Study the Effect of Variable Viscosity and Thermal Conductivity of Micropolar Fluid in a Porous Channel

Transcription

1 Study the Effet of Variable Visosity and Thermal Condutivity of Miropolar Fluid in a Porous Channel Gitima Patowary 1, Dusmanta Kumar Sut 1 Assistant Professor, Dept. of Mathematis, Barama College, Barama, Assam, gitimapatowary@gmail.om Assistant Professor, Dept. of Mathematis, Jorhat Institute of Siene and Tehnology, Jorhat, Assam, sutdk1@yahoo.om Abstrat To study the effet of thermal radiation on unsteady boundary layer flow with variable visosity and thermal ondutivity due to strething sheet in porous hannel in presene of magneti field, a numerial model has been developed. The Rosseland diffusion approximation is used to desribe the radiative heat flux in the energy equation. The governing equations redued to similarity boundary layer equations using suitable transformations and then solved using shooting method. A parametri study illustrating the influene of the radiation R, variable visosityε, Dary number Da, porous media inertia oeffiient γ, thermal ondutivity κ, unsteady A parameters on skin frition and Nusselt number and Magneti field parameter M. Key words: unsteady flow, radiation, strething sheet, variable visosity, variable thermal ondutivity, porous hannel. 1. Introdution The heat transfer from a strething surfae is of interest in many pratial appliations. Speially in the manufaturing proess of plasti and rubber sheets where it often neessary to blow a gaseous medium through the unsolidified material this situation arise. Extrusion proesses, glass blowing, ontinuous oating and spinning of fibers also involve the flow due to a strething surfae. Crane[1] was the first who studied the momentum boundary layer for linear strething of sheet. The temperature field in the flow over strething surfae subjet to a uniform heat flux was studied by Grubka and Bobba[13], while Elbashbeshy[6] onsidered the ase of strething surfae with a variable surfae heat flux. Elbashebeshy and Bazid[7] have found similarity solutions of the boundary layer equations, whih desribes the unsteady flow and heat transfer over a strething sheet. Unsteady flow and heat transfer over a strething sheet in visous and inompressible fluid was also studied by Sharidan et al.[3]. Pop and Tsung[1] studied unsteady flow past a strething sheet. Unsteady boundary layer flow in the region of the stagnation point on a strething boundary layer flow in the region of the stagnation point on a strething was studied by Nazar et al.[]. Yurusoy[19] also investigated unsteady boundary layer flow of power law fluid on the strething sheet surfae. Effet of visous dissipation on heat transfer in a non Newtonian liquid film over an unsteady strething sheet was studied by Chen[5]. Dandapat et al.[4] investigated the effet of variable fluid properties and thermo apillarity on the flow of a thin film on an unsteady strething sheet. Chiam[4] onsidered the effet of a variable thermal ondutivity on the flow and heat transfer from a linearly strething sheet. Most of the previous studies were onerned with MHD fluid. Effets of variable visosity and thermal ondutivity on MHD flow and heat transfer in visoelasti fluid over a strething sheet studied by Salem[3]. Effet of variable visosity and thermal 147

2 ondutivity on a unsteady two dimensional laminar flow of visous inompressible onduting fluid past a semi infinite vertial porous moving plate taking into aount the effet of a magneti filed in the presene of variable sution was studied by Seddeek et al.[16]. Odda and Farhan[] was studied the effets of variable visosity and variable thermal ondutivity on heat transfer from a strething sheet by onsidering fluid visosity and the thermal ondutivity to vary as inverse linear funtions of temperature. Natural onvetion with variable visosity and thermal ondutivity from a vertial wavy one and steady two dimensional laminar fored flow and heat transfer of a visous inompressible fluid having temperature dependent visosity and thermal ondutivity past a wedge with a uniform surfae heat was studied by Hossain et al.[14,15]. Chamkha[] has studied unsteady hydromagneti flow and heat transfer from a nonisothermal strething immersed in a porous media. Hassanien et al.[9] studied variable visosity and thermal ondutivity effets on ombined heat and mass transfer in mixed onvetion over a UHF/UMF wedge in porous media in the entire regime. The effet of thermal radiation on mixed onvetion flow of two immisible fluids in a vertial porous stratum was studied by Umavathi[11]. Considering the effet of thermal radiation on Forhheimer natural onvetion over vertial flat plate in a fluid saturated porous medium was studied by El-Amin et al.[17]. El-Kabeir et al.[1] have investigated the effet of thermal radiation and a transverse magneti field with surfae mass transfer in free onvetion on a vertial strething surfae with sution and blowing. The effet if thermal radiation on free onvetion flow with variable visosity and uniform sution veloity along a uniformly heated vertial porous plate embedded in a porous medium in ht epresene of a uniform transverse magneti field is analyzed by Modather and El-Kabeir[18]. The effet of thermal radiation on unsteady boundary layer flow with temperature dependent visosity and thermal ondutivity due to strething sheet through porous media was studied by Abdou[1]. In the present work We propose to extend the work of Abdou[1] and study the effet of radiation on unsteady boundary layer flow with variable visosity and thermal ondutivity due to strething sheet through porous hannel in presene of magneti field. The governing redued to similarity boundary layer equations using suitable transformations and then solved using shooting method. Numerial results are presented in terms of loal skin frition oeffiient and rate of heat transfer for various values of the variable visosity ε, Dary number Da, porous media inertia oeffiient γ, thermal ondutivity κ, unsteady A parameters on skin frition and Nusselt number and Magneti field parameter M. The effet of variation in ε, κ, A, Da, R, γ and M are presented graphially.. Mathematial formulation We onsider the unsteady two-dimensional laminar flow of a visous inompressible miropolar fluid with variable visosity and thermal ondutivity past a semi infinite strething sheet in the region y> in presene of a magneti field as shown in Fig. 1. Keeping the origin fixed, the equal and opposite fores are suddenly applied along the x-axis, whih results in strething of the sheet and hene, flow is generated. At the same time, the wall temperature T w ( x, t) of the sheet is suddenly raised from T to T w ( > T ). Under these assumptions, the basi unsteady boundary layer equations governing the flow and heat transfer due to strething sheet are given by The equation of ontinuity is u u + = x (1) The equation of momentum is u u 1 u µ + u + v = µ u Cu t x ρ K () The equation of energy is T T T 1 T r + u + v = k q t x ρ p (3) u σ B ρ u 148

3 In the above equations t is the time, u and v are the omponents of fluid veloity in the x and y diretions respetively, ρ - the density of ambient fluid, T is the fluid temperature in the boundary layer region, B is external magneti field, µ and k are respetively the dynami visosity and the thermal ondutivity, following are given as below: T T µ = µ T T (4a) and T T k = k 1 + T T (4b) In (4) µ is the visosity and k is the thermal ondutivity of the ambient fluid, T is the temperature of the ambient fluid, T is some referene temperature and, 1 are onstant. Clearly = 1 and = represent that the dynamial visosity and the thermal ondutivity be uniform. Solutions of the above equations have to satisfy the following boundary onditions [1] t < ; u = v = ; T = T for any x, y t ; u = uw v = T = Tw (5a) u, T T as y We assume now that the veloity of the sheet u w and the sheet temperature T w have the following form: 1 uw = x(1 δt), Tw = T + (1 δt) vx (5b) where is the strething rate being a positive onstant, δ is a positive onstant, whih measures the unsteadiness. The quantity q r on the right-hand side of equation (3) represents the radiative heat flux in the y diretion. In order to redue the omplexity of the problem and to provide a means of omparison with future studies, a more detailed representation for radiative heat flux, for optially thik radiation limit, is onsidered in the present analysis. Thus the radiative heat flux term in the energy equation is simplified by utilizing the Rosseland diffusion 3 approximation (Sparrow and Cess [8]) for an optially thik boundary layer as follows: 4 r 4σ T q = 3χ( a + σ s ) (5) Where σ is the Stefan-Boltzman onstant, χ is the mean absorption oeffiient, a is the Rosseland mean absorption oeffiient and σ s is the sattering oeffiient. This approximation is valid at point optially far from the bounding surfae layer. If temperature differenes withing the flow are suffiiently small suh that T 4 may be expressed as a linear funtion of the temperature, then the Taylors series for T 4 about T, after negleting higher order term s, is given by : [1] T 4TT 3T We introdue now the following variables ψ ψ u =, v = x v = y y, ψ = xf ( ), T = T + t v (1 δt) 1 δt v x (6) where ψ is the stream funtion satisfying the ontinuity equation, f ( ) is the dimensionless stream funtion, is the pseudo-similarity variable, θ () is the dimensionless temperature of the fluid in the boundary layer region, v = µ / ρ is the stream kinematis visosity. Substituting the transformation given in (6) into (), (3) one obtains the following non-similar system equations governing the flow: (7) R 1+ k θ θ + k θ Pr 3 (8) 1+ εθ Da ( 1 δ ) 3 / θ ( ) ( + εθ ) f + ( f + εθ ) f ( 1+ γ ) f A f + f + M f = ( ) + f θ + θ f ( 3 θ + θ ) = The transform boundary onditions are f ( ) =, f () = 1 θ () = 1 (9) A 149

4 Where δ A =, f ( ) = θ ( ) = µ Pr = k p, T T ε = 1, T T k = T T T T σb ρ kv p k M 1 =, R = R =, Da = 3 ρ 4σT µ v Where ε is the visosity variation, k is the thermal ondutivity parameter, A is a non dimensional onstant whih measures the flow and heat transfer unsteadiness and Pr is the Prandtl number, M 1 is the magneti parameter, R is the radiation parameter, Da is the Dary number. The quantities of physial interested, namely, the loal skin frition C f and the rate of heat transfer in terms of loal Nussel number Nu x are presribed by: τ w C f = ( 1/ ) ρu qwx Nu x = k ( Tw T ) Where τ w is the skin frition and q w is the heat transfer from the sheet are given by: u τ = w = µ µ f (), y= q w T = k y= 1/ 1/ C = (), / Re = θ f Re f Nu x x () Where Re = u w x v is the loal Reynolds number. x / 3. Results and Disussion Equations (7), (8) with the boundary onditions (9) are solved numerially using shooting method. Calulations were arried out for the value of Prandtl number 1., the visosity variation parameter ε ranged from. to 1., unsteady parameter A ranged from. to., radiation parameter R ranged from. to 4., Dary number Da range from 1. to and porous media inertia oeffiient γ range from. to 1.. Figs. 1-9 display the dimensionless of veloity f and temperature θ profiles for various values of Dary number parameter Da, unsteady parameter A, the thermal ondutivity variation parameter k, and porous media inertia oeffiient γ, magneti parameter M 1, radiation parameter R, visosity parameter ε respetively, while the others parameters are fixed. It an be seen that veloity and temperature of the fluid inreases as Dary number parameter Da, thermal ondutivity variation parameter k and magneti parameter M 1 inrease, while the opposite trend is observed for the effet of A and porous media inertia oeffiient γ. This behaviour is in agreement with the results of Elbashbeshy and Bazid [7] and Sharidan et al. [3]. The same result was found by Abdou [1] in non magneti ases. Moreover, the boundary layer thikness dereases with an inrease in A whih in turn inreases the skin frition oeffiient and Nussel number. In addition, the effet of k on the boundary layer separation is not very pronouned ompared to the effets of A. 4. Conlusion In the present work we onsider the effet of thermal radiation on unsteady boundary layer flow with variable visosity and thermal ondutivity due to a strething sheet through porous media in presene of external magneti field. The governing equations redued to similarity boundary layer equations using suitable transformations and then solved using the shooting method. Numerial results are presented in terms of Dary number parameter Da, unsteady parameter A, the thermal ondutivity variation parameter k, and porous media inertia oeffiient γ, magneti parameter M 1, radiation parameter R, visosity parameter ε respetively. We notie the values of Nusselt number inreases with inreasing of radiation parameter R and thermal ondutivity parameter k, while the skin fition has the opposite behaviour, on the other hand as porous media inertia oeffiient γ inreases the shear stress dereases and Nusselt number inreases. REFERENCES 1. Abdou, M.M.M., Effet of radiation with temperature dependent visosity and thermal ondutivity on unsteady a strething sheet through porous media, Nonlinear Analysis: 15

5 Modeling and Control, Vol. 15, No. 3, pp (1). A.J. Chamkha, Unsteady hydromagneti flow and heat transfer from a non-isothermal strething sheet immersed in a porous medium, Int. Commun. Heat Mass, Vol. 5 (6), pp (1998). 3. A. M. Salem, Variable visosity and thermal ondutivity effets on MHD flow and heat transfer in visoelasti fluid over a strething sheet, Phys. Lett. A, Vol. 369 (4), pp (7). 4. B. S. Dandapat, B. Santra, K. Vajravelu, The effets of variable fluid properties and thermoapilliarity on the flow of a thin film on an unsteady strething sheet, Int. J. Heat Mass Tran, Vol. 5 (5-6), pp (7). 5. Ch. H. Chen, Effet of visous dissipation on heat transfer in a non-newtonian liquid film over an unsteady strething sheet, J. Non- Newton. Fluid, Vol. 135(-3), pp , (6). 6. E. M. A. Elbashebeshy, Heat transfer over a strething surfae with variable surfae heat flux, J. Phys. D Appl. Phys., Vol. 31, pp (1998) 7. E. M. A. Elbashebeshy, M. A. A. Bazid, Heat transfer over a unsteady strething surfae, Heat Mass Transfer, Vol. 41, pp. 1-4 (4) 8. E.M. Sparrow, D.R. Cess, Radiation Heat Transfer Augmented Edition, Hemisphere, Publishing Corporation, Washington, D.C. (1978). 9. I.A. Hossanien, A.H. Essawy, N.M. Moursy, Variable visosity and thermal ondutivity effets on ombined heat and mass transfer in mixed onvetion over a UHF/UMF wedge in porous media: the entire regime, Appl. Math. Comput. Vol. 145(-3), pp (3). 1. I. Pop, T. Y. Na, Unsteady flow past a strething sheet, Meh. Res. Commun, Vol. 3(4), pp (1996) 11. J.C. Umavathi, A. J. Chamkha, M. H. Manjula, A. AI-Mudhaf, Radiative heat transfer of a two fluid flow in a vertial porous Stratum, J. Fluid Mehanis Researh, Vol. 35(6), pp (8). 1. L. J. Crane, Flow past a strething plate, Z. Angew. Math. Phys., Vol. 1, pp (197) 13. L. J. Grubka, K. M. Bobba, Heat transfer harateristi of a ontinuous strething surfae with variable temperature, J. Heat Transf., Vol. 17, pp (1985) 14. M. A. Hossain, M. S. Munir, D.A.S. Rees, Flow of visous inompressible fluid with temperature dependent visosity and thermal ondutivity past a permeable wedge with uniform surfae heat flux, Int. J. Therm. Si., Vol. 39, pp (). 15. M. A. Hossain, M. S. Munir, I. Pop, Natural onvetion with variable visosity and thermal ondutivity from a vertial wavy one, Int. J. Therm. Si., Vol. 4, pp (1). 16. M. A. Seddeek, F.A. Salama, The effets of temperature dependent visosity and thermal ondutivity on unsteady MHD onvetive heat transfer past a semi-infinite vertial porous moving plate with variable sution, Comp. Mater. Si., Vol. 4(), pp (7). 17. M. F. EI-Amin, I. Abbas, R. S. R. Gorla, Effet of thermal radiation on a natural onvetion in a porous medium, Int. J. Fluid Mehanis Researh, Vol. 34(), pp (7). 18. M. Modather, M. Abdou, S.M.M. EL-Kabeir, Magneto hydrodynamis and radiative effet on free onvetion flow of fluid with variable visosity from a vertial plate through a porous medium, J. Porous Media, Vol. 1(5), pp (7). 19. M. Yurusoy, Unsteady boundary layer flow of power-law fluid on strething sheet surfae, Int. J. Eng. Si., Vol. 44 (5-6), pp (6).. R. Nazar, N. Amin, D. Filip, I. Pop, Unsteady boundary layer flow in the region of the stagnation point on a strething sheet, Int. J. Eng. Si., Vol. 4(11-1), pp (4) 1. S.M.M. EL-Kabeir, M. Modather, M. Abdou, R.S.R. Gorla, Hydromagneti ombined onvetion from radiating vertial strething sheet with surfae mass transfer, Int. J. Fluid 151

6 Mehanis Researh, Vol. 35(6), pp (8).. S. N. Odda, A.M. Farhan, Chebyshev finite differene method for the effets of variable visosity and variable thermal ondutivity on heat transfer to a miro-polar fluid from a non-isothermal strething sheet with sution and blowing, Chaos Solution Frat., Vol. 3(4), pp (6). 3. S. Sharidan, T. Mahmood, I. Pop, Similarity solution for the unsteady boundary layer flow and heat transfer due to a streathing sheet, Int. J. Appl. Meh. Meh. Eng., Vol. 11(3), pp (6). 4. T.C. Chiam, Heat transfer in a fluid with variable thermal ondutivity over a linearly strething sheet, Ata Meh., Vol. 19, pp (1998). Fig. 1. Veloity profiles for various values of the unsteady parameter A Fig.. Veloity profiles for various values of the thermal ondutivity parameter k. 15

7 Fig. 3. Veloity profiles for various values of the unsteady parameter Da Fig. 4. Veloity profiles for various values of the magneti parameter M 1. Fig. 5. Veloity profiles for various values of permeability parameter γ. 153

8 Fig. 6. Temperature profiles for various values of the unsteady parameter A Fig. 7. Temperature profiles for various values of magneti parameter M 1. Fig. 8. Temperature profiles for various values of visosity parameter ε. 154

9 Fig. 9. Temperature profiles for various values of the radiation parameter R. 155