Hall Effect on Transient MHD Flow Past. an Impulsively Started Vertical Plate in a Porous. Medium with Ramped Temperature, Rotation

Transcription

1 Applied Mahemaical Sciences, Vol. 7, 3, no. 5, HIKARI Ld, Hall Effec on Transien MHD Flow Pas an Impulsively Sared Verical Plae in a Porous Medium wih Ramped Temperaure, Roaion and Hea Absorpion N. Ahmed and K. Kr. Das Deparmen of Mahemaics, Gauhai Universiy, Guwahai saheel_nazib@yahoo.com kishoredas969@yahoo.in Copyrigh 3 N. Ahmed and K. Kr. Das. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac An exac soluion o he problem of an unseady MHD free convecive flow pas an impulsively sared verical plae wih Ramped emperaure in a porous medium wih roaion and hea absorpion aken in o accoun he Hall effec is presened. I is assumed ha he enire sysem roaes wih a uniform angular velociy Ω r abou he normal o he plae and a uniform ransverse magneic field is applied along he normal o he plae direced in o he fluid region.the magneic Reynolds number is assumed o be so small ha he induced magneic field can be negleced. The non dimensional equaions governing he flow are solved by Laplace ransform echnique in closed form. The expression for primary and secondary velociy fields, emperaure field, and skin fricions a he plae due o he primary and secondary velociy fields and Nussel number a he plae are obained for boh ramped emperaure and isohermal plaes. The velociy fields, and skin fricions a he plae are demonsraed graphically and he effec of magneic parameer (M), Hall parameers (m), on hese fields are discussed. Some of our resuls for m= has been compared wih he resuls of Seh e al. (Seh e al.) and found in good agreemen. Keywords: Hall Effec, MHD, Roaion, Skin-fricion

2 56 N. Ahmed and K. Kr. Das. INTRODUCTION Many engineering problems are suscepible o MHD analysis.the sudy of MHD flow problems has achieved remarkable ineres due o is applicaion in MHD generaors, MHD pumps and MHD flow meers ec. The sudy of effecs of magneic field on free convecion flow is imporan in liquid meals, elecrolyes and ionized gases.geophysics encouners MHD phenomena in ineracion on conducing fluids and magneic fields. The roaing flow of an elecrically conducing fluid in presence of magneic field has go is imporance in Geophysical problems. The sudy of roaing flow problems is also imporan in he solar physics dealing wih he sunspo developmen, he solar cycle and he srucure of roaing magneic sars. I is well known ha a number of asronomical bodies possess fluid ineriors and magneic fields. Changes ha ake place in he roaion, sugges he possible imporance of hydro magneic spin-up. The general heory of roaing fluids has received growing ineres during las decade because of is applicaion in cosmic and geophysical science. In his regard we may cie down he works done by Rapis (983). MHD in he presen form is due o pioneer conribuion of several noable auhors like Alfven (94), Cowling (957). I was emphasized by Cowling ha when he srengh of he applied magneic field is sufficienly large, Ohm s law needs o be modified o include Hall curren. The Hall effec is due merely o he sideways magneic force on he drafing free charges.the elecric field has o have a componen ransverse o he direcion of he curren densiy o balance his force. In many works of plasma physics, i is no paid much aenion o he effec caused due o Hall curren. However, he Hall effec can no be compleely ignored if he srengh of he magneic field is high and number of densiy of elecrons is small as i is responsible for he change of he flow paern of an ionized gas. Hall effec resuls in a developmen of an addiional poenial difference beween opposie surfaces of a conducor for which a curren is induced perpendicular o boh he elecric and magneic field. This curren is ermed as Hall curren. I was discovered in 979 by Edwin Herber Hall while working on his docoral degree a he Johns Hopkins Universiy in Balimore, Maryland, USA. Pop (97), Dua e al. (976), Malique and Saar (5) and Ahmed e al. (), have presened some model sudies on he effec of Hall curren on MHD convecion flow because of is possible applicaion in he problem of MHD generaors and Hall curren. Recenly Ahmed and Sarmah () have carried ou an invesigaion of MHD ransien flow pas an impulsively sared infinie horizonal porous plae in a roaing sysem wih Hall curren. Due o imporance of sudying MHD flow problems in roaing fluid wih Hall curren, we have proposed in he presen paper o invesigae Hall effec on an unseady MHD free convecive flow pas an impulsively sared verical plae wih ramped emperaure in a porous medium wih roaion and hea absorpion.the presen work is an exension o he work sudied by G.S.Seh e al. (G.S.Seh e al. ()) o consider he Hall effec on he flow and ranspor characerisics.

3 Hall effec on ransien MHD flow 57. BASIC EQUATIONS We now consider an unseady hydromagneic laminar naural convecion boundary layer flow of a viscous incompressible elecrically conducing and heaabsorbing fluid pas an impulsively sared infinie verical plae embedded in a porous medium aking in o accoun of Hall curren in presence of a uniform magneic field. X u = w =, u = U,w = T= T fory, a y = for f î u w y T= T + ( Tw T ) ay= for p T= T ay= for f B r j w o Transverse magneic field ˆk Z Ω r g v Secondary flow direcion Primary direcion u w T T O Fig- Flow configuraion We inroduce he coordinae sysem ( x, y, z) wih X axis 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 as well as he plae is in a sae of rigid body roaion wih a uniform angular velociy Ω r abou Y-axis. Iniially a ime, boh he fluid and he plae were a res and a a uniform emperaure T. A ime >, he plae sars moving in X direcion wih uniform velociy U and he emperaure of he plae is raised or lowered o T + ( Tw T ) when and here afer i is mainained a uniform emperaure T w when > ( being he characerisic ime ). Le q r = $ iu+ ˆ jv+ kw $ be he fluid velociy, J = J i+ J j+ J k$ P x,y,z, and $ $ denoe he curren densiy a poin x y z r ĵ j Y

4 58 N. Ahmed and K. Kr. Das r B= B $ j be he applied magneic field, $$ i,j,k$ being he uni vecors along uuur uuur uuur OX, OY and OZ respecively. As he plae is infinie in X and Z direcion, all physical quaniies excep possibly he pressure are funcions of y and only. v The equaion of coninuiy. q r = is rivially saisfied wih u = u( y, ), w = w( y, ) () The equaions governing he flow are u u υu σ B u+ mw + Ω w = υ + g ( T T ) y K ρ + m ( ) w w σb mu w υw Ω u = υ + y ρ + m K T T Q =α ( T T ) (4) y ρcp The iniial and boundary condiions for he fluid flow problem are: u = w =, T = T for y and u = U, w = a y = for > T = T + ( Tw T ) ay = for< (5) T = Tw ay = for > u, w, T T as y for > In order o express he governing equaions () o (4) and iniial boundary condiions (5) in dimensionless form, he following non dimensional quaniies and parameers are inroduced. y Ωυ σb υ υ η=, ( u,w) = ( u,w ), =, K =, M =, P r =, U U U ρu α (6) T T KU gυ( Tw T ) υ υq T =, K =, G r =,P 3 r =,and Q= Tw T υ U α ρcpu In view of (6), he equaions () o (4) in non dimensional form, reduce o u u M u + K w = mw+ u + G r T (7) η + m K w w M w K u = ( mu w) ( m ) K η + T T = QT P η r () (3) (8) (9)

5 Hall effec on ransien MHD flow 59 According o he above non dimensionalizaion process, he characerisic ime υ can be defined as = () U Making use of (6) and (), he iniial boundary condiions (5), in non dimensional form become u = w =, T = for η and u =, w =, a η= for > T = a η= for < () T = a η= for > u,w,t asη for > 3. METHOD OF SOLUTION We inroduce a new complex variable f defined by f = u+ i v where i= The non dimensional form of he equaions governing he flow can be rewrien as follows: f f M ( im) = ik + f + G r T () η ( + m ) K T T = QT (3) Pr η The iniial and boundary condiion in combined form are f =, T = for η and f = a η= for > T = a η= for < (4) T = a η= for > f, T as η for > On aking he Laplace-Transforms of he equaions () and (3), he following differenial equaions are obained d f ( λ+ s) f = Gr T (5) d η d T P r ( s + Q ) T = (6) d η subjec o he boundary condiions: f = a η= and f = a η (7) s

6 53 N. Ahmed and K. Kr. Das s T = ( e ) a η= and T = a η (8) s M ( im) where, λ= ik +,T = L{ T ( η,)}, f = L{ f ( f,)} + m K The soluion of he equaions (5) and (6) under he condiions (7) and (8) are s e ( s+ Q) Pr η T = e (9) s G r s ( λ + s) η f = + + ( e ) e + s Pr s s ( s ) G r ( s ) ( s+ Q) Pr η + e e () Pr s s s ( Pr Q λ) Where, = Pr Taking inverse Laplace ransforms of he equaions (9) o () we derive he fluid emperaure and fluid velociy fields as follows: T η, = f f () f( η,) = φ α Ω Ω () Gr Where, α= Pr Q λ f = f Pr,Q, η,, f = f Pr,Q, η, H, φ =φ λ, η,, e Ω = φ3 φ4 φ φ e Ω = φ φ φ φ ( f f ) ( f f ) 3 4 ( Q,,Pr,), ( Q,,Pr, ) H( ), (,,,) φ =φ η φ = φ η φ =φ +λ η 3 φ = e φ +λ, η,, H, φ =φ + Q, η,pr,, 3 4 φ 4 = e φ + Q, η,pr, H f = f, λ, η,, f = f, λ, η, H (3)

7 Hall effec on ransien MHD flow 53 xzy y z xzy y z φ ( x, y,z, ) = e erfc x e erfc x + + f ( x,y,z,) z x xyz z x + e erfc y y + + = z x xyz z x e erfc y y In he equaions () o (), erfc(x) is complemenary error funcion and H (-) is he Heaviside uni sep funcion. 4. SOLUTION IN CASE OF ISOTHERMAL PLATE In order o highligh he effecs of he ramped emperaure disribuion near a verical plae, i may be imporan o compare he effecs of he isohermal emperaure disribuion for he fluid flow. The emperaure and he velociy for he fluid flow near a isohermal plae can be expressed as T η, =φ (4) f( η,) =φ +α φ φ αe φ φ (5) SKIN FRICTION AND NUSSELT NUMBER The expressions for he skin fricion and Nussel number, which are he measures of shear sress and rae of hea ransfer a he plae respecively, are presened in he following form for ramped emperaure ( τ + i τ ) = ω α ξ ξ (6) x y Nu = ψ ψ (7) where, ψ =ψ( Pr,Q, ), ψ =ψ( Pr,Q, ) H( ), ψ = ψ(, λ,), ψ = ψ, λ, H, ω =ω, λ,, ω =ω, λ, H, (,, ), e (,, ) H( ) ( Pr,Q, ), e ( Pr,Q, ) H( ) ( Pr,Q,) ( Pr,Q, ) H( ) ω =ω λ+ ω = ω λ+ ω =ω + ω = ω ω =ω ω =ω 4 4

8 53 N. Ahmed and K. Kr. Das e ξ = ω ω ω ω ψ ψ 3 4 e ξ = ω ω ω ω ψ ψ 3 4 x x ψ ( x, y, ) = erf ( y ) + x y erf ( y ) + e y π x y ω ( x,y,) = xyerf( y) + e π where τ x and τ y are respecively, primary and secondary skin fricions and Nu is Nussel number. y 6. RESULT AND DISCUSSIONS In order o ge physical insigh in o he problem we have carried ou numerical calculaions for non-dimensional velociy field for boh ramped emperaure and isohermal plaes and skin fricion a he plae for ramped emperaure for differen values of physical parameers involved and hese values have been demonsraed in graphs. Our invesigaion is resriced o =.7 and Pr =.7 and oher parameers namely Hall parameer (m), roaion parameer ( K ), permeabiliy parameer ( K ), Grashof number (Gr) and hea absorpion coefficien (Q) have been considered arbirarily. I is noiced from figures o 5 ha for boh ramped emperaure and isohermal plaes he primary velociy u and secondary velociy w aain a disincive maximum value in he viciniy of he plae surface and hen decrease properly wih increase in boundary layer co-ordinae η o approach he free seam value. Figures and 3 exhibi he variaion of primary velociy u and secondary velociy w under he influence of Hall parameer (m) and i is seen ha for boh ramped emperaure and isohermal plaes he primary and secondary moions are acceleraed due Hall curren. Figures 4 and 5 display he effec of magneic parameer (M) on primary and secondary velociies. I is seen from hese figures ha he primary as well as secondary velociy falls when M increases. Tha is he primary or secondary fluid moion is rearded due o applicaion of ransverse magneic field. This phenomenon clearly agrees o he fac ha Lorenz force ha appears due o ineracion of he magneic field and fluid velociy resiss he fluid moion. Figures 6 o 7 demonsrae he behaviors of primary and secondary skin fricion versus Hall parameer (m) for ramped emperaure under he influence of magneic parameer (M).

9 Hall effec on ransien MHD flow 533 Figure 6 shows ha when he srengh of applied magneic field is increased he primary skin fricion τ x increases and falls due o Hall effec. I is observed from figure 7 ha iniially, he magniude of secondary skin fricion τ y decreases under he effec of magneic parameer (M). However for larger values of Hall parameer (m), he above rend is reversed i.e τy increases due o he increasing values of magneic parameer (M). In oher words he imposiion of he applied magneic field causes he magniude of he drag force o decrease iniially and he behavior akes a reserve rend for higher values of Hall parameer (m). 7. CONCLUSIONS Our resuls of invesigaion may be summarized o he following imporan conclusions i) The primary and secondary moion is rearded under he effecs of ransverse magneic field whereas his moion is acceleraed under Hall effec for boh ramped emperaure and isohermal plaes. ii) The skin fricion τ x (drag force due o primary velociy) rises wih increasing values of magneic parameer (M) for ramped emperaure. iii) The magniude of he secondary skin fricion τ y (drag force due o secondary velociy) shows a growh for increasing magneic parameer (M). u Ramped Isohermal----- m=.5,.5, η - w.3 Ramped.5 Isohermal m=.5,.5, η.5.5 Fig- Primary velociy profiles when M=4, K =,Gr = 4,K =.,Q = Fig-3 Secondary velociy profile when K =,Gr = 4,K =.,Q =,M = 4

10 534 N. Ahmed and K. Kr. Das u Ramped Isoherma M=4,6,8.5 η.5.5 w Ramped Isohermal M=4,6, η Fig-4 Primary velociy profiles when K =,Gr = 4,K =.,Q =, m =.5 Fig-5 Secondary velociy profiles when K =,Gr = 4,K =.,Q =, m = > τ x M=4,6, m τ -.64 y m M=4,6,8 Fig-6 Primary skin fricion τ x versus m m for Q=, Gr=4, K,K. = = m for Q=, Gr=4, Fig-7 Secondary skin fricion τ y versus K =,K =. 8. NOMENCLATURE ρ, Fluid densiy ; p, Pressure ; B, Srengh of applied magneic field ; g, Acceleraion due o graviy ;, The volumeric coefficien of solual expansion;,the volumeric coefficien of hermal expansion ; μ Co-efficien of viscosiy ; K,Permeabiliy of porous medium ; C p,specific hea a consan pressure ; Q, Hea absorpion coefficien ;

11 Hall effec on ransien MHD flow 535 k, Thermal conduciviy ; σ,elecrical conduciviy ; Pr, Prandl number ; m, Hall parameer ; M, Magneic parameer ; K, Non dimensional permeabiliy parameer ; Q, Non dimensional hea absorpion parameer ; K, Non dimensional roaional parameer ; G r,grashof number for hea ransfer ; Q, Non dimensional hea absorpion ; υ, Kinameic viscosiy ;, Characerisic ime ; T, Non dimensional fluid emperaure ; ( x, y, z ), ( x,y,z ) The coordinaes in hree dimension ; ( u,v,w ), ( u,v,w ) The velociy componens along ( x, y, z ) /( x,y,z ); U Plae velociy ; T w Reference emperaure ; T Temperaure far away from he plae ; α, Fluid hermal diffusiviy. REFERENCES [] Rapis, A. (983): Mass ransfer and free convecion hrough a porous medium by he presence of a roaing fluid, In. Comm. Hea and Mass ransfer, (), 4. [] Alfven, H. (94): Discovery of Alfven Waves, Naure 5, 45. [3] Cowling, T. G. (957): Magne hydrodynamics, Wiley Iner Science, New York. [4] Pop, I. (97): The effec of Hall currens on hydromagneic flow near an acceleraed plae, J. Mah. Phys. Sci, 5, [5] Daa, N. and Jana, R.N.(976): Oscillaory magneo hydrodynamics flow pas a fla plae wih Hall effecs, J. Phys. Soc. Japan, 4, 469. [6] Malique, M.A., and Saar, M.A. (5): The effecs of variable properies and Hall curren on seady MHD laminar convecive fluid flow due o a porous roaing disk, In. J. Hea and Mass Transfer, 48(3-4), [7] Ahmed, N., Kalia, H. and Baruah, D.P.(): Unseady MHD free convecive flow pas a verical porous plae immersed in a porous medium wih Hall curren, hermal diffusion and hea ransfer, In. J. Engg. Science and Technology, (6), [8] Ahmed, N. and Sarmah, H.K. (): MHD ransien flow pas an impulsively sared horizonal porous plae in a roaing sysem wih Hall curren, In. J. of. Appl. Mah and Mech, 7(), -5.. [9] Seh, G.S., Nandkeolyar, R. and Ansari, M.S. () : Effec of roaion on unseady hydromagneic naural convecion flow pas an impulsively moving verical plae wih ramped emperaure in a porous medium wih hermal diffusion and hea absorpion, In. J. of Appl. Mah and Mech, 7(), 5-69,. Received: January 7, 3