International Journal of Advancements in Research & Technology, Volume 4, Issue 2, February ISSN

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1 Itertiol Jourl of Advcemets i Reserch & Techology, Volume 4, Issue, Februry -0 4 ISSN Stedy flows i pipes of equilterl trigulr cross sectio through porous medium with mgetic field Dr. Ad Swrup Shrm Associte rofessor, Dept. of Applied Scieces, Idel Istitute of Techology, Ghzibd (U.) Idi. Emil: shrm.s09@gmil.com ABSTRACT: I this pper we hve ivestigted the Stedy flow i pipes of equilterl trigulr cross sectio through porous medium with mgetic field. We hve obtied the velocity, volumetric flow d vortex lies. KEY WORDS: Stedy flow, Equilterl trigulr cross sectio, icompressible fluid, pipes, porous medium d mgetic field. NOMENCLATURE u =velocity compoet log x xis v = velocity compoet log y xis w (x, y) = velocity i x-y ple t = the time = the desity of fluid = the fluid pressure K= the therml coductivity of the fluid INTRODUCTION = Coefficiet of viscosity = Kiemtic viscosity Q = the volumetric flow = Vorticity compoet i x - directio y We hve ivestigted the Stedy flow i pipes of equilterl trigulr cross sectio through porous medium with mgetic field. Attempts hve bee mde by severl reserchers i.e.. Egui, J. Zueco, E. Grd & D. tio [] NSM solutio for ustedy MHD Couette flow of dusty coductig fluid with vrible viscosity d electric coductivity. A. Elcrt, B. Forberg, M. Hor & K. Miller [] some stedy vortex flows pst circulr cylider. J. W. Elder [] Trsiet covectio i porous medium. S. M. M. EL- Kbeir, A. M. Rshd & S. R. G. Rm [4] ustedy MHD combied covectio over movig verticl sheet i fluid sturted porous medium with uiform surfce het flux. K. Ellgee & H. O. Abdul [] the effect of surfce tesios the wve growth d trsitio to slug flow. E. Erturk & C. Gokcol [6] fourth order compct formultio of Nvier-stokes equtios d drive cvity flow t high Reyolds umbers. E. Erturk, T. C. Corke & C. Gokcol [7] umericl solutios x = Vorticity compoet i y - directio z = Vorticity compoet i z - directio of -D stedy icompressible drive cvity flow t high Reyolds umbers. E. Erturk, T. C. Corke [8] boudry lyer ledig-edge receptivity to soud t icidece gles. E. Erturk, O. M. Hddd & T. C. Corke [9] umericl solutios of lmir icompressible flow pst prbolic bodies t gles of ttck. A. T. Eswre & G. Nth [0] ustedy o-similr two-dimesiol d xisymmetric wter boudry lyer with vrible viscosity d rdti umber. M. A. Ezzt & A. A. EI-Bry [] spce pproch to the hydromgetic flow of dusty fluid through porous medium. M. A. Ezzt [] mgeto hydrodymic ustedy flow of o-ewtoi fluid pst ifiite porous plte. H. Fisst & B. Eckhrdt [] Trsitio from the Couette Tylor system to the ple Couette system. D. Flore [4] o ssocited elstics viscoplstic model for bitumious cocrete. I this pper we hve ivestigted the velocity, volumetric flow d vortex lies. Copyright 0 SciResub.

2 Itertiol Jourl of Advcemets i Reserch & Techology, Volume 4, Issue, Februry -0 ISSN FORMULATION OF THE ROBLEM Let z - xis be tke the directio of flow log the xis of the pipe. The u = 0, v = 0 for stedy d icompressible fluid the velocity compoet is idepedet of z. The equtio of cotiuity. (, ) y-xis C u v w 0...() x y z w But u 0, v () (-, 0) A O (, 0) x (o, o) N z z w w ( x, y)...() y = x + y =- x + B (, - ) AB BC CA, AN Figure- i.e. w is idepedet of z The Nvier-Stokes equtios of i the bsece of body forces. w w 0 z x y K p d p z d z 0 = 0...(4) y B w 0...() It is cler from () & (4) is idepedet of x & y i.e. p is the Fuctio of z SOLUTION OF THE ROBLEM: p = p (z) Costt B w w dp w w let B B w B w...(6) K x y dz x y h ' ',.. xh y D D B w C F e where h & h' re relted by h h' B 0 d. I. w (x, y) = e Where h h B D D B B B Cse -I: h ' xhy ' ' usig boudry coditios t y = x+ B e ' x hxh ' x hxh...(7) x & t y...(8) e B From (7) & (8) e e 0 e e h ' ' ' ' xh x h xh x x x h xh h xh Copyright 0 SciResub.

3 Itertiol Jourl of Advcemets i Reserch & Techology, Volume 4, Issue, Februry -0 6 ISSN ' ' ' ' h h h h h h ' 0 ' & ' 0 t x e e h h h B h B h B x B B B( x) e 0 e e w ( x, y) e B B B B Cse - II: w ( x, y) 0 t (-, 0) & w( x, y) 0 t (, ) B e h...(9) B e h h'...(0) h h' h h' h h' h B B h' h B 4 h B h & h' Bx y B e w ( x, y) e B B Cse III: w ( x, y) 0 t (, 0) & w( x, y) 0 t (, ) e B h...() O solvig: The volumetric Flow: Copyright 0 SciResub. B e h h' B B B h, h' & e w ( x, y) e B B Bx By Bx By w( x, y) e. e e e e B Bx B y By e e e e B B( x) B( x) w x y Bx Cosh e B e...() B x y B y B( x) (, )... ()

4 Itertiol Jourl of Advcemets i Reserch & Techology, Volume 4, Issue, Februry -0 7 ISSN yx Bx By Bx Q w ( x, y) dx dy Cosh e e dx dy x yx Bx B x By Bx Cosh e e dy dx B 0 Bx x 4 B x x B B B x Sih e e dx Bx Bx Bx Bx B x e e Let I e Sih. dx e dx B x B Bx e e e dx x B B B B e B B B B e B x B x x Bx Bx. B B B Let I e dx e e dx B x B e B e e B e B B B B B B B B Let I dx x x B B Q I I I. e B B e B B B B B B 9 B 6 Q e e B B B B B B B 9 e B B B B B B B e B 9 Q e e B B B B B B ( )... (4) dx dy dz The equtio of vortex lie: where x, y & z re vorticity compoets x y z Copyright 0 SciResub.

5 Itertiol Jourl of Advcemets i Reserch & Techology, Volume 4, Issue, Februry -0 8 ISSN Bx B y Bx Let q ui vj wk Cosh e e k B Bx Bx w v B y B y x B Sih e e Sih y z B B y B x u w B y Bx B Cosh e Be z x B Bx By Bx Cosh e e & z 0 B dx dy dz Bx Bx 0 By By Bx e Sih Cosh e e B B Tkig I st Two dx Bx Bx By By Bx e Sih Cosh e e Bx Bx By dx Sih dy C Bx By Cosh e e e B xx By By Cosh dx e dx. Cosh C B Bx 4 B x By Cosh dx e dx Cosh C B Bx4 4 B x B x By B x By CB Sih e Cosh C or Sih e Cosh A B B B B x4 dy B y B y the first vortex lie is e Sih Cosh A... () tkig lst two dz 0 the secod vortex lie z B... (6) Tbles for velocity: Cse- I Copyright 0 SciResub.

6 Itertiol Jourl of Advcemets i Reserch & Techology, Volume 4, Issue, Februry -0 9 ISSN B0 B0 Let,.,, re sme, B K K K B xy,, 0 B0, K 9, 6, Tble - (for velocity), 8,, 4 4, 7, w x y w x, y w x y Cse-II: B0 B0 B0 Let,.,, re sme, Let d B K K K Tble - (for velocity) xy, 9, 6,, 8,, 4, 4 7, K, w x y B 0 B0 K w x, y w x, y Cse- III: Copyright 0 SciResub.

7 Itertiol Jourl of Advcemets i Reserch & Techology, Volume 4, Issue, Februry ISSN B0 B0 B0 Let,.,, re sme, Let d B K K K B0 K xy, 9, 6 Tble - (for velocity),, 8,, 4, 4 7, w x, y wx, y B0, K w x y CONCLUSION AND DISCUSSION I this pper, we hve ivestigted the velocity by the tble- of equtio () betwee velocity d 9,,but the vlue of velocity i porous poit. The velocity i porous medium d 6 B0 medium t is greter th the mgetic field t is less th K K correspodig vlue of velocity i mgetic field the vlue of velocity i porous with mgetic B B0 t 0 d t porous with mgetic field t field t t poit 9, K 6 but the vlue of velocity i porous medium d B0 i the itervl B0 K mgetic field t is greter K, xy, 7, th the correspodig vlue of velocity i. Agi by the tble- the velocity i mgetic B0 porous with mgetic field t K B0 field t is less th the correspodig i the itervl, xy, 7,. vlue of velocity i porous medium t Agi by the tble- the velocity i porous K d is lso less th the correspodig vlue of medium t is less th the velocity i porous with mgetic field t K correspodig vlue of velocity i mgetic field B0 t poit K 9,,but the B 6 t 0 d is lso less th the B0 vlue of velocity i mgetic field t correspodig vlue of velocity i porous with B0 is greter th the correspodig vlue of velocity mgetic field t t poit K Copyright 0 SciResub.

8 Itertiol Jourl of Advcemets i Reserch & Techology, Volume 4, Issue, Februry -0 6 ISSN i porous medium t with mgetic field t d t porous K B0 i the K itervl, xy, 7,. Also we hve ivestigted the volumetric flow d vortex lies by the equtios (4), () d (6) respectively. REFERENCES [] Egui., Zueco J., Grd E. & tio D., NSM solutio for ustedy MHD Couette flow of dusty coductig fluid with vrible viscosity d electric coductivity. Applied Mthemticl Modellig,, pp 0-6, (0). [] Elcrt A., Forberg B., Hor M. & Miller K., some stedy vortex flows pst circulr cylider. J. Fluid Mech. 409 pp 7, (000). [] Elder J. W., Trsiet covectio i porous medium. the J. of Fluid Mechics, vol. 47, pp ,(967). [4] EL-Kbeir S. M. M., Rshd A. M. & Rm S. R. G., ustedy MHD combied covectio umbers. It. J. for umericl methods i fluids 0, pp 4-46, (006). [7] Erturk E., Corke T. C. & Gokcol C., umericl solutios of -d stedy icompressible drive cvity flow t high Reyolds umbers. It. J. for umericl methods i fluids 48, pp , (00). [8] Erturk E., Corke T. C., boudry lyer ledig-edge receptivity to soud t icidece gles. J. Fluid Mech.; 444, pp 8 407, (00). [9] Erturk E., Hddd O. M. & Corke T. C., umericl solutios of lmir icompressible flow pst prbolic bodies t gles of ttck. AIAA jourl 4, pp 4-6, ( 004). [0] Eswre A. T & Nth G., ustedy osimilr two-dimesiol d xisymmetric wter boudry lyer with vrible viscosity d rdti umber. It. J. Egg. Sci. Yo., No., pp 87-79, (994). [] Ezzt M. A. & EI-Bry A. A., spce pproch to the hydro-mgetic flow of dusty fluid through porous medium. J. of computer d mthemtics with pplictio. Vol.-9, pp , (00). [] Ezzt M. A., mgeto hydrodymic over movig verticl sheet i fluid sturted porous medium with uiform surfce ustedy flow of o-ewtoi fluid pst ifiite porous plte. Idi J. ure Appl. het flux. Mthemticl d Computer Mths, Yo. No. 6, pp 6-664, (994). Modelig, Vol. 46, Issues -4, pp 84-97, (007). [] Ellgee K. & Abdul H. O., the effect of surfce tesios the wve growth d trsitio to slug [] Fisst H. & Eckhrdt B., Trsitio from the Couette Tylor system to the ple Couette system. hys. Rev. E, 6, pp 77 70, (000). flow. J. of Fluids Egg. Vol. 7, No., pp [4] Flore D., o ssocited elstics 89-9, (99). [6] Erturk E. & Gokcol C., fourth order compct formultio of Nvier-stokes equtios d viscoplstic model for bitumious cocrete. It. J. Egg. Sci. Vol., No.. pp 8-9, (994). drive cvity flow t high Reyolds Copyright 0 SciResub.