Numerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter In The Porous Medium

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1 IOSR Joural of Egieerig (IOSRJEN) ISSN (e): 5-3, ISSN (p): Vol. 5, Issue 4 (April. 5), V PP Numerical Study o MHD Flow Ad Heat rasfer With he Effect Of Microrotatioal Parameter I he Porous Medium R.K.Modal ** B. M. Jewel Raa, R. Ahmed * Mathematics Disciplie Khula iversity, Bagladesh. Mathematics Disciplie Khula iversity, Bagladesh. Mathematics Disciplie Khula iversity, Bagladesh. Abstract: he umerical studies are performed to examie the Micropolar fluid flow past a ifiite vertical plate. Fiite differece techique is used as a tool for the umerical approach. he micropolar fluid behaviour o two- dimesioal usteady flow have bee cosidered ad its osimilar solutio have bee obtaied. Nosimilar equatios of the correspodig mometum, agular mometum ad cotiuity equatios are derived by employig the usual trasformatio. he dimesioless osimilar equatios for mometum, agular mometum ad cotiuity equatios are solved umerically by fiite differece techique. he effects o the velocity, micro rotatio ad the spi gradiet viscosity of the various importat parameters eterig ito the problem separately are discussed with the help of graphs. Keywords: Micropolar fluid, usteady, Explicit fiite differece method, Porous medium. I. INRODCION Because of the icreasig importace of materials flow i idustrial processig ad elsewhere ad the fact shear behavior caot be characterized by Newtoia relatioships, a ew stage i the evaluatio of fluid dyamic theory is i the progress. Erige(966) proposed a theory of molecular fluids takig ito accout the iteral characteristics of the subtractive particles, which are allowed to udergo rotatio. Physically, the micropolar fluid ca cosists of a suspesio of small, rigid cylidrical elemets such as large dumbbell-shaped molecules. he theory of micropolar fluids is geeratig a very much icreased iterest ad may classical flows are beig re-examied to determie the effects of the fluid microstructure. he cocept of micropolar fluid deals with a class of fluids which exhibit certai microscopic effects arisig from the local structure ad micromotios of the fluids elemets. hese fluid cotai dilute suspesio of rigid macromolecules with idividual motios that support stress ad body momets ad are iflueced by spi iertia. Micropolar fluids are those which cotai micro-costituets that ca udergo rotatio, the presece of which ca affect the hydrodyamics of the flow so that it ca be distictly o-newtoia. It has may practical applicatios, for example aalyzig the behavior of exotic lubricats, the flow of colloidal suspesios, polymetric fluids, liquid crystals, additive suspesios, huma ad aimal blood, turbulet shear flow ad so forth. Peddisio ad McNitt(97) derived boudary layer theory for micropolar fluid which is importat i a umber of techical process ad applied this equatios to the problems of steady stagatio poit flow, steady flow past a semi-ifiite flat plate. [] Erige (97) developed the theory of thermo micropolar fluids by extedig the theory of micropolar fluids.he above metioed work they have exteded the work of El-Arabawy (3) to a MHD flow takig ito accout the effect of free covectio ad micro rotatio iertia term which has bee eglected by El-Arabawy (3) [3]. However, most of the previous works assume that the plate is at rest. Free-covectio flow with thermal radiatio ad mass trasfer past a movig vertical porous plate have aalyzed by Makide, O. D [9]. steady MHD free covectio flow of a compressible fluid past a movig vertical plate i the presece of radioactive heat trasfer have bee discussed by Mbeledogu, I., Amakiri, A.R.C ad Ogulu, A, []. Numerical Study o MHD free covectio ad mass trasfer flow past a vertical flat plate has bee discussed by S. F. Ahmmed []. I our preset work, we have studied about umerical study o Numerical study o micropolar fluid flow through vertical plate.he goverig equatios for the usteady case are also studied. he these goverig equatios are trasformed ito dimesioless mometum, eergy ad cocetratio equatios are solved umerically by usig explicit fiite differece techique with the help of a computer programmig laguage Compaq visual FORRAN 6.6. he obtaied results of this problem have bee discussed for the differet values of well-kow parameters with differet time steps. he tecplot is used to draw graph of the flow. II. MAHEMAICAL FORMLAION Let us cosider the micropolar fluid flow through vertical plate. We also cosider the x-axis be directed upward alog the plate ad y-axis ormal to the plate. Agai let u ad v be the velocity compoets alog the x-axis ad y - Iteratioal orgaizatio of Scietific Research 8 P a g e

2 Numerical Study o MHD Flow Ad Heat rasfer With he Effect Of Microrotatioal Parameter axis respectively, u v x y u u u u ub u v g( ) t x y y y u u v t x y j y j y u v t x y C y k he boudary coditios for the problem are: at ; u ; v ; ; every where p ' u ; v ; ; at x (5) at u ; v ; ; at y u ; v ; ; at y It is required to make the give equatios dimesioless. For this itetio we itroduce the followig dimesioless quatities u v X x, y,, V, t,, w Fially, we obtai V (6) X ' V B V G r (7) X V (8) X V (9) X Pr () () (3) (4) he correspodig boudary coditios for the problem are reduced i the followig from at,,, V O,, everywhere, V,, at X at,, V,, at, V,, at () III. NMERICAL SOLIONS May physical pheomea i applied sciece ad egieerig whe formulated ito mathematical models fall ito a category of systems kow as o-liear coupled partial differetial equatios. Most of these problems ca be formulated as secod order partial differetial equatios. A system of o-liear coupled partial differetial equatios with the boudary coditios is very difficult to solve aalytically. For obtaiig the Iteratioal orgaizatio of Scietific Research 9 P a g e

3 Numerical Study o MHD Flow Ad Heat rasfer With he Effect Of Microrotatioal Parameter solutio of such problems we adopt advaced umerical methods. he goverig equatios of our problem cotai a system of partial differetial equatios which are trasformed by usual trasformatios ito a odimesioal system of o-liear coupled partial differetial equatios with iitial ad boudary coditios. Hece the solutio of the problem would be based o advaced umerical methods. he fiite differece Method will be used for solvig our obtaied o-similar coupled partial differetial equatios. From the cocept of the above discussio, for simplicity the explicit fiite differece method has bee used to solve from equatios (6) to (9) subject to the coditios give by ().o obtai the differece equatios the regio of the flow is divided ito a grid or mesh of lies parallel to X ad axis is take alog the plate ad - axis is ormal to the plate. Here the plate of height X ( ) max i.e. X varies from to ad regard ( 5) max as correspodig to i.e. varies from to 5. here are m= ad = grid spacig i the X ad directios respectively. It is assumed that ΔX ad Δ are costat mesh sizes alog X ad directios respectively ad take as follows, X. x.5 x 5 with the smaller time-step, Δt=.5. Now usig the fiite differece method we covert our goverig equatios i the followig form i, j i, j Vi, j Vi, j X ',,,,,,, i j i j i j i j i j i j,,,, i j i j i j i j Vi j X ( ) i, j i, j Gr M i, j () () ' i, j X ( ) i, j i, j i, j i, j i, j i, j i, j i, j i, j Vi, j i, j i, j ' i, j i, j i, j i, j i, j i, j i, j i, j i, j i, j Vi, j X Pr ( ) Ad the iitial ad boudary coditios with the fiite differece scheme are i, j i, j i, j i, j, V,,, j,, j,, j,, j V i,, i,, i,, i, V i, L, i, L, i, L, i, L V Here the subscripts i ad j desigate the grid poits with x ad y coordiates respectively. (3) (4) (5) IV. RESLS AND DISCSSION he effects of usteady micropolar fluid behavior o a heated plate have bee ivestigated usig the fiite differece techique. o study the physical situatio of this problem, we have computed the umerical values by fiite differece techique of velocity, micrirotatio ad temperature effect at the plate. It ca be see that the solutios are affected by the parameters amely, Microrotatio parameter ( ), Spi gradiet viscosity parameter (Λ), Magetic parameter (M), the vortex viscosity parameter (), Grashof Number (G r ) ad Pradtl umber (P r ). he mai goal of the computatio is to obtai the steady state solutios for the o-dimesioal Iteratioal orgaizatio of Scietific Research P a g e

4 Numerical Study o MHD Flow Ad Heat rasfer With he Effect Of Microrotatioal Parameter velocity, microrotatio Γ ad temperature for differet values of Microrotatio parameter ( ), Spi gradiet viscosity parameter (Λ), the vortex viscosity parameter (), Grashof Number (G r ) Pradtl umber (P r ). For these computatios the results have bee calculated ad preseted graphically by dimesioless time = up to = 8. he results of the computatios show little chages for = to = 6. But while arisig at = 7 ad 8 the results remai approximately same but microrotatio. hus the solutio for = 8 are become steady-state. Moreover, the steady state solutios for trasiet values of, Γ ad are show i figures - 4), for time =,, 3, 4, 5, 6, 7, 8 respectively. he values of Microrotatio parameter ( =.) is fixed. Whereas, figures (-8) show the velocity profile for differet values of Grashof Number (G r =.,.4,.6) at time =,, 3,4, 5, 6, 7, 8 respectively. From this figures it is observed that the velocity profile icrease with the icrease of Grashof Number(G r ), ad the velocity profiles are goig upward directio. While arisig at = 7 ad 8 the solutios become steady-state. Other importat effects of microrotatios are show i figures (9-6) for differet values of Spi gradiet viscosity parameter (Λ) at time =,, 3, 4, 5,6, 7, 8 respectively. It is see from this figures that the microrotaio icreases with the icrease of Spi gradiet viscosity parameter (Λ) ad is goig to the upward directio from the horizotal wall with the icrease of time. While arisig at = 7 ad 8 the results also icreasig with time. Other effects of temperature are show i figures (7-4) for differet values of Pradtl umber (P r =.7,., 7.) at time =,, 3, 4, 5, 6, 7, 8 respectively. Oe the other had, at salt water the Pradtl umber is(p r =. ). It is see from this figures that the temperature distributio is decreases with the icrease of Pradtl umber ad the flow patter is directed to the outer wall with the icrease of time. While arisig at = 7 ad 8 the flow becomes steady state.5 G r =.,.4,.6.8 G r =.,.4, Fig. : Velocity Profile for differet values of Grashoff Number (G r ) ad =., M =. at time τ = Fig. : Velocity Profile for differet values of Grashoff Number (G r ) ad =., M =. at time τ = Iteratioal orgaizatio of Scietific Research P a g e

5 Numerical Study o MHD Flow Ad Heat rasfer With he Effect Of Microrotatioal Parameter G r =.,.4,.6 G r =.,.4, Fig. 3: Velocity Profile for differet values of Grashoff Number (G r ) ad =., M =. at time τ = 3 Fig. 4: Velocity Profile for differet values of Grashoff Number (G r ) ad =., M =. at time τ = 4 G r =.,.4,.6 G r =.,.4,.6 Fig. 5: Velocity Profile for differet values of Grashoff Number (G r ) ad =.,M =. at time τ = 5 Fig. 6: Velocity Profile for differet values of Grashoff Number (G r ) ad =., M =. at time τ = 6 Iteratioal orgaizatio of Scietific Research P a g e

6 Numerical Study o MHD Flow Ad Heat rasfer With he Effect Of Microrotatioal Parameter G r =.,.4,.6 G r =.,.4,.6 Fig. 7: Velocity Profile for differet values of Grashoff Number (G r ) ad =., M =. at time τ = 7 Fig. 8: Velocity Profile for differet values of Grashoff Number (G r ) ad =., M =. at time τ=8 Λ=.,.4,.6 Λ=.,.4, Fig. 9: Microrotatio Profile for differet values of Spi Gradiet viscosity parameter(λ) ad =., M =. at time τ = Fig. : Microrotatio Profile for differet values of Spi Gradiet viscosity parameter(λ) ad =., M =. at time τ = Iteratioal orgaizatio of Scietific Research 3 P a g e

7 Numerical Study o MHD Flow Ad Heat rasfer With he Effect Of Microrotatioal Parameter.8 Λ=.,.4,.6 Λ=.,.4, Fig. : Microrotatio Profile for differet values of Spi Gradiet viscosity parameter(λ) ad =., M =. at time τ = 3 Fig. : Microrotatio Profile for differet values of Spi Gradiet viscosity parameter(λ) ad =., M =. at time τ = 4 Λ=.,.4,.6 Λ=.,.4, Fig. 3: Microrotatio Profile for differet values of Spi Gradiet viscosity parameter(λ) ad =., M =. at time τ = 5 Fig. 4: Microrotatio Profile for differet values of Spi Gradiet viscosity parameter(λ) ad =., M =. at time τ = 6 Iteratioal orgaizatio of Scietific Research 4 P a g e

8 Numerical Study o MHD Flow Ad Heat rasfer With he Effect Of Microrotatioal Parameter Λ=.,.4,.6 3 Λ=.,.4,.6 Fig. 5: Microrotatio Profile for differet values of Spi Gradiet viscosity parameter(λ) ad =., M =. at time τ = 7 Fig. 6: Microrotatio Profile for differet values of Spi Gradiet viscosity parameter(λ) ad =., M =. at time τ = 8 Pr=.7,.,7. Pr=.7,., Fig. 7: emperature Profile for differet values of Pradtl Number (P r ) ad M =. at time τ = Fig. 8: emperature Profile for differet values of Pradtl Number (P r ) ad M =. at time τ = Iteratioal orgaizatio of Scietific Research 5 P a g e

9 Numerical Study o MHD Flow Ad Heat rasfer With he Effect Of Microrotatioal Parameter Pr=.7,.,7. Pr=.7,., Fig. 9: emperature Profile for differet values of Pradtl Number (P r ) ad M =. at time τ = 3 Fig. : emperature Profile for differet values of Pradtl Number (P r ) ad M =. at time τ = 4 Pr=.7,.,7. Pr=.7,., Fig. : emperature Profile for differet values of Pradtl Number (P r ) ad M =. at time τ = 5 Fig.: emperature Profile for differet values of Pradtl Number (P r ) ad M =. at time τ = 6 Iteratioal orgaizatio of Scietific Research 6 P a g e

10 Numerical Study o MHD Flow Ad Heat rasfer With he Effect Of Microrotatioal Parameter Pr=.7,.,7. Pr=.7,., Fig. 3: emperature Profile for differet values of Pradtl Number (P r ) ad M =. at time τ = 7 Fig. 4: emperature Profile for differet values of Pradtl Number (P r ) ad M =. at time τ = 8 REFERENCES [] Erige, A.C., (966), Joural of Mathematical Mechaics,6, [] Erige, A.C., (97), Joural of Mathematical Aalysis Applied,38,48 [3] El-Arabawy,H.A.M.,(3)., Iteratioal Joural of Heat ad Mass rasfer,46,47 [4] Peddiese,j., ad McNitt,R.P., (97),Recet Advaced Egieerig Sciece,5,45 [5] Callaha,G.D. ad Marer,W.J.(976).It.J.Heat Mass rasfer,9,65 [6] Soudalgekar ad Gaesa(98).,Iteratioal Joural of Egieerig adsciece,8,87 [7] P.Chadra, N.C.Sacheti ad A.K.Sigh, steady free covectio flow with heat flux ad accelerated motio. J. Phys. Soc. Japa 67,4-9, 998. [8] V. M. Soudalgekar, B. S. Jaisawal, A. G. plekar ad H. S. akhar, he trasiet free covectio flow of a viscous dissipative fluid past a semi-ifiite vertical plate. J. Appl. Mech. Egg.4, 3-8, 999. [9] O. D Makide Free-covectio flow with thermal radiatio ad mass trasfer past a movig vertical porous plate, It. Comm. Heat Mass rasfer, 3, pp. 4 49, 5 [] Mbeledogu, I., Amakiri, A.R.C ad Ogulu steady MHD free covectio flow of a compressible fluid past a movig vertical plate i the presece of radioactive heat trasfer, It. J. of Heat ad Mass rasfer, 5, pp , 7. [] S. F. Ahmmed, S. Modal ad A. Ray. Numerical studies o MHD free covectio ad mass trasfer flow past a vertical flat plate. IOSR Joural of Egieerig (IOSRJEN). ISSN 5-3, vol. 3, Issue 5, pp 4-47, May. 3. Iteratioal orgaizatio of Scietific Research 7 P a g e