NUMERICAL STUDY OF HEAT TRANSFER OF A MICROPOLAR FLUID THROUGH A POROUS MEDIUM WITH RADIATION

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1 THERMAL SCIENCE: Year 8, Vol., No. B, pp NUMERICAL STUDY OF HEAT TRANSFER OF A MICROPOLAR FLUID THROUGH A POROUS MEDIUM WITH RADIATION by Fakhrodin MOHAMMADI a* and Mohammad Mehdi RASHIDI b,c* a Deparmen of Mahemaics, Universiy of Hormozgan, Bandarabbas, Iran b Shanghai Key Laboraory of Vehicle Aerodynamics and Vehicle Thermal Managemen Sysems, Tongji Universiy, Shanghai, China c ENN-Tongji Clean Energy Insiue of Advanced Sudies, Shanghai, China Inroducion Original scienific paper hps://doi.org/.98/tsci5985m An efficien Specral Collocaion mehod based on he shifed Legendre polynomials was applied o ge soluion of hea ransfer of a micropolar fluid hrough a porous medium wih radiaion. A similariy ransformaion is applied o conver he governing equaions o a sysem of non-linear ordinary differenial equaions. Then, he shifed Legendre polynomials and heir operaional marix of derivaive are used for producing an approximae soluion for his sysem of non-linear differenial equaions. The main advanage of he proposed mehod is ha he need for guessing and correcing he iniial values during he soluion procedure is eliminaed and a sable soluion wih good accuracy can be obained by using he given boundary condiions in he problem. A very good agreemen is observed beween he obained resuls by he proposed Specral Collocaion mehod and hose of previously published ones. Key words: micropolar fluid, porous medium, specral collocaion mehod, shifed Legendre polynomials Since he ime of Fourier, orhogonal funcions and polynomials have been used in he analyic sudy of differenial equaions and heir applicaions for numerical soluion of ODE refer, a leas, o he ime of Lanczos (938). I is well known ha he eigenfuncions of cerain singular Surm-Liouville problems such as Legendre or Chebyshev orhogonal polynomials allow he approximaion of funcions C [a, b] where runcaion error approaches zero faser han any negaive power of he number of basic funcions used in he approximaion, as ha number (order of runcaion N) ends o infiniy. This phenomenon is usually referred o as specral accuracy [-3]. The collocaion approach appears o have been firs used by Slaer and by Kanorovic (934) in specific applicaions. This approach is especially aracive whenever i applies o variable-coefficien and even non-linear problems [4]. Some major advanages of he collocaion mehods are: Since no inegraion is required, he consrucion of he final sysem of equaions is very efficien. The funcions mus be evaluaed only a he collocaion nodes in conras o oher mehods. * Corresponding auhors', s: f.mohammadi6@homail.com; mm_rashidi@ongji.edu.cn

2 558 THERMAL SCIENCE: Year 8, Vol., No. B, pp Compuaional cos of calculaing non-linear erms is reasonably low wih good numerical accuracy. The specral collocaion mehod has been applied for numerical soluion of differen kind of differenial and inegral equaions. For example, i has been used for deriving approximae soluion of sochasic Burgers equaion [4], Burgers-ype equaion [5], Navier-Sokes equaions [6], wo-poin boundary value problem in modelling viscoelasic flows [7], Poisson equaion in polar and cylindrical co-ordinaes [8], Volerra inegral equaions [9, ], compressible flow, -D and axisymmeric boundary-layer problems [], hypersonic boundary-layer sabiliy [], Helmholz and variable coefficien equaions in a disk [3] and Burgers-Huxley equaion [4]. The Legendre polynomials [] are well known family of orhogonal polynomials on he inerval [, ] of he real line. These polynomials presen very good properies in he approximaion of funcions. Therefore, Legendre polynomials appear frequenly in several fields of mahemaics, physics and engineering. Specral mehods based on Legendre polynomials as basis funcions for solving numerically differenial equaions have been used by many auhors, (see for example [5-7]). The problem of micropolar fluids pas hrough a porous media has received much aenion in several indusrial and engineering processes such as porous rocks, foams and foamed solids, aerogels, alloys, polymer blends and microemulsions. The simulaneous effecs of a fluid ineria force and boundary viscous resisance upon flow and hea ransfer in a consan porosiy porous medium were analyzed by Vafai and Tien [8]. Rapis [9] invesigaed boundary-layer flow of a micropolar fluid hrough a porous medium. Abo-Eldahab and El Gendy [] considered he convecive hea ransfer pas a sreching surface embedded in non-darcian porous medium in he presence of magneic field. Abo-Eldahab and Ghonaim [] sudied he radiaion effec on hea ransfer of a micropolar fluid pas on unmoving horizonal plae hrough a porous medium. The DTM in [] was applied successfully o find he analyical soluion of hea ransfer of a micropolar fluid hrough a porous medium wih radiaion. Rashidi and Mohimanian [3] presened complee analyic soluion o hea ransfer of a micropolar fluid hrough a porous medium wih radiaion. y ν u Boundary-layer Figure. Geomery of he problem and co-ordinae sysem x Flow analysis and mahemaical formulaion Consider a seady -D flow of an incompressible micropolar fluid hrough a porous medium pas a coninuously semi-infinie horizonal plae (fig. ). The governing equaions of boundary-layer o micropolar fluid hrough a porous medium are given [-3]. u v + = x u u u σ νϕ + = ν + + ( ) + ϕ ( ) u v K U u C U u x K σ u G σ = 3y () () (3)

3 THERMAL SCIENCE: Year 8, Vol., No. B, pp subjec o he following boundary condiions: u q T u k T r + v = x ρcp ρcp u=, v=, σ =, T=T as y= u= U, σ =, T=T as y w (4) (5) where k = ρs and n = (µ + S) / ρ. By using he Rosselan approximaion, we have: 4 4σ T qr = * 3k where σ* is he Sefan-Bolzman consan and k* he mean absorpion coefficien. If he flux is sufficienly small, T 4 can be expanded as a Taylor series abou T. By neglecing higher order erms of he Taylor series, we have: By subsiuion from eqs. (6) and (7) in eq. (4) we have: (6) T 4TT 3T (7) u + v = + x ρc 3ρck * T T k T 6σ T p * p By inroducing he following similariy ransforms [, 4, 5] U ψ ν σ ν x ( xy, ) = Uxf ( ), = Ug ( ) U T T y, θ ν x T T = = where ψ is he sream funcion and is defined as u = ψ/ and ν = ψ / x, he governing eqs. ()-(4) are reduced o he following sysem of ODE : f + ff + g + ( f ) + N( f ) = () M subjec o he boundary condiions: where w (8) (9) Gg ( g + f ) = () (3R+ 4) θ + 3RPrθ f = () f() =, f () =, g() =, θ () = f ( ) =, g( ) =, θ ( ) = ρν c k k ν GU k* k p Pr = (Pranl number), = (Coupling consan), G = (Microroaion paramear), R = (Radiaion paramear), 3 ν x 4 σ * T KU = (Permeabiliy paramear), = ϕν x (Ineria coefficien paramear) ϕν x M N C (3) (4)

4 56 THERMAL SCIENCE: Year 8, Vol., No. B, pp Shifed Legendre polynomials and heir properies The well known Legendre polynomials are defined on he inerval and can be deermined wih he aid of he following recurrence formulae [, 6-8]: m+ m Lm ( ) = Lm( ) Lm ( ) m =,, m+ m+ where L () =, L () =. In order o use Legendre polynomials on he inerval [, ] we define he so-called shifed Legendre polynomials by inroducing he change of variable = x. The orhogonaliy condiion for hese polynomials is: for m = n Pm( xp ) n( X)d x= m + for m n (6) A funcion f () defined over [, ] may be expanded in he erms of shifed Legendre polynomials: (5) f() = cp() (7) k k k = where c k = [f (), P k ()], in which (...) denoes he inner produc. If he infinie series in eq. (7) is runcaed, hen i can be wrien: N T f() = cp k k() = CΦ() (8) k = where C and Φ() are (N + ) vecors given by: T C = [ c, c..., c ] Φ ( ) = [ P ( ), P()... P()] (9) N N In he nex heorem we derived a relaion beween shifed Legendre polynomials and heir derivaives ha is very imporan for deriving he operaional marix of derivaive for shifed Legendre polynomials. Theorem. Le Ψ() be he Legendre polynomial vecor defined [6-8]: Φ ( ) = [ P ( ), P ( ),..., P( )] () N he derivaive of his vecor can be expressed by: d Φ () = D Φ () () which D is (N + ) (N + ) marix and is (i, j) h elemen is defined: Mehod of soluion D i, j ( j ) j =,..., i and ( i+ j) odd = () o.w. Consider he coupled non-linear differenial eqs. (7) and (8) subjec o boundary condiions (). By using change of variable: =, F () = f( ), H () = g ( ), ϑ() = θ( ) (3) we have he following non-linear differenial sysems in he inerval [, ],

5 THERMAL SCIENCE: Year 8, Vol., No. B, pp F F H F F M + = 3 d d d 3 d 3 d F N 3 d H d F G H + = d ϑ dϑ (3R+ 4) + 3 RPr F = (6) d d he boundary condiions become F() =, F () =, H() =, ϑ() = (7) F () =, H() =, ϑ() = Now we expand he unknown funcion F(), H() and ϑ() by he shifed Legendre polynomial ino inerval [, ] as: T T T (4) (5) F() C Φ(), H() C Φ(), ϑ() C Φ() (8) where C, C, and C 3 are he unknown shifed Legendre polynomial coefficien vecors defined in eq. (9). By using he operaional marix derived in eq. () we ge: 3 df T d F T d F T 3 C DΦ(), C D Φ(), C 3 D Φ() (9) dh T d H T C DΦ(), C D Φ() (3) subsiuing eqs. (9)-(3) in eqs. (4)-(6), we have: dϑ T d ϑ T C3 DΦ(), C 3 D Φ() (3) C D Φ () + [ C DΦ()][ C D Φ ()] + C DΦ () + T 3 T T T [ 3 T 3 ( )] {[ [ T + C DΦ + N C DΦ ( )] } = (3) M GC D Φ() [ C Φ () + C D Φ ()] = (33) T T T (3R+ 4) C D Φ () + 3 RPr C Φ() C DΦ () = (34) T T T 3 3 Moreover, boundary condiions (7) resul: C Φ () =, C DΦ () =, C () =, C Φ () = T T T T 3 C DΦ () =, C Φ () =, C Φ () = T T T 3 To find he approximae soluion of he non-linear sysem (4)-(6), we use he ypical collocaion mehod and collocae eqs. (33) and (34) a (M ) differen poins and eq. (3) a (M ) differen poins in he inerval [, ]. For choosing suiable collocaion poins, we use he firs roos of shifed Legendre P M+ (). These equaions ogeher wih eqs. in (35) generae 3(M + ) non-linear equaions. The well-known Newon-Raphson have been used for approximae soluion of derived non-linear sysems. Afer finding he soluion of his non-linear sysems we obain unknown vecors C, C, and C 3. By subsiuing hese vecors in eq. (8) he soluion (35)

6 56 THERMAL SCIENCE: Year 8, Vol., No. B, pp funcions F(), H(), and ϑ() can be approximaed. Now, he change of variable in eq. (3) resuls approximaion of funcions f (), g(), and θ(). Numerical resuls In his secion, he Legendre Collocaion mehod presened in previous secion was applied o approximae soluion of he non-linear differenial eqs. ()-() subjec o he boundary condiions (3). In order o verify he resuls of his sudy, he resuls have been compared wih previously published resuls from he lieraure. Table shows he values of f (), g () and θ () for differen values of permeabiliy parameer, M coupling consan, Δ radiaion parameer, R and G =, N =., Pr =.7. The resuls reveal ha as he permeabiliy parameer increase he wall emperaure gradien θ () and rae of change g () are increase while he shear sress decrease. Table. Variaion of f (), g () and θ () for differen values of M, Δ, R and G =, N =., and Pr =.7 f () g () θ () M Δ R Presen Ref. [] Presen Ref. [] Presen Ref. [] θ ( ) R =. R =. R = θ ( ) Δ = Δ = Δ = (a) (b) Figure. Variaion of he dimensionless emperaure; (a) M = Δ =.5, N =., G =, Pr =.7 and differen values of R, (b) M =.5, R = N =., G =, Pr =.7 and differen values of Δ Moreover, as he radiaion parameer decrease, he wall emperaure gradien θ () increases and shear sress f () and rae of change g () have no changes. Figs. (a) and 4(a) show numerical soluion derived by he Legendre Collocaion mehod for various values of permeabiliy, vorex-viscosiy, and radiaion parameers. The effec of radiaion on he dimensionless emperaure θ profile are presened in fig. (a).

7 THERMAL SCIENCE: Year 8, Vol., No. B, pp f ( ) g( ).4 Δ = Δ = Δ =. Δ = Δ = Δ = (a) (b) Figure 3. Variaion of he dimensionless velociy; (a) M =.5, R = N =., G =, Pr =.7 and differen values of Δ, (b) M =.5, R = N =., G =, Pr =.7 and differen values of Δ θ ( ) M =.5 M =.7 M = (a) f ( ) (b) M =.5 M =.7 M = Figure 4. (a) Variaion of he dimensionless emperaure Δ =.5, R = N =., G =, Pr =.7 and differen values of M, (b) variaion of he dimensionless velociy for Δ =.5, R = N =., G =, Pr =.7 and differen values of M Figures (b) and 3(b) show he effecs of he vorex-viscosiy parameer Δ on he velociy of he fluid, dimensionless emperaure and angular velociy of he microsrucures. I is easily concluded from his figures ha as Δ increase, he fluid velociy and emperaure of he microsrucure increase. The variaion of permeabiliy parameer M and is effecs on he velociy of fluid, emperaure disribuion and angular velociy of microsrucures are ploed in figs. 4(a) and 5. Conclusions In his sudy, we have inroduced an efficien Specral Collocaion mehod based on he shifed Legendre polynomials o ge soluion of hea ransfer of a micropolar fluid hrough a porous medium wih radiaion. This proposed approach is simple in applicabiliy, as i does no require iniial values during he soluion procedure. Moreover, by using he given boundary condiions in he problem, a sable soluion wih very good resuls can be obained. The effecs of permeabiliy, vorex-viscosiy and radiaion parameers are examined on he velociy of fluid, emperaure disribuion and angular velociy of microsrucures. A very good agreemen is ob g( ) M =.5 M =.7 M =.9 Figure 5. Variaion of he dimensionless angular velociy g for Δ =.5, R = N =., G =, Pr =.7 and differen values of M

8 564 THERMAL SCIENCE: Year 8, Vol., No. B, pp served beween he obained resuls of by he proposed Specral Collocaion mehod and hose of previously published ones. Nomenclaure C, C, C 3 Shifed Legendre polynomial coefficien c p specific hea D operaional marix of derivaive f dimensionless velociy funcions G microroaion consan g dimensionless microroaion angular velociy K permeabiliy of he porous medium k * mean absorpion coefficien L m Legendre polynomial P m shifed Legendre polynomial Pr Pranl number q r he radiae hea flux S consan characerisic o he fluid T emperaure disribuion uniform sream velociy U References [] Canuo, C., e al., Specral Mehods in Fluid Dynamics, Springer, New York, USA, 988 [] Babolian, E., Hosseini, M. M., A Modified Specral Mehod for Numerical Soluion of Ordinary Differenial Equaions wih Non-Analyic Soluion, Appl. Mah. Compu., 3 (), -3, pp [3] Mohammadi, F., e al., Legendre Wavele Galerkin Mehod for Solving Ordinary Differenial Equaions wih Non-Analyic Soluion, In. J. Sys. Sci., 4 (), 4, pp [4] Kamrani, M., Hosseini, S. M., Specral Collocaion Mehod for Sochasic Burgers Equaion Driven by Addiive Noise, Mah. Compu. Simul., 8 (), 9, pp [5] Khaer, H. A., e al., A Chebyshev Specral Collocaion Mehod for Solving Burgers-Type Equaions, J. Compu. Appl. Mah., (8),, pp [6] Malik, M. R., e al., A Specral Collocaion Mehod for he Navier-Sokes Equaions, J. Compu. Phys., 6 (985),, pp [7] Karageorghis, A., e al., Specral Collocaion Mehods for he Primary Two-Poin Bondary Value Problem in Modelling Viscoelasic Fows, In. J. Numer. Mehods. Eng., 6 (988), 4, pp [8] Chen, H., e al., A Direc Specral Collocaion Poisson Solver in Polar and Cylindrical Co-ordinaes, J. Compu. Phys., 6 (),, pp [9] Chen, Y., Tang, T., Convergence Analysis of he Jacobi Specral-Collocaion Mehods for Volerra Inegral Equaions wih a Weakly Singular Kernel, Mah. Compu., 79 (), 69, pp [] Nemai, S., e al., Numerical Soluion of a Class of Two-Dimensional Non-Linear Volerra Inegral Equaions Using Legendre Polynomials, J. Compu. Appl. Mah., 4 (3), Apr., pp [] Prue, C. D., Sree, C. L., A Specral Collocaion Mehod for Compressible, Non-Similar Boundary Layers, In. J. Numer. Mehods. Fluids., 3 (99), 6, pp [] Malik, M. R., Numerical Mehods for Hypersonic Boundary Layer Sabiliy, J. Compu. Phys., 86 (99),, pp [3] Bialecki, B., Karageorghis, A., Specral Chebyshev-Fourier Collocaion for he Helmholz and Variable Coefficien Equaions in a Disk, J. Compu. Phys., 7 (8), 9, pp [4] Darvishi, M. T., e al., Specral Collocaion Mehod and Darvishi s Precondiionings o Solve he Generalized Burgers-Huxley Equaion, Commun. Non-Linear. Sci., 3 (8),, pp. 9-3 [5] Khalil, H., Ali Khan, R., A New Mehod Based on Legendre Polynomials for Soluions of he Fracional Two-Dimensional Hea Conducion Equaion, Compu. Mah. Appl., 67 (4),, pp [6] Rong-Yeu, C., Maw-Ling, W., Shifed Legendre Funcion Approximaion of Differenial Equaions, Applicaion o Crysallizaion Processes, Compu. Chem. Eng., 8 (984),, pp. 7-5 [7] Saadamandi, A., Dehghan, M., A New Operaional Marix for Solving Fracional-Order Differenial Equaions, Compu. Mah. Appl., 59 (), 3, pp u ν x y velociy in x-direcion velociy in y-direcion disance along he surface disance normal o he surface x- and y-axes Greek symbols σ angular velociy σ * Sefan-Bolzman consan µ dynamical viscosiy ρ densiy of fluid θ dimensionless emperaure ν kinemaics viscosiy φ porosiy ψ sream funcion Ψ shifed Legendre polynomial vecor

9 THERMAL SCIENCE: Year 8, Vol., No. B, pp [8] Vafai, I. K., Tien, C. L., Boundary and Ineria Effecs on Fow and Hea Transfer in Porous Media, In. J. Hea Mass Trans., 4 (98),, pp [9] Rapis, A., Boundary Layer Fow of a Micropolar Fuid hrough a Porous Mediu, J. Porous Media, 3 (),, pp [] Abo-Eldahab, E. M., El Gendy, M. S., Convecive Hea Transfer Pas a Coninuously Moving Plae Embedded in a Non-Darcian Porous Medium in he Presence of a Magneic Feld, Canadian Journal of Physics, 79 (), 7, pp [] Abo-Eldahab, E. M., Ghonaim, A. F., Radiaion Effec on Hea Transfer of a Micropolar Fuid hrough a Porous Medium, Applied Mahemaics and Compuaion, 69 (5),, pp. 5-5 [] Rashidi, M. M., Abbasbandy, S., Analyic Approximae Soluions for Hea Transfer of a Micropolar Fuid hrough a Porous Medium wih Radiaion, Communicaions in Nonlinear Science and Numerical Simulaion, 6 (), 4, pp [3] Rashidi,M. M., Mohimanian Pour, S. A., A Novel Analyical Soluion of Hea Transfer of a Micropolar Fuid hrough a Porous Medium wih Radiaion by DTM-Pade, Hea Transfer-Asian Research, 39 (), 8, pp [4] Willson, A. J., Boundary Layers in Micropolar Liquids, Camb. Phil. Soc,. 67 (97),, pp [5] Nield, D., Bejan, A., Convecion in Porous Media, Springer-Verlag, Berlin, 999 [6] Mohammadi, F., e al., Numerical Soluions of Falkner-Skan Equaion wih Hea Transfer, Sudies in Non-Linear Sciences, 3 (), 3, pp [7] Mohammadi, F., Rashidi, M. M., Specral Collocaion Soluion of MHD Sagnaion-Poin Flow in Porous Media wih Hea Transfer, Journal of Applied Fluid Mechanics, 9 (6),, pp [8] Mohammadi, F., Rashidi, M. M., An Efficien Specral Soluion for Unseady Boundary Layer Flow and Hea Transfer Due o a Sreching Shee, Thermal Science, (7), 5, pp Paper submied: Sepember 8, 5 Paper revised: April 6, 7 Paper acceped: April 9, 7 7 Sociey of Thermal Engineers of Serbia Published by he Vinča Insiue of Nuclear Sciences, Belgrade, Serbia. This is an open access aricle disribued under he CC BY-NC-ND 4. erms and condiions