Unsteady Transient Couette and Poiseuille Flow Under The effect Of Magneto-hydrodynamics and Temperature

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1 J. Appl. Envion. Biol. Sci., 5(7) , 5 5, TexRoad Publicaion ISSN: Jounal of Applied Envionmenal and Biological Sciences Unseady Tansien Couee and Poiseuille Flow Unde The effec Of Magneo-hydodynamics and Tempeaue Taza Gul, Mubaak Jan, Zahi Shah, S. Islam, M.A. Khan Mahemaics Depamen, Abdul Wali Khan Univesiy Madan, KPK Pakisan. Mahemaics Depamen, ISPaR/Bacha Khan Univesiy Chasadda, KPK Pakisan. Received: Mach 9, 5 Acceped: May 9, 5 ABSTRACT Cuen aicle deal wih he sudy of non-newonian Couee and Poiseuille flow beween wo peiodically and paallel oscillaing veical plaes. The unseady consiuive equaions of diffeenial ype have been used. Unifom magneic field is applied pependiculaly o he flow field beween paallel plaes. The effec of empeaue is also involved in he flow field. Two diffeen vaieies of flow poblems have been modelled in ems of non-linea paial diffeenial equaions wih some physical condiions. Opimal Homoopy Asympoic Mehod (OHAM) and Adomian Decomposiion Mehod (ADM) have been used o obain he analyical soluion of he modelled paial diffeenial equaions. These mehods ae used usually due o is emendous esuls fo solving nonlinea diffeenial equaions aise in vaious applied and engineeing sciences. In his wok he excellen ageemen of hese wo mehods is analyzed numeically and gaphically. Effec of diffeen modeled paamees on velociy and empeaue fields has been sudied numeically and gaphically. KEYWORDS: Unseady Second Gade Fluid, MHD, Tempeaue, Veicals plaes, (OHAM) and (ADM). INTRODUCTION Non-Newonian fluids have go gea impoance gea in he field of eseach, especially in applied and bio mahemaics, indusy and engineeing poblems. Examples of such ypes of fluids ae plasic ade, food pocessing, movemen of biological fluids, wie and fibes coaing, pape poducion, gaseous diffusion anspiaion cooling, dilling mud, hea pipes ec. Seveal complex fluids such as polyme mels, pain, shampoo, mud, kechup, blood, ceain oils and geases, and many emulsions ae involved in he class of non-newonian fluids. Due o indusial and echnological usages non-newonian fluids have become is significan pa. Tha s why eseaches ake a gea inees in i. Islam e al. [] sudied Couee and Poiseuille flow and hee genealized fom unde he effec of hea analysis. Fo he soluion of he poblem hey used OHAM. Haya e al. [] woked on he MHD seady flow of oldoyd-6 consan fluid. HAM mehod was used in his wok fo he nonlinea diffeenial equaion of hee diffeen ypes of flows. Aia [3] examined he MHD non Newonian unseady couee and poisuille flows. The effec of Hall em and physical paamees ae discussed fo velociy and empeaue disibuions. Aiyesimi e al. [4-5] calculaed he soluion of MHD Couee flow, Poiseuille flow poblems of velociy and empeaue pofile by using egula peubaion mehod. Danish e al. [6] sudied Poisuille and Coueee and poisuille flow of hid gade fluid.. Rajagopal, e al.[7], examined non-newonian fluids beween wo paallel and veical plaes in he fom of a Naual Convecion Flow.Bhagava e al. [8], invesigaed Numeical soluion of fee convecion MHD mico pola fluid flow beween wo paallel poous veical plaes. They have discussed he effec of vaious physical paamees. In ecen Gul e al. [9-3] woked ou on diffeenial ype fluids in vaiaions of aicles. They discussed he effec of vaious physical paamees on flow fields. Volume flux, skin ficion, aveage velociy, and he empeaue disibuion acoss he film wee shown in hee sudied. In mos of hei wok hey used wo analyical echniques (OHAM and ADM) o obained bes esuls.dileep and kuma [4] invesigaes he unseady second gade fluid in a poous channel. They give he effec of physical paamees on he fluid moion duing poous and clea egion. Salah e al. [5], examined he flow of second gade fluid in a poous and oaing fame. Consan and acceleaed fluid flows cases ae sudied in hei woks.nemai e al.[6], sudied he unseady hin film flow of non-newonian fluid ove a moving bel. The appoximae soluions of velociy pofile have been shown by using HAM. Ifikha [7], examined he unseady bounday laye flow of a second gade fluid affeced by an impulsively seching shee. HAM mehod is used o ge he analyical soluion and he effecs of he physical paamees ae discussed hough gaphs. Chauhan and Kuma [8], examined he unseady shea flow of a second gade fluid beween wo hoizonal paallel plaes.in hei wok Laplace ansfom mehod is applied o find he soluion of he flow poblem. Abbas e al.[9], discussed he unseady hin liquid film of second gade fluid hough seching suface. HAM mehod was used fo analyical soluion. Kumai and Pasad [], discussed he hea effec in Sokes second poblem in unseady cas unde he effec of magneic field. analyical esul is shown fo empeaue field.hameed and Ellahi [], woked on hin film flow in a non-newonian fluid on a veical moving bel.feecau *Coesponding Auho: Taza Gul, Mahemaics Depamen, Abdul Wali Khan Univesiy Madan, KPK Pakisan. 339

2 Taza Gul e al., 5 [], examined he longiudinal and osional oscillaions of second gade fluid cicula cylinde. Tan e al.[3] sudied Soke fis poblem fo a second gade fluid in a poous half space wih heaed bounday. They epoed some good esuls. A vaiey of analyical echniques have been used by he eseaches fo he soluion of diffeenial equaions. In he ecen yeas he Adomian decomposiion Mehod (ADM) and Opimal Homoopy Asympoic Mehod (OHAM) ae he wo analyical echniques eceiving moe aenion. The ADM was evised wih some new esuls by Adomian [4]. Wazwaz [5] and Siqqiqui e al [6] used Adomian decomposiion mehod in hei wok o ge aacive esuls. Applicaion of he opimal homoopy asympoic mehod fo solving nonlinea equaions aising in hea ansfe was invesigaed by Mainca e al. [7-9]. In anohe invesigaion Mainca e al. [3] have used opimal homoopy asympoic mehod fo he seady flow of a fouh-gade fluid pas a poous plae. Govening Equaion The MHD and hea equaion govening he poblem (momenum, mass and second ode fluid equaion) can be wien as. U =, () DU ρ =.Τ + J B + f. () g D DΘ ρ c = k Θ + ( T.L ), p (3) D D Hee U epesen velociy of he flow, ρ is flow densiy, is he maeial ime deivaive, and D f g is body foce due o gaviy. Thus,he Loenz foce peuni volume is =, u,, (4) Whee B = (, B,) is he unifom magnaic filed, B is he applied magneic field and σ is he elecical conduciviy. The cuen densiy J is ( E+ U B), B µ J. J = σ = (5) Hee, µ is he magneic pemeabiliy, E is an elecic field which we ignoe in his wok, and The cauchy sess enso, Τ is Τ = pi + S, (6) Whee Sis he exa ess enso, p I is he isoopic sess.fo second gade fluid S = µ A + α A + α A, (7) T A = I, A = L+ L, L = gad U, (8) T D = ( ) + ( ), = An- T A u u A + An- ( u) + ( u) An-, n D (9) A is he Rivlin Eicksen sess enso and µ is he viscosiy cofficien. Fomulaion of he Couee ype flow Poblem Conside wo Veical and paallel plaes such ha one of hem is oscillaing and moving wih consan velociy U and he ohe plae kep oscillaing only. The oal hickness of he fluid beween he plaes assumed o be h.moving and oscillaing plae caies wih iself a liquid of widh h.the configuaion of fluid flow is along he Y-axis and pependicula o x-axis. A ansvese magmaic field applied o he bel. Gaviaional foce and magneics foce causes he fluid moion. We assume ha he flow is un-seady, lamina and incompessible. 34

J. Appl. Envion. Biol. Sci., 5(7)339-353, 5 Figue:.

3 J. Appl. Envion. Biol. Sci., 5(7) , 5 Figue:. Geomey of he Couee flow poblem The velociy field and bounday condiions fo he poblem is given as U =, u x,,, And Θ = Θ (x, ) () ( ( ) ) U (h,) = U + UCosω, U (-h,) = UCosω, () Θ (h, ) = Θ, Θ( h, ) = Θ, () ω is fo he Expessions he fequency of he oscillaing plaes. Consuming () in () and (3) educed o he fom ρ u = T u, xy g B σ (3) Θ Θ ρcp = k, + T xy (4) Fom (7) Cauchy sess enso componens S is calculaed as u T xx = P + ( α + α ), (5) u u T xy = µ + α, (6) u Tyy = P + α, (7) Inseing of equaion (4) in equaion () and (3) give us u u u ρ = µ + ρ g σ B α u, (8) Θ u u u = k Θ + +, ρ c p µ α (9) Using non-dimensional paamees ( u ( x ( µ ( Θ Θ µ U σ Bδ u =, x =, =, Θ =, B,, = M = U h ρδ Θ Θ k ( Θ Θ ) µ µ c p ( ωδ ρ δ ρ g α h p P =, ω =, S =, α =, Ω = k µ µ U ρδ µ U () 34

4 Taza Gul e al., 5 Whee is he non-dimensional vaiable, is he Binkman numbe, is Soke numbe and is he Pandl numbe, ω is he oscillaing paamee, M is he magneic paamee. Using he above dimensionless paamee in equaion (8,9) and educing bas we obain u u u = + + α S, Mu () Θ Θ u u u P = + B, + α () And he bounday condiions ae u(,) = + Cosω, u(-,) = Cosω, (3) Θ(, ) =, Θ(, ) =, (4) ADM Soluion fo he Couee flow poblem To obained he analyical soluion fis we apply he ADM mehod using he above bounday condiions. The zeo, fis and second ode poblems and hee is soluion is given below in ode u (x,) P : = S, (5) Θ(x,) =, (6) u (x,) u P : = α[ A ] + Mu, (7) Θ (x,) Θ = B ([ B ] + α[ C ]), (8) u (x,) u P : = α[ A ] + Mu, (9) Θ (x,) Θ = P B ([ B ] + α[ C ]), (3) The Adomian polynomials ae defined as, u u A =, A =, (3) u, u u u u B = C =, B =, x u u u u C = +, (3) The soluion of he above componens is, P : u ( x, ) = ) cos[ ] ( ) ( + x + ω + x. (33) θ ( x, ) = ( + x). (34) [ ω ] ω [ ω ] 3 M ( 6 x + 6 x + x ) + MCos ( x ) Sin c ( + x ) ( ) = ωsin [ ω ]( x ) + SM (5-6 x + x ) P : u x, ( ) = (3 B ( x ) 4 xb ( x ) B ( x 4 ). θ x, + S + S 4 (35) (36) 34

5 J. Appl. Envion. Biol. Sci., 5(7) , u ( x, ) = M {(75 + 7x 9x x + 5 x + 3x ) + M cos [ ω] (5 8x + 3 x ) M ω cos ω (5 8x + 3 x ) + 36αω cos ω x Mω sin ω (3 + 36x + 3 x ) + [ ] [ ]( ) + [ ] 4 [ ω] x + S x x + x 6 36 αω sin ( ) M ( )}. 3 4 θ ( x, ) = {5MB ( + 4 x + x 4 x x ) + B Mx Cos [ ω ] ( x ) + 6 B xαω 7 cos x B x sin x 6M B x sin x M B S (6 56x [ ω ]( ) ω [ ω] ( ) + αω [ ω] ( ) [ ω] + αω [ ω] x [ ω ]( x ) + 6 M S αω [ ω ]( x + M S + x x + 8 x 6x 4 x ) S B MCos ( x ) 6S B cos ( ) + S B ω sin B sin ) B ( ). The seies soluions of velociy pofile is obained as ( x, ) ( x, ) ( x, ) ( x, ) u = u + u + u (39) u 4 4 ( x ) ωsin [ ω ] c ( + x ) + ωsin [ ω ]( x ) + SM ( 5-6x + x )} + M 7 (75 x x x x x M x x M os 3 ( x, ) = ( + x) + cos [ ω ] + ( x ) + { M ( 6 x + 6 x + x ) + MCos[ ω ] [ ω ] 5 ) [ ω ] ) + 36αω [ ω ]( ) ω [ ω ] 4 [ ω ] x S x x + x { cos ( ) + ω c (5 8 x x + 36αω sin ( ) + M ( )}. 6 The seies soluions of empeaue pofile is obained as θ ( x, ) ( x, ) ( x, ) ( x, ) ( x, ) 4 cos x + M sin (3 + 36x + 3 x ) θ = θ + θ + θ (4) 4 = ( + x) + {3B ( x ) + 4S xb ( x ) + S B ( x )} + {5MB ( + 4x x 4 x x ) + B MxCos [ ω ]( x ) + 6 B xαω cos[ ω ]( x ) B xω sin[ ω ] ( ) + 6 B αω sin[ ω] ( ) + B ( x 6 4 ) + B Cos[ ω ]( x ) + 6S B αω cos[ ω ]( x ) + S B ω si [ ω ]( x ) + 6 M Sαω [ ω ]( x ) + M B S ( 5 + 6x 8 x ).. x M x x M S x x x S M (37) (38) (4) n B sin (4) OHAM Soluion Couee flow poblem Now hee we apply OHAM mehod o ge he equied soluion. Zeo and fis componen poblem fo velociy and empeaue pofiles ae u (x,) P : = S, Θ(x,) =, u(x,) u u u u P : = S ( + c ) + c + c αc Mc u, x x (43) (44) (45) 343

6 Taza Gul e al., 5 Θ(x, ) Θ Θ Θ u u u P c3 c 3 B c3 α Bc3 = + + +, x x u(x, ) u u u u u u P : = S c + c + + c c c + αc x x x u + αc Mc u Mc u, Θ(x, ) Θ Θ Θ Θ Θ u = P c4 P c3 + + c 4 + c 3 + B c4 x u u u u u u u u + α Bc4 + Bc3 + α Bc3 + α Bc3. x x x (48) Soluions of zeo, fis and second componens poblem using bounday condiions fom equaion (4,5)in equaions (43-48) ae given as P : u ( x, ) = ( + x) + cos[ ω ] + ( x ) S (49) θ ( x, ) = ( + x). (5) 3 Mc ( 6 x + 6 x + x ) + Mc Cos [ ω] ( x ) P : u ( x, ) = + cω Sin[ ω] ( x ) + S ( x ) +, 4 4 c S ( x ) + M Sc (-5+6 x x ) (5) 4 θ ( x, ) = B c 3 {3( x ) + 4 S x( x ) + S ( x )}. 4 (5) is and exemely lag so we ignoe o wie.we can expess i only gaphicaly Fomulaion of he Poiseuille ype flow Poblem In his Poblem, we conside he second gade fluid in beween wo veical plaes y=h and y=-h. We assumed ha boh plaes ae oscillaing and he flow of fluid is due o he consan pessue gadien. (46) (47) Figue:. Geomey of he Poiseuille ype flow Poblem. When pessue gadien is involved hen he momenum and hea equaion is 344

7 J. Appl. Envion. Biol. Sci., 5(7) , 5 u p u u ρ = + µ + ρ g σ B α u, (53) Θ u u u = k Θ + +, ρ c p µ α (54) Using he dimensionless paamee (9) in () we obained u u u = Ω + + α S, Mu (55) Θ Θ u P = + B + u u, α (56) And he bounday condiions ae u(,) = Cosω, u(-,) = Cosω, (57) Θ(, ) =, Θ(, ) =, (58) ADM Soluion fo he Poiseuille ype flow Poblem The adomian polynomials of boh poblems ae same P : u ( x, ) = cos [ ω ] + ( x ) (S + Ω). (59) θ ( x, ) = ( + x). (6) 4 [ ] M (S + Ω)(5 6 x + x ) + MCos ω ( x ) P : u ( x, ) = ωsin [ ω ]( x ) + SM ( 5-6x + x ) (6) P (S ) B ( x 4 x, ). ( ) θ = + Ω (6) M (S + Ω )(6+ 75x 5 x + x ) + M cos [ ω ](5 8x + 3 x ) : u ( x, ) = + ω cos [ ω ]( 5 8x 3 x ) + 36αω cos[ ω ]( x ) + Mω sin[ ω ] 7 ( x 6 x ) 36 αω sin [ ω ]( x ). + 4 [ ω] αω [ ω] [ ω ] M [ ω] 4 4 3cos (S + Ω) B ( x ) + 5 cos (S + Ω) B (x ) + θ ( x, ) = ω αω M (S + Ω) B ( x x ) sin (S + Ω) B (x ) + 5 sin (S + Ω) B (x ) +. The seies soluions of velociy pofile is obained as ( x, ) ( x, ) ( x, ) ( x, ) (63) (64) u = u + u + u (65) The seies soluions of empeaue pofile is obained as ( x, ) ( x, ) ( x, ) ( x, ) θ = θ + θ + θ (66) θ, Ω + + Ω ( x ) = ( x) (S ) B ( x ) {3cos [ ω ](S ) B ( x ) 5αω cos[ ω ] [ ] [ ] (S + Ω) B (x ) + 3ω sin ω S + Ω) B (x ) + 5Mαω sin ω (S + Ω) B (x ) 4 6 (S ) B ( 3 5 x x + M + Ω + ).. (67) The OHAM Soluion of Poiseuille ype flow Poblem. 345

8 Taza Gul e al., 5 Hee we apply OHAM mehod o ge he equied soluion. Soluions of zeo, fis and second componens poblem using same bounday P : u ( x, ) = ( + x) + cos[ ω ] + ( x ) S (68) θ ( x, ) = ( + x). (69) 3 Mc ( 6 x + 6 x + x ) + McCos [ ω] ( x ) + P : u ( x, ) = cω Sin [ ω] ( x ) + S ( x ) +, (7) 4 4 cs ( x ) + M Sc (-5+6 x x ) θ ( x, ) = ( + x). (7) 4 θ ( x, ) = B c 3 {3( x ) + 4 S x( x ) + S ( x )}. (7) 4 is and exemely lag so we ignoe o wie.we can expess i only gaphicaly.the seies soluions of velociy pofile is obained as Table Compaison of OHAM and ADM fo he velociy pofile in case of Couee flow, when ω =., α =., M =.5, =, S =.5, B =.4, c = -.669, c = x OHAM ADM Absolue Eo Table Compaison of OHAM and ADM fo he velociy pofile, when ω =., α =., M =.5, =.5,, B =.4, c = , c = x OHAM ADM Absolue Eo

J. Appl. Envion. Biol. Sci., 5(7)339-353, 5 Figue 3 Gaphical Compaison of OHAM and ADM fo he velociy pofile in Couee flow poblem, when ω =., α =., M =.5, =, S =.5, B =.4, c = -.669, c = -.

9 J. Appl. Envion. Biol. Sci., 5(7) , 5 Figue 3 Gaphical Compaison of OHAM and ADM fo he velociy pofile in Couee flow poblem, when ω =., α =., M =.5, =, S =.5, B =.4, c = -.669, c = , Figue 4: Compaison gaph of OHAM and ADM fo he velociy pofile of Poiseuille ype flow Poblem, when ω =., α =., M =.5, =.5,, B =.4, c = , c = Figue 5: 3D gaphs fo he fluid flow duing diffeen ime level in Couee flow poblem. When ω =., α =., M =.5, S, Ω =.3, 347

3, Figue 7: 3D gaph fo empeaue disibuion in Couee flow

10 Taza Gul e al., 5 Figue 6: Velociy disibuion gaphs duind oscillaion a diffeen ime level in Couee flow poblem. When ω =., α =., M =.5, S, Ω =.3, Figue 7: 3D gaph fo empeaue disibuion in Couee flow poblem. When ω =., α =., M =.5,S =.3,B = 4, P =.6; Figue 8: empeaue disibuion a diffeen ime level in Couee flow poblem. When ω =., α =., M =.5,S =.3,B = 4, P =.6; 348

11 J. Appl. Envion. Biol. Sci., 5(7) , 5 Figue 9: The fluid flow duing diffeen ime level of Poiseuille ype flow Poblem. When ω =., α =., M =.5, S, Ω =.3, Figue : Effec of velociy pofile of Poiseuille ype flow Poblem. When ω =., α =., M =.5, S, Ω =.3, Figue : 3D Effec of empeaue in Poiseuille ype flow Poblem. When ω =., α =., M =.5,S =.3,B = 4, P =.6; 349

12 Taza Gul e al., 5 Figue : Effec of empeaue disibuion a diffeen ime level inpoiseuille ype flow Poblem. When ω =., α =., M =.5,S =.3,B = 4, P =.6; Figue 3: Physical illusaion of Magneic paamee M fo Poiseuille flow when ω =., α =., M =.5,S =.3, P =.6; Figue 4: Physical illusaion of Magneic paamee M fo Coueee flow when ω =., α =., M =.5,S =.3, P =.6; 35

13 J. Appl. Envion. Biol. Sci., 5(7) , 5 Figue 5: Physical illusaion of Magneic paamee M fo poiseuille flow when ω =., α =.,S =.3, =, Ω=.4 Figue 6: Physical illusaion of Magneic paamee M fo Couee flow when ω =., α =.,S =.3, == 5. Figue 7: Physical illusaion of Soke numbe S fo poiseuille flow poblems flow Whee ω =., α =., M =.5, Ω =.4; = 5 35

14 Taza Gul e al., 5 Figue 8: Physical illusaion of Soke numbe S fo Couee flow when ω =., α =., M =.5, Ω =.4; = 5 RESULTS AND DISCUSSION Figues and show he geomey of Couee and poiseuille flow poblems. Tables, and Figues 3,4 ae ploed fo he compason of ADM and OHAM mehods. Figues 5- show he effec of velociy and empeaue fields a diffeen ime level fo boh Couee and poiseuille flow poblems especually. The influence of diffeen dimensionless physical paamees (Sock numbes,binkman numbe B,and some ohe paemaes ae descibed in Figues.3 8. The effec of Binkman numbe B fo boh poblems have been shown in Figues 3,4. Inceasing Binkman numbe inceases fluid moion. The eason is ha he cohesive foces educes wich incease he fluid moion. The effec of sock numbe and magneic field fo boh boblems have been shown in Figues 5-8. Inceasing hese paamees educes fluid moion. Because he esisance foce inceases wich decease he fluid moion. Conclusion In his aicle, he modelled paial diffeenial equaions have been solved analyically by using ADM and OHAM mehods. The compaison of hese mehods analysed numeically and gaphically. We have concluded excellen ageemen of hese wo mehods. REFERENCES [] Islam S.,A. M., Shah A. R. and Ali I. (),Opimal homoopyasymoic soluions of couee and poiseuille flows of a hid gade fluid wih hea ansfe analysis, in. J. of Nonlinea Sci. and Numeical Sci., : [] Haya T., Khan M. and Ayub M., (4), Couee and poiseuille flows of an oldoyd 6-consan fluid wih magneic field, J. Mah. Anal. Appl., 98: [3] Aia A. H. (8), Effec of Hall cuen on ansien hydomagneic Couee Poiseuille flow of a viscoelasic fluid wih hea ansfe, Applied Mahemaical Modelling, 38: [4] Aiyesimi Y. M., Okedayo G. T. and Lawal O. W. (4) Effecs of Magneic Field on he MHD Flow of a Thid Gade Fluid hough Inclined Channel wih Ohmic Heaing. J. Appl Compua Mah 3(): -6. [5] Danish M., Kuma S. and Kuma S. () Exac analyical soluions fo he Poiseuille and Couee-Poiseuille flow of hid gade fluid beween paallel plaes. Comm. in Nonlinea Science and Numeical Simulaion. 7(3): [6] R. Bhagava, L. Kuma, H.S. Takha,"Numeical soluion of fee convecion MHD micopola fluid flow beween wo paallel poous veical plaes"4 (3) 336. [7] Taza Gul, Rehan Ali Shah, Saeed Islam, andmuhammad Aif"MHD Thin Film Flows of a Thid Gade Fluid on a Veical Bel wih Slip Bounday Condiions"(3),ID [8] T, Gul S, Islam RA, Shah,.(4) Hea Tansfe Analysis of MHD Thin Film Flow of an Unseady Second Gade Fluid Pas a Veical Oscillaing Bel. PLoS ONE 9(): doi:.37 jounal. pone

15 J. Appl. Envion. Biol. Sci., 5(7) , 5 [9] Taza Gul, S.Islam, R. A. Shah, I.Khan, S. Shaidan, Thin film unseady flow of a second gade fluid on a veical oscillaing bel wih MHD, Hea Tansfe. Engineeing Science and Technology,an Inenaional Jounal. doi:.6.j.jesch.4..3 [] Taza Gul, Iman Khan, M. A. Khan, S. Islam, T. Akha, S.Nasi. Unseady Second ode Fluid Flow beween Two Oscillaing Plaes. J. Appl. Envion. Biol. Sci., 5()5-6, 5. [] Taza Gul, S. Islam, R. A. Shah, I. Khan, L.C.C. Dennis Tempeaue Dependen Viscosiy of a Thid Ode Thin Film Fluid Laye on a Lubicaing Veical Bel. J of Absac and Applied Analysis Volume (4). Aicle ID [] Dileep S. Chauhan and Vikas Kuma "Unseady flow of a non-newonian second gade fluid in a channel paially filled by a poous medium"advances in Applied Science Reseach,, 3 (): [3] Faisal Salah, Zainal Abdul Aziz, and Dennis Ling Chuan Ching "On Acceleaed MHD Flows of Second Gade Fluid in a Poous Medium and Roaing Fame".IAENG Inenaional Jounal of Applied Mahemaics, 43:3, IJAM [4] H, Nemai.M, Ghanbapou.M, Hajibabayi. and Hemmanezhad (9) Thin film flow of non-newonian fluids on a veical moving bel usinghomoopy Analysis Mehod, Jounal of Engineeing Science and Technology Review (): 8-. [5] I,Ahmad.(3) On Unseady Bounday Laye Flow of a Second Gade Fluid ove a Seching Shee, Adv. Theo. Appl. Mech., 6(): [5] D.S Chauhan. and V.Kuma. () Unseady flow of a non-newonian second gade fluid in a channel paially filled by a poous medium, Advances in Applied Science Reseach, 3 (): [6] Abbas,T. Haya.M, Sajid. and S Asgha. (8) Unseady flow of a second gade fluid film ove an unseady seching shee, Mahemaical and Compue Modeling, 48: [7] B.A. Kumai and K.P (4) Effecs of magneic field and hemal in Sokes'second poblem fo unseady second gade fluid flow, Inenaional Jounal of Concepions on Compuing and Infomaion Technology, (): [8] FeecauC. and FeecauC. (5)Saing soluions fo some unseady unidiecional flows of a second gade fluid, Inenaional Jounaof Engineeing Science, 43(): [9] M. Hameed, R. Elahi, "Thin film flow of non-newonian MHD fluid on a veically moving bel", In. J. Nume. Meh. Fluid., 66(), [] C. Feecau, C. Feecau, Saing soluions fo he moion of a second gade fluid due o longiudinal and osional oscillaions of a cicula cylinde, In.J. Eng. Sci. 44 (6) [] W.C. Tan, T. Masuoka, Soke fis poblem fo a second gade fluid in a poous half space wih heaed bounday. In. J. Non-Linea Mech. 4(5)55-5. [] G. Adomian, A eview of he decomposiion mehod and some ecen esuls fo nonlinea equaions, Mah Compu. Model. 3 (99) 87?99. [3] A. M. Siddiqui, M. Hameed, B. M. Siddiqui, Use of Adomian decomposiion mehod in he sudy of paallel plae flow of a hid gade fluid, Communicaions in Nonlinea Science and Numeical Simulaion, 5, (9), 388?399,. [4] A. Wazwaz, A compaison beween vaiaional ieaion mehod and Adomian decomposiion mehod, Jounal Compuaional and Applied Mahemaics, 7,(), 9?36, 7. [5] Mohammad Mehdi Rashidia, Esmaeel Efania, Behnam Rosam "Opimal Homoopy Asympoic Mehod fo Solving Viscous Flow hough Expanding o Conacing Gaps wih Pemeable Walls" Tansacion on IoT and Cloud Compuing () [6] M. Faooq, M. T. Rahim, S. Islam, A. M. Siddiqui "wihdawal and dainage of genealized second gade fluid on veical cylinde wih slip condiions" 9(3), [7] A. M. Siddiqui, A. Ashaf, Q. A. Azim and B. S. Babcock"Exac Soluions fo Thin Film Flows of a PTT Fluid Down an Inclined Plane and on a Veically Moving Bel "7 (3) [8] S. Islam and C. Y. Zhou, Ceain invese soluions of a second-gade magneohydodynamic aligned fluid flow in a poous medium,jounal of Poous Media, vol., no. 4, pp. 448, 7. [9] G. Adomian, A eview of he decomposiion mehod, in applied mahemaics, J. Mah. Anal. Appl. 354 (988) [3] Fazle Mabood "The Applicaion of Opimal Homoopy Asympoic Mehod fo One-Dimensional Hea and Advecion"Inf. Sci. Le., No., 57-6 (3). 353

STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE WEIBULL DISTRIBUTION

Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE