ELLIPTIC VARIATIONAL INEQUALITY OF THE SECOND KIND FOR THE FLOW OF A VISCOUS, PLASTIC FLUID IN A PIPE.

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1 ELLIPTIC VARIATIONAL INEQUALITY OF THE SECOND KIND FOR THE FLO OF A VISCOUS, PLASTIC FLUID IN A PIPE. Shr Muhsen Jbr Dept. of Mth., College of Eduction, Bbylon University Abstrct The elliptic vritionl inequlity of the second kind for the flow of viscous, plstic fluid in pipe is considered. This elliptic vritionl inequlity is relted to second order prtil differentil opertor. The physicl nd theticl interprettion nd soe properties of the solution re proved. - Introduction The vritionl inequlity is n iportnt nd very useful clss of non-liner probles rising fro echnics, physics etc. the EVI hs two clsses, nely EVI of the first kind nd EVI of the second kind. In this pper we shll study the existence, uniqueness nd properties of the solutions of EVI of the second kind. -: Nottions: V: rel Hilbert spce with sclr product (.,. ) nd ssocited nor. V * : The dul spce of V. (.,.): V V Â is biliner, continuous nd V- elliptic for on V V. A biliner for (.,.) is sid to be V-elliptic if there exists positive constnt v, v ³ v. such tht ( ) In generl we do not ssue (.,.) to be syetric, since in soe pplictions non-syetric biliner fors y occur nturlly []. L : V Â continuous, liner functionl. K : is closed, convex, non-epty subset of V. j (). :V Â=ÂÈ{ } is convex, lower sei- continuous (l.s.c) nd proper functionl. (j(.) is proper if j(v)>- nd j¹ )..: EVI of First Kind To find such tht u is solution of the proble : ì( u, u) ³ L( u), " vîk P... í îuî k.3: EVI of second kind To find such tht u is solution of the proble : ì( u, u) + j( v) - j( u) ³ L( u)," vîv P... í î.4: Existence nd Uniqueness Results for EVI of Second Kind.4.: A Theore of Existence nd Uniqueness Theore.4. []: The proble P hs one nd only one solution.

2 - Uniqueness Let u nd u be two solutions of (P ). Then we hve: u, u ) + j( v) - j( u ) ³ L( u ),u ÎV..() ( u, u ) + j( v) - j( u ) ³ L( u ),u ÎV.() Since j(.) is proper p there exists v o ÎV such tht - <j(v o )<. Hence for i=, - <j(u i ) j(v o )-L(v o -u i )+(u i,v o -u i )...(3) This shows tht j(u i ) is finite for i=,. Hence by substituting u for v in () nd u for v in () nd dding we obtin u - u ( u - u, u - u )..(4) Hence u = u. Existence: For ech nd r> we ssocite proble ( ) below:- To find wîv such tht:- P u r of type (P ) defined s ( w, w) + rj( v) -rj( w) ³ ( u, w) + rl( w) -r( u, w) u ì ( Pr )... í îwîv...(5) The dvntge of considering this proble overt the proble (P ) is tht the biliner for ssocited with ( P u r ) is the inner product of V which is syetric. Let us first ssue tht ( P u r ) hs unique solution for ll nd r>. For ech r define the p f r : V V by f r ( u) = w where w is the unique solution of ( P u r ). e shll show tht f r is uniforly strict contrction pping for suitbly chosen r. Let u, u ÎV nd w i = fr( u i), i=,. since j(.) is proper we hve j(u i ) finite which cn be proved s in (3). Therefore we hve w, w - w + rj w -rj w ³ u, w - w + rl w - w -r u, w - w,...(6 ( w, w - w) + r j( w) -rj( w) ³ ( u, w - w) + rl( w - w) -r( u, w - w),...(7) Adding these inequlities we obtin fr ( u ) f ( u ) w w - r = - ( I -r A)( u - u ), w - w ) I -r A u - u w - w.(8) Hence: ( ) ( ) ( ) ( ) ( ) ( ) ) ( u )- f ( u ) I A u u r -r fr - It is esy to show tht I -r A < when < r<. A

3 This proves tht f r is uniforly strict contrcting pping nd hence hs unique fixed point u. This u turns out to be the solution of (p ) since f r (u)=u iplies. (u, u)+rj(v)- rj(u)³(u, u)+ rl(u)-r(u, u) "vîv. Therefore (u,u)+j(v)-j(u) ³L(u) "vîv.(9) Hence (P ) hs unique solution. - An Exple of EVI of the second kind The Flow of A viscous, plstic Fluid in A pipe..: Nottions * : bounded doin in Â. * G:. * x={x, x } generic point of. ï ì ï ü * Ñ= í, ý x x ïî ïþ C : Spce of -ties continuously differentible rel vlued functions for * ( ) which ll the derivtive up to order re continuous in. * C = { vîc : supp (v) is copct subset of } p * v = å D, p, v for v ÎC æ ö L where =(, ) ;, non- ç ø negtive integers, = + nd D p * p * w, : copletion of, * H = w,, * H = w. w, : copletion of ( ) è = X X C in the nor defined bove. C in the obove nor..: The continuous Proble: Existence nd Uniqueness results []. Let be bounded doin of  with sooth boundry G. e define V = H ( u v) òñ, L j = u Ñv dx * ( v) =< f, v>, f ÎV ( v) = òñv dx Let, g be two positive preters, then Theore..: The vritionl inequlity ( u, u) + gj( v) - gj( u) ³ L( u), ()..

4 hs unique solution. In order to pply theore (.4.), we only hve to verify tht j(.) is convex, proper nd l.s.c. It is obvious tht j(.) is convex nd proper. Let u,v ÎV, then j v - j u es. u- v, () ( ) ( ) ( ) V hence j(.) is l.s.c. This proves the theore. Rerks. If we tke g= in () we recover the vritionl forultion of the Dirichlet proble - Du= f in u= on G. since (.,.) is syetric, the solution u of () is chrcterized s the unique solution of the iniiztion proble J( u) J( v) ()... where J( v) = ( v, v) + gj( v) -L( v).3: Physicl Motivtion If L v = c vdx (for instnce c>), it is proved in [3] tht () odels the ( ) ò linr, sttionry flow of Bingh fluid in cylindricl pipe of cross-section, u(x) being the velocity t xî[4], [5]. The constnt c is the liner decy of pressure nd, g re respectively. The viscosity nd plsticity yield of the fluid. The bove + = xî= x > nd { } ediu behves like viscous fluid (of viscosity ) in ( ) like rigid ediu in = { x Î=( x) = }, [6], [7]..4: Existence of Multipliers Let us define A by: A= q : qî L L, q x q { ( ) ( ) ( ).e. } ( x) = q ( x) q ( x), then we hve + Theore.4. The solution u of () is chrcterized by the existence of p such tht (3). ( u, v) gò + p. Ñvdx=< f, v>,

5 (4)... p. Ñ u=. e., pîa e shll prove tht (3), (4) iply (). It follows fro (3) tht u, u + g p. Ñ u dx= u, u + g p. Ñvdx- (5)... ( ) ò ( ) ( ) ò - gò p. dx=< f, u> It follows fro (4) tht p. Ñxdx= dx (6) ò ò nd fro the definition of A tht p. dx p Ñv dx Ñvdx (7) ò ò ò Then fro (3), (5)-(7) we obtin tht (u,u)+gj(v)-gj(u)³<f,u> "vîv, Thus (3) nd (4) iplies (). References [] : Lions J. L., Stpcchi G., vritionl Inequlities, co.. Pure Applied Mth., XX, (967), PP. (493-59). []: Chipot M., Michille G., Uniqueness results nd onotonicity Properties for the solutions of soe vritionl Inequlities, contents of the Proceedings FBP, 987, Gerny. [3]: Duvut G., Lions J. L., The Equtions in Mechnics of Physics, Dunod, Pris, (97). [4]: Frnzini J. B., Finneore E. J., Fluid Mechnics with Engineering Applictions, McGrw-Hill cop. Inc [5]: Prger., Introduction to Mechnics of continuu, Ginn nd copny, Boston, 96. [6]: Mosolov, P. P., Misuikov V. P., vritionl ethods in the Theory of the Fluidity of viscous plstic Mediu, J. Mech. And Appl. Mth. (P. M. M.), vol.9, (965), 3, pp. (468-49). [7]: Mosolov P. P., Misnikov V. P., on qulittive singulrities of the flow of viscous plstic Mediu in pipes, J. Mech. And Appl. Mth. (P.M.M.), vol.3, (967), pp. (58-585). الخلاصة تم اعتبار المتباینة التغایریة الناقصیة من النوع الثاني لجریان ماي ع لزج في انبوب. هذه المتباینة التغایریة الناقصیة تعود الى مؤثر تفاضلي جزي ي من الرتبة الثانیة. وقد تم برهنة التفسیر الریاضي و الفیزیاي ي وبعض خواص الحل.