SHAPE DIFFERENTIABILITY OF DRAG FUNCTIONAL AND BOUNDARY VALUE PROBLEM SOLUTIONS FOR FLUID MIXTURE EQUATIONS
Journal: SCIENCE EVOLUTION
Volume 1 № 2
, 2016
Rubrics: MATHEMATICAL SCIENCES
Authors:
1.
Kemerovo State University
Russian Federation
Russian Federation
2.
Kemerovo State University
Russian Federation
Russian Federation
Type:
Article
Pages:
from 41
to 56
Status:
Published
Received:
30.12.2016
Accepted:
30.12.2016
Published:
30.12.2016
Subject area:
GRNTI 29.01 Общие вопросы физики
GRNTI 27.01 Общие вопросы математики
GRNTI 31.01 Общие вопросы химии
GRNTI 34.01 Общие вопросы биологии
GRNTI 27.01 Общие вопросы математики
GRNTI 31.01 Общие вопросы химии
GRNTI 34.01 Общие вопросы биологии
Language:
English
Keywords:
boundary value problem, mixture of viscous compressible fluids, conjugate problem, flowing around an obstacle, material derivative, shape derivative
Abstract (English):
Problems of optimal design of various elements of technical structures stimulate mathematical statements of new problems of continuum mechanics and hydrodynamics in particular. This study refers to problems of shape optimization of profiles in a flow of fluid or gas. The paper deals with properties of solutions and their functional of inhomogeneous boundary value problem for nonlinear composite type partial differential equation system, simulating the a mixture of viscous compressible fluids (gases) flowing around an obstacle. Methods of the theory of partial differential equations, functional analysis and, in particular, the results on the solvability of boundary value problems for transport and Stokes equations established the well-posedness of a linear boundary value problem with singular coefficients (the problem of the original problem solution difference). This result allowed to obtain the uniqueness theorem to determine the nature of solutions dependence on the shape of the flow range and to prove domain differentiability of the solutions considered. Domain differentiability of the solution functional reflecting the force of the obstacle resistance to the incident flow is proved. A formula to equate this derivative as a sum of two summands, one of which clearly depends on mapping setting the domain shape, and the other can be expressed in terms of the so-called adjoint state, depending only on the solution of the original problem in a non-deformed domain. The functional derivative formulas may be used as the basis for building a numerical algorithm for finding the optimal shape of the body in a flow of mixture of viscous compressible fluids.
Problems of optimal design of various elements of technical structures stimulate mathematical statements of new problems of continuum mechanics and hydrodynamics in particular. This study refers to problems of shape optimization of profiles in a flow of fluid or gas. The paper deals with properties of solutions and their functional of inhomogeneous boundary value problem for nonlinear composite type partial differential equation system, simulating the a mixture of viscous compressible fluids (gases) flowing around an obstacle. Methods of the theory of partial differential equations, functional analysis and, in particular, the results on the solvability of boundary value problems for transport and Stokes equations established the well-posedness of a linear boundary value problem with singular coefficients (the problem of the original problem solution difference). This result allowed to obtain the uniqueness theorem to determine the nature of solutions dependence on the shape of the flow range and to prove domain differentiability of the solutions considered. Domain differentiability of the solution functional reflecting the force of the obstacle resistance to the incident flow is proved. A formula to equate this derivative as a sum of two summands, one of which clearly depends on mapping setting the domain shape, and the other can be expressed in terms of the so-called adjoint state, depending only on the solution of the original problem in a non-deformed domain. The functional derivative formulas may be used as the basis for building a numerical algorithm for finding the optimal shape of the body in a flow of mixture of viscous compressible fluids.
Keywords:
boundary value problem, mixture of viscous compressible fluids, conjugate problem, flowing around an obstacle, material derivative, shape derivative
boundary value problem, mixture of viscous compressible fluids, conjugate problem, flowing around an obstacle, material derivative, shape derivative
INTRODUCTION Search for the optimal shape (with the lowest drag) ofthe obstacle in a flow ofa mixture of viscous compressible fluids (gases) is eventually associated with the problem of derivation ofthe shape functional, expressing the force of the obstacle resistance to the incident flow. This problem, in turn, requires a study of well-posedness of inhomogeneous boundary value problem for the corresponding equations and study of dependence ofthis boundary value problem solutions on the flow region shape. Most of the known results for Navier-Stokes equations for viscous compressible fluids and moreover for equations ofmixtures ofsuch media concerns flows in are as bounded by impenetrable walls, while the results of study ofinhomogeneous boundary value problems remain fairly modest. Among the papers dealing with the last issue, we'd like to mention [l], which proves existence theorem for non-stationary Navier-Stokes equations fro viscous compressible fluid with constant boundary value conditions, and [2], which establishes existence of a weak solution ofbarotropic viscous gas flow equations in convex domains with the outlet, independent on the time variable. Local strong solutions (close to a uniform flow) of stationary problems with inhomogeneous boundary value conditions were studied in [3-5] for two-dimensional domains on the hypothesis that the velocity field at the boundary of the flow region is close to a prescribed constant. Important results relating to the existence of strong solutions of inhomogeneous boundary value problems for stationary Navier-Stokes equations in case of small Reynolds and Mach numbers were obtained in [6-8]. Results on the well-posedness ofan inhomogeneous boundary value problem for the equations of mixtures of viscous compressible fluids were obtained in [9]. The shape optimization theory is a section of variational calculus, where the functional arguments are shapes of geometrical and physical objects. A classic example of the shape optimization problem is the isoparametric Newto n's problem of the body of least resistance. Description of the general theory and bibliography on this subject are available in [10-15]. The first global result concerning dependence on the solution region of compressible Navier-Stokes equations was obtained by Feireisl [16], and was further developed Please cite this article inpress as: Kucher N.A., Zhalnina A.A. Shape differentiability of drag functional and boundary value problem solutions for fluid mixture equations. Science Evolution, 2016, vol. 1, no. 2, pp. 41-56. doi: 10.21603/2500-4239-2016-1-2-41-56. Copyright © 2016, KemSU. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), allowing third parties to copy and redistribute the material in any medium or format and to remix, transform, and build upon the material for any purpose, even commercially, provided the original work is properly cited and states its license. This article is published with open access at http:// science-evolution.ru/ Science Evolution, 2016, vol. 1, no. 2 in a series of papers of Plotnikov, Ruban, Sokolowski [7, 8, 17-20). These studies also provide an algorithm for calculation of the drag functional derivatives determined in the collection of domains. The problem statement is as follows. Range of flow of viscous compressible fluid mixture is a domain n= B \S of Euclidean space IR 3 of points x= (x1 ,x2 ,x3 ) , external with respect to an obstacle S (which is assumed to be a compact set) and bounded by a closed surface L. Let us assume that x � T(x) denotes the vector field of class c2 (IR3) , equal to zero in the neighborhood of the boundary L. Let us define a mapping x � y= fs (x) = x + &T(x) which defines the perturbation of the obstacle S . For small & , mapping x � Ts (x) is a diffeomorphism of the flow region n onto n& = B\S& ' where SE = t(S) is perturbed obstacle in a flow. Stationary motion of the mixture of viscous compressible fluids in the region ns is described by the following equations [21): - m ;l"'e , 1 - - m ;l"'e , 1 - , , , , 2 2 (-(})) R (-(i) V)-(i) Re V ( ) J-(i)- 0. '""' .- 1 2 (1) ue + --2 ue + --2 P;e P;e Pie Pie + + °""'�Lii ue + epie ue · j=I Ma div(P;e u!i))= 0 in Qe , i= 1,2, (2) A-(}) A-(}) ( ( where u?), u?) represent the velocity fields of the ReynoldsandMachnumbers, respectively; Ly ,i,j= 1,2, mixture components; Pis , p2s are the component refers to the second-order differential operators Pis = Pis CPis ), i= 1,2, Pis = Pis CPis ), i= 1,2, are assumed to be sufficiently are assumed to be sufficiently density functions, and corresponding pressures Lij(u-(}))---µ!i !J.U - Aj + /l,� !i )Vd"cvu-(}),.1,1.- 1,2, where constant (dimensionless) viscosity coefficients smoothfunctionsoftheirdensities; Re and Ma denotethe Aj , A'!i satisfy the conditions µ11 > 0, 4µ11 . µ22 -(µ12 + µ21 ) 2 > 0, "-11 + 2µ11 > 0, 4("-11 + 2µ11 )("-22 + 2µ22 )-("-12 + 2µ12 + "-21 + 2µ21 ) 2 > 0. Summands JU)= (-) 1(i)a(u?)- ui1)), a= canst > 0, i= 1,2, describe intensity of the momentum exchange between the mixture components [22,23]. Equations (1) and (2) represent the laws of conservation of momentum and mass of the mixture components, respectively. QU) (b) Fig. 1. Diagrams of the flow of the jth component of the mixture around the obstacle: (a) three-dimensional flow; plane section. For statement of boundary value conditions let us use the vector fields ijU),j= 1,2, of class C\IR 3 ), vanishing in a neighborhood of the set S . Let us use vector functions ijU) on the boundary L of the domain B to allocate "inflow" areas: L{n = {x EL: ijU) -ii< O}, j= 1,2 , and "outflow" areas: L�ut = {x EL: ijU) · n > O}, j= 1,2 (see Fig. 1). Let us assume that the following conditions are satisfied. f f Condition 1. Sets rj = clL{n n(L\L{n ), j= 1,2, ("characteristic" surface areas) are closed one-dimensional varifolds, such that L= Lin ur1 UL�ut , and, among other things: ij(J) ·nsd= 0, j= 1,2; ijU) · V(U(j) ·n) > C > Oon r1, j= 1,2, where C > 0 is a constant. L Pjs = PJe on Lin,j= 1,2, Pjs = PJe on Lin,j= 1,2, ii£})= 0 on ass , ii£})= 0 on ass , uij)= ijU) on L, uij)= ijU) on L, Adjoin the following boundary value conditions to equations (1), (2) (3) where pJc , j= 1,2, are prescribed positive constants. 42 Science Evolution, 2016, vol. 1, no. 2 The force ofdrag to incident flow from the obstacle S8 is expressed by the formula Jn(Ss)= -u00 I,,=1 asf. [f1=1[µii (VusO)+�u2f)+liidivu2)1]- i:2 P;{ps)IJ·nds, (4) optimal shape ofthe obstacle. optimal shape ofthe obstacle. rate at "infinity". The problem of minimization this rate at "infinity". The problem of minimization this n for one-parameter family of differential equationswi rb ts e, et n for one-parameter family of differential equationswi rb ts e, et where U00 is the constant vector that simulates the flow functional is solution to the problem of selection the The problem (1)-(3) can be conveniently reduced to a boundary value problem in the unperturbed domain in accordance with the following formulas: in accordance with the following formulas: innttrhodpuecretutheedfunccoteioffincsie;;nUl. aFnodr Pth;i,si =p1u,rp2 odsefinled uins 8 8 ;;.,i = 1,2, where N(x) = (detM(x) M-1(x) , M(x) = I+cDT(x) , DT(x) = {oT;ax(x)} is theJacobimatrix ofthe mapping x f-7 T(x). 2 2 i i ; 2 2 2 i i ; 2 As a result ofthis transformation, the problem (1)-(3) is transformejd into the problem ii ii j=l j=l j=l j=l �} AuUl -Vq; = :�:)iiA(uUl ;N)+ReB(p;,u< ),u< l ;N)+(-1) S(u< l -uO\N) in n, (5) A(ii;N) = Au -(N A(ii;N) = Au -(N r r )- )- 1 1 div(g-1 div(g-1 NN NN r r v(Nv(N- 1�), 1�), Where g = g(x;N) =�dteN(x) ;linear operators A, S and non-linear mapping B are defined according to the formulas B(p,u,w;N) = p(Nr)-1� v{N-1 w)), S(ii;N)= g ·a (Nr)-1 N-1 ii; 2 1 Re 1 j=I j=I qi = -Lg- (Aj+l;1)divu;[0 ])dx, (i) (i) n-div(ii(i ) .,P \iln)+, .. .r (i) =, .. inPn n' J=I=0 onL(i) i ;· =1 2 ':, J 11 ':, J JI g , ':, J out, , , · ':, J 11 ':, J JI g , ':, J out, , , · ':, J 11 ':, J JI g , ':, J out, , , · ':, J 11 ':, J JI g , ':, J out, , , · Here, the constant parameters 1:;1 are related in a known manner with the viscosity coefficients 43 Science Evolution, 2016, vol. 1, no. 2 a;n[d0], 'P;p[Bres] cribedare bountdhaery kvnaoluwens such that 0 E vs,r X xs,r m- E �,':,;-z�}) E xs,r ' where , 1 , 1 functions of the vector B=(v(l),v(2); 1Z"1,1Z"2,'P1,'P2) functions of the vector B=(v(l),v(2); 1Z"1,1Z"2,'P1,'P2) <1> Theorem 1. Let the surface L and vector Theorem 1. Let the surface L and vector wcohmerpeotnheenetsx,isetxenpcreestshioeonrsemofiswphriocvhedaraes wgievlel.n in [9], , 1 , 1 3 3 parameters s E (0,1 ), r E (1, oo) satisfy the conditions T T s · r > 3, 2s --r < 1. Thus vector 0- belongs to a ball Br of radius centered at the zero point of the space Here, , refer to spaces Here, , refer to spaces r are such numbers a• > 1 and r• E (0,1) that if are such numbers a• > 1 and r• E (0,1) that if fields U(I), u<2) satisfy the condition 1. Then there vs,r xxs, . III III Nll 2( ) Nll 2( ) - n - n �r , r E(0, �r , r E(0, r*],p (p )EC (0, ),j =1, r*],p (p )EC (0, ),j =1, 2 2 , , 2 2 3 3 oo oo the matrix N is selected based on the condition vs,r xs,r xs,r =ws,r (0.) n wl,2 (0.) vs,r =ws+l r (0.) n w2,2 (0.) wishaere W1·P(O.) (I is baynoSn.-Ln.egSaotbivoeleivnte[2g4er],, 1� p < oo ) fi fi 1 1 1 1 and the problem parameters are such that -A1 ::; ,2 , of msetaansduarardblespaicne 0. functions having gecnoenrsailsitzinedg oo oo Re�.2 •;; 2:: a.,z.= 1,2, wdeirthivaantivinedseixn o0.f poowf eurp pto. Fl oorrrdeearl isncElu(0si,v1e) l,yr, sEum(1m, e)d 0- - 0- r E (0, r*] , then the problem (6) has a solution - (V-(!),V-(Z),· 1Z"1, 1Z"2,'P1,'P2 ), m;,':,;-;(}), Z.,J. -- 1'2 ' iantfuerpncotliaotnioanl mspeathceod ws,r (b0.etw) eiesnorbta(0.ine)danbdywthe (0.rea)l, lu(x)-u(y)Ifur nctions finite -3 -3 and consists of measu[r2a5b]le with a 1 nr orm. and consists of measu[r2a5b]le with a 1 nr orm. llul�s,r cn) =II ull1: cnt Iu ls ,r,n, I u ls ,r ,n= nfxn Ix-YI ( Ix-y I s J dxdy. functions with a finiteenfionrem. functions with a finiteenfionrem. =liullw =liullw iIsngaennienrtaelg,tehreasnpdacies wd 1+s,r (d0.as), 0th= fv(x)·u(x)d x is a continuous determined by the re[Wal0o (0.) wJ (O.) s r with the norm v n � s ::; 1, 1 < r < r r' � s ::; 1, 1 < r < r r' (0.) (0.) Let us assume that O interpolation. oo , -1 +-1 =1 . Let ,r (0.) denote completion of r(0.) in the norm uEWo'' (Q) Q uEWo'' (Q) Q SU� fU ·Vdx. llul�0,,,(0.) = 1 functional in the space ws,r (0.), continuously embedded in r' . As is known [7] that if n is a bounded domain in �.3 with a boundary of class C1 , cws,r' ' cws,r' ' then w-s,r (0.) is algebraically and topologically isomorphic to the dual space (O.)) ' and can be identified with it. In addition, let us introduce functional spaces us,r =wH r (Q)X ws,r (Q)X �, vs,r =ws+l r (Q)X ws,r (Q)X �, zs ,r =ws-l r (0.) n L2 (0.) £s,r = zs,r xxs,r x�, :Fs,r = vs ,r xxs,r x�, 2 3 2 3 2 3 2 3 and at the same time Fi E Wi., F E W , F E W . If and at the same time Fi E Wi., F E W , F E W . If product of spaces W is to be understood inthe product of spaces W is to be understood inthe with reference to the paameter 8. For this purposei ces a y tu d e o h s s with reference to the paameter 8. For this purposei ces a y tu d e o h s s Membership of tW1hexvectoxrWmagnitude F of the direct or vseencstoert)h, asteFparisatmedadbeyupseomf tihcroeleoncomFp=on(eF;nt;sF(s;cFala)r 2 2 3 3 2 2 3 3 problem (6) and differentiability of its solution range deformation. range deformation. otnisthnee mastrirx tNo scodmypdleetpeelyn deentcermifnetdesbey tohleutflioonw Wi =W =W =W , write F EW and separate the At this stage, the matrix N structure is irrelevant vector c2ompo3nents by comma. DOMAIN DEPENDENCE OF SOLUTIONS studTyheismtoostpirmpoveorttahnet sutnaigqeueonfetshse oshfaspoeluotpiotinmiozfatitohne 2 1 2 1 foanrd tahnereaforbrietrathrye rsemsuolotsthobmtaaitnreixd-vbaelluoewd afurencvtailoind qo-q1, qo-q1, N(x), x En. N(x), x En. The problem for differences. For difference 1 2 , , 2,'P1,'P2,..,1; ,..,2; ,':,J; ,':>2 2,'P1,'P2,..,1; ,..,2; ,':,J; ,':>2 , , 1 1 -q; -_ cV-;< 1 ),v-;< ). 1Z"1i, 1Z" i i i ,,..c ) ,,..( ) ,,..(2) ,,..c ;). m i, m2i),z. _- 0,1, 44 Science Evolution, 2016, vol. 1, no. 2 of solutions to the problem (6) obtained according to Theorem1 and corresponding to different matrices NO and N1, let us introduce the following notations w-(}) --v-(}) -v-(j) , m --1r0 -1rI , lf/ --<,'I';• -_ OonL; ,-,"1o)• -_ OonL;; n,IIw;• -_ W;•,l.,J.-_ 1,2. (19) (20) Linear operators 'H; and M; are defined by the following formulas 'H;(h) = p}V(N·,;1u&i))N·,;1 1z -(Nr)-1 div(pJuti) ® (Nr;1h)), M;(h) = (ii&i) V(No1 u&0) · No1h,i = 1,2. The further content of this section is devoted to proving existence and uniqueness of strong and weak solutions of the conjugate problem (15)-(20). These results enable us to derive estimates of norms for the * , i ,.,, i , ':,] , ':>2 , i 'J.Y f"(i)*.n*)EV f"(i)*.n*)EV s s , , r r sat1's•l:.•1'ng the sat1's•l:.•1'ng the h h -(i) =(w-(i).m• -(i) =(w-(i).m• ]"(i)* ]"(i)* .,,• .,,• inequality (21) d1'ffierences w-(i) , m; , If/;,<:,]";(}) ,n; . If we apply a more restrictive condition to the Theorem 2. Let the conditions of the existence theorem are satisfied and parameters s,r are such that right-hand member J , namely J E [,s,r , the solution 2 2 -1 < s < 1, (1-s)r > 3. Then there are numbers c,a-c h belongs to the class ;:s,r and at the same time the following inequality is valid. and re depending on the objective and parameters s,r such that if min{r1 ,r2 }>a-c and O<-r�rc, then for every vector junction J=(j(l),J<2)), 11h11rs,r �cllJil'£s,r · Teorem 2 proof scheme (22) j), Equations (15)-(19) of the conjugate problem can be represented in a symbolic form A[h.]-B[h.]=F,h.=(1z!1l,1zPl} h}i)=(iWl;m;,lf/;,;?l*,;fl*;n;) - - - - - where the integral-differential operators A and B are determined by the formulas A[h.] = (A1 [h.],A2 [h.]), A;[h.]= - - - - - - B[h.] = (B [h.],B [h.]), (23) 1 2 46 1 1 Science Evolution, 2016, vol. 1, no. 2 2 2 1 IµjiAo(wi )) + ReH;(wYl ) + (-li S0(w£ l -w£ l)-\JI; · V(j)} j=I j=I B;[h.] = ReM;(w£il)+\JI ;divii6i) + I[akw; + nZxZa1J B;[h.] = ReM;(w£il)+\JI ;divii6i) + I[akw; + nZxZa1J II�Jlo'V; + t.(n; xtil;, + µ•�iro;)] 2 k=l Let us consider the following0 value problem by connecting to the system of equations the boundary conditions boundary A[h.] = F (24) It is easy to see win= 0 on an, \JI;•= 0 on L�u" s?J• = 0, s?· =0 on L;", IIro;• =ro;',i =1,2. (25) F- -- (F-(I) ,F-(2) ) , F-th(ai)t =fo(Hr-(ei)a.,cGh ;,F;rig,hMt-ha,nMd m;,se;m) boerf After this, the components w£i> , m;,i =1,2, are found li 2 s r as solutions to the Stokes-type problems, i.e. wiil = 0 on 80, IIm;* =m;*, i =1,2. wiil = 0 on 80, IIm;* =m;*, i =1,2. tdhievisdpeadceinuto s, e,vtehrealboinudnedpaernydveanltuleinperoarblbeomun(2d4a)r,y(2v5al)uies I2 µ1 ;(Awi1l -Vro;) = ffUl -I2 e;;?. VI;;°* inn, problems, namely, functions 1;;°*,i, j = l,2, are defined J-l divii,iil =IIG; inJ-l0, trainsport equation trainsport equation independently as solutions to the following problem for j j - - C • �lhllFs,r · (28) C • �lhllFs,r · (28) U-o< lnY'-J� (i)* +Tih� jut -- Mji l.n:,nt.,'-o� j(i)* - 0 On,:,.�;in'l.,j. -- 1 '2 . sTohleutinoenxttosthepe bisoudnedfianriytivoanluoefpcroomblpeomnents If/.; as the (T2h5e) pernodcsewduithre fifonrdbinugildthinegcsoonlsuttainotns ton;thaeccporordbilnegmto(2t4h)e, IIBhlls,r �C • �lhl�s,n (27) n; = n; = f(; f(; Iµ Iµ 1 1 ;co; ;co; +Y;\JI;] dx+s;,i =1,2. +Y;\JI;] dx+s;,i =1,2. formulas 1l, u<2l and parameters r,s, such 1l, u<2l and parameters r,s, such On the basisnof�kn;o=w1 n results on transport equations and leixniesaternScteookfest-hteypceonpsrtoanbltesmc [a7n,2d6,CJ"9c] ,wdeepceanndiansgseornt tthhee is uniquely so1 lva2ble in scpaces vs,r and :Fs,r and the is uniquely so1 lva2ble in scpaces vs,r and :Fs,r and the tphraotbliefmmdina{tar n,r, u} <� CJ" , then the problem (24), (25) followinllgh..iln�esqru�alictlyl isllvsalid Here, the constan11Bt hcl1£ds,erp�ends only on n, vector fields Now we can complete the proof of existence and Now we can complete the proof of existence and -{j(ll,0(2) and parameters r,s. -{j(ll,0(2) and parameters r,s. Let us assume that conditions of Theorem 2 are Let us assume that conditions of Theorem 2 are uniqueness of the conjugate problem solution. wsartiitstfieenda.s tThheeopceornajtuogr aetqeuaptiroonblem (15)-(20) can be operators A-1 : s r s r operators A-1 : s r s r data and parameters . On the basis of the conditions data and parameters . On the basis of the conditions Due to (26), (27) li(nJe-arA-1B)h =£1 f u , � v , and , F ,r , llh..l�s,r �cllFl�s,r · (26) A-1B: vs,r � vs,r a'BreIIboun'de�d, m·orre,over (29) The foregoing can be interpreted in the form ofexistence 11£ L(Vs' ) cv 1 : s r s r 1 : s r s r ( A- £ , � :F , ) which assigns a unique solution ( A- £ , � :F , ) which assigns a unique solution that the inequ1 a,lit2ie)s .aLreetvalid 'c that the inequ1 a,lit2ie)s .aLreetvalid 'c of (inverse) bounded linear operator A-1 : us,r � vs,r s r ( r ) s r ( r ) to the element FE u , FE e, . to the element FE u , FE e, . h-. =£I [F-] to the boundary value problem (24), (25) Let us now tum to the integral-differential operator where the constant cv depends only on the problem i0n 3 _!_2 < <1, (1- > 3 pBropgeivrteiens.by the formulas (23), and note the following From (29)O;-(ik) , belongm. g to the same spaces W1 '2 (0) . This implies that equations (8)-(1 1),(1 3) are y y e so u10n e so u10n w. ,OJ; ,lf/; ,':>i w. ,OJ; ,lf/; ,':>i ,':> 2 ,n; ,z- , , o ,':> 2 ,n; ,z- , , o 1 t. 1 t. • • • • b b th th -(i) -(i) ;:(i)* ;:(i)* ;:(i)* • ;:(i)* • · · -1 2 t -1 2 t satisfied in the strong sense.Multiplying these equations the conjugate problem (15)-(20), corresponding to the �[sr- \ �[sr- \ i i solution (lv!i ),OJ;,IJl;,c;ti )*,; ;)•,n;),i=l,2, to the element ]eLr (O)xW1r (O)xJR, and integrating by conjugate problem (15)-(20) belongs to the class parts,we obtain the identity f:t \t_V(i)H- (i) +co;G;+'l';F;+� (i)Mli +� (i)M2 ;Jdx+n;s;]-- 1 2 0 (3 2) -�L.i w. ('D;+(-1);E)+d 8�;'11;.+�L.i('C ji � (i)·+n•;x !io8-j +µ/>/o;•))]dx. 1=1 n 1=1 n _ s[-(i) ( ;=1 j Let us note that vector functions (ifCi)·, OJP. If/p· ':,;:lCi) ,1:,;:2Ci)) and (w(i );G;,F;,MJi ,M2;) belong to dual spaces (r > 3, 1 r' _r_ r ) imply that these functions r -l < = < < = < ws-1r, (0) X w-sr, '(0) and Wci-sr, ' (0) X wsr, (0)' belong to w-sr, '(0) . respectively. Indeed, iJU) Ews-lr, (0) Further, the vector field w(i) E vsr, and hence because fIU) EC(0) , and there is a limited embedding C(0) in ws-i ,r (0) . Scalar fields w;G;,F;,M!i ,M2;;s;) from a narrower functional "2 f-ui-ui ( J "2 f-ui-ui ( J (32) can be written in terms ofduamlietymfoberrms and thus, the space r(O)x wl r(O)x JR, i.e, we may write i;n i;n 2;n 2;n w ) +(ro;, G; )o+('1';,F'; )o+(s i , M L.it.;:ICni)*•'=>;:2(in)*., n;,• ), l. _- l ,2, l. S (L3e4m) mwae2 o1.sbtparionvethde. identity(30) stated in Lemma 2. the "sroilguhtito-hnatnodtmheemprboebrl"emf((i)1,5)=-(12,0)·, Fcroormres(p3o1n),ddinuge ttoo Identity (30) proved in Lemma 2 means that the n i 2 n i 2 i 2 i 2 Theorem 2, we have that pdriffobelreemnce(8)ij-(01-3)ij.1 is a very weak solution to the linear hn = (hp>,h�2) � h = (h(l),h(2) n vs, r , n � 00• (34) Lemma 3. Let the conditions of Lemma 2 are Passing to the) limit as � oo )in (33), due to (31), satisfied. Then the following inequality is valid n t(11wCi> 1�6-s,r' +llw;llw-s, r ' +llm;llw-s,r' +lls?> llw-s,r' +llsfllw-s,r'+In; 1) � r,s r,s � cGldlk1(n)+IID1 lk1(n)+IID2 IIL'(n)+ll£lk1(n)) (35) . . where the constant c depends on the problem data and parameters r,s Proof. Turning to the equality (30), we note that since the coefficients O;, O;, O; are bounded ipnrombloedmuludsatabyanadconsta(nsteede(p1e4n))d,inthgenontlhyeornigthhteha avnadlume ember of this identitycan be estimated by I�lwY> ltcnJ+IIIf;ltcnJ+llm;ltcnJ+ll;?l *lhn)+ll;Jil *lbnJ+In; i)·{lldlk'(n)+IID1 ILI '(n)+IID2 lk'(n)+11£1k'(n)). c (36) In addi=iltion, due to embedding theorem and estimate ( 1) we have 2 2 2 i=l i=l "L.i(llw.('') ltcn)+Ill/I;• Ileen)+llm;• ltcn)+ll,;1('')* ltcn)+ll,;2('')* ltcn)+In;·)I � cllh- l�s,r � cll.f1-�s,r From (36) and (37) we obtain (37) "2 { - (i) (i) i) (i) 2 i=l i=l i=l i=l L.i (H , w- ),+(m;, G;)o+(f// ;,F';)o+(,;1( , Mu)o+(,;2 , M2;)0 +n;s;} � c"L.i��ldlk'(n)+IID;ILI '(n)+11£11L'(n))llf1ls,r(38) problem (6) is a Lipschitz solution in a weak norm. problem (6) is a Lipschitz solution in a weak norm. LTehme imneaq3uaisliptyro(v3e8d).obviously implies the estimate (35). soluTtihoenintoeqthuealpirtoyb(l3e5m) i(m6)p.lIinesa,dfdiristtiloyn, ,th(3e5u)niimqupelineesstshoaft wthiethmsaopluptiinognasqsotcoiathtienginthhoemmoagtreinxe-ovuaslubeodunfudnacrtyiovnalNue MATERIAL DERIVATIVES MATERIAL DERIVATIVES So-called material derivatives of solution pTrhoivse secttihoen isdradgevotefud nctotionthael ir sdtiuffdeyr.enStioalbuitliiotyn. to the problem (6) is a function of spatial variable = O and parameter & • Let us assume that = O and parameter & • Let us assume that ij = (v(I),v(2);n'i ,Jr2 , <2> > 1 <2> < 1> <2> > 1 <2> to the problem (6) is an important tool to scpoamcpeoonfefuntnsctijio)nsasofa xfu.nction of & with values in the 1 1 Material derivative ij'(O) = (w , w ;mi, m2 ,f// ,f//2 ,,;p , ,;J >,.g1 ,,;?\n1 ,n2 ) of , solution ij(c)0= (v<1l (c\ v<2> (cfo},l1rlo1w(cs:),1r2 (c),'A (c), rp2 (c\;p>(c),i;;-JI> (c),;f> (c),;f> (c);m1 {c),m2 (c)) to the problem (6) with w(J) w(J) &-+0 &-+0 , J &-+0 ]Ii 6 , J &-+0 ]Ii 6 , , J &-+0 JG , J &-+0 JG ' J &-+0 JG l, j = '2 • J &-+0 JG , J &-+0 JG ' J &-+0 JG l, j = '2 • (39) (39) will =-c-1 (vU> (o)- vU> (c)} m16 =-c-1 (1ri0)-1ric)} f//J6 =-c-1 (rpiO)-rp ic)} will =-c-1 (vU> (o)- vU> (c)} m16 =-c-1 (1ri0)-1ric)} f//J6 =-c-1 (rpiO)-rp ic)} t: = is defined as = lim w(j) m. = lim m. ff/. = lim f//. ;(;) = lim ;Vl n. = limn. . . l ,;;� = -t:-1{s;o(o)- ;y>(t:)} n16 = -t:-1 (mio)-mit:)} i,J = 1,2, provided that the limits in (39) exist in some sense. 49 Science Evolution, 2016, vol. 1, no. 2 Equations for material derivatives can be easily obtained by formal use of 0=B(s) and N = N(s) in equations ( 6) and differentiating results according to L w( i * i v; L w( i * i v; 2 µiiA j)-y'(J) = ,c (\j/;,w( ))+ +j=l s at zero. This formal procedure gives the following system of equations and boundary conditions for material derivatives. ( )l i �*+ a�(2)_ w(I) ))inn, 2 2 2 2 u-(i)�l.o)· nv \jl +-c u\lf;----w(i) · nv cp �l.o)+L""".°'·A . \lf i+"'"'L,Al' *roi+YA.; n;+u[;*d * m. Hr,, (40) i i j=l ii ii j=I -div�(i)()O l;?�H;;/;y) =div�(i)s;°(o))H}id * in 0, w-u)-- o on an' \jl;-- o onL.,� iin• ffi;-- rrffi;, ):":,j(i) -- o onL.,� oi ut, n; _-�Xi0k f[8-k·d *+�(\U-k}·\lf 1+ P-kJ. roJ+ y-kJ·l;J(k))�dx,.1,.1 _-1,2. i:t_ n i::_ J These equations use the following notations 2 v; =-Rep;()�Cil(}v(n»iil(o))+n» 1r uul(} vuui () )+Lµ ldiv('ll'vuUl(»o +w 1r AuUl()o+A(n»uul(»), \If \If \lf \lf £·( ;,we;�= Re( ;uc;J (o-) vuu)(+o) j=I wCi wCi P;(o) ).vuui ()o+ P;uCi)(o)-vwu�; (41) h • h h d h • h h d • • • • • • • • fii • fii • b b f f • • £* =a11'�( 2)()-uO)(»; d * = divT; H»=divTI-DT; 'll'=divTI-Df-nrY. -. -. -. -. A . A . A. A . A . A. A. -· A. -· -· -· C C oe c1ents aii,/Ju,r; ,8; ,aiJ ,/Ju,r; ,8; ,au,/JiJ ,riJ,8; ,i,j=1,, on solution to the problem ( 6),corresponding to c =0 . mt e ng t- an mem ers o equati•ons (40) depend To prove existence of a very weak solution of the problem 40)-41) let us introduce the conjugate problem in the following formulation: for given vector fields fl(i),scalar fields G;,F;,Mli,M2; and constants s;,i = ,2,it is necessary to find vector fields wYl, scalar fields w;, If/;,<;;U) * constants n;, i, j= 1,2,such that i. · i. · j-1 j-1 j-1 j-1 �2>/AwYl -vro;)-ReH;(win)+ -f1 +1 a(wfl-wi'�+-\jl Vcp;()+�2);;)°(O)· Vl;y>" =iI(i) in o., J kj J J kj J - dl.V1\'lf;·U-u)t.�o:J\J"\+'t;;'If;•-- ReM·i f\-Wu• i\) + �';:1.[UA·k/I\· + �f:t.\1nk•xkj0 a-·i + µ a •iffik·)j� + Fi l·n :."l., ,:; Ul (o)\7�yi· +'t;;�yi· = r;In:xzi +Mji inn, k-1 42) 0ur 0ur • - • - ._,; ._,; II II w-.Ul--0 onu"'"':,", \If;*--0On£..._,; , '-J): J(i)*--0 On£..;n, ro ;-ro ;*,., .J--,1 2. Linear operators 1{; and M; are defined by the following formulas H;(h)=p;()v(uul()o) h-div{r';()o uUl ()o @h), M;(h)=�Cil(o)vu1 ,":l1 ' l O w� ")l '.J, '.J, ' ' ' ' � '.J Lemma 5. There are numbers & > 0 and c, c that wEO) � w(i) weakly B -s,r' 1n' i = 12 O � O � 1 1 , (m; ,lf/; ,<;;�),c;;�))�(m; ,If/; ,c;/1 l,,;/2 l) depending on llfllc2 (n such that for any & E [O;c1 ], 8 8 weakly ) the following representations exist 2 ± ±1 2 ± (44) in w-s,r' (n), i = 1,2, n;8 � n; a IR at c � 0. In ; ; addition,forany (H( );G;,F;,M1 ;,M2 ;;e; )EU s,r ,i = 1,2, N(ct =l±cIIll+c IDi ,9(c) =l±cdivT+c g the identity is valid (45) (41). (41). Here h. = (£(1 ),J;.(2)} ,J;) = (w!i );m;,If/;,,;?l*,,;ii)• Here h. = (£(1 ),J;.(2)} ,J;) = (w!i );m;,If/;,,;?l*,,;ii)• £* £* ;n;) E vs,r is the solution to problem (42) defined in Lemma 4; v;, Waned dco*nafirneedeofiunresdelbvyesthetofolramuslashort scheme of 6 6 ' & ' 6 6 ' & ' (I (I proof of Theorem 3. Establishing the boundedness of sequence of functions h = (1z ) 1z<2)) such that wii ) �wv(N-1;; Vl )\ +(;m ,A)0+(\f/;,;C )0), (54) as where i=l w(i), ;m ,\V;,i= 1 ,2,are given i i i i (53) by the mThaeteorrieaml d3er,ivatives 00 00 B;= (µ1; +µ2).-111U00 + Rep;(O)�< )(o).v'TJ) U - Rep;(o)vu< )o).U00TJ, (55) A=V11 ·U00 , C;=-RerJ(uCi> o and Le takes the form 2 ).vu(o) · v11)IDu (oY+vG;u> (o))'-d;vG;w (o))�)(prfv�dx _ - t,u.it,µ, (11•vG;u, (o))-v6,uu> (oY+{11•�u, (o))-v6,uu> (oYY)- q, (o)11') v�dx. whePrero]I])of=ofdTvihTeoI-remDT4.. Substituting relationships 2 2 (56) u-(i)(e)- u-(i)o )-- 8111-(6;) , P; &)- P;(o)-- 8;\f/ 6,q;e( )- q;(o)-- &; '°' ;1n61,·z--1,2, in (51 ),we obtain the difference relationship which can be represented as: 6 +&L.J fl j=l j=l (57) where LE ,U = -I2 ReUOO J11 �;(i:)wf)y7z7(i)(i:)+\j/ iez7(i)(o)v(i)(i:)+P;(o)ii(i)(o)vwf)]c1x i=l n i=l n -t.u.J[t.µ,(vwP +v{wj;l)'- d;,4�l)1))v�dx +t.u.jw,, +t.�,•.}�dx (58) '¾, =tu.J{Re>i�,€,)u0l(•)v{i-N€,J-}u0l(,j)]+[t,µ,(v{uUl €,)) + v{uUl(,f- d;v�0l(,))i)-q,(,++ Note that z7 Cil e),ii(il o),wf) ,i= 1,2 are continuously W1 • 2 (n). Thus,taking into account that wfl,i= 1,2 differentiable in n and belong to the class W2•2 (n). vanishon an andequations dvi (P;(O)· z7 (i)o))=O,i= 1,2 In turn,p;(.,),;(O),\f/;6 , q;e1q;(O),;m 6 , i = 1,2 are in n are satisfied,we make transformation of the continuous in the domain n and belong to the class following integrals. JRe P;(o)u £..out> W-.u) = 0 on ar.��, "'I''i* -- 0 on -..,i A, B;, C; A, B;, C; T T Here the coefficients and given functions are defined through the state variables as in the problem (40) and (42); ae defined by the formulas (55). Therefore, the adjoint state is completely defined by us note that B; E us note that B; E and A,C; and A,C; E E w w s s , , r r (n). (n). w w s s -l, -l, r r (n) (n) the state variables, and does not depend on . Let Consequently, according to Theorem 2, the boundary �/C)* =OonI1.n,IIro;* * ,i,j=l,2. =ro; =ro; linear form of T . linear form of T . value problem for adjoint state has a unique solution of class ws+i r(n)xW8 ·'(n)xlR. Therefore, the adjoint state is well-defined and unique. The following theorem shows that Lu can be represented as an integral of a the Tfohlleoowrienmg 5fo.r1muUlnadiesrvtahleidassumptions of Theorem 4, where h. =cliJ1),11.(2)), J;J;)=(w£il, m;,If/;, c;p)•, c;fl*,n;) is the adjoint state (solution to the problem (67)) and v; =-Rep;�(o .v�uury+IDJ°II'ii(il .vuury+L2 µldiv(1I'vuury+Il])'II' ,1.iiul +.1.�uury), j=I j=I [*= a1!'�(2)-iiQ� d* =½rr(Il]))= divT ·1, TIJ!= divT ·1-DT, 1['= divTI-Df-(prY ' '' '' ' '' '' '' '' '' '' ' ' ' ' '' ' ' ' ' ' '' ' ' ' ' ' an app y1ng ' ' an app y1ng eorem eorem , we o tam t e ormu a , we o tam t e ormu a Proof. Choos1.ng (H-(i) G. F M1. M2. s- )--(B-(i) A C. 0 0 0) z. --1 2 d 1 · Th 3 b · h £ 1 Thus we shave studied properties of the boundary value problem (6) solutions depending on the shape of the obstacle in a flow of fluid, which allowed the study of the properties ofthis problem solution functional. Formulas for shapederivative ofthis functional were obtained, whichcan be the basis of building and implementation of numerical calculations to find the optimal shape of the obstacle in a flow of mixture ofviscous compressible fluids. 55 Science Evolution, 2016, vol. 1, no. 2
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