N. Sandeep 1 and V. Sugunamma 2

Transcription

1 Journal of Applied Fluid Mechanics, Vol. 7, No., pp , 4. Available online a ISSN , EISSN Radiaion and Inclined Magneic Field Effecs on Unseady Hydromagneic Free Convecion Flow pas an Impulsively Moving Verical Plae in a Porous Medium N. Sandeep and V. Sugunamma Assisan Professor, Deparmen of Mahemaics, SAS, Vellore Insiue of Technology, Vellore, India Associae Professor, Deparmen of mahemaics, S.V.Universiy, Tirupa,India Corresponding Auhor nsreddy.dr@gmail.com (Received Febraury 5, ; acceped July 9, 3) ABSTRACT We analyse he effecs of radiaion and roaion on unseady hydromagneic free convecion flow of a viscous incompressible elecrically conducing fluid pas an impulsively moving verical plae in a porous medium by applying inclined magneic field, Under Boussinesq approximaion, assuming ha he emperaure of he plae has a emporarily ramped profile. An exac soluion of he governing equaions, in dimensionless form is obained by Laplace ransform echnique. To compare he resuls obained in his case wih ha of isohermal plae and exac soluion of he governing equaions are also obained for isohermal plae and resuls are discussed graphically in boh ramped emperaure and isohermal cases. Keywords: MHD, Roaion, Radiaion, Inclinaion, Free convecion, Ramped emperaure.. INTRODUCTION The sudy of convecive hea ransfer from a solid body wih differen geomeries embedded in a fluid sauraed porous medium has varied and wide applicaions in many areas of science and engineering such as geohermal reservoirs, drying of porous solids, chemical caalyic reacors, hermal insulaors, nuclear wase reposiories, hea exchanger devices, enhanced oil and gas recovery, underground energy ranspor ec. Keeping he above facs Kumar and Varma () have sudied he radiaion effecs on MHD flow pas an impulsively sared exponenially acceleraed verical plae wih variable emperaure in he presence of hea generaion. The combined effecs of roaion and radiaion on MHD flow pas an impulsively sared verical plae wih variable emperaure was sudied by Rajpu and Kumar (). Chebbi and Bouzaiane () discussed he effecs of roaion on he passive scalar and kinemaic fields of homogeneous sheared urbulence. Jha and Ajibade () have sudied he unseady free convecive Couee flow of hea generaing/absorbing fluid. The unseady MHD hea and mass ransfer free convecion flow of polar fluids pas a verical moving porous plae in a porous medium wih hea generaion and hermal diffusion has been analyzed by Saxena and Dubey (). The effecs of hermal radiaion and free convecion currens on he unseady Couee flow beween wo verical parallel plaes wih consan hea flux a one boundary have been sudied by Narahari (). Effecs of hall curren and roaion on unseady MHD couee flow in he presence of an Inclined Magneic field was analyzed by Seh e al ().Vijayalakshmi () have sudied Radiaion effecs on free-convecion flow pas an impulsively sared verical plae in a roaing fluid. The effec of a uniform ransverse magneic field on he unseady ransien free convecion flow of an incompressible viscous elecrically conducing fluid beween wo infinie verical parallel plaes wih consan emperaure was sudied by Rajpu and Sahu (). Besman and Adjepong (988) discussed he magneo hydrodynamic free convecion flow, wih radiaive hea ransfer, pas an infinie moving plae in roaing incompressible, viscous and opically ransparen medium. Arpaci (968) analysed effecs of hermal radiaion on he laminar free convecion from a heaed verical plae. Sandeep and Sugunamma (3) analyzed aligned magneic field and chemical reacion effecs on flow pas a verical oscillaing plae hrough porous medium The presen paper deals wih he effecs of radiaion and roaion on unseady hydromagneic free convecion

2 N. Sandeep and V. Sugunamma / JAFM, Vol. 7, No., pp , 4. flow of a viscous incompressible elecrically conducing fluid pas an impulsively moving verical plae in a porous medium by applying inclined magneic field, Under Boussinesq approximaion, assuming ha he emperaure of he plae has a emporarily ramped profile. An exac soluion of he governing equaions, in dimensionless form is obained by Laplace ransform echnique. To compare he resuls obained in his case wih ha of isohermal plae and exac soluion of he governing equaions are also obained for isohermal plae.. MATHEMATICAL FORMULATION We consider an unseady hydromagneic free convecion flow of a viscous incompressible elecrically conducing fluid pas an impulsively moving infinie verical plae embedded in a porous medium. And consider he coordinae sysem in such a way ha he x-axis is aken along he plae in he upward direcion, y-axis normal o he plane of he plae in he fluid and z-axis perpendicular o xy plane. The fluid is permied by an inclined magneic field applied along he direcion of y- axis. Boh he fluid and plae roae in unison wih a permi uniform angular velociy abou y-axis. Iniially, a ime, boh he fluid and plae are a res and a a consan emperauret. A ime, he plae sars moving in x direcion wih uniform velociy U and he emperaure of he plae is raised or lowered o w T T T when, and hereafer, for, i is mainained a he consan emperaure T. Since he plae is infinie in x and z w direcions and is elecrically non-conducing wih all physical quaniies, excep pressure will be funcions of y and only. Taking ino consideraion he assumpions made above, he governing equaions for laminar free convecion flow of a viscous incompressible elecrically conducing fluid pas a verical plae in a uniform porous medium wih radiaive hea ransfer, under Boussinesq approximaion, in a roaing frame of reference are u u w g T T u BSin u y K () T k T q r cp y cp y where T, g,,,,, k, K, (3) c p and q r are respecively, emperaure of he fluid, acceleraion due o graviy, volumeric coefficien of hermal expansion, kinemaic coefficien of viscosiy, elecrical conduciviy, fluid densiy, hermal conduciviy, permeabiliy of porous medium, specific hea a consan pressure and radiaive flux vecor. Here applied magneic field. The iniial and boundary condiions are B B is u w, T T for y and, u U, w a y for, T T Tw T (4) a y for, T T y w a for,,, as for. u w T T y We now use Rosseland approximaion which leads o he value of radiaive hea flux q r as q r 4 3k T y 4, (5) Where k is mean absorpion coefficien and is Sefan-Bolzmann consan. I may be noed ha by using Rosseland approximaion we limi our analysis o opically hick fluids. Assuming small emperaure differences beween fluid emperaure T and free sream emperaure T, he Eq. (5) is linearized by 4 T in Taylors series abou free sream expanding emperaure T, afer neglecing second and higher order erms in T T i akes he form T 4T T 3T (6) Making use of Eqs. (5) and (6), Eq. (3) becomes T k T 6 T T c y c 3k y 3 p p Inroduce he following non-dimensional variables (7) w w y K u w BSin w () y / U, u u / U, w w / U, / and T T T Tw T, (8) The Eqs. (), () and (3) in he non-dimensional form are 76

3 N. Sandeep and V. Sugunamma / JAFM, Vol. 7, No., pp , 4. u u K w G T M u r K w w K K u M w R T T P r Where K / U is roaion parameer, M ' B / U is magneic parameer, M M ' Sin is inclined magneic field. K K U Is porosiy parameer, (9) 3 is Grashof number, G r g T w T U Pr cp k is Prandl number and R6 T 3 3kk Applying Laplace ransform echnique, he Eqs. () and () (4) wih he help of (5a) () reduces o d f sm ik f G T () r, d K () is radiaion parameer. According o above non s dimensionalizaion process, he characerisic ime o can f, s f, e d and be defined as s, T, s T, e d U. () () Making use of () he iniial and boundary condiions (5), in non-dimensional form, reduces o u w, T for and, u, w a for, (3) T a for, T a for, u, w, T as for. combining (9) and () in he form of f u iw, we obain f f. (4) (4) s ik f Mf f G rt e K f, s e e e s s s The iniial and boundary condiions (3) in combined form are f, T for and (5a) f a for (5b) M ik, K () T a for, (5c) (5c) Gr a, a. T a for, (5d) (5d) f, T as for. (5e) (), he exac soluion (5e) for he fluid emperaure 3. SOLUTION OF THE PROBLEM I is eviden from he Eqs. () and (4) ha he energy Eq. () is uncoupled from he Eq. (4). Therefore, firs we can obain he soluion for he fluid emperaure T, by solving Eq. () and hen using i in Eq. (4) he soluion for f, can be obained. dt ast (6) d, (7) where a Pr R and f, s and T, s are Laplace ransforms of, T, respecively defined by f and ( s Being Laplace ransform parameer). The boundary condiions (5b) o (5e) becomes s f / s, T e s a, (8) f, T as. The soluions of Eqs. (6) and (7), subjec o he boundary condiions (8) are given by s e as e, (9) s, T s where s s as, () Taking he inverse Laplace ransform of Eqs. (9) and T, and fluid velociy f, u, iw, are obained and expressed in he following form as 77

4 N. Sandeep and V. Sugunamma / JAFM, Vol. 7, No., pp , 4.,,, T P H P () f, Where e erfc e erfc,, F H F, (3) a a a P, erfc e (4) a 4 e erfc e F, e erfc e erfc 4 e erfc 4 e a a a e erfc e a a erfc erfc erfc a a a e a 4. (5) In he Eqs. () o (5), erfc x is he complemenary error funcion and H is he Heaviside uni sep funcion. 4. SOLUTION IN CASE OF ISOTHERMAL PLATE The analyical soluion for he fluid emperaure and velociy, represened by Eqs.() and (3) respecively, are obained for an unseady hydromagneic free convecion flow of a viscous incompressible elecrically conducing fluid near a verical moving plae wih ramped emperaure. In order o highligh he effecs of ramped emperaure disribuion wihin he plae on he fluid flow, i may be worhwhile o compare such a flow wih he one, near a moving plae wih uniform emperaure. Taking ino accoun he assumpions made in he presen sudy, he soluion for he fluid emperaure and velociy for he fluid flow near a verical moving isohermal plae is obained and expressed as T, f, a erfc (6) e erfc e erfc e e erfc e erfc a erfc e a a e erfc a a e erfc (7) Where /. 5. RESULTS AND DISCUSSION To sudy he effecs of radiaion, magneic field, roaion, porosiy of medium, inclined angle and ime on he flow-field he numerical values of fluid velociy are displayed graphically versus boundary layer coordinae in Figs. o for various values of magneic parameer M, roaion parameer K, Grashof number Gr, radiaion parameer R, porosiy parameer K,inclined angle and ime aking P r.7 (ionized air).here inclined angle is aken as velociy and emperaure profiles. for discussing I is noiced from Fig. ha an increase in he magneic parameer M leads o a decrease in he velociy for ramped emperaure and isohermal plaes. Bu iniially a isohermal plae velociy akes reverse acion. i.e. increase in magneic field causes an increase of velociy afer wards velociy decreases gradually in increase magneic field. This is due o he fac ha he applicaion of an inclined magneic field o an elecrically 78

5 F(),f() N. Sandeep and V. Sugunamma / JAFM, Vol. 7, No., pp , 4. conducing fluid gives rise o a resisive force which is known as Lorenz force. From Fig. I is eviden ha, for boh ramped emperaure and isohermal plaes, an increase in roaion parameer K leads o a decrease in he velociy near he plae. From Fig.3 i is found ha he velociy of he fluid a ramped emperaure increases an increase in Grashof number Gr bu in an isohermal plae, velociy varies in cerain ime and follows ramped profile. Fig.4 ha he velociy increases by increase in he radiaion parameer R,bu a ramped emperaure velociy decreases wih an increase in he radiaion. Fig. 5 shows ha he effec of he fluid velociy a ramped emperaure and an isohermal plaes, for differen values of porosiy parameer K. I is observed ha an increase in porosiy parameer causes an increase in velociy. I is revealed from Fig.6 ha he velociy decreases wih an increase in he ime a isohermal plae bu a ramped emperaure, velociy increases by increasing he ime. Figs. 7 and 8 show he effec of inclined magneic field over velociy. From hese i is observed ha a ramped emperaure, velociy of he fluid decreases by an increase in inclined angle. In he similar manner a isohermal plae also, velociy varies iniially and hen i is decreased by increase in an inclined angle. In order o have physical view of fluid emperaure, he profiles of fluid emperaure are drawn versus boundary layer coordinae in Figs. 9 o for various values of radiaion parameer R, Prandl number Pr and ime.i is eviden from Fig. 9 ha he fluid emperaure T increases by an increase in he radiaion parameer a ramped emperaure bu i is reversed in case of isohermal plae. From Fig., i is observed ha he fluid emperaure T increases on increasing Prandl number Pr for boh ramped emperaure and isohermal plaes. I is eviden from Figs. and shows ha an increase in he ime causes he decrease he fluid emperaure in boh cases..9.8 Ramped Temperaure... Isohermal M=,3,5,7 M=,3,5,7.. M=,3,5, Fig.. Velociy field for differen values of M. When K =, G r =5, R=3, K =. and = 79

6 N. Sandeep and V. Sugunamma / JAFM, Vol. 7, No., pp , 4.. Ramped Temperaure... Isohermal. 8 F(),f(). 6 K =9,6,5, Fig.. Velociy field for differen values of K.When M=4, G r =5, R=3, K = and = Gr=,3,5,8 Ramped Temperaure... Isohermal F(),f( Gr=,3,5,8 Gr=,3,5, Fig. 3. Velociy field for differen values of Gr. When K =, M=, R=, K =. and = 8

7 F(),f() N. Sandeep and V. Sugunamma / JAFM, Vol. 7, No., pp , 4.. Ramped Temperaure... Isohermal R=,4,6,8 R=,4,6, Fig. 4. Velociy field for differen values of R. When K =, G r =5, M=, K =. and =.9 Ramped Temperaure... Isohermal F(),f() K Fig. 5. Velociy field for differen values of K. When K =4, M=5, R=, G r = and = 8

8 N. Sandeep and V. Sugunamma / JAFM, Vol. 7, No., pp , 4..9 Ramped Temperaure... Isohermal =.8,.6,.4,.3 F(),f() =.3,.4,.6, Fig. 6. Velociy field for differen values of. When K =4, M=, R=, K =. and G r =5.5.95,,, F() Fig. 7. Velociy field for differen values of θ.when K =, M=, R=3, K =., G r =5 8

9 T N. Sandeep and V. Sugunamma / JAFM, Vol. 7, No., pp , ,,, f() Fig. 8. Velociy field for differen values of θ.when K =, M=, R=3, K =., G r =5 and = Ramped emperaure... Isohermal R=,4,6, R=8,6,4, Fig. 9. Temperaure field for differen values of R. When K=4, M=, R=, K=., Gr=3 and =.5 83

10 T T N. Sandeep and V. Sugunamma / JAFM, Vol. 7, No., pp , 4.. Ramped Temperaure... Isohermal Pr=.4,3,7, Fig.. Temperaure field for differen values of Pr.When K =4, M=, R=, K =., G r =3 and = Ramped Temperaure =.,.,.3, Fig.. Temperaure field for differen values of in Ramped emperaure.when K =4, M=, R=, K =., G r =3 84

11 T N. Sandeep and V. Sugunamma / JAFM, Vol. 7, No., pp , Isohermal.6.5 =.,.,.3, Fig.. Temperaure field for differen values of in Isohermal plae.when K =4, M=, R=, K =., G r =3 6. CONCLUSIONS The presen sudy invesigaes he unseady hydromagneic free convecion boundary layer flow of a viscous incompressible elecrically conducing fluid pas a ramped emperaure impulsively moving plae in a roaing porous medium in he presence of inclined magneic field and hermal radiaion. The significan findings are summarized as: For boh ramped emperaure and isohermal plaes: Magneic field ends o reduce fluid flow in boh he ramped and isohermal cases. Increase of radiaion roaion decreases he fluid flow in he isohermal plae whereas i increases he fluid flow in he ramped emperaure. Roaion, Inclined magneic field angle and ime decrease he fluid flow in boh cases. Prandl number has endency o increase he fluid emperaure. Radiaion parameer casus he decrease of emperaure in ramped emperaure case bu i reversed in isohermal case. Here inclinaion angle causes he increase and decrease of magneic field effec so a i acs like ransverse magneic field. REFERENCES Arpaci V.S (968). Effecs of hermal radiaion on he laminar free convecion from a heaed verical plae, In. J. Hea Mass Transfer,Vol., pp Besman A.R. and Adjepong S.K (988).Unseady hydromagneic free-convecion flow wih radiaive hea ransfer in a roaing fluid, Asrophysics and Space Science, Vol. 43, pp Chebbi. B and M. Bouzaiane (). On he Effecs of Roaion on he Passive Scalar and Kinemaic Fields of Homogeneous Sheared Turbulence. Journal of Applied Fluid Mechanics, Vol. 5, No., pp , Jha B.K., A.O. Ajibade (). Unseady free Convecive Couee flow of hea Generaing/absorbing fluid.in. J. Energy and Tech., (), 9. Kumar A.G.V., S.V.K Varma (). Radiaion effecs on MHD flow pas an impulsively sared exponenially acceleraed verical plae wih variable emperaure in he presence of hea generaion.in. J. Eng. Sci.Tech.,3(4),p 897. Narahari M., (). Effecs of hermal radiaion and free convecion currens on he Unseady Couee flow beween wo verical parallel plaes wih consan hea flux WSEAS Transacions on Hea and Mass Transfer, 5(), -3 Rajpu U.S.,S. Kumar (). Combined effecs of roaion and radiaion on MHD flow pas an impulsively sared verical plae wih variable emperaure In. J. Mah. Analysis,5(4),

12 N. Sandeep and V. Sugunamma / JAFM, Vol. 7, No., pp , 4. Rajpu U.S., P.K. Sahu ().Effec of a uniform ransverse magneic field on he unseady ransien free convecion flow of an incompressible viscous elecrically conducing fluid beween wo infinie verical parallel plaes wih consan emperaure In. J. Mah. Analysis,5(34), Sandeep and Sugunamma (3) Aligned Magneic Field And Chemical Reacion Effecs On Flow Pas A Verical Oscillaing Plae Through Porous Medium Communicaions in applied sciences, Vol.,Number,3,8-5. Seh G.S.R. Nandkeolyar and Md.S. Ansari ().Effecs of Hall Curren and Roaion on Unseady MHD Couee Flow in he Presence of aninclined Magneic Field. Journal of Applied Fluid Mechanics, Vol. 5, No.,pp , Vijayalakshmi A.R(). Radiaion effecs on freeconvecion flow pas an impulsively sared verical plae in a roaing fluid Theore. Appl. Mech., Vol.37, No., pp.79 95, Saxena S. S, G.K. Dubey ().Unseady MHD hea and mass ransfer free convecion flow of polar fluids pas a verical moving porous plae in a porous medium wih hea generaion and hermal diffusion Adva. Appl. Sci. Res., (4),