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 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" =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)vu (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