech. Phs.,, 4, No.4, [] hamkha A J., Int.ommn.Heat mass transfer,,, N., 4-4. [] Ram.P., Int. J. of energ research, 995, 9,7-76. [] Mohamed A. Seddeek and Emad M. Aboel D, IJMMS 6: 4,, [4].hadhar R. and Jain. P,Theoret. Appl. Mech., 6, Vol., No.4, pp [5] Ahmed N, Kalita H and. Bara D P., International Jornal of Engineering, Science and Technolog,, Vol., No. 6, pp [6] hadhar R, Jain.P., Ukr. J. Phs. 7. V. 5, N.pp. -. [7] Abdlla I A., Thermal science, 9, Vol., No., pp [8] Raj.M., Redd N.A, Varma S.V.K., Thermal science,, Vol. 5, No., pp [9] Mthcmarswam R., Acta Mech., 55, [] Manivannan K, Mthcmarswam R, Thangaraj V., Thermal science, 9, Vol., No., pp [] Sharma P R, Kmar N, Sharma P., Appl. Math. Sciences,, Vol.5, No.46, [] Mahaptra N, Dash G, Panda S, Achara M., J.Engg. Phs. and Thermo phsics,, Vol. 8, No.. [] Rajasekhar K, Ramana Redd G V,Prasad B D N, Advances in applied science Research,, (5), [4] Kishan N, Srinivas M, Advances in applied science Research,,(), [5] Anjalidevi S P, David A M G, Advances in applied science Research,, (), 9-4. [6] Kishan N, Deepa G, Advances in applied science Research,, (), [7] Gaikwad S N, Rahldev B M, Advances in applied science Research,, (), Pelagia Research Librar 49

MHD Flow of a Newtonian Fluid Through a Porous Medium in Planer Channel

MHD Flow of a Newtonian Fluid Through a Porous Medium in Planer Channel Global Jornal of Pre and Applied Mathematics. ISSN 973-768 Volme 3, Nmber 7 (7), pp. 38-383 Research India Pblications http://www.ripblication.com MHD Flow of a Newtonian Flid Throgh a Poros Medim in Planer