International Journal of Scientific & Engineering Research, Volume 5, Issue 3, March ISSN

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1 International Jornal of Scientific & Engineering Research, Volme 5, Isse 3, March-4 83 ISSN Doble Dispersion effects on free convection along a vertical Wavy Srface in Poros Media with Variable Properties R. Bhvanavijaya, B. Mallikarjna Abstract In the present paper, we analyzed doble diffsive free convection past a vertical wavy srface embedded in a flid satrated poros medim with variable properties. The Darcy law is assmed to describe the homogenos flid satrated poros medim. The temperatre dependent variable properties (variable viscosity and variable thermal condctivity) are considered. The flid flow, momentm, energy and soltal governing eqations are transformed into bondary layer non-dimensional nonlinear ordinary differential eqations with specified transformation and then solved with nmerical techniqe. The reslts are reported for varios physical parameters; variable viscosity, variable thermal condctivity, thermal dispersion and soltal dispersion and amplitde of the wavy srface on hydrodynamic velocity, temperatre and concentration distribtions as well as rate of heat (Nsselt nmber) and mass (Sherwood nmber) transfers. The nmerical reslts obtained in the present method compared with previosly pblished reslts and fond to be in good agreement. Inde Terms Vertical Wavy Srface, Doble Dispersion Effects, Variable Properties, Darcy Poros Media, Free convection. INTRODUCTION n recent years, the stdy of natral convective flow, heat and concentration in a poros medim. Iand mass transfer in poros media has received considerable interest in the literatre. The interest for sch Recently, Shalini and Rathish Kmar [5] investigated the stdies is motivated by grain storage inslations, nclear inflence of variable heat fl on natral convection along a waste disposal, oil etraction, grond water polltion, resin corrgated wall in poros media. Mohamed et.al [6] stdied transfer modeling, dispersion of chemical contaminants combined radiation and free convection from a vertical wavy throgh water satrated soil, fibros inslations, packed srface embedded in poros media. Elgazery and Elazem [7] beds and geo thermal systems. Comprehensive reviewers of investigated the effects of variable properties on MHD the convection throgh Darcy poros media have been nsteady natral convection heat and mass transfer over a reported by Nield and Bejan [] and by Ingham and Pop []. vertical wavy srface. Rathish kmar and Krishan Mrthy [8] Darcy s law states that the volme averaged velocity is analyzed Soret and Dfor effects on doble diffsive free proportional to the pressre gradient. The present stdy deals convection from a corrgated vertical srface in a non-darcy with free convective flow on a vertical wavy srface poros medim. Neag [9] analyzed free convective heat and embedded in a satrated poros medim. Natral convection mass transfer indced by a constant heat and mass fles from wavy srfaces is a topic of fndamental importance in vertical wavy wall in a non-darcy doble stratified poros heat transfer devices, sch as flat-plate solar collectors and medim. Narayana et.al. [] stdied doble diffsive flat-plate condensers in refrigerators. Mainly, roghness convection and cross diffsion effects on a horizontal wavy elements distrbs the flow and alters the rate of heat and mass srface in a poros medim. Parveen and Alim [] transfer, this is type of irreglarities mostly occr in investigated Jole heating and MHD free convection flow manfactring. At first, Rees and Pop [3] investigated free along a vertical wavy srface with viscosity and thermal convection along a vertical wavy srface in a poros medim. condctivity dependent on temperatre. Cheng [4] stdied natral convection heat and mass transfer The hydrodynamic miing is called dispersion, which near a vertical wavy srface with constant wall temperatre is the secondary effect of a poros medim on the flid flow takes place in the reslt of miing and recirclation of local flid particles throgh tortos paths formed by the poros B. Mallikarjna, Department of Mathematics, JNT University, Anantapr, medim solid particles. There has been renewed interest in PIN mallikarjna.jnta@gmail.com stdying doble diffsive convection de to the effect of R. Bhvanavijaya, Department of Mathematics, JNT University, Anantapr, PIN-55. bhvanarachamalla@gmail.com thermal and soltal dispersions; these are additional energy (* Corresponding athor: B. Mallikarjna) and concentration mass transport process. In certain thermal and soltal dispersion applications sch as those involving oil 4

2 International Jornal of Scientific & Engineering Research, Volme 5, Isse 3, March-4 84 ISSN reservoir and geothermal engineering applications sch as the conservation of mass, momentm, energy and ceramic processing, sensible heat storage beds and petrolem concentrations are: recovery etc., In view of the aforesaid applications, many v + = y athors have analyzed the effects of thermal and soltal () dispersion on convective heat and mass transfer throgh µ T C = v ± ρg βt + βc poros media. Abbas et.al. [] stdied effects of thermal y K K y y (3) dispersion on free convection in a flid satrated poros medim. Kairi and Mrthy [3] considered the doble + v = T T y dispersion effects to stdy mied convection heat and mass y α + α y y (4) transfer in a non-newtonian flid satrated non-darcy poros medim. Pathak and Ghiaasiaan [4] investigated convective C C T T + v = heat transfer and thermal dispersion dring laminar plsating D + Dy (5) y y y flow in poros media. The effects of MHD and doble dispersion on free convection in a non-darcy poros medim The corresponding bondary conditions are satrated with power law flid are investigated by π Srinivasacharya et.al [5]. Ramreddy [6] has been stdied the =, v =, T = Tw, C = Cw, at y = σ ( ) = a sin l (6) effects of doble dispersion on convective flow over a cone., T T, C C as y In view of the above application, the athors are envisage to investigate free convection along a vertical wavy srface embedded in a flid satrated poros medim with variable properties and doble dispersion effects. The governing bondary eqations for flow mass, momentm, energy and concentration are transformed into nondimensional nonlinear ordinary differential eqations by sing appropriate transformation and then solved by sing nmerical method. The reslts are reported graphically for varios physical parameters for flow velocity, temperatre and concentration distribtions as well as Nsselt nmber and Sherwood nmber. The present reslts are compared with previosly eisting reslts and obtained a very good agreement. FORMULATION OF THE PROBLEM We consider the steady, two dimensional laminar, viscos incompressible flid over a vertical wavy plate embedded in a satrated poros medim. The configration of the model and coordinate system is shown in fig.. We assme that the wavy srface is given by π y = σ ( ) = asin l () Where a represents amplitde of the wavy srface and l represents the characteristics of wavy length. The plate is maintained with constant temperatre T w and concentration C w, which are higher than the ambient flid temperatre T and concentration C. The Darcy law can be sed to describe the poros medim. In addition, we consider thermal and soltal dispersion effects. In view of the above assmptions and invoking the bondary layer and Bossinesq approimations, the governing bondary layer eqations for 4 where, and v are velocity components in and y directions respectively. μ is the kinematic viscosity, K is the permeability of the poros medim, ρ is the density of the flid, β t is the thermal epansion coefficient, β c is the soltal epansion coefficient, g is the acceleration de to gravity, α, D and α y, D y are the effective thermal and soltal diffsivities respectively, have the contribtion of both moleclar diffsion and hydrodynamic dispersion, these can be described as (see Telles and Trevisan [7]) α = α + γdv, αy = α + γd (7) D = D + ζdv, Dy = D + ζd where α is the thermal condctivity, D is the moleclar diffsivities, γ is the coefficient of thermal dispersion and ζ is the soltal dispersion. The flid properties namely, viscosity and thermal condctivities are assmed to be vary as an inverse linear and linear fnction of the temperatre respectively and these can be written as (see [8-]) ( δ = + ( T T )) or bt ( T r ) µ µ µ = and α = αo ( + ET ( T )) where δ b =, and Tr = T. δ µ Both b and T r are constants and their vales depend on the reference state and the thermal property of the flid i.e. δ, αo is the thermal diffsivity at the wavy srface temperatre T w and E is a constant depending on the natre of the flid. In general, b> for liqids and b< for gases. It is worth mentioning here that E is positive for flids sch as air and E is negative for flids sch as lbrication oils. r, which is defined by

3 International Jornal of Scientific & Engineering Research, Volme 5, Isse 3, March-4 85 ISSN Tr T r = = (8) µ ν βc Cw = is the kinematic viscosity of the flid, N = Tw T δ ( Tw T ) ρ βt Tw α is the boyancy ratio, Le = o is the Lewis nmber, and is constant. The parameter was first introdced by Ling D and Dybbs [] It is worth mentioning here that for δ (i.e. gkβt( Tw T ) d µ = µ =constant) then r, the effect of viscosity is negligible. The variable thermal condctivity can be written in the non-dimensional form (see []) as α = α + β (9) o ( ) Rad = is the pore diameter dependent αν o ( C ) ( T ) Rayleigh nmber which describes the relative intensity of the boyancy force, sch that d is the pore diameter. The associated bondary conditions are given by where β=e(t w-t ) is the thermal condctivity parameter. The variation of β can be taken in the range. β for lbrication oils, β. for water and β 6for air. We define the stream fnctionψ, which is to be satisfied the continity eqation () sch that ψ* =, =, ϕ =, on y = asin( ), * ψ y, as y, Let s consider the following transformations (4) ψ ψ =, v = y To convert the governing bondary layer eqations in nondimensional form, we introdce the following dimensionless variables redces into the following bondary layer eqations: y a σ =, y =, a =, σ =, ψ ψ l l l l ( + a cos ( )) + ( + a cos ( )) () r ψ T T C C ψ*, = =, ϕ = / α Tw T Cw C ψ = N + r By sing eqs. (7) (), the eqs. (3) (5) redces to / ψ ψ = ( + a cos ( )) ( + β ) ψ * ψ * ψ * ψ * ϕ = Ra + N r y y y () r y y + β( + a cos ( )) + Ds ( + a cos ( )) + ( β ) ψ * ψ * = β y y y y γd ψ * ψ * ψ * ψ * + + l y y y y () ψ * ϕ ψ * ϕ ϕ ϕ Le = + y y y ζd ψ * φ ψ * φ ψ * φ ψ * φ + + l y y y y (3) ( ) gkβt Tw T l where Ra = is the modified - Rayleigh nmber, αν o y asin( ) /, = =, ψ * = Ra ψ. (5) / / Ra Invoking the eq (5) and letting Ra into eqs. () - (4) (6) ψ ψ (7) / ψ ϕ ψ ϕ ϕ Le = ( + a cos ( )) ψ φ ψ φ + Dc( + a cos ( )) + where Ds = γ. Rad is the thermal dispersion parameter Dc = ζ. Rad is the soltal dispersion parameter. To transform Eqs. (6) - (8) into a set of ordinary differential eqations, we introdce the following similarity transformations (8) 4 ˆ / =, ψ= f ( ˆ ), = ( ˆ) and ϕ= ϕ ( ˆ) + a cos ( ) (9)

4 International Jornal of Scientific & Engineering Research, Volme 5, Isse 3, March-4 86 ISSN Ths, we obtain concentration distribtions is presented in figs. () (4). It is noticed from fig. () that an increase in variable viscosity f + f = + Nϕ r r ( ) ( + a ( ) 3 cos 3 ( )) ( ) f + Ds ( f + f ) = ( + a cos ( )) β β () () Table-: Comparison of the rate of heat and mass transfer for a=, β=, τ= and at N=, and Le=. N Ra Sh Ra / N Ra / Sh Ra / / 3 3 ( + a cos ( )) ϕ + f ϕ + Dc ( f φ + φ f ) = Le ( + a cos ( )) where prime denotes differentiation with respect to ˆ. The associated bondary conditions are f =, =, and ϕ = at ˆ = f, and ϕ as ˆ () (3) Le N Cheng [] Cheng [] Present Present The engineering design qantities of physical interest inclde Nsselt nmber and Sherwood nmbers which are defined as / f () () Ra N = + Ds / ( a cos ( )) + ( + a cos ( )) (4) / f () ϕ () Ra and Sh = + Ds parameter / ( + a cos ( )) r reslted in depreciation in velocity distribtion ( + a cos ( )) near the plate p to reach certain vale and then increase the velocity profile ntil approaches a constant vale (zero) at oter bondary layer regime. From figs. (3) and (4) we conclde that 3 RESULTS AND DISCUSSION increasing variable viscosity parameter, clearly sbstantially The problem of free convection along a vertical wavy srface enhances the temperatre and concentration distribtions. embedded in flid satrated Darcy poros media sbject to the The variation of variable thermal condctivity (β) on variable viscosity, variable thermal condctivity and doble velocity, temperatre and concentration distribtions is dispersion effects has been investigated. A simple coordinate illstrated in figs. (5) (7). The velocity profile reslts for transformation is employed to redce the governing non-linear different vales of β are given by fig. (5), these reslts are bondary eqations into non-linear ordinary differential having similar behavior as shown in fig. (). From fig. (6) it is eqations and then employed Rnge-Ktta method with evident that temperatre profiles is more prononced with shooting techniqe. We restrict the physical parameter vales increasing vales of β. Conversely, a strong decrease in < 5, β 5, <Ds, <Dc, and with the fied concentration distribtion as shown in Fig. (7); occrs with vales N=, Le= and a=. In order to validate the present increasing vales of β. method the nmerical reslts obtained sing the Rnge Ktta The effect of thermal dispersion parameter (Ds) on the nondimensional velocity, temperatre and concentration is method with shooting method are compared with Cheng [] reslts. Table- shows the comparison reslts in the absence of depicted in figs. (8) (). From fig. (8) we conclde that the variable properties and doble dispersion effects (i.e. Ds= and reslts of velocity profile redce near the srface for larger Dc=) over vertical wavy srface with Cheng [] and the vales of thermal dispersion parameter and opposite reslts are reslts are fond to be in good agreement. observed as the radial distance moves far away from the srface We have fond the nmerical soltions for non dimensional with increase in thermal dispersion parameter. The presence of velocity, temperatre and concentration distribtions as well as thermal dispersion in the energy eqation gives thermal rate of heat and mass transfer coefficients as shown graphically condction more dominance. It is observed from fig. (9) that in Figs. () (4). The variation of variable viscosity parameter increasing thermal dispersion parameter tends to enhance the () on non-dimensional velocity, temperatre and temperatre distribtion. i.e. thermal dispersion enhances the transport of heat along radial direction to the plate. It is noticed 4

5 International Jornal of Scientific & Engineering Research, Volme 5, Isse 3, March-4 87 ISSN from fig. () that the soltal bondary layer thickness is redced as increase in thermal dispersion parameter. The set of figs. () (3) are plotted for the variation of nondimensional velocity, temperatre and concentration distribtions across the bondary layer for different vales of soltal dispersion parameter (Dc). From fig. () we noticed that an increase in Dc is seen to significantly enhance the momentm bondary thickness. It is observed from fig. () that temperatre profile redced with increase in Dc. It can be evident from this figre that as Dc increases thermal bondary layer thickness increases. It is observed from fig. (3) that increasing the soltal dispersion parameter (Dc), accelerates the concentration of the flid. Hence the concentration bondary layer thickness increases with an increase in soltal dispersion parameter (Dc). Figs. (4) (6) illstrate the velocity, temperatre and concentration distribtions for different vales of -location. It can be fond from fig. (4) that velocity profile is increased with increase in -location. Hence the hydrodynamic bondary layer thickness increases as increase in -location. We noticed from figs. (5) and (6) that the similar behavior of temperatre and concentration profile, in comparison with velocity distribtion as shown in fig. (4). It is an important to note that it qickly reaches similarity soltions not far away from the leading edge. The variation of rate of heat and mass transfer (Nsselt nmber and Sherwood nmber) with streamwise coordinate at the wall are shown in figs. (7) (8) for different vales of variable viscosity parameter (). It is noticed from these figres that both Nsselt nmber and Sherwood nmber decreases with increase in. Hence, it is clear that increase in r reslts an depreciation in the amplitde of the Nsselt nmber and Sherwood. Figs (9) () represent the variation of Nsselt nmber and Sherwood nmber for different vales of variable [4] C.Y. Cheng, Natral convection heat and mass thermal condctivity parameter (β). Figs. (9) () transfer near a vertical wavy srface with constant demonstrates that Nsselt nmber and Sherwood nmber redces considerably for larger vales of β. Figs. () () reveals that enhancement of thermal dispersion parameter reslts enhancement in the amplitde of the Nsselt nmber and Sherwood nmber. The variation of Nsselt nmber and Sherwood nmber for different vales of soltal dispersion parameter (Dc) is given in figs. (3) (4). Figs. (3) (4) ehibits the similar behavior of Nsselt nmber and Sherwood with streamwise coordinate, in comparison with what observed in figs. () (). Figs. (5) (6) illstrates the variation of Nsselt nmber and Sherwood nmber for different vales of the amplitde of the wavy srface. For a = the vertical wavy srface redces to vertical flat srface. It is noticed from figs. (5) and (6) that increasing the amplitde of the wavy srface, clearly sbstantially enhances the amplitde of the Nsselt nmber and Sherwood nmber with streawise coordinate. 3 CONCLUSSIONS The effect of thermal and soltal dispersion on free convection along a vertical wavy srface with temperatre dependent viscosity and thermal condctivity analyzed and the governing bondary non-dimensional eqations are solved by 4 employing Rnge-Ktta method with shooting techniqe. The main reslts in this stdy are as follows:. Increasing variable viscosity parameter leads to decrease velocity distribtion, Nsselt nmber and Sherwood nmber while temperatre and concentration distribtions are increases.. Increasing thermal dispersion parameter tends to increase velocity and temperatre distribtions as well as Nsselt nmber and Sherwood nmber while opposite reslts are noticed for concentration profile. 3. Velocity and concentration distribtions as well as Nsselt nmber and Sherwood nmber are increased with increase in soltal dispersion parameter while we noticed that opposite reslts are reported for temperatre profile. 4. Increase in the amplitde of the wavy srface reslts an enhancement in Nsselt nmber and Sherwood nmber. ACKNOWLEGMENTS: One of the athors Mr. B. Mallikarjna wishes to thank to the Department of Science and Technology, New Delhi, India for providing financial spport to enable condcting this research work nder Inspire Fellowship Program. REFERENCES [] D.A. Nield and A. Bejan, Convection in poros media, Newyork, Springer, 99. [] D.B. Ingham and I. Pop, Transfer phenomena in poros media, Oford; Pergamon; 998 [3] D.A.S. Rees and I. Pop, A note on free convection along a vertical wavy srface in a poros medim, Jornal of Heat Transfer, Vol. 6, pp , May 994. wall temperatre and concentration in a poros medim, International Commnication in Heat and Mass Transfer, Vol-7, No. 8, pp ,. [5] Shalini and B.V. Rathish Kmar. Inflence of variable heat fl on natral convection along a corrgated wall in poros media, Commnications in Nonlinear Science and Nmerical Simlation, Vol, pp , 7. [6] R.A. Mohamed, A. Mahdy, and F.M. Hady, Combined radiation and free convection from a vertical wavy srface embedded in poros media, International Jornal of Applied Mathematics and Mechanics, Vol 4, No-, pp , 8. [7] N.S. Elgazery and N.Y.A. Elazem, The effects of variable properties on MHD nsteady natral convection heat and mass transfer over a vertical wavy srface, Meccanica, Vol 44, pp , 9. [8] B.V. Rathish kmar and S.V.S.S.N.V.G. Krishan Mrthy, Soret and Dfor effects on doble diffsive free convection from a corrgated vertical srface in a non-darcy poros medim, Transport in Poros media, Vol 85, pp. 7 3,. [9] M. Neag, Free convective heat and mass transfer indced by a constant heat and mass fles vertical

6 International Jornal of Scientific & Engineering Research, Volme 5, Isse 3, March-4 88 ISSN wavy wall in a non-darcy doble stratified poros medim, International Jornal of Heat and Mass Transfer, Vol 54, pp. 3-38,. [] M. Narayana, P. Sibanda, S.S. Motsa and P.G. Siddheswar, On doble diffsive convection and cross diffsion effects on a horizontal wavy srface in a poros medim, Bondary vale Problems, Vol 88, No-, pp. -,. [] N. Parveen and M.A. Alim, Jole heating and MHD free convection flow along a vertical wavy srface with viscosity and thermal condctivity dependent on temperatre, Jornal of Naval Architectre and Marine Engineering, Vol, pp. 8-98, 3. [] I.A. Abbas, M.F. El-Amin, and A. Salama, Effects of thermal dispersion on free convection in a flid satrated poros medim, International Jornal of Heat and Flid Flow, Vol-3, pp. 9-36, 9. [3] R.R. Kairi, and P.V.S.N. Mrthy, The effects of doble dispersion on mied convection heat and mass transfer in a non-newtonian flid satrated non- Darcy poros medim, Jornal of Poros Media, Vol-3, No-8, pp ,. [4] M.G. Pathak and S.M. Ghiaasiaan, Convective heat transfer and thermal dispersion dring laminar plsating flow in poros media, International Jornal of Thermal Sciences, Vol-5, pp ,. [5] D. Srinivasacharya, J. Pranitha, and Ch. Ramreddy, Magnetic and doble dispersion effects on free convection in a non-darcy poros medim satrated with power law flid, International Jornal of Comptational Methods in Engineering Science and Mechanics, Vol-3, No-3, pp. -8,. [6] Ch. Ramreddy, Effects of doble dispersion on convective flow over a cone, International Jornal of Nonlinear Science, Vol 5, No-4, pp. 39-3, 3. [7] R.S. Telles, and O.V. Trevisan, Dispersion in heat and mass transfer natral convection along vertical bondaries in poros media, International Jornal of Heat and Mass Transfer, Vol-36, No-5, pp , 993. [8] EC.Lai, FA.Klacki, Effects of variable viscosity on convective heat transfer along a vertical srface in a.6 satrated poros medim, International Jornal of Heat and Mass Transfer, Vol-33, pp , [9] M.S.Seddeek, A.M.Salem, Laminar mied convection. adjacent to vertical continosly stretching sheets. with variable viscosity and variable thermal diffsivity, Heat and Mass Transfer, Vol-4, pp , 5. [] J.C.Slattery, Momentm, energy and mass transfer in contina Mc. Graw-Hill, New York, 97. [] J.X. Ling and A. Dybbs, Forced convection over a flat plate sbmersed in a poros medim: variable viscosity case ASME Paper 87-WA/HT Fo-3, American society of Mechanical Engineers, New York, 987. [] C.Y. Cheng, Natral convection heat and mass transfer near a vertical wavy srface with constant 4 wall temperatre and concentration in poros medim, International Commnications in Heat and Mass Transfer, Vol-7, No-8, pp , Fig.. Velocity profile for different vales of variable viscosity parameter for N=, Le=, Ds=.3, Dc=.3, a=,β=, and =. =.5 = = 5 =.5 = = Fig.. Temperatre profile for different vales of variable viscosity parameter for N=, Le=, Ds=.3, Dc=.3, a=, β=, and =. φ =.5 = = Fig.3. Concentration profile for different vales of variable viscosity parameter for N=, Le=, Ds=.3, Dc=.3, a=, β=, and =.

7 International Jornal of Scientific & Engineering Research, Volme 5, Isse 3, March-4 89 ISSN β = β = β = 3.5 Ds =. Ds =.3 Ds = Ds = Fig.5. Velocity profile for different vales of variable thermal condctivity parameter for N=, Le=, Ds=.3, Dc=.3, a=, r=.5, and = Fig.8. Velocity profile for different vales of Ds for N=, Le=, Ds=.3, Dc=.3, a=, r=.5, β=, and = β = β = β = Fig.6. Temperatre profile for different vales of variable thermal condctivity parameter for N=, Le=, Ds=.3, Dc=.3, a=, r=.5, and =. Ds =. Ds =.3 Ds = Ds = Fig.9. Temperatre profile for different vales of Ds for N=, Le=, Ds=.3, Dc=.3, a=, r=.5, β=, and =. φ β = β = β = Fig.7. Concentration profile for different vales of variable thermal condctivity parameter for N=, Le=, Ds=.3, Dc=.3, a=, =.5, and =. φ Fig.. Concentration profile for different vales of Ds for N=, Le=, Ds=.3, Dc=.3, a=, r=.5, β=, and =. Ds =. Ds =.3 Ds = Ds = 4

8 International Jornal of Scientific & Engineering Research, Volme 5, Isse 3, March-4 9 ISSN Dc =. Dc =.3 Dc = Dc = = = = Fig.. Velocity profile for different vales of Dc for N=, Le=, Ds=.3, a=, r=.5, β=, and = Fig.4. Velocity profile for different vales of Dc for N=, Le=, Ds=.3, Dc=.3, a=, r=.5, β= Dc =..6 Dc =.3 Dc = Dc = Fig.. Temperatre profile for different vales of Dc for N=, Le=, Ds=.3, a=, r=.5, β=, and =. = = = Fig.5. Temperatre profile for different vales of Dc for N=, Le=, Ds=.3, Dc=.3, a=, r=.5, β=. φ.6.4 Dc =. Dc =.3 Dc = Dc = φ.6.4 = = = Fig.3. Concentration profile for different vales of Dc for N=, Le=, Ds=.3, a=, r=.5, β=, and = Fig.6. Concentration profile for different vales of Dc for N=, Le=, Ds=.3, Dc=.3, a=, r=.5, β=. 4

9 International Jornal of Scientific & Engineering Research, Volme 5, Isse 3, March-4 9 ISSN =.5 = r = 3 4 β = β = β = 3 N Ra - /.45 Sh Ra - / Fig.7. Aial distribtion of the Nsselt nmber for different vales of variable viscosity for N=, Le=, Ds=.3, Dc=.3, a=, =, β= Fig.. Aial distribtion of the Sherwood nmber for different vales of variable thermal condctivity for N=, Le=, Ds=.3, Dc=.3, a=, =, =.5. Sh Ra - / =.5 r = r = 3 r.6 N Ra - / Ds =. Ds =.3 Ds = Fig.8. Aial distribtion of the Sherwood nmber for different vales of variable viscosity for N=, Le=, Ds=.3, Dc=.3, a=, =, β= Fig.. Aial distribtion of the Nsselt nmber for different vales of Ds for N=, Le=, β=, Dc=.3, a=, =, =.5. N Ra - /.6 β = β = β = 3 Sh Ra - / Ds =. Ds =.3 Ds = Fig.9. Aial distribtion of the Nsselt nmber for different vales of variable thermal condctivity for N=, Le=, Ds=.3, Dc=.3, a=, =, = Fig.. Aial distribtion of the Sherwood nmber for different vales of Ds for N=, Le=, β=, Dc=.3, a=, =, =.5. 4

10 International Jornal of Scientific & Engineering Research, Volme 5, Isse 3, March-4 9 ISSN N Ra - / Dc =. Dc =.3 Dc = Sh Ra - / a =. a =. a =.4 a = Fig.3. Aial distribtion of the Nsselt nmber for different vales of Dc for N=, Le=, β=, Ds=.3, a=, =, =.5. Fig.6. Aial distribtion of the Sherwood nmber for different vales of a for N=, Le=, β=, Ds, =.3, Dc=.3, a=, =, =.5.. Dc =. Dc =.3 Dc = Sh Ra - / Fig.4. Aial distribtion of the Sherwood nmber for different vales of Dc for N=, Le=, β=, Ds=.3, a=, =, =.5. N Ra - / a =. a =. a =.4 a = Fig.5. Aial distribtion of the Nsselt nmber for different vales of a for N=, Le=, β=, Ds, =.3, Dc=.3, a=, =, =.5. 4