y=h B 2h Z y=-h ISSN (Print) Dr. Anand Swrup Sharma
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1 Scolars Journal of Enginring and Tcnology (SJET) Sc. J. Eng. Tc., 5; 3(A):4-54 Scolars Acadmic and Scintific ublisr (An Intrnational ublisr for Acadmic and Scintific Rsourcs) ISSN 3-435X (Onlin) ISSN (rint) Stady lan oisuill flow of iscous incomprssibl Fluid btwn Two orous aralll lats troug orous Mdium wit Magntic Fild Dr. Anand Swrup Sarma Associat rofssor, Dpt. of Applid Scincs, Idal Institut of Tcnology, Gaziabad Dr. Anand Swrup Sarma Abstract: In tis papr w a instigatd t stady plan poisuill flow of iscous incomprssibl fluid btwn two porous paralll plats troug porous mdium wit magntic fild. W a studid t locity, arag locity, saring strss, skin frictions a, t olumtric flow, drag cofficints and stram lins. ywords: Stady poisuill flow, iscous paralll plats, incomprssibl fluid, porous mdium and magntic fild.. NOMENCLATURE: u =locity componnt along x axis = locity componnt along y axis t = t tim = t dnsity of fluid = t fluid prssur = t trmal conductiity of t fluid = Cofficint of iscosity = inmatic iscosity Q= t olumtric flow INTRODUCTION: W a instigatd stady plan poisuill flow of iscous incomprssibl fluid btwn two porous paralll plats troug porous mdium wit magntic fild. Attmpts a bn mad by sral rsarcrs i.. M. Aydin & R.T. Fnnr [] boundary lmnt analysis of drin caity flow for low and modrat Rynolds numbrs M and modrat Rynolds numbrs N. Baloul & Boutana and. Vassur [] Doubl-diffusi and sort-inducd conction in a sallow orizontal porous layr. V. t al. Barbu [3] xact controllability magnto-ydrodynamic quations. D. Barkly & L. S. Tuckrman [4] Stability analysis of prturbd plan Coutt flow. D. Barkly & L. S. Tuckrman [5] Turbulnt-laminar pattrns in plan Coutt flow. D. Barkly & L. S. Tuckrman [6] Man flow of turbulnt-laminar pattrns in plan Coutt flow. D.Barkly, L. S. Tuckrman [7] Turbulnt-laminar pattrns in plan Coutt flow. E. Barragy & G. F. Cary [8] Stram function-orticity drin caity solutions using p finit lmnts. G..Batclor [9] a proposal concrning laminar waks bind bluff bodis at larg Rynolds numbr. G.. Batclor [] on stady laminar flow wit closd stramlins at larg Rynolds numbr. C. Baytas and I. op [] fr conction in obliqu nclosurs filld wit a porous mdium. R.M. Bam & R.F. warming [] an implicit factord scm for t comprssibl Nair-stoks quations. Bant Sing & Canprt Sing [] Analysis of Vortx Motion in orous Mdia Journal of Elctronics Cooling and Trmal Control. In tis papr w a instigatd t locity, arag locity, saring strss, skin frictions a, t olumtric flow, drag cofficints and stram lins. FORMULATION OF ROBLEM: y A y= B C Z O X y=- D Fig. 4
2 Sarma AS., Sc. J. Eng. Tc., 5; 3(A):4-54 Lt us considr two infinit porous plats AB & CD sparatd by a distanc. T fluid ntrs in y-dirction. T locity componnt along x-axis is a function of y only. T motion of incomprssibl fluid is in two dimnsion and is stady tn u u ( y), w & t T quation of continuity for incomprssibl fluid u w u and put w =, & x y z x y is indpndnt of y but motion along y-axis. So w can say is constant locity i.. T fluid ntrs t flow rgion troug on plat at t sam constant locity Also Nair-Stok's quations for incomprssibl fluid in t absnc of body forc wn flow is stady du p d u u B u... () dy x dy p... () y SOLUTION OF THE ROBLEM: Equation () Sows tat t prssur dos not dpnd on y nc p is a function of x only and so () rducs to Bo u dx dy dy dp d u du u Wr dp dx = Constant = - d u du u Bou B o D D u dy dy B 4 B o B A.E m m m y C. F. ccos Ay csin Ay I. B B B Lt A and B y u( y) ccos Ay csin Ay B Using boundary conditions: u at y - and u U at y ccos A csin A... (3) B U ccos A csin A... (4) B or ccos A csin A, U ccos A csin A B B 43
3 Sarma AS., Sc. J. Eng. Tc., 5; 3(A):4-54 & c U c U Cos A B B Sin A B B y Cos Ay u( y) U Cos A B B y Sin Ay U Sin A B B B ( y) ( y) u ( y Sin A y Sin A y ) U B Sin A Cos A B Sin A Cos A B y y u y U SinA y SinA( y )... (5) Sin A B B B lan oisuill flow: In tis cas bot plats ar at rst so U = y y u y Sin A y Sin A( y ) Sin A B B B y ( y) u( y) Sin A( y ) Sin A( y ) Sin A... (6) B Sin A Saring strss at any point du y ( y) Sin A y A Cos A( y ) dy B Sin A ( y) ( y) Sin A ( y ) A Cos A ( y ) ( y ) ( y ) SinA( y ) SinA( y ) B SinA A ( y ) ( y ) CosA( y ) CosA( y )...(7) B SinA Skin friction at lowr & uppr plats Sin A A Cos A y B Sin A Sin A ACos A A... (8) y B Sin A Sin A A Cos A y B Sin A Sin A ACos A A... (9) y B Sin A 44
4 Sarma AS., Sc. J. Eng. Tc., 5; 3(A):4-54 T arag locity distribution in poisuill flow: ua u( y) dy y ( y) Sin A( y ) Sin A( y ) Sin A d y B Sin A ( y ) Now Lt I Sin A( y ) dy o o y A y y A y y A y ( y ) A( y ) dy A A A A A A A A B A A o A A A A A B I Sin A ACos A A B ( y) I Sin A( y ) dy Sin A ACos A A B I3 Sin A dy Sin A ua I I I3 B Sin A B Sin A Sin A A Cos A A Sin A A Cos A A B Sin A T olumtric flow Q ua A Cos Cos A BSin A... () B Sin A u a A Cos Cos A BSin A... () B Sin A ' T Drug cofficints: C & C at y & y C f f f Sin A ACos A A y B Sin A u a A Cos Cos A BSin A...() 45
5 Sarma AS., Sc. J. Eng. Tc., 5; 3(A):4-54 C f ' Sin A ACos A A y B Sin A u a A Cos Cos A BSin A dx dy dz T stram lin in t plan poisuill flow: u w dx dy dz y ( y ) o Sin A( y ) Sin A( y ) Sin A B Sin A Taking first two y y B Sin A x Sin A ( y ) Sin A ( y ) Sin A dy C ( y ) Lt I Sin A( y ) dy I ( y) ( y) A( y) ( y) A( y) A A A( y) A( y) A A Sinc A B B ( y) Sin A( y ) ACos A( y ) B o ( y) ( y ) ( ) ( ) ( ) I Sin A y dy Sin A y ACos A y B I3 Sin A. dy y. Sin A First stram lin. B Sin A x I I I 3 C B Sin A ( y) Sin A( y ) A Cos A( y ) x Clarly t curl q B ( y)...(3) Sin A( y ) A Cos A( y ) y Sin A C... (4) B Scond stram lin z c... 5 t fluid is Rotational 46
6 Sarma AS., Sc. J. Eng. Tc., 5; 3(A):4-54 Tabl for locity: B B 9,.5,.5, 6,, Wn 6 A Tabl- (for locity) y u(y) B u(y) B 3 u(y) Fig. - 47
7 Sarma AS., Sc. J. Eng. Tc., 5; 3(A):4-54 Tabl for Skin friction B B 9,.5,.5, 6,, Wn 6 A Tabl- (for skin friction) y B 6 B Fig.- 48
8 Sarma AS., Sc. J. Eng. Tc., 5; 3(A):4-54 Tabl for locity: B B 9,.5,.5, 6,, Wn p B 5 A Tabl-3(for locity) y u(y) u(y) B B 3 u(y) Fig.-3 49
9 Sarma AS., Sc. J. Eng. Tc., 5; 3(A):4-54 Tabl for Skin friction B B 9,.5,.5, 6,, Wn p B 5 A Tabl-4(for skin friction) y B B Fig.-4 5
10 Sarma AS., Sc. J. Eng. Tc., 5; 3(A):4-54 Tabl for locity: B B 9,.5,.5, 6,, Wn f 5 A Tabl-5 (for locity) y u(y) B B 3 u(y) u(y) Fig.-5 5
11 Sarma AS., Sc. J. Eng. Tc., 5; 3(A):4-54 Tabl for Skin friction B B 9,.5,.5, 6,, Wn f 5 A Tabl-6(for skin friction) y B B Fig.-6 5
12 Sarma AS., Sc. J. Eng. Tc., 5; 3(A):4-54 CONCLUSION AND DISCUSSION: In tis papr, w a instigatd t locity by t grap of tabl- of quation (5). T locity in porous mdium and magntic fild at fild at B 6 is lss tan t corrsponding alu of locity in porous wit magntic B 3 in t intral. y.3 But t alu of locity is qual to zro in all mdiums at y.5. is lss tan t corrsponding alu of B and is also lss tan t corrsponding alu of locity in porous mdium wit Again from t tabl-3 t alu of locity in porous mdium at locity in magntic fild at B magntic fild at 3 in t intral y.4. T alu of locity in all mdiums is qual to zro at y.5 and alu of locity (ngatily) in porous mdium is lss tan t alu of locity (ngatily) in magntic fild and also is lss tan t locity (ngatily in porous mdium wit magntic fild at y.6. Again from t tabl-5 t alu of locity in magntic fild at B is lss tan t corrsponding alu of locity in porous mdium at and is also lss tan t corrsponding alu of locity in porous mdium wit B magntic fild at 3 in t intral y.4.t alu of locity in all mdiums is qual to zro at y.5 and alu of locity (ngatily) in magntic fild is lss tan t alu of locity (ngatily) in porous mdium and also is lss tan t locity (ngatily in porous mdium wit magntic fild at y.6. Again w a instigatd t skin friction by t grap of tabl- of quation (8). T skin friction in porous mdium B and t magntic fild at 6 is lss tan t corrsponding alu of skin friction in porous mdium wit magntic fild at and magntic fild at B 3 in t intral y.3 and also skin friction (ngatily) in porous mdium mdium wit magntic fild at B 6 is lss tan t corrsponding alu of skin friction (ngatily) in porous B 3 in t intral.4 y.6. B Again from t tabl-4 t alu of skin friction in porous mdium at skin friction in magntic fild at magntic fild at is lss tan t corrsponding alu of and is also lss tan t corrsponding alu of skin friction in porous wit B 3 in t intral. y.3 and skin friction (ngatily) in porous mdium at is lss tan t corrsponding alu of skin friction (ngatily) in magntic fild at B and is also 53
13 Sarma AS., Sc. J. Eng. Tc., 5; 3(A):4-54 lss tan t corrsponding alu of skin friction (ngatily) in porous mdium wit magntic fild at in t intral.4 y.6. Again from t tabl-6 t alu of skin friction in magntic fild at B B 3 is lss tan t corrsponding alu of skin friction in porous mdium at and is also lss tan t corrsponding alu of skin friction in porous wit B magntic fild at 3 in t intral y.3 and skin friction (ngatily) in magntic fild at B is lss tan t corrsponding alu of skin friction (ngatily) in porous mdium at and is also B lss tan t corrsponding alu of skin friction (ngatily) in porous mdium wit magntic fild at 3 in t intral.4 y.6. Also w a instigatd saring strss, t olumtric flow, drag cofficints and stram lins by t quations (7), (9), (), (),(3), (4) and (5) rspctily. T fluid is rotational. REFERENCES. Aydin M, Fnnr RT; Boundary lmnt analysis of drin caity flow for low and modrat Rynolds numbrs, Int. J. Numr. Mt. Fluids, ; 37: Baloul A, BoutanaN, Vassur ; Doubl-diffusi and sort-inducd conction in a sallow orizontal porous layr, t Journal of Fluid Mcanics, 3; 49: Barbu V, Haârnanu T, opa C, Sritaran SS; Exact controllability magnto-ydrodynamic quations, Communications on ur and Applid Matmatics, 3; 56(6): Barkly D. & Tuckrman L.S. Stability analysis of prturbd plan Coutt flow, ys. Fluids, (999), pp Barkly D, Tuckrman LS; Turbulnt-laminar pattrns in plan Coutt flow, IUTAM, 5; Barkly D, Tuckrman LS; Man flow of turbulnt-laminar pattrns in plan Coutt flow, J. Fluid Mc, 7;576: Barkly D, Tuckrman LS; Turbulnt-laminar pattrns in plan Coutt flow IUTAM, 5; Barragy E, Cary GF; Stram function-orticity drin caity solutions using p finit lmnts Computrs and Fluids, 997; 6: Batclor G; A proposal concrning laminar waks bind bluff bodis at larg Rynolds numbr J. Fluid Mc., 956;: Batclor G; On stady laminar flow wit closd stramlins at larg Rynolds numbr J. Fluid Mc, 956;: Baytas C, op I; Fr conction in obliqu nclosurs filld wit a porous mdium,int. J. of Hat and Mass Transfr, 999; 4(6): Bam RM, Warming RF; An implicit factord scm for t comprssibl Nair-stoks quations AIAA J, 978; 6: Sing B, Sing C; Analysis of Vortx Motion in orous Mdia Journal of Elctronics Cooling and Trmal Control J. of Elctronics Cooling and Trmal Control, ; :
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