EFFECT OF HEAT AND MASS TRANSFER ON MHD OSCILLATORY FLOW WITH CHEMICAL REACTION AND SLIP CONDITIONS IN ASYMMETRIC WAVY CHANNEL

Transcription

1 EFFECT OF HEAT AND MASS TRANSFER ON MHD OSCILLATOR FLOW WITH CHEMICAL REACTION AND SLIP CONDITIONS IN ASMMETRIC WAV CHANNEL J. Sasikmar and A. Govindarjan Department of Mathematics, SRM Universit, Kattanklathr, India ABSTRACT In this std, the effects of slip parameter on hdro magnetic oscillator flow combined with heat and mass transfer in asmmetric wav channel with species concentration throgh poros medim having non niform wall temperatre are investigated. De to the plsatile flow natre, the dimensionless flow governing eqations on continit, momentm and energ are solved based on some simplifing assmptions. Analtic soltions are obtained for temperatre distribtion, mass concentration and velocit fields. The featres of flow characteristics are analzed graphicall. Reslts of radiation parameter, Peclet nmber, Hartmann nmber and geometric parameters on flow and heat / mass transfer characteristics are discssed. It is observed that velocit profiles increase with an increase in the cold wall slip parameter and decrease for an increase in poros medim parameter. Kewords: navier slip, MHD, poros, oscillator, heat and mass transfer, chemical reaction, asmmetric channel. INTRODUCTION The slip effect on MHD oscillator flow of flid in a poros channel with heat and mass transfer and chemical reaction has applications in the fields of engineering, geophsics, agricltre etc. These applications are geothermal reservoirs, thermal inslation, oil recover, cooling of nclear reactor. Man chemical engineering processes like polmer extrsion processes involve cooling of a molten liqid being stretched into a cooling sstem. In this cooling sstem, better electromagnetic properties are normall sed as cooling liqid as their flow can be reglated b external magnetic fields in order to improve the qalit of the final prodct. Oscillator flow is a periodic flow that oscillates arond a zero vale. Oscillator flow is alwas important for it has man practical applications for example in the aerodnamics of helicopter rotor or in flttering airfoil and also in a variet of bio engineering problems. Flows throgh poros media are freqentl sed in filtering gasses, liqid and dring of blk materials. This also pla an important role in hman bod particlarl the breathing and discharge of excretes throgh poros skin. In the field of agricltral engineering, poros media heat transfer plas an important role in germination of seed. A chemical reaction in this paper involves in the breaking of bonds in the reactive sbstances and formlation of bonds to form different prodcts. Several researches have stdied and have related literatres on the slip effect on nstead MHD oscillator flow of flid in a poros channel with heat and mass transfer and chemical reaction. The std of electricall condcting flid flows bonded b wav wall is of special interest de to its application to transpiration cooling of re-entr vehicles and rocket boosters, cross-hatching on ablative srfaces and film vaporization in combstion chambers. In view of these applications, Shankar and Sinha [1] have made a detailed std of the Raleigh problem for a wav wall. The have conclded that at low Renolds nmbers the waviness of the wall qickl ceases to be of importance as the liqid is dragged along b the wall, whereas, at large, the effects of Renolds nmbers on viscosit are confined to thin laers close to the wall and known potential soltion emerges in time. Vajravel and Sastr [2] have investigated the problem of free convection in an incompressible viscos flid bonded b a long vertical wav wall and a parallel flat wall. The have given a special attention to the effect of the wall waviness on the flow and the heat transfer characteristics of the flid when the motion is generated b a difference in the wall temperatres. Ramachandra Rao et al. [5] have considered the MHD oscillator flow of blood throgh channels of varing cross section. It was observed that the separation does not occr in the presence of a strong applied magnetic field. Ching-ang Cheng [6] reported a std of the phenomenon of natral convection heat and mass transfer near a vertical wav srface embedded in a flid satrated poros medim. Recentl, Makinde and Mhone [8] have discssed the problem of heat transfer to MHD oscillator flow in a channel filled with poros medim. Gria and Jana [1] stdied hdrodnamic flows throgh vertical wav channel with travelling thermal waves embedded in poros medim. Rebhi Mebrok et al. Das et al (212) analzed the effect radiative heat and mass transfer on nstead natral convection coqette flow of a viscos incompressible flid throgh a poros medim in the slip flow regime in the presence of sction and radiative sorce. The Effect of slip condition on nstead MHD oscillator flow in a channel filled with poros medim in the presence of transverse magnetic field and radiative heat and mass transfer is stdied b Nitananda Senapati 1164

2 and Rajendra Kmar Dhal (213). Makinde and Mhone (25) investigated the combine effect of transverse magnetic field and radiative transfer to nstead flow of a condcting opticall thin flid throgh a channel filled with satrated poros medim and non niform walls temperatre. Sekhar et al (212) stdied the nstead MHD mixed convective oscillator flow of an electricall condcting opticall thin flid flow thogh a planar channel filled with satrated poros medim. The effect of boanc, heat sorce, thermal radiation and chemical reaction are taken into accont embedded with slip bondar condition, varing temperatre and concentration. An analsis of first order homogeneos chemical reaction and heat sorce on MHD oscillator flow of visco elastic flid throgh a channel filled with satrated poros medim are reported b Devika et al (213). The present std investigates the slip effect on MHD oscillator flow of flid in a poros medim with heat and mass transfer and chemical reaction in a asmmetric channel. The temperatres prescribed at the plates are non-niform and asmmetric. Mathematical Formlation of the Problem Consider the nstead laminar slip flow of an incompressible, viscos and electricall condcting thin flid in an asmmetric wav channel filled with poros medim nder the inflence of an externall applied magnetic field and radiative heat transfer as shown below. The channel wall eqations are given b (A) The walls of channel are maintained at the temperatre T1 and T respectivel which is high enogh to indce radiative heat transfer. Magnetic field and Renolds nmber are assmed to be ver small and hence indced magnetic field is negligible. Assming Bossinesq approximation,the governing eqations of the flow i.e momentm eqation, energ eqation and concentration eqation are formlated as follows: (1) (2) (3) with the bondar conditions, 1165

3 (B) Where are slip parameters de to poros medim. The flid is opticall thin with a relativel low densit and radiative heat flx q satisfies The non dimensional qantities are defined b a= ; b= ; d= The bondar in non-dimensional form becomes The dimensionless eqations, from (1), (2) (3) are (4) (5) Nomenclatre a 1, b 1 amplitdes of the wav walls a, b amplitde ratios B (= μ eh ) electromagnetic indction c p specific heat at constant pressre d 1 + d 2 width of the channel d mean half width of the channel D a Darc nmber g gravitational force G r Grashof nmber H intensit of magnetic field M Hartmann nmber K Thermal condctivit k Poros medim permeabilit coefficient N Radiation parameter P e Peclet nmber P Pressre q Radiative heat flx Re Renolds nmber S Poros medim shape factor S c Schmidt nmber t Time U Flow mean velocit w Axial velocit N 1 Nsselt nmber at the wall = h 1 N 2 Nsselt nmber at the wall = h 2 Greek Smbols θ Flid temperatre β Coefficient of thermal expansion μe Magnetic permeabilit σc Condctivit of the flid ρ Flid densit ν Kinematics viscosit coefficient λ Wave length ω Freqenc of the oscillation α Mean radiation absorption coefficient τ1 Skin friction at the wall = h 1 τ2 Skin friction at the wall = h 2 Soltion of the Problem For prel oscillator flow, we assme sitation of the form with the bondar conditions (6) (7) Taking pressre gradient of the form, B sing (7) we obtain the following ordinar differential eqations from (4) (5) & (6) (8) 1166

4 (9) Where and Sbject to the bondar conditions, (1) (11) Solving (8), (9) and (1) sing (11) and (12) we obtain, (13) (12) (14) Where and N =1 N = 2 N = 3 N = 4 RESULTS AND DISCUSSIONS The nmerical reslts are analsed b drawing graphs for Velocit profiles Temperatre profiles Concentration profiles Figre-1. Velocit profile for different N-radiation parameter vales. 1167

5 M = 1 M = 2 M = 3 M = 4 = 1 =2 =3 =4 U Figre-2. Velocit profile for different M vales Harttman vales. Figre-6. Velocit profile for different ω vales Gr =1 Gr =2 Gr =3 Gr =4 - U Sc = 1 Sc = 2 Sc = 3 Sc = Figre-3. Velocit profile for different Gr vales vales. Figre-7. Velocit profile for different Sc vales Gc =1 Gc = 2 Gc = 3 Gc = 4 U K = 1 K = 2 K = 3 K = Figre-4. Velocit profile for different Gc vales Figre-8. Velocit profile for different Kc vales. Pe = Pe =.7 Pe =1.5 Pe = 2 =.1 =.2 =.3 = Figre-5. Velocit profile for different Pe vales Figre-9. Effect of cold wall velocit slip parameter Gamma. 1168

6 N = 1 N = 1.5 N =2 N = =.1 =.2 =.3 = Figre-1. Effect of heated wall velocit slip parameter sigma Figre-14. Temperatre profiles for the vales of N. -1 SPECIES CONCENTRATION Sc = 1 Sc = 1.5 Sc = 2 Sc = Pe = 1 Pe = 2 Pe = 3 Pe = Figre-11. Concentration profile for Schmidt nmber Sc. C = = =.5 = Figre-12. Concentration profile for omega vales and t = 1. C.5 Kc = 1 Kc = 1.5 Kc = 2 Kc = Figre-15. Temperatre profiles for the vales of Pe. CONCLUSIONS The nmerical calclations are carried ot for different vales of radiation parameter,hartmann nmber,peclet nmber,grashoff nmber, freqenc of oscillations ω on velocit,temperatre and species concentration. Velocit increases when radiation parameter N increases and velocit decreses as phase angle increases. The effect of increasing magnetic field strength is to dampen the velocit which is the Harttman reslt and the same has been verified. As the Peclet nmber,grashoff nmber Gr and modified grashoff nmber increases, the velocit profile increases between the bondaries. With an increase in radiation parameter N,throgh absorption of heat the flid temperatre increases. Mass concentration increases with the increase in chemical reaction parameter and decreases with increase in freqenc of oscillations,schmidt nmber. Increasing heated wall slip parameter cases reversal flow towards the heated wall Figre-13. Concentration profile for chemical parameter K. 1169

7 REFERENCES [1] P.N. Shankar, V.N. Sinha, The Raleigh problem for a wav wall (implsive motiongeneration of viscos flid flow), J. Flid Mech. 77 (1976) [2] K. Vajravel, K.S. Sastr, Free convective heat transfer in a viscos incompressibleflid confined between a long vertical wav wall and a parallel flat wall, J. FlidMech. 86 (1978) [3] N. Bhaskara redd, D. Bathaiah, Magneto hdrodnamic flow of a viscosincompressible flid between a parallel flat wall and a long wav wall, Def. Sci. J.34 (1981) [4] P. Saknthala, D. Bathaiah, Magneto hdrodnamic flow of a dst viscos flidbetween a parallel flat wall and a long wav wall, Def. Sci. J 34 (1981) [5] A. Ramachandra Rao, K.S. Deshikachar, MHD oscillator flow of blood throghchannels of variable cross section, Int. J. Eng. Sci. 1 (1985) [6] Jer-Han Jang, Wei-Mon an, Hi-Chng Li, Natral convection heat and mass transfer along a vertical wav srface, Int. J. Heat Mass Transfer 46 (23) [7] O.D. Makinde, P.. Mhone, Heat transfer to MHD oscillator flow in a channelfilled with poros medim, Rom. J. Phs. 5 (25) [8] Samel Olmide Adesana, Olwole Daniel Makinde, MHD Oscillator slip flow and heat transfer in channel filled with poros media, U.P.B.Sci.Bll,Series A, Vol 76, Iss. 1, 214. [9] R.Mthraj, S.Srinivas, A note on heat transfer to MHD oscillator flow in an asmmetric wav channel, International Commnications in Heat and Mass Transfer, 37(21) [1] A. Ogl, T.M. Abbe, Simlation heat transfer on an oscillator blood flow in anindented poros arter, Int. Commn. Heat Mass Transfer 32 (25) [12] J. Prakash, A. Ogl, A std of plsatile blood flow modeled as a power law flidin a constricted tbe, Int. Commn. Heat Mass Transfer 34 (27) [13] T. Malath, S. Srinivas, Plsating flow of a hdromagnetic flid betweenpermeable beds, Int.comm. Heat Trans. 35 (28) [14] M. Hssain, T. Haat, S. Asghar, C. Feteca, Oscillator flows of second grade flidin a poros space, Nonlinear Analsis: Real World Applications 11 (21), [15] A. Ogl, A.R. Bestman, Deep heat mscle treatment a mathematical model- I,Acta Phs. Hng. 73 (1993) [16] M. Mishra, A. Ramachandra rao, Peristaltic transport of a Newtonian flid in anasmmetric channel, ZAMP 54 (23) [17] O. D. Makinde, P.. Mhone, Heat transfer to MHD oscillator flow in a channel filled withporos medim, Rom. Jorn. Phs., 5 (25) [18] A. Mehmood, A. Ali, The effect of slip condition on nstead MHD Oscillator flow of aviscos flid In a planer channel, Rom. Jorn. Phs., 52 (27) [19] A. K. Abdl Hakeem, K. Sathianathan, An analtic soltion of an oscillator flow throgha poros medim with radiation effect, Nonlinear Analsis: Hbrid Sstems 3 (29) [2] B. K. Jha and A. O. Ajibade, Free convective flow between vertical poros plates withperiodic heat inpt, Z. Angew. Math. Mech., (21) 1 9 [21] B. K Jha, Ajibade A. O. Free convective flow of heat generating/absorbing flid betweenvertical poros plates with periodic heat inpt. Int Commn Heat Mass Trans 36 (29) [22] J. C. Mistra, B. Pal, A Mathematical Model for the Std of the Plsatile Flow of BloodUnder an Externall Imposed Bod Acceleration, Mathematical and Compter Modelling 29 (1999) [11] M. Gria, R.N. Jana, Hdrodnamic flows throgh vertical wav channel withtravelling thermal waves embedded in poros medim, Int. J. Appl. Mech. Eng. 3 (26)