LINEAR STABILITY ANALYSIS OF A PLANE-POISEUILLE HYDROMAGNETIC FLOW USING ADOMIAN DECOMPOSITION METHOD

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1 .P.B. Si. Bull., Series A, Vol. 75, Iss., 13 ISSN LINEAR STABILITY ANALYSIS OF A PLANE-POISEILLE HYDROMAGNETIC FLOW SING ADOMIAN DECOMPOSITION METHOD Samuel O. ADESANYA 1 I this paper, the small-disturbaes stability of plae-poiseuille flow of a eletrially odutig fluid i the presee of a trasverse mageti field is studied. sig the mode approah, the fourth order differetial equatio for the flow is derived ad solved aalytially usig Adomia deompositio method (ADM) ad the result obtaied showed a good agreemet with previously obtaied result by Multidek asymptoti tehiques. Keywords: ADM, Mageto-Hydrodyamis, Stability Aalysis, Hartma s Number 1. Itrodutio Adomia deompositio method [1, 5, 1-16] was developed i the 8 s. It possesses great potetial i solvig differet kids of differetial equatios. The major stregth of the method is that it avoids liearizatio, trasformatio ad disretizatio. The mai objetive of this paper is to apply Adomia deompositio method (ADM) to study the temporal stability aalysis for hydromageti Plae Poiseuille flow. Now osider the stadard operator Lu + Ru + Nu = g, (1.1) Where u is the ukow futio, L is the highest order derivative, whih is assumed easily ivertible, R is a liear differetial operator of order less tha L, Nu represets the oliear terms, ad g is the soure term. Applyig the iverse 1 operator L to both sides of (1.1) ad usig the give oditios we obtai 1 1 ( y) L ( Ru) L ( Nu) u = h, (1.) where h ( y) represets the terms arisig from itegratig the soure term g ad from the boudary oditios, The stadard ADM defies the solutio u by the series = u u, (1.3) = moreover, the oliear term series 1 PhD., Departmet of Mathematial Siees, Redeemer s iversit Redemptio Cit Nigeria, adesayaolumide@yahoo.om

2 1 Samuel O. Adesaya where = Nu A, (1.) = A are the Adomia polyomials determied formally from the relatio 1 d! = i A = N λ ui. (1.5) dλ i λ= If the oliear term is expressed as a oliear futio f(u), the Adomia polyomials are arraged ito the form A = f u ( ) () 1 f ( u ) A1 = u1 () 1 1 ( A ( ) ) = u f u + u1 f ( u ) (1.6)! () 1 ( ( ) ) 1 3 ( 3 A ( ) ) 3 = u3 f u + u1u f u + u1 f ( u ) 3!... The ompoets u u,,... are the determied reursively by usig the relatio ( y), 1 u u = h, (1.7) 1 1 uk + 1 = L Ruk L Ak, k where u is referred to as the zeroth ompoet. The partial sum of the series is thus obtaied as k ( y ) = u ( y ) u = (1.8) The overgee results for the Adomia deompositio method has bee studied extesively i the works by Cherraultz et al [6, 7 ad 8].. Mathematial formulatio Cosider the flow of a eletrially odutig visous iompressible fluid through a two dimesioal hael uder the ifluee of a trasverse mageti field. The goverig equatios are []; u + = (.1) u u u p 1 u u + u + v = + H u t x y x Re + x y (.)

3 Liear stability aalysis of a plae-poiseuille hydromageti flow 11 p 1 v v + u + v = + + (.3) t Re together with the boudary oditios u ( + 1 ) =, v( + 1) = (.) The flow quatities i equatio (.1)-(.) have bee o-dimesioalized as follows: u u v x y p σ B e =, v =, x =, y =, p =, t =, Re =, H = (.5) a a ρ a ν ρ Where x ad y are the streamwise ad ormal oordiates respetivel u ad v are the streamwise ad ormal veloity respetivel t-time, p pressure, Re Reyolds umber ad H is the Hartma s umber, is the harateristi veloity of the fluid, a is the harateristi half width of the hael, υ is the kiemati fluid visosity ad ρ is the fluid desit σ e is the fluid eletrial odutivity. The equatio ad the boudary oditios for the basi flow are d H = A; ( ± 1) = (.6) dy By Adomia deompositio, equatio (.6) admits a series solutio of the form ( y) ( y) = = usig (.7) i (.6) we obtai the zeroth ompoet as y y ( y) = a t a a (.7) Adydy (.8) While other ompoets a be easily obtaied usig the reursive relatio y y + 1 ( y) = H dydy (.9) obtaiig few terms of (.9) we get

4 1 Samuel O. Adesaya A ( y) = a y y y 1( y) = ah AH! (.1) 6 y y ( y) = ah AH! 6! y 6 y 3( y) = ah AH. 6! 8! Summig up (.1) leads to the partial sum 3 = ( y ) (.11) as the approximate solutio. sig () 1 =, the ukow ostat is determied to be 6 1 H H H A + + +!! 6! a = (.1) 6 H H 1+ H + +! 6! Therefore, the series overges to the exat solutio obtaied i [] A Cosh ( ) ( Hy) ( ) y = 1 (.13) H Cosh H If we assume that < H << 1 the (.13) leads to A AH ( H << 1) = ( 1 y ) + ( 5 + 6y y ) (.1) AH ( 61 75y + 15y y ) + O( H ) 7 3. Computatioal Approah By Squire Theorem [3,, 9, 17], we impose a -Dimesioal disturbae i the form u x, t = y + u' x, t, v ( ) ( ) ( ) t) = + v' t), t) = P( x) + p' t), p (.15)

5 Liear stability aalysis of a plae-poiseuille hydromageti flow 13 is the solutio of the basi flow equatio (.13) ad u ', v', p' are the small disturbaes, substitutig (.15) ito (.1)-(.3) ad egletig all quadrati terms, we get Where ( y) u' ' + = u' u' p' 1 u' u' + + v' = + + H u' (.16) t Re ' ' p' 1 v' v' + = + + t Re We ow seek a mode solutio i the form ( x y t i ( x t ) i ( x Rt ) It ) ( y ) e α ( y ) e α e α Ψ,, = φ = φ (.17) Where = R + ii is omplex valued futio ad α is real, it is lear from (.17) that whe I > the disturbae grows ad the flow beome ustable. For I < the disturbae deays ad the flow beome stable ad eutrally stable whe I =. Additioally R > ehaes the flow stabilit So that the veloity ompoets a be obtaied as iα x t u' x, t = φ' y e ( ) ( ) ( ) iα ( x t ) v' t) = iαφ( y) e iα ( x t p' t) = h( y) e ). Puttig (.18) i (.16) ad elimiatig p ' t) ordiary differetial equatio iv φ = α iα Re φ'' + iα + H (.18), we obtai the fourth order ( ) ( ) Reφ'' 3 + ( iα Re α ) φ iα Re( α + '')φ (.19) subjet to the boudary oditios φ ( 1 ) = φ' ( 1) = (.) φ '() 1 = φ() 1 = (.1) I the limitig ase as H, equatio (.19) redues to the well-kow Orr- Sommerfield equatio. By ADM the solutio of (.19) - (.1) a be writte as

6 1 Samuel O. Adesaya ( ) ( y) + 1 y y y y 1111 y y y y y ϕ y = bdydy+ bdydydy ϕ = dϕ dϕ α iαr H iα R iαr α ϕ iαr α ϕ dydydydy dy dy ( ) ( + + ) + ( 3 ) ( + '') (.) where the ukow ostats are to be evaluated usig the boudary oditio (.1). To obtai the eigevalues of the approximate solutio, the partial sum φ k ( y ) = ( y ) φ = is solved usig the boudary oditios (.1). This returs two equatios as futios ofb adb 1. sig Mathematia, the two ostats are elimiated, ad we obtai the followig results for the wave speed () whe k =. The umerial results of (.) are show as Tables 1-3 for differet parameter values.. Results ad Disussio Table 1 shows the effet of a irease i Hartma s umber o the flow stability. The result shows that the value of i redues with a irease i Hartma s umber i a quadrati maer, this is true due to the retardig effet of Loretz fores o the flow applied aross the hael. Therefore, ireasig mageti field itesity ehaes the flow stability. This behaviour validates the previously obtaied result by []. Table 1 Computatio showig variatios i wave speed α = 1, Re = 1 H r i i i i i i i i i i i i

7 Liear stability aalysis of a plae-poiseuille hydromageti flow 15 Computatio showig variatios i wave speed α = 1, H = 1 Re 1, i, i 3, i, i 5, i 6, i 7, i 8, i 9, i 1,, i 1,,, i r i Table Computatio showig variatios i wave speed H = 1, Re = 1 α r i i i i i i i i i i i i Table 3 I Table, it is observed that as the Reyolds umber ireases there is irease i the r while there is derease i the value of i this brigs about istability. Fiall Table 3 shows that both r ad i osillates with irease i the wave umber α. 5. Colusio I this paper, the ADM is used to study the temporal developmet of small disturbaes i hydromageti fluid flow. The riteria for the oset of istability have bee preseted theoretially ad ofirmed aalytially. It is observed that irease i Hartma s umber stabilizes the flow while the Reyolds umber has destabilizig effet o the flow.

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