MHD free convection and mass transfer flow of a micro-polar thermally radiating and reacting fluid with time dependent suction

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1 Indian Jounal of Pue & Alied Physics Vol. 46, Octobe 8, MHD fee convection and mass tansfe flow of a mico-ola themally adiating and eacting fluid with time deende suction J Pakash *, A Ogulu 1 & E Zhandie 1 * Deatme of Mathematics, Faculty of Science Univesity of Botswana, Pivate Bag, Gaboone, Botswana 1 School of Physics, estville Camus Univesity of KwaZulu-Natal Duban 4, South Afica s: * akashj@moii.ub.bw, 1 ogulua@ukzn.ac.za, 1 zhandiee@ukzn.ac.za Received Febuay 7; acceted 17 Januay 8 The oblem of fee convection of a chemically eacting mico-ola fluid with mass tansfe has been studied in the esence of a unifomly alied tansvese magnetic field and vaiable suction. The govening, couled, non-linea atial diffeeial equations ae solved emloying a fouth ode Runge-Kutta algoithm. The esults show that the velocity inceases as the suction aamete is inceased, the chemical eaction ate deceases; while the angula velocity inceases with inceasing values of the suction aamete u to a oi aound y = 1.5, befoe the tend is evesed. Keywods: Fee convection, Mico-ola fluid, Vaiable suction 1 Ioduction The oblem of fee convection flow of a chemically eacting fluid has been studied fo its alication in engineeing fluid dynamics. Muthucumaaswamy and Ganesan 1 oied out that chemically eacting flows ae classified as heteogeneous o homogeneous deending on whethe they occu at an ieface o as a single hase volume eaction. Singh and Singh have studied the tansie MHD fee convection in a otating system but did not include the effect of chemical eaction. Ibahim, Hassanien and Bak 3 eoted the esults of a study of unsteady magneto-hydodynamic mico-ola fluid flow and heat tansfe ove a vetical oous late though a oous medium in the esence of themal and mass diffusion with a consta heat souce. Heat tansfe to unsteady magneto-hydodynamic flow ast an infinite moving vetical late with vaiable suction was the focus of an ealie study eoted by Ogulu and Pakash 4 whee it was shown that inceasing the late velocity has the effect of inceasing the flow velocity with this incease been moe onounced fo highe values of the fee convection. Ogulu and Motsa 5 eoted the study on adiative heat tansfe to magneto-hydodynamic couette flow with vaiable wall temeatue. Ratis and Pedikis 6 have studied the viscous flow ove a non-linealy stetching sheet in the esence of a chemical eaction. El-Amin 7 had studied magneto-hydodynamic fee convection and mass tansfe flow in mico-ola fluid with consta suction and the ese study is the extension of El-Amin 7. Fomulation of Poblem Let us conside a steady, two-dimensional (lane), fee convection with mass tansfe flow of a mico-ola fluid, occuying a semi-infinite egion of the sace bounded by an infinite vetical oous limiting sace. The x-axis is taken along in the uwad diection and the y-axis nomal to it. A magnetic field of unifom stength is alied tansvesely to the diection of the flow. The magnetic Reynolds numbe of the flow is taken to be sufficiely small enough, so that the induced magnetic field can be neglected. The axial heat conducting effect is neglected. The fluid is assumed to have consta oeties excet that the effect of the density vaiation with temeatue is consideed only in the body foce tem. Howeve, the effect of the density vaiation in othe tems of the momeum and enegy given in Eqs (3) and (4), esectively and the vaiation of the exansion coefficie with temeatue ae negligible. The hysical vaiables ae functions of y only and theefoe, the govening equations oosed hee ae:

2 68 INDIAN J PURE & APPL PHYS, VOL 46, OCTOBER 8 Coinuity equation: The aoiate bounday conditions ae: v = (1) u y u N m T T T T e n t = : =, =, = + ε ( ), y Momeum equation: u χ u χ N ρ ρ v = υ gβ T T T β gβc ( C C ) σ u ρ Angula momeum equation: v N γ N x u ρ j ρ j () = + Enegy equation: v µ χ 1 q = α + ρ ρ T T + u c c Mass tansfe equation: C C v = D m K C N (3) (4) (5) whee u and v ae the comones of the velocity in x and y diections, esectively, g is the acceleation due to gavity, β and T β ae the coefficies of c volume exansion due to temeatue and conceation, esectively, T and C ae the temeatue and conceation esectively, ρ is fluid density, β the magnetic induction, σ the electic conductivity of the fluid, α the themal diffusivity, D the solutal diffusivity, c m the secific heat at consta essue while K is the chemical eaction ate consta. γ, j and χ eese the sin gadie viscosity, mico-inetia e unit mass and votex viscosity, esectively. q is the adiative flux vecto. n t C C ε C C e y v = + = : =, N, T T, C C (6) whee T and C ae the temeatue and conceation on the limiting suface, esectively. e see fom Eq. (1) that the suction velocity at the late suface should be a function of time so that we can assume it in the fom: ( ε ) 1 n t v V Ae = + (7) whee A is a ositive eal consta, V is a scale of suction velocity, ε is small and hence, ε A<<1, n is consta and t is time. By using the Rossland aoximation fo the adiative flux in the enegy equation, Ogulu and Motsa 5, we have: * 4 4σ T q = * 3K (8) * * with σ is Stefan-Boltzmann consta and K is the mean absotion coefficie. e make the assumtion that the temeatue diffeence within the flow is 4 sufficiely small such that T can be exessed as a linea function of temeatue. e can, theefoe, 4 exand T in a Taylo seies about the fee steam temeatuet, neglecting highe ode tems, to obtain: T 4T T 3T (9) Combining Eqs (4), (8) and (9), we have: * 3 µ + χ 16σ T = α + * ρ 3 ρ T T u T v (1) c K c e ioduce the following non-dimensional vaiables and aametes ν y NV T T y =, u = uv, N =, θ =, V T T ν

3 PRAKASH et al.: MHD FREE CONVECTION AND MASS TRANSFER FLO 681 C C q V K V φ = = = C C, q, K, ν ν ν t nv gβ ν t =, n =, G = T T, 3 V ν V c 3 gβ ν χ µ Gc = ( C C ), a =, ν =, V µ ρ ν γ ν ν,, c jv jµ Dm α b = e = S = P = (11) On the stength of Eqs (11), ou govening equations now become: u u N + a + + Ae + a ( 1 ) ( 1 ε ) + θ + ϕ = (1) M u G Gc N N u e + ( 1 + ε Ae ) a. b + N = θ ( 1+ Ra ) + P ( 1+ ε Ae ) φ u + EcP ( 1+ a) = ( ε ) φ θ + S 1 c + Ae ScK φ = (13) (14) (15) It is wothy to note hee that by setting A and K equal to zeo, ou oblem educes to that eoted in El-Amin 7. The tansfomed bounday conditions ae now: assigning numeical values to these aametes. To be ealistic, we take the Pandtl numbe, P =. 71 which, coesonds to ai at o C and 1 atmosheic essue, the Schmidt numbe, Sc is chosen fo the secies at low conceation in ai at o C and 1 atmosheic essue (see Table 1), The Gashof numbe G and the Ecket numbe Ec take ositive values coesonding to cooling of the late by fee convection cues. The othe values ae chosen at andom. Unless othewise stated, we have chosen a =.5, b =.1, t = 1, M =, ε =. 1 Numeical esults ae dislayed gahically in Figues 1-1. e discuss only the effects of the suction aamete A, and chemical eaction, k as the othe aamete effects ae discussed in El-Amin 7. In Figs 1 and, we deict the effects of suction aamete A and chemical eaction ate aamete k on the flow velocity. In geneal, we see that the velocity ises Table 1 Values of Schmidt numbe at o C and 1 atmosheic essue Secies Schmidt numbe, Sc Ethyl benzene.1 Benzene 1.76 Ethe 1.66 Ethyl alcohol 1.3 Methanol.97 CO.94 NH 3.78 O.75 ate vaou.6 H. du u =, N = m, 1 e, dy θ = + ε φ = 1+ εe at y = u, N, θ, φ as y (16) 3 Results and Discussion To give a good insight of the hysical oblem addessed in this study, we discuss the effect of the aametes of the oblem on the flow velocity, temeatue conceation and skin-fiction by Fig. 1 Effect of suction aamete A on the velocity distibution

4 68 INDIAN J PURE & APPL PHYS, VOL 46, OCTOBER 8 Fig. Effect of chemical eaction on the velocity distibution aidly fom zeo at the late, attains a maximum befoe deceasing steadily fa away fom the late. The velocity inceases fo ositive values of A (suction) and deceases fo negative values (blowing/injection) due to the conceation buoyancy effect, while we obseve that the velocity deceases fo destuctive chemical eaction (k > ) and inceases fo geneative chemical eaction (k < ) in ageeme with Muthucumaaswamy and Ganesan 1. Fom Figs 3 and 4, we obseve that as the suction at the late inceases thee is a coesonding incease in the late temeatue, but incease in the eaction ate aamete has no effect on the temeatue distibution. In Figs 5 and 6, we deict the effect of the chemical eaction and suction on the conceation. e obseve a decease in the conceation fo destuctive chemical eaction k > and an incease in the conceation fo geneative chemical eaction k < Suction has the same effect on the conceation as it has on the temeatue, that is, as the suction aamete A inceases thee is an incease in the conceation at the late. hen we comae Fig. 3 with Fig. 6, we obseve that the suction effect is moe onounced on the conceation than it is on the temeatue. Figues 7 and 8 deict the effect of suction and chemical eaction on the angula velocity. e obseve fom Fig. 7 that incease in the suction aamete A leads to an incease in the angula velocity nea the late but a decease in the angula velocity fa away fom the late (y > 1.5). The angula velocity deceases fo destuctive chemical eaction but inceases fo geneative chemical eaction. Fig. 3 Effect of suction on the temeatue distibution Fig. 4 Effect of chemical eaction aamete on the temeatue distibution Fig. 5 Effect of chemical eaction on the conceation distibution

5 PRAKASH et al.: MHD FREE CONVECTION AND MASS TRANSFER FLO 683 Fig. 6 Effect of suction aamete on the conceation ofles Fig. 9 Shea stess vesus suction aamete, A Fig. 7 Effect of suction on the angula velocity distibution Fig. 1 Rate of mass tansfe vesus suction aamete Figues 9 and 1 show lots of the shea stess and ate of secies tansfe, esectively. Fom Fig. 9, we obseve that the shea stess inceases as the suction aamete inceases, but deceases as the Schmidt numbe inceases; wheeas the ate of secies tansfe at the late deceases as the suction aamete inceases and also deceases as the chemical eaction ate aamete inceases. Fig. 8 Effect of chemical eaction on the angula velocity distibution 4 Conclusions Fom ou esults we conclude that (i) the velocity and the conceation fields decease as the chemical eaction ate aamete inceases; (ii) the chemical eaction ate does not affect the temeatue distibution; (iii) as the suction aamete A, inceases both the temeatue field and the velocity field incease; (iv) the shea stess inceases as the suction

6 684 INDIAN J PURE & APPL PHYS, VOL 46, OCTOBER 8 aamete is inceased but deceases as the Schmidt numbe is inceased and (v) the ate of mass tansfe deceases as the chemical eaction ate consta and the suction aamete ae inceased. Refeences 1 Muthucumaaswamy R & Ganesan P, J Engng Phys and Themo-hys, 75(1) () 113. Singh A K & Singh J N, Astohys and Sace Sci, 16 (1989) Ibahim F S, Hassanien I A & Bak A A, Canadian J. Phys, 8 (4) Ogulu A & Pakash J, Physica Scita, 74 (6) 3. 5 Ogulu A & Motsa S, Physica Scita, 71(1-4) (5) Ratis A & Pedikis C, I J Non-linea Mech, 41 (6) El-Amin M F, J Mag & Mag Mate, 34 (1) 567.