Effect of Radial Magnetic Field on Free Convective Flow over Ramped Velocity Moving Vertical Cylinder with Ramped Type Temperature and Concentration

Transcription

1 Joural of Applied Fluid Mechaics, Vol. 9, No. 6, pp , Available olie at ISSN , EISSN Effect of Radial Magetic Field o Free Covective Flow over Ramped Velocity Movig Vertical Cylider with Ramped Type Temperature ad Cocetratio V. Vaita ad A. Kumar Departmet of Mathemmatics, Cetral Uiversity of Rajastha, Ajmer, Idia Correspodig Author aaadbhu@gmail.com (Received December 30, 2015; accepted April 15, 2016) ABSTRACT A umerical study has bee performed to aalyze the effect of radial magetic field o free covective flow of a electrically coductig ad viscous icompressible fluid over the ramped movig vertical cylider with ramped type temperature ad cocetratio cosidered at the surface of vertical cylider. The goverig partial differetial equatios which describe the flow formatio have bee solved umerically by usig implicit fiite differece method of Crak-Nicolso type. The simulatio results of the cosidered model have bee show graphically. Oe of the iterestig result of our aalysis is that the local as well as average ski-frictio, Nusselt umber ad Sherwood umber have icreasig tedecy i time iterval (0,1), thereafter these quatities decrease. We have also compared the case of ramped type boudary coditios with that of costat boudary coditios with help of table. The advatage of takig ramped type boudary coditios is that iitial heat trasfer rate ad mass trasfer rate are miimum i this case. Keywords: Ramped velocity; Ramped temperature; Ramped cocetratio; Radial magetic field; Magetic parameter; Vertical cylider. NOMENCLATURE B 0 radial magetic field C species cocetratio C dimesioless species cocetratio D mass diffusio coefficiet g acceleratio due to gravity Gr Thermal Grashof umber Gc Mass Grashof umber M Magetic parameter Nu Local Nusselt umber Nu av Average Nusselt umber Pr Pradtl umber 0r characteristic radius r radial coordiate R dimesioless radial coordiate Sc Schmidt umber Sh Sherwood umber Sh av average Sherwood umber T temperature T dimesioless temperature t time t dimesioless time t characteristic time 0 u velocity compoet i x-directio u 0 characteristic velocity v velocity compoet i r-directio U dimesioless velocity compoet i x- directio V dimesioless velocity compoet i r- directio x axial coordiate X dimesioless axial coordiate α thermal diffusivity β T volumetric coefficiet of thermal expasio β C volumetric coefficiet of solutal expasio ν ρ τ τ av coductivity kiematic viscosity desity local ski-frictio average ski-frictio subscripts w at the wall at free stream

2 1. INTRODUCTION Free covectio flow ivolvig heat ad mass trasfer simultaeously, have show wide appearace i practical as well as i idustrial situatios like evaporatio, codesatio, formatio ad dispersio of fog, ocea circulatios, thermal isulatio, ehaced oil recovery etc. I idustries, may trasport processes are observed i which simultaeous heat ad mass trasfer occurs as a result of combied buoyacy effects of temperature differece ad cocetratio differece. Sice, cyliders have bee used iuclear waste disposal, udergroud eergy extortio, coolig of uclear reactors ad catalytic bed reactors, therefore the covective heat ad mass trasfer about cylidrical bodies has gaied the attetio of may researchers. Sparrow ad Gregg (1956) were the first, who studied the heat trasfer from vertical cyliders. Yag (1960) performed a aalysis of the usteady lamiar boudary-layer equatios for free covectio o vertical plates ad cyliders ad he derived ecessary coditios for feasibility of similarity solutios. The combied heat ad mass trasfer i atural covectio alog vertical cylider for both coditios of uiform wall temperature or cocetratio ad uiform heat or mass flux was ivestigated by Che ad Yuh (1980). Lie et al. (1985) have cotemplated the isothermal ad costat heat flux cases for the free covective flow past a impulsively movig ifiite vertical circular cylider. Gorla (1989) preseted a umerical solutio of combied forced ad free covectio i the boudary layer flow of a micropolar fluid o a cotiuous movig vertical cylider. He oticed that the wall shear stress ad surface heat trasfer rate icrease with icreasig buoyacy force ad icreasig trasverse curvature of the surface. Velusamy ad Grag (1992) gave a umerical solutio for the trasiet atural covectio over a heat geeratig vertical cylider. Gaesa ad Rai (1998) cosidered the trasiet atural covectio alog vertical cylider with heat ad mass trasfer ad they cocluded that with icrease i Schmidt umber steady state reaches later while buoyacy ratio parameter N put opposite effect. Takhar et al. (2000) aalyzed the combied heat ad mass trasfer alog a vertical movig cylider with a free stream ad deduced that the Pradtl umber affects the surface heat trasfer while Schmidt umber affects the mass trasfer. Further, Gaesa ad Logaatha (2001, 2002) studied differet problems of atural covective flow past the movig vertical cylider. Abdallah ad Zeghmati (2011) preseted the aalysis of opposig buoyacies oatural covectio heat ad mass trasfer i the boudary layer alog a vertical cylider. They deduced that whe Sc < Pr, the cocetratio layer is much thicker tha that of thermal layer ad for Pr > Sc the result is exactly opposite. Fig. 1. Velocity profiles at a fixed cross-sectio X = 0.5 for differet values of M, Gr, Gc ad Sc. 2856

3 Fig. 2. Temperature Profiles at a fixed crosssectio X = 0.5 for differet values of M, Gr, Gc ad Sc. Fig. 3. Cocetratio Profiles at a fixed crosssectio X = 0.5 for differet values of M, Gr, Gc ad Sc. 2857

4 (2010) discussed the dissipatio effects o MHD oliear flow ad heat trasfer past a stretchig porous surface uder a trasverse magetic field. They cocluded that with icrease i Eckert umber thickess of Thermomagetic layer icreases. Also, Magetic parameter ad suctio parameter put retardig effect o the ski frictio coefficiet at the wall. Fig. 4. Local Ski-frictio Profiles for differet values of time parameter. The problem of free covectio uder the ifluece of a magetic field has proved to be sigificat due to its applicatio i geophysics, astrophysics ad petroleum idustry. I techological processes MHD covectio flow problems are importat i the field of aeroautics, stellar ad plaetary magetosphere, chemical egieerig ad electroics. Sastry ad Bhadram (1987) have aalysed the combied free ad forced covective flow ad heat trasfer i vertical aulus by takig ito accout radial magetic field. The effect of MHD free covectio ad mass trasfer o the flow past a oscillatig ifiite coaxial vertical circular cylider was examied by Raptis ad Agarwal (1991). Gaesa ad Rai (2000) studied flow behavior o the MHD flow past a vertical cylider with heat ad mass trasfer ad discussed the usteady effects of material parameters o the velocity, temperature ad cocetratio. Postelicu (2004) applied DarcyBoussiesq model to aalyze the simultaeous heat ad mass trasfer by atural covectio from a vertical flat plate embedded i electrically coductig fluid saturated porous medium with Soret ad Dufour effect ad he showed effect of goverig parameters o the heat ad mass trasfer. Reddy ad Reddy (2009) ivestigated the study of radiatio ad mass trasfer effects o usteady MHD free covectio flow of a icompressible viscous fluid past a movig vertical cylider. They deducted the behavior of the velocity, temperature, cocetratio, ski-frictio, Nusselt umber ad Sherwood umber with variatios i the goverig thermophysical ad hydro-dyamical parameters. Devi ad Gaga The trasiet free covective MHD flow past a ifiite cylider was studied by Deka ad Paul (2013) ad they foud that the trasiet velocity icreases with Grashof umber but decreases with magetic field parameter. Kumar ad Sigh (2013) scrutiized the effect of iduced magetic field oatural covectio i vertical cocetric auli heated or cooled asymmetrically ad gave coclusio that there is rapid decrease i fluid velocity ad iduced magetic field with icrease i the value of Hartmaumber by cosiderig oe of the cyliders as coductig compared with the case whe both the cyliders are o-coductig. Rai ad Reddy (2013) studied the ifluece of soret ad dufour effects o trasiet double diffusive free covectio of couple-stress fluid past a vertical cylider. They deliberated that icreasig values of So or decreasig values of Du icrease the average values of ski-frictio ad heat trasfer rate. Reddy (2014) ivestigated the radiatio effects o MHD flow alog a vertical cylider embedded i a porous medium with variable surface temperature ad cocetratio. He deduced that with gai i stregth of magetic field parameter M, the trasiet velocity decreases while Gr ad Gc have opposite effect. Choudhury ad Dass (2014) obtaied expressios for trasiet velocity, temperature, species cocetratio ad o-dimesioal ski frictio at the plate by ivestigatig MHD free covective flow of viscoelastic fluid through porous media i presece of radiatio ad chemical reactio. Javaherdeh et al. (2015) studied the atural covectio heat ad mass trasfer i MHD fluid flow past a movig vertical plate with variable surface temperature ad cocetratio i a porous medium ad preseted the dimesioless velocity, temperature ad cocetratio profiles as well as gave umerical solutio for the local Nusselt umber ad Sherwood umber. His study emphasized o the sigificace of the relevat parameters. Rajesh et al. (2016 ) performed the fiite differece aalysis to study the effect of chemical reactio ad temperature oscillatio o usteady MHD free covective flow past a semi-ifiite vertical cylider. They derived graphs for velocity, local as well as average skifrictio, Nusselt umber ad Sherwood umber idicatig effects of differet physical parameters. Sice, free covective flow alog vertical cylider with heat ad mass trasfer has wide rage of applicatios i the field of geothermal power geeratio, emergecy coolig of a uclear fuel elemet by forced circulatio i case of power failure, ocea circulatios due to heat curret ad differece i saliity drillig operatios etc. I glass ad polymer idustries, hot filamets, which are cosidered as a vertical cylider, are cooled as they pass through the surroudig eviromet. 2858

5 started boudary coditio for the temperature, cocetratio ad motio of cylider are take as ramped like fuctio. The goverig o-liear partial differetial equatios have bee solved umerically by usig the implicit fiite differece method of Crak-Nicolso type ad the results obtaied by this study are preseted graphically. 2. PROBLEM FORMULATION Fig. 5. Local Nusselt umber Profiles for differet values of time parameter. Iature, motio starts with costat velocity i.e. u = 1, that meas whe time is icreased from zero, the flow gais full velocity which is ot possible. Therefore, some researchers start motio by takig u = t which shows that velocity icreases with icrease i time. But i practical situatios, after a log time velocity does ot icrease with time because it will result i a ustable system. Therefore, we have to deliberate a way i betwee these two situatios which has bee iveted by takig ramped structures. A sigificat cotributio i the study of ramped velocity was give by Kumar ad Sigh (2010). The same cocept is applied for temperature. Whe a fluid starts heatig, temperature gradually icreases but after boilig poit temperature becomes costat. This pheomeo is commoly see i coolig systems like air-coditioer, refrigerators. Kumar ad Sigh (2011) ivestigated trasiet MHD atural covectio past a vertical coe havig ramped temperature o the curved surface. They preseted umerical results for the velocity, temperature, ski-frictio ad Nusselt umber with help of graphs. Recetly Das et al. (2014) ad Seth et al. (2016) have doe sigificat studies by cosiderig ramped like temperature profiles. Similar cocept is adopted for species cocetratio. I preset study, we have discussed the aalysis of the effect of radial magetic field o the free covective flow of a electrically coductig ad viscous icompressible fluid past a vertical movig cylider with heat ad mass trasfer, where the Cosider the trasiet lamiar free covective flow of a electrically coductig ad viscous icompressible fluid over a movig vertical cylider of radius r 0 i presece of foreig species. The x-axis is take alog the axes of vertical cylider ad r-axis is choseormal to it. A radial magetic field of the form Br 00 / r, which is assumed to be applied trasversely to the vertical cylider ad fixed relative to the fluid. The magetic Reyolds umber of the flow is take to be small eough so that the iduced magetic field ca be eglected. Further, the cylider is assumed to be electrically o-coductig. At the begiig, for t 0, the temperature ad foreig species cocetratio at the cylider are assumed to be T ad C respectively. It is supposed that whe time t > 0, the cylider temperature ad species cocetratio at the cylider are istataeously raised or lowered to T ( Tw T ) t / t0 ad C ( Cw C ) t / t0 by ijectio/sublimatio respectively ad the cylider is assumed to be moved i the upward directio with velocity ut 0 / t 0 up to time t t 0. Further, for t t 0, these are maitaied at costat temperature, species cocetratio ad the velocity of the cylider. For small cocetratio level, the Soret-Dufour effects ca be eglected i the eergy equatio. Uder these assumptios, the physical variables are fuctios of (x,r,t ) oly. The uder usual Boussiesq approximatio, the goverig equatios are derived as follows: ( ru) ( rv) 0, x r u u u u v gβ T ( T T) t x x 2 2 v u B β ( ) ( ) 0r0u g C CC r r r r 2 r T T T α T u v r t x r r r r (1) (2) (3) C C C D C u v r (4) t x r r r r The iitial ad boudary coditios for the cosidered problem are as follows: t0 : u 0, v0, TT, C C for all x ad r u 0, v 0, T T, C C, at x

6 t0 : u u0 f( t), TT ( Tw T ) f( t), C C ( Cw C ) f( t ), at r r0 u 0, TT, C C as r where t/ t0 t t0 f( t) 1 t t0 (5) The o-dimesioal variables used are as follows: r t TT C C R, t, T, C r0 t0 Tw T Cw C 2 (6) u vt, 0 x r U V X, t 0 0 u0 r0 u0t0 v By usig o-dimesioal quatities the goverig equatios are reduced to followig form: ( RU ) ( RV ) 0, X R U U U 1 U U V R t X R R R R M GrT GcC u 2 R (8) T T T 1 1 T U V R t X R pr RR R (7) (9) C C C 1 1 C U V R (10) t X R Sc RR R The iitial ad boudary coditios iodimesioal form are obtaied as follows: t0: U 0, V 0, T 0, C0 for all X ad R U 0, V 0, T 0, C 0, at x 0 t 0 : U f(), t T f(), t C f(), t at R1 U 0, T 0, C 0 as R where (11) t t 1 f() t 1 t 1 I the o-dimesioal process, we have obtaied the followig o-dimesioal parameters 2 2 gβ Tr0 ( Tw T ) gβ Cr0 ( Cw C ) v Gr, Gc, Sc, vu0 vu0 D 2 2 v B, 0r Pr M 0. α ρv 3. NUMERICAL SOLUTION PROCEDURE The trasport Eqs. (7)-(10) are highly o-liear i ature ad their solutios subject to the boudary coditio (11) have bee obtaied umerically. The solutios of the trasformed equatios have bee solved by implicit fiite differece method of Crak- Nicolso type. The fiite differece equatios correspodig to Eqs. (7)-(10) are as follows: 1 1 Ui, j Ui 1, jui, jui 1, j 2X 1 1 Vi, j Vi 1, j Vi, j Vi 1, j 2R 1 Vi, j 0 [1 ( j1) R] (12) 1 Ui, j Ui, j Ui, j 1 Ui, j Ui1, jui, jui1, j t 2X Vi, j 1 1 Ui, j1 Ui, j1 Ui, j1 Ui, j1 4R Ui, j1 2Ui, j Ui. j1 Ui, j1 2Ui, ju i, j1 2 2( R) 1 1 Ui, j 1 Ui, j 1Ui, j1 U i, j1 4[1 ( j1) R] R 1 1 Ti, j T i, j Ci, j C i, j Gr Gc Ui, j U i, j M 2 2[1 ( j1) R] 1 i, j i, j i, j 1 Ti, j Ti1, jti, jti1, j T T U t 2X i, j 1 1 Ti, j1 Ti, j1 Ti, j1 Ti, j1 V 4R T T T T T T 2( R) 1 1 Ti, j1 Ti, j1 Ti, j1t i, j 1 4[1 ( j1) R] R 1 1 i, j1 2 i, j 1 i, j1 i, j1 2 i, j i, j1 2 1 ij, ij, ij, 1 Cij, Ci1, jcij, Ci1, j C C U t 2X ij, 1 1 Cij, 1 Cij, 1 Cij, 1 Cij, 1 V 4R (13) (14) 2860

7 1 1 1 Ci, j1 2Ci, j Ci, j1 Ci, j1 2Ci, jc i, j1 2 2( R) 1 1 Ci, j1 Ci, j1 Ci, j1 C i, j1 4[1 ( j 1) R] R (15) Fially, the Eqs. (12)-(15) are coverted ito the liear algebraic system ad expressed tridiagoally the solved by Thomas algorithm. I the computatioal procedure, the physical domai is coverted ito computatioal domai as a rectagle frame of lies idicatig Xmi 0, Xmax 1, Rmi 0 ad Rmax 20 where R max correspods to R which lies very far from the boudary layers. The mesh sizes i the X ad R directio are take as X = ad R = 0.05 respectively with time step t = The steady state umerical solutios have bee obtaied for the velocity, cocetratio ad temperature fields whe the followig covergece criterio is satisfied 1 ij, ij, 5 10, 1 (16) ij, ij, where ij, stads for either the temperature, velocity or cocetratio field. The superscripts deote the values of the depedet variables after the th ad (+1)th iteratios of time t(= t) respectively, whereas the subscripts i ad j idicate grid locatio i XR plae. Where X = i X ad R = j R with X, R ad t the mesh size i X, R ad t directios respectively for the Eqs. (12)-(16). Other importat result of practical importace is the ski-frictio, Nusselt umber ad Sherwood umber. By usig the computed values of the velocity field, the local skifrictio ad the average ski-frictio iodimesioal form are obtaied as follows: U τ=, (17) R R 1 1 U τ av = dx. 0 R (18) I egieerig applicatios, oe of the importat characteristic of the flow is the rate of heat ad mass trasfer over the coe surface. This is estimated by the values of the Nusselt umber Nu ad Sherwood umber Sh. T Nu X, R R1 C Sh X, R R1 (19) (20) 1 T Nuav = dx. 0 R 1 C Shav = dx. 0 R (21) (22) 4. RESULT AND DISCUSSION Fig. 6. Local Sherwood umber Profiles for differet values of time parameter. Durig ay oe time step, the computed values of the previous time step have bee used for the coefficiets U, T ad C appearig i Eqs. (12)- (15). At the ed of each time step, first we have computed the temperature field ad the computed the cocetratio field ad fially the evaluated values are employed to obtai the velocity compoets i X ad R directios respectively.the usteady values of the compoets of velocity, temperature field ad cocetratio field for a desired time have bee obtaied by takig required umber of iteratios. With a view to see ito the physical isight of the problem, umerical computatios are carried out for physical parameters ivolvig magetic parameter M, thermal Grashof umber Gr, mass Grashof umber Gc, Schmidt umber Sc ad time parameter t ad the umerical solutios are displayed with help of graphs. Fig. 1 represets variatio i velocity profiles for differet values of the Magetic Parameter (Fig. 1a), thermal Grashof umber Gr (Fig. 1b), mass Grashof umber Gc (Fig. 1c) ad Schmidt umber Sc (Fig. 1d) respectively at a fixed cross-sectio X =0.5. Fig. 1a depicts that with icrease i value of M, the velocity profiles decrease. This is due to the fact that whe value of M is icreased, a resistive type of force is produced kow as Loretz force. This force opposes the motio of the fluid as a cosequece velocity 2861

8 decreases. Fig. 1b delieates that with icrease i value of Gr velocity icreases. This happes due to the reaso that icreasig value of Gr will result i icrease i Buoyacy force, which accelerates the fluid motio. Therefore, velocity profiles icrease. It is observed from Fig. 1c that icrease i value of Gc will icrease the velocity profiles. Whe the value of Sc is icreased, velocity profiles get reduced which is show i Fig. 1d. of M, cocetratio profiles icreases. Further, steady state is achieved earlier with icrease i M. It is observed from Fig. 3b that cocetratio profiles get reduced as the value of Gr rises up. The species cocetratio is observed to be high ear the surface of the cylider, decreases cotiuously with icrease i the value of Gr ad becomes miimum at the ed of the boudary layer. From Fig. 3c, we caotice that ifluece of Gc is qualitatively similar to those of Gr. As the value of Schimdt umber icreases, the mass trasfer icreases ad hece the cocetratio profiles decreases which is show i Fig. 3d. Also, the steady-state is achieved earlier with icrease i all parameters. I Fig 4 (a)-(b) we have show the variatio i local ski-frictio profiles. Fig. 4a reveals that with icrease i value of M ad Sc, local ski-frictio icreases. Fig. 4b shows the ifluece of Gr ad Gc o local ski-frictio profiles. From the Fig. 4b it ca be oticed that local ski-frictio decreases with icrease i Gr ad Gc. Fig 5(a)-(b) presets variatio i local Nusselt umber. It is observed from the Fig 5a that the local Nusselt umber de-creases with a icrease i both M ad Sc. From Fig. 5b, we deduce that effect of Gr ad Gc is to ehace the local Nusselt umber ad hece the heat trasfer rate icreases. Fig 6a represets the ifluece of M ad Sc o Sherwood umber. Fig. 7. Average profiles for differet values of M, Gr, Gc ad Sc w.r.t. time. The temperature profiles for differet M, Gr, Gc ad Sc ad at a fixed cross-sectio X = 0.5 are plotted i Fig. 2. It is oticed from Fig. 2a that temperature profiles icrease with icrease i magetic parameter M. Further, time take to reach the steady-state decrease. There is suppressio i temperature profiles with icreasig values of Gr ad Gc which is clearly visible i Fig. 2b ad 2c. Also, steady-state time decreases with icrease i Gr ad Gc. Fig. 2d describes that with icrease i value of Sc, temperature profiles icrease. Fig. 3 illustrates cocetratio profiles for differet values of M, Gr, Gc ad Sc. Fig. 3a exhibits that with icreasig value Table 1 Compariso of average Nusselt Number ad average Sherwood Number for ramped case ad costat case t Ramped Case Costat Case Nu_{av} Sh_{av} Nu_{av} Sh_{av} Fig 6a depicts that with icrease i value of Sc Sherwood umber icreases while effect of M is exactly opposite. It is oticed from Fig 6b that 2862

9 Sherwood umber icreases with icrease i both Gr ad Gc. Here, the effect is clearly visible for higher values of time. Fig. 7 (a)-(c), depict graphs of average values of ski-frictio, Nusselt umber ad Sherwood umber respectively. Fig. 7a reveals that average value of ski-frictio icreases with icrease i M ad Sc but decreases with icrease i Gr ad Gc. Fig. 7b delieates that average heat trasfer rate get reduced with icrease i Sc ad M but ehaced with icrease i Gr ad Gc. Fig.7c depicts that average values of Sherwood umber decrease with icrease i magetic parameter M but all other parameters put adverse effect. Oe iterestig result of our cosideratio is that all average values of skifrictio, Nusselt umber ad Sherwood umber have icreasig tred i time iterval (0,1) ad thereafter they decreases ad fially merge with steady state. The reaso behid the icreasig tred of these values i iterval (0,1) is the ramped ature of boudary coditios. I order to check the ifluece of ramped type profiles, we have compared the average values of Nusselt umber ad Sherwood umber correspodig to ramped type boudary coditios for velocity, temperature ad cocetratio with the case of costat boudary coditio for velocity, temperature ad cocetratio, which is show i Table 1. From Table 1, it is clear that i case of ramped type boudary coditios, the values of average Nusselt umber ad average Sherwood umber are low as compared to case of costat boudary coditios. But the steady state values are almost same, achieved at the same time i both the cases. This shows the advatage of takig ramped type boudary coditios, as the heat-trasfer rate ca be lowered which helps i keepig the system stable. This type of pheomeo ca be used i coolig the systems ad equipmet. 5. CONCLUSIONS The trasiet free covective flow past a movig vertical cylider uder the ifluece of radial magetic field has bee studied. The dimesioless goverig equatios have bee solved by usig a implicit fiite-differece method of Crak-Nicolso type ad the results have bee preseted with help of graphs. By our study, we cocluded the followig results: (i) The effect of M ad Sc o velocity profiles is to reduce it as a cosequece mometum boudary layer thickess decreases while effect of Gr ad Gc is to ehace it accordigly mometum boudary layer thickess icreases. (ii) The thermal boudary layer expaded with icrease i M ad Sc while shrik with icrease i Gr ad Gc. (iii) Cocetratio profiles rise up with rise i M while fall dow with icrease i all other parameters. (iv) Local ski-frictio icrease with icrease i M ad Sc while decrease with icrease i Gr ad Gc. (v) The effect of M ad Sc is to reduce the heat trasfer rate while effect of Gr ad Gc is to ehace it. (vi) With icrease i value of M Sherwood umber decreases while all other parameters have opposite effect. (vii) Average value of ski-frictio declie with icrease i magetic parameter ad Schimdt umber but thermal Grashof umber ad mass Grashof have opposite effect. (viii) Average value of Nusselt umber reduces with icrease i M ad Sc ad while ehaces with icrease i Gr ad Gc. (ix) Average value of Sherwood umber decreases with icrease i magetic parameter M but have exactly opposite behavior with icrease i Gr, Gc ad Sc. (x) With icrease i time parameter local ad average ski-frictio, Nusselt umber ad Sherwood umber icrease i time iterval (0,1). REFERENCES Abdallah, M. S. ad B. Zeghmati (2011). Natural covectio heat ad mass trasfer i the boudary layer alog a vertical cylider with opposig buoyacies. J. Appl. Fluid Mech. 4(4), Che, T. S. ad C. F. Yuh (1980). Combied heat ad mass trasfer iatural covectio alog a vertical cylider. It. J. Heat Mass Trasf. 23, Choudhury, R. ad S. K. Das (2014). Viscoelastic MHD free covective flow through porous media i presece of radiatio ad chemical reactio with heat ad mass trasfer. J. Appl. Fluid Mech. 7(4), Das, M., B. K. Mahatha, R. Nadkeolyar, B. K. Madal ad K. Saurabh (2014). Usteady hydromagetic flow of a heat absorbig dusty fluid past a permeable vertical plate with ramped temperature. J. Appl. Fluid Mech. 7(3), Deka, R. ad A. Paul (2013). Trasiet free covective MHD flow past a ifiite vertical cylider. Theo. Appl. Mech. 40, Devi, S. A. ad B. Gaga (2010). Dissipatio effects o MHD oliear flow ad heat trasfer past a porous surface with prescribed heat flux. J. Appl. Fluid Mech. 3(1), 1-6. Gaesa, P. ad H. P. Rai (1998). Trasiet atural covectio alog vertical cylider with Heat ad Mass trasfer. Heat Mass Trasf. 33, Gaesa, P. ad P. Logaatha (2001). Effects of mass trasfer ad flow past a movig vertical cylider with costat heat flux. Acta Mach. 2863

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