A new form of governing equations of fluids arising from Hamiltons principle

A new form of go v erning equations of fluids arising from Hamilto n’s p rinciple S. Ga vr i lyuk a and H. Gouin a a L ab or atoir e d e Mo d´ elisation en M´ ec anique et Thermo dynam ique , F a cult´ e des Scienc es, Universit´ e d’A ix - Marseil le III, Case 322, Avenue Esc adril le Normandie-Niemen, 13397 Marseil le Ce dex 20, FRANCE Abstract A new form of gov ern ing equations is derived from Hamilton’s principle of least action for a constrained Lagrangian, dep ending on conserv ed quan tities and their deriv ativ es with resp ect to the ti me-space. This form yields conserv ation la ws b oth for non-d isp ersiv e case (Lagrangian d ep ends only on conserv ed quanti ties) and dis- p ersive case (Lagrangian dep ends also on their deriv ativ es). F o r n on-disp ersive case the s et of conserv ation laws allo ws to rewr ite the go v erning equations in the sym- metric form of Go dunov-F riedric hs-Lax. The linear stabilit y of equilibrium states for p oten tial motions is also studied. In particular, the disp ersion relation is obtained in terms of Hermitian matrices b oth for non-disp ersiv e and disp ersive case. Some new results are extended to the t w o-fluid non-disp ersiv e case. Key wor ds: Hamilton’s p rinciple; Symmetric f orms; Disp ersion relat ions 1 In tro duct ion Hamilton’s principle of least action is frequen tly used in cons erv ativ e fluid mec hanics [1-3 ]. Usually , a giv en Lag rangian Λ is submitted to constraints represen ting conserv ation in the time-space of collinear v ectors j k Div j k = 0 , k = 0 , . . ., m (1 . 1) where Div is the dive rgence op erator in the time-space. Equation (1.1 ) means the conserv atio n of mass, en trop y , concen tratio n, etc. Lagrangian app ears as a function o f j k and t heir deriv ativ es. T o calculate the v ariat io n of Hamilto n’s action w e don’t use Lagrange m ultipliers to tak e into accoun t the constraints (1.1). W e use the same metho d as Serrin in [2] where Preprint submitted to Elsevier 2 No vem b er 2018 the v ariat ion of the densit y ρ is expressed directly in terms of the virtual displacemen t o f the medium. This approac h yields an an tisymmetric form for the gov erning equations m X k =0 j ∗ k ∂ K k ∂ z − ∂ K k ∂ z ! ∗ ! = 0 (1 . 2) where z is the time-space v ariable, ”star” means the transp osition, and K k is the v ariational deriv ativ e of Λ with resp ect to j k K ∗ k = δ Λ δ j k (1 . 3) Equations (1.1) - (1.3 ) admit particular class of solutions called p otential flo ws j k = a k j 0 , a k = const , k = 1 , . . ., m K ∗ 0 = ∂ ϕ 0 ∂ z where ϕ 0 is a scalar function. W e shall study the linear stability of constant solutions fo r p oten tial flows. In Section 2, w e presen t the v ariations of unkno wn quan tities in terms o f vir- tual displacemen ts of the con tinuum. In Section 3, w e o bt a in the gov erning system (1.2) us ing Hamilton’s principle of least action. Conserv ation laws ad- mitted by the system ( 1.1) - (1.3) a r e obtained in section 4. F or non-disp ersiv e case, w e obtain the equations (1.1) - (1.3) in the symmetric fo rm of Go duno v- F riedric hs-Lax [4,5]. Section 5 is dev oted to the linear stabilit y of equilibrium states (constan t solutions) for p o ten tial motions. W e obtain disp ersion rela- tions in terms o f Hermitian matrices b oth for disp ersiv e and non-disp ersiv e flo ws and prop ose simple criteria of stability . In Section 6-7, we generalize our results for tw o-fluid mixtures in the non-disp ersiv e case. Usually , the theory of mixtures considers t w o differen t cases of con tin uum media. In homo gene ous mixtures suc h a s binary gas mixtures , each comp onent o ccupies the whole v olume of a ph ysical space. In he ter o gene ous mixtures suc h as a mixture of incompressible liquid con taining gas bubbles, each comp onen t occupies a part of the v olume of a phys ical space. W e do not distinguish the tw o cases b ecause they ha v e the same f orm for the gov erning equations. W e obtain a simple sta- bilit y criterion (criterion of h yp erb olicit y) for small relative v elo city o f phases. In App endix, w e prov e non-straig htforw ard calculations. Recall that ”star” denotes c onjugate (or tr an sp o s e ) ma pping o r co v ectors ( line v ectors). F or any v ectors a , b w e shall use the notation a ∗ b for their sc alar pr o duct (the line v ector is m ultiplied b y the column v ector) and ab ∗ for their tensor p r o duct (the column v ector is mu ltiplied b y the line vec tor) . The pro duct of a mapping A b y a ve ctor a is den oted by A a , b ∗ A means 2 co v ector c ∗ defined b y the rule c ∗ = ( A ∗ b ) ∗ . The div ergence of a linear transformation A is the cov ector Div A suc h that, for an y constan t v ector a , Div ( A ) a = Div ( A a ) The iden tical transformation is denoted by I . F or div ergence a nd gradient op erators in the time-space we use resp ectiv ely sym b ols Div and ∂ ∂ z , where z ∗ = ( t, x ∗ ), t is the time a nd x is the space. The gradient line (column) op erator in the space is denoted b y ∇ ( ∇ ∗ ), a nd t he divergenc e op erator in the space b y div. The elemen ts of the matrix A a r e denoted b y a i j where i means lines and j columns. If f ( A ) is a scalar function of A , matrix B ∗ ≡ ∂ f ∂ A is defined b y the form ula ( B ∗ ) j i = ∂ f ∂ A ! j i = ∂ f ∂ a i j The rep eated latin indices mean summation. Index α = 1 , 2 refers to the parameters of comp onen ts densities ρ α , v elo cities u α , etc. 2 V ariations of a con tin uum Let z =  t x  b e Eulerian co ordinates of a particle o f a con tin uum and D ( t ) a v olume of the ph ysical space o ccupied by a fluid at time t . When t b elongs to a finite in terv a l [ t 0 , t 1 ], D ( t ) generates a four- dimensional do ma in Ω in the time-space. A pa r ticle is lab elled b y its p osition X in a reference space D 0 . F or example, if D ( t ) contains alw ays the same particles D 0 = D ( t 0 ), and w e can define the motion of a con tinu um as a diffeomorphism from D ( t 0 ) in to D ( t ) x = χ t ( X ) (2 . 1) W e generalize (2.1) b y defining the motion as the diffeomorsphism from the reference space Ω 0 in to the time-space Ω o ccupied b y the medium in the follo wing parametric form      t = g ( λ , X ) x = φ ( λ , X ) (2 . 2) where Z =  λ X  b elongs to a reference space Ω 0 . The mappings      λ = h ( t , x ) X = ψ ( t , x ) (2 . 3) 3 are the in v erse of (2.2). Definitions (2.2) imply t he follo wing expres sions f o r the differen tials dt and d x  dt d x  = B  dλ d X  (2 . 4) where B =      ∂ g ∂ λ , ∂ g ∂ X ∂ φ ∂ λ , ∂ φ ∂ X      F orm ulae (2.2), (2 .4) assume the form          dt = ∂ g ∂ λ dλ + ∂ g ∂ X d X d x = ∂ φ ∂ λ dλ + ∂ φ ∂ X d X (2 . 5) F rom equation (2.5) we obtain d x = u dt + F d x where v elo city u and deformation gr a dien t F are defined by u = ∂ φ ∂ λ ∂ g ∂ λ ! − 1 , F = ∂ φ ∂ X − ∂ φ ∂ λ ∂ g ∂ X ∂ g ∂ λ ! − 1 (2 . 6) Let      t = G ( λ , X , ε ) x = Φ ( λ , X , ε ) (2 . 7) b e a one- pa rameter family o f virtual motions of the medium suc h t ha t G ( λ , X , 0) = g ( λ , X ) , Φ ( λ , X , 0) = φ ( λ , X ) where ε is a scalar defined in the vicinit y of zero. W e define Eulerian displace- men t ζ = ( τ , ξ ) asso ciated with t he virtual motion (2.7) τ = ∂ G ∂ ε ( λ , X , 0) , ξ = ∂ Φ ∂ ε ( λ , X , 0) (2 . 8) W e note that ζ is na turally defined in Lagrangian co ordinates. Ho w ev er, w e shall supp ose t hat ζ is represen ted in Eulerian co ordinates by means of (2.3). Let us now consider any tensor quan tit y represen ted by f ( t, x ) in Eulerian co ordinates and ◦ f ( λ , X ) in Lagrangia n coo rdinates. Definitions (2 .2 ), (2.3) in v olv e ◦ f ( λ , X ) = f  g ( λ, X ) , φ ( λ, X )  (2 . 9) 4 Con v ersely , f ( t , x ) = ◦ f  h ( t, x ) , ψ ( t, x )  (2 . 10) Let ∼ f ( λ, X , ε ) a nd ∧ f ( t, x , ε ) be tensor quantities asso ciat ed with the virtual mot io ns, suc h that ∼ f ( λ, X , ε ) ≡ ∧ f ( t, x , ε ) where λ , X , t , x are connected by relatio ns (2.7) s atisfying ∼ f ( λ , X , 0) = ◦ f ( λ , X ) or equiv alen tly ∧ f ( t , x , 0) = f ( t , x )). W e then obtain ∼ f ( λ , X , ε ) = ∧ f  G ( λ , X , ε ) , Φ ( λ , X , ε ) , ε  (2 . 11) Let us define Eulerian and Lagrangian v ariations of f ∧ δ f = ∂ ∧ f ∂ ε ( t , x , 0) and ∼ δ f = ∂ ∼ f ∂ ε ( λ , X , 0) Differen tiating relation (2.1 1) with resp ect to ε at ε = 0, w e get ∧ δ f = ∼ δ f − ∂ f ∂ z ζ (2 . 12) 3 Go v erning equations Consider a four - dimensional ve ctor j 0 satisfying conserv ation law Div j 0 = 0 (3 . 1) Actually , (3.1) repres en ts the mass conserv at io n la w, where j 0 = ρ v , ρ is the densit y and v ∗ = (1 , u ∗ ) is the four-dimensional ve lo cit y v ector. Let a k b e scalar quan tities suc h as the sp ecific en trop y , the n um b er of bubbles p er unit mass, the mass concen tration, etc., whic h are conserv ed along the tr a jectories. Consequen tly , if j k = a k j 0 , Div j k ≡ ∂ a k ∂ z j 0 = 0 , k = 1 , . . ., m (3 . 2) Hence j k , k = 1 , . . . , m form a set of solenoidal v ectors collinear to j 0 . Hamilton’s principle needs the kno wledge of Lagrangian of the medium. W e tak e the Lagrangian in the form L = Λ  j k , ∂ j k ∂ z , . . . , ∂ n j k ∂ z n , z  (3 . 3) 5 where j k are submitted to the constraints (3.1), (3.2) r ewritten as Div j k = 0 , k = 0 , . . . , m (3 . 4) Let us consider three examples. a – Gas dynamics [1-3] Lagrangian of the fluid is L = 1 2 ρ | u | 2 − ε ( ρ, η ) − ρ Π ( z ) , where ε is the in ternal energy p er unit v olume, η = ρs is the entrop y p er unit v olume, s is the sp ecific entrop y and Π is a n external p oten tial. Hence, in v ariables j 0 = ρ v , j 1 = ρs v , z , Lagrangian take s the form L = 1 2  | j 0 | 2 l ∗ j 0 − l ∗ j 0  − ε ( l ∗ j 0 , l ∗ j 1 ) − l ∗ j 0 Π( z ) = Λ ( j 0 , j 1 , z ) where l ∗ = ( 1 , 0 , 0 , 0). b – Thermo capillary fluids [6,7] Lagrangian of the fluid is L = 1 2 ρ | u | 2 − ε ( ρ , ∇ ρ , η , ∇ η ) − ρ Π( z ). Since ∇ ρ = ∇ ( l ∗ j 0 ) and ∇ η = ∇ ( l ∗ j 1 ), w e obtain Lagr angian in the for m (3.3) L = Λ j 0 , ∂ j 0 ∂ z , j 1 , ∂ j 1 ∂ z , z ! c – One-v elo city bubbly liquids [8- 10] L = 1 2 ρ | u | 2 − W  ρ , dρ dt , N  , with dρ dt = ∂ ρ ∂ z v where ρ is now the av erage densit y of the bubbly liquid a nd N is the n um b er of iden tical bubbles p er unit volume of the mixture. W e define again j 0 = ρ v and j 1 = N v . By using dρ dt = ∂ ρ ∂ z v = ∂ ( l ∗ j 0 ) ∂ z j 0 l ∗ j 0 and N = l ∗ j 1 , w e obtain Lag rangian in the fo rm L = Λ  j 0 , ∂ j 0 ∂ z , j 1  The Hamilton principle reads: for e a c h field of virtual disp l a c em ents z ∈ Ω − → ζ such that ζ a n d its derivatives ar e z e r o on ∂ Ω, δ Z Ω Λ d Ω = 0 (3 . 5) 6 Since v ariation ∧ δ is indep enden t of domain Ω and measure d Ω, the v ariation of Hamilton action (3.5) with the zero b oundary conditions for j k and its deriv ative s yields δ Z Ω Λ d Ω = Z Ω m X k =0 δ Λ δ j k ∧ δ j k d Ω = 0 (3 . 6) where δ δ j k refers to the v ar ia tional deriv ativ e with resp ect to j k . In particular, if Lagrangian ( 3 .3) is L = Λ  j k , ∂ j k ∂ z , z  w e g et δ Λ δ j k = ∂ Λ ∂ j k − D iv     ∂ Λ ∂  ∂ j k ∂ z      W e ha v e to emphasize that in (3.6), the v ariatio ns ∧ δ j k should tak e in to accoun t constrain ts (3.4). W e use the same metho d as in [2, p. 145] for the v ariat io n of densit y . This metho d do es not use Lagrange multiplie rs sinc e the constraints ( 3.4) are satisfied automat ically . T he calculation o f ∧ δ j k , k = 0 , . . . , m is p erformed in t w o steps. First, in App endix A w e calcu- late Lagra ngian v ariations ∼ δ v , ∼ δ ρ and ∼ δ a k (expressions ( A.2), (A.5) and (A.6), resp ectiv ely). Second, b y using (2.12) w e obtain in App endix B Eulerian v ariat io ns ∧ δ j k , k = 0 , . . . , m (see (B.3)) ∧ δ j k =  ∂ ζ ∂ z − (D iv ζ ) I  j k − ∂ j k ∂ z ζ Let us now define the fo ur- dimensional co v ector K ∗ k = δ Λ δ j k (3 . 7) T aking in to a ccoun t conditions (3.4 ) and the fact that ζ and its deriv a t ives are zero on the b oundary ∂ Ω, equations (3.6), (B.3) yield δ Z Ω Λ d Ω = Z Ω m X k =0 K ∗ k ∂ ζ ∂ z − (D iv ζ ) I ! j k − ∂ j k ∂ z ζ ! d Ω = Z Ω m X k =0 − Div ( j k K ∗ k ) + j ∗ k ∂ K k ∂ z ! ζ d Ω = Z Ω m X k =0 j ∗ k ∂ K k ∂ z −  ∂ K k ∂ z  ∗ ! ζ d Ω = 0 7 Hamilton’s principle yields the go v erning equations in the form m X k =0 j ∗ k ∂ K k ∂ z −  ∂ K k ∂ z  ∗ ! = 0 (3 . 8) where K k are giv en b y definition (3.7 ). The system (3.4), ( 3 .8) represen ts m + d + 1 partial differential equations for m + d unkno wn functions u , ρ , a k , k = 1 , . . . , m , where d is the dimension of the Ω-space. Since the matrix R k = ∂ K k ∂ z −  ∂ K k ∂ z  ∗ (3 . 9) is an tisymmetric and all the vec tors j k , k = 0 , . . . , m are collinear, w e obtain that j ∗ k R k j 0 ≡ 0. Consequen tly m X k =0 j ∗ k R k j 0 ≡ 0 (3 . 10) and the o v erdetermined system (3.4) , (3.8) is compatible. In the case a k = const , k = 1 , . . . , m a nd Λ = Λ j 0 , ∂ j 0 ∂ z ! , the system (3.4), (3.8) can b e rewritten in a simplified form        j ∗ 0 ∂ K 0 ∂ z −  ∂ K 0 ∂ z  ∗ ! = 0 Div j 0 = 0 (3 . 11) W e call p otential motion suc h solutions of (3.11) that K ∗ 0 = ∂ ϕ 0 ∂ z (3 . 12) where ϕ 0 is a scalar function. In this case, the gov erning sy stem for the p oten tial motion is in the form        δ Λ δ j 0 = ∂ ϕ 0 ∂ z Div j 0 = 0 (3 . 13) F or the gas dynamics mo del, the system (3.12), (3 .1 3) reads                    ∂ ϕ 0 ∂ t = −  1 2 | u | 2 + ∂ ∂ ρ e ( ρ ) + Π  , e ( ρ ) = ε ( ρ, ρs e ) ∇ ϕ 0 = u ∗ ∂ ρ ∂ t + div ( ρ u ) = 0 8 Eliminating the deriv ative ∂ ϕ 0 ∂ t , w e obtain the classical mo del for p oten tial flo ws [2 ]                    ∂ u ∗ ∂ t + ∇  1 2 | u | 2 + ∂ ∂ ρ e ( ρ ) + Π  = 0 u ∗ = ∇ ϕ 0 ∂ ρ ∂ t + div ( ρ u ) = 0 (3 . 14) 4 Conserv ation laws Equations (3.4), (3.8 ) can b e rewritten in a div ergence form. The demonstra- tion is p erfor med for Lagra ngian Λ whic h dep ends only on j k , ∂ j k ∂ z and z . The following result is prov ed in App endix C. Theorem 4.1 L et L = Λ  j k , ∂ j k ∂ z , z  . The fol lowin g ve ctor r ela tion is an identity Div m X k =0 K ∗ k j k I − j k K ∗ k + A ∗ k ∂ j k ∂ z ! − Λ I ! + ∂ Λ ∂ z + m X k =0 K ∗ k Div j k − m X k =0 j ∗ k R k ≡ 0 (4 . 1) wher e the m atric es R k ar e given by (3.9) an d A ∗ k = ∂ Λ ∂ ∂ j k ∂ z ! . In particular, w e get Theorem 4.2 The governing e quations (3.4), (3.8) ar e e quivalent to the system o f c on s e rva- tion laws Div m X k =0 K ∗ k j k I − j k K ∗ k + A ∗ k ∂ j k ∂ z ! − Λ I ! + ∂ Λ ∂ z = 0 Div j k = 0 , k = 0 , . . . , m 9 In the sp ecial case of p oten tial flo ws (3.13) t he gov erning equations admit additional conserv ation law s Div  cK ∗ 0 − ( K ∗ 0 c ) I  = 0 (4 . 2) where c is an y constant v ector. Indeed, Div ( cK ∗ 0 ) = c ∗  ∂ K 0 ∂ z  ∗ , D iv ( K ∗ 0 c I ) = ∂ ∂ z ( K ∗ 0 c ) = c ∗  ∂ K 0 ∂ z  Since for t he p otential flo ws ∂ K 0 ∂ z =  ∂ K 0 ∂ z  ∗ w e obtain (4 .2 ). In particular, if w e t a k e c = l for the ga s dynamics equa- tions, w e get conserv ation laws ( 3.14). Multiplying (4.1) b y j 0 and taking in to accoun t the iden tit y (3.10) , w e obtain the followin g theorem Theorem 4.3 The fol lowing sc alar r e la tion is an algebr aic identity Div m X k =0 ( K ∗ k j k I − j k K ∗ k + A ∗ k ∂ j k ∂ z ) − Λ I !! j 0 + ∂ Λ ∂ z j 0 + m X k =0 ( K ∗ k j 0 ) Div j k ≡ 0 (4 . 3) This theorem is a general repres en tatio n of the Gibbs iden tity expressing that the ”energy equation” is a conseque nce of the conserv ation of ”mass”, ”mo- men tum” and ”en tropy”. Examples of t his iden tit y for thermo capillary fluids and bubbly liquids w ere obtained previously in [6,10]. Iden tit y (4.3) yields an imp ortant consequence. Let us r ecall the Go duno v- F riedric hs-Lax metho d of symmetrisation of quasilinear conserv ation laws [4- 5] (see also different applications and g eneralizations in [1 1-12]). W e supp ose that the system of conserv atio n law s for n v ariables q has the form ∂ f i ∂ t + div F i = 0 , f i = f i ( q ) , F i = F i ( q ) , i = 1 , . . . , n (4 . 4) Let us also assume that (4.4) a dmits an additio nal ”energy” conserv ation la w ∂ e ∂ t + div E = 0 , e = e ( q ) , E = E ( q ) 10 whic h is obtained b y m ultiplying eac h equation of (4.4) b y some functions p i and then by summing ov er i = 1 , . . . , n ∂ e ∂ t + div E ≡ p i ∂ f i ∂ t + div F i ! (4 . 5) In particular, if w e consider e , E and F i as functions of f i , w e obtain from (4.5) ∂ e ∂ f i = p i , ∂ E ∂ f j = p i ∂ F i ∂ f j = ∂ e ∂ f i ∂ F i ∂ f j (4 . 6) Let us intro duce functions N and M suc h tha t N = f i p i − e, M = F i p i − E (4 . 7) Consequen tly , f r o m equations (4.6 ) and (4.7) we find that ∂ N ∂ p i = f i , ∂ M ∂ p i = F i (4 . 8) Hence, substituting (4.8) in to (4.4), we get a symme tric system in the f o rm ∂ 2 N ∂ p i ∂ p j ∂ p j ∂ t + tr ∂ 2 M ∂ p i ∂ p j ∂ p j ∂ x ! = 0 (4 . 9) If matrix N ij = ∂ 2 N ∂ p i ∂ p j is p ositiv e definite then the symmetric system (4.9) is t - hyp erb olic s ymm etric in the sense of F rie drichs . Ob viously , matrix ∂ 2 N ∂ p i ∂ p j is p ositive definite if and only if matrix e ij = ∂ 2 e ∂ f i ∂ f j is p ositive definite, since e ij N j p = δ i p , where δ i p is the Kro neck er sym b ol. No w, let us rewrite identit y (4.3) in the non-disp ersive case Div m X k =0 ( K ∗ k j k I − j k K ∗ k ) − Λ I !! j 0 + m X k =0 ( K ∗ k j 0 ) div j k ≡ 0 (4 . 1 0 ) Iden tit y (4.10) is exactly of the same t yp e as iden t ity (4.5 ). It means that the system (3.4), (3.8) can b e alw a ys rewritten in a Go dunov-F riedric hs-Lax symmetric for m. Actually , if w e tak e e = m X k =0 K ∗ k j k − ( j k K ∗ k ) 1 1 − Λ a nd E = − P j k K ∗ where P =    0 1 0 0 0 0 1 0 0 0 0 1    11 the conjugate v ariables p are p =      u − 1 ρ K ∗ k j 0      , k = 0 , . . . , m Consequen tly , the G ibbs iden tit y (4.10) gives directly a set of conjugate v ari- ables. Therefore, we ha v e pro ve d the following theorem Theorem 4.4 If L = Λ ( j k ) , the system (3.4) , (3 . 8 ) is always symmetrisable . The prop ert y of conv exit y , needed fo r the h yp erb olicity of the gov erning sys- tem, should b e verifie d for eac h part icular case. F or example, in the gas dy- namics, the energy of the system is give n b y the formu la e = 1 X k =0 K ∗ k j k − 1 X k =0  j k K ∗ k  1 1 − Λ = ε ( ρ, η ) + ρ | u | 2 2 Since K ∗ 0 j 0 = ρ | u | 2 2 − ∂ ε ∂ ρ ! , K ∗ 1 j 0 = − ρ ∂ ε ∂ η the conjugate v ariables are (see also [4]) p =           u ∂ ε ∂ ρ − | u | 2 2 ∂ ε ∂ η           =         u µ − | u | 2 2 T         where µ is the G ibbs p oten tial and T is the temp erature. Ob viously , if ε ( ρ, η ) is con v ex, the t o tal energy e is conv ex with r esp ect to ρ u , ρ and η . 5 Stabilit y of equilibrium states for pot en tial flows. W e assume that L is a function of j k and ∂ j k ∂ z b esides it do es no t depend on z , i.e. L = Λ j k , ∂ j k ∂ z ! 12 Let us giv e some definitions. An e quilibrium state is a solution of equations (3.4), ( 3.8) such that j k = j k e = const . Let ν be a real unit v ector. An y v ector β can b e represen ted in the fo rm β = ω ν + β σ , where ν ∗ β σ = 0 The equilibrium state j k e 6 = 0 is line arly stable in the dir e ction ν if and only if all non-trivial solutions of the form J k e i β ∗ z of the system (3.4), (3.8) linearized at the equilibrium state j k e are such t hat ω is real f o r an y real β σ . 5.1 Non-disp ersive c ase W e note that in a non- disp ersiv e case the stabilit y means h yp erb olicit y of go v erning system [13-14]. W e omit the index “0“ and rewrite sys tem (3.13 ) in the form        K ∗ ≡ ∂ Λ ∂ j = ∂ ϕ ∂ z Div j = 0 (5 . 1) where Λ = Λ( j ). The Legendre transformat io n of Λ( j ) is ∆( K ) = K ∗ j − Λ( j ) (5 . 2) If the matrix Λ ′′ ( j ) = ∂ ∂ j ∂ Λ ∂ j ! ∗ ! is non-degenerate, (5.2) inv olves the form ula j ∗ = ∂ ∆ ∂ K (5 . 3) Hence, relations ( 5 .1) - (5.3) yield ∂ j ∂ z = ∂ ∂ K ∂ ∆ ∂ K ! ∗ ! ∂ K ∂ z = ∆ ′′ ( K ) ∂ K ∂ z = ∆ ′′ ( K ) ϕ ′′ ( z ) (5 . 4) where ∆ ′′ ( K ) = ∂ ∂ K ∂ ∆ ∂ K ! ∗ ! , ϕ ′′ ( z ) = ∂ ∂ z ∂ ϕ ∂ z ! ∗ ! The second equation (5.1) a nd ( 5 .4) inv olve tr  ∆ ′′ ( K ) ϕ ′′ ( z )  = 0 If w e replace ϕ b y i Φ e i β ∗ z , w e g et immediately t he disp ersion r elation β ∗ ∆ ′′ ( K e ) β = 0 (5 . 5) 13 where index ”e” corresp onds to t he equilibrium state. The matrix ∆ ′′ ( K e ) can b e easily calculated in terms of Lagrangian Λ( j ). Indeed, I = ∂ j ∂ j = ∂ j ∂ K ∂ K ∂ j = ∆ ′′ ( K ) Λ ′′ ( j ) It follows that ∆ ′′ ( K e ) =  Λ ′′ ( j e )  − 1 (5 . 6) Relations (5.5) - (5.6) imply the following result. Theorem 5.1 ( criterion of stability for p otential non-d i s p ersi v e motions ) If the symmetric matrix G = Λ ′′ ( j e ) has the signatur e ( − , + , + , +) , the e quili b ri um state j e is stable in any dir e ction ν b elonging to the interse ction of two c ones C 1 = n ν | ν ∗ G − 1 ν < 0 o and C 2 = n ν | ν ∗ G ν < 0 o Pro of . Since C 1 and C 2 con tain the eigen v ector corresp onding to the neg- ativ e eigen v alue of G , C 1 ∩ C 2 6 = ∅ . W e can find orthogona l co ordinates ( t, x 1 , x 2 , x 3 ) suc h t ha t the disp ersion relation (5 .5) takes the form − t 2 + 3 X i =1 λ i x 2 i = 0 , with λ i > 0 In co ordinat es ( t, x 1 , x 2 , x 3 ) the cone C 2 is defined by the inequalit y − t 2 + 3 X i =1 x 2 i λ i < 0 Let us represen t β in the form β =      t x 1 x 2 x 3      = ω      1 n 1 n 2 n 3      +      − n 1 y 1 − n 2 y 2 − n 3 y 3 y 1 y 2 y 3      The disp ersion relation β ∗ G − 1 β = 0 implies − 3 X i =1 n i y i − ω ! 2 + 3 X i =1 λ i ( ω n i + y i ) 2 = 0 It inv olve s ω 2 − 1 + 3 X i =1 λ i n 2 i ! + 2 ω 3 X i =1 (1 + λ i ) n i y i + 3 X i =1 λ i y 2 i − 3 X i =1 n i y i ! 2 = 0 14 Due to the follow ing inequalit y , 3 X i =1 λ i y 2 i − 3 X i =1 n i y i ! 2 = 3 X i =1 λ i y 2 i − 3 X i =1 n i √ λ i q λ i y i ! 2 ≥ 3 X i =1 λ i y 2 i − 3 X i =1 n 2 i λ i ! 3 X i =1 λ i y 2 i ! = 3 X i =1 λ i y 2 i ! 1 − 3 X i =1 n 2 i λ i ! ω is real if ν b elongs simultaneously to C 1 and C 2 . The t heorem is prov ed. F or the ga s dynamics mo del, if the v olume energy is a con v ex function of ρ and η = ρs , the matrix G satisfies the conditions of Theorem 5.1. 5.2 Disp ersive c ase If L = Λ  j , ∂ j ∂ z  , the g o v erning system is          K ∗ ≡ δ Λ δ j = ∂ Λ ∂ j − D iv   ∂ Λ ∂  ∂ j ∂ z    = ∂ ϕ ∂ z Div j = 0 The linearised system in a co ordinate form is Λ k s j s + Γ p k s j s ,p − D mp k s j s ,mp = ϕ, k , and j s ,s = 0 (5 . 7) where the comma denotes the deriv ativ e with respect to z k while Λ k s , Γ p k s , D mp k s , calculated at p oint j e , are defined b y the relations Λ k s = ∂ 2 Λ ∂ j k ∂ j s , Γ p k s = ∂ 2 Λ ∂ j k ∂ j s ,p − ∂ 2 Λ ∂ j s ∂ j k ,p , D mp k s = ∂ 2 Λ ∂ j k ,m ∂ j s ,p (5 . 8) The following sym metry relations result fro m (5.8) Λ k s = Λ sk , Γ p k s = − Γ p sk , D mp k s = D pm sk (5 . 9) F or the solution of (5.7) in the form j s = J s e iβ p z p , ϕ = i Φ e iβ p z p , w e get the following dispersion relation det    G β β ∗ 0    ≡ − β ∗ Adj ( G ) β = 0 (5 . 10) where Adj ( G ) is the adjoin t matrix to G , and the elemen ts G k s of the matrix G are defined b y G k s = Λ k s + i Γ p k s β p + D mp k s β m β p (5 . 11) 15 Equations (5.9) - (5.11) imply that G is hermitian matrix. In particular, if det G 6 = 0 t hen Adj ( G ) = det( G ) G − 1 and the dispersion relation (5 .10) is equiv alen t to β ∗ G − 1 β = 0, whic h is a generalization of (5.5 ) for the disp ersiv e case. W e get the fo llo wing obvious result: Theorem 5.2 ( c riterion of stability for p otential disp ersive motions ) If Γ p k s = 0 and the symm etric matrix G d e fine d by (5. 8 ), (5.11) has the signatur e ( − , + , + , +) for β = 0 , the e quilib ri um state j e is stable for smal l β in any d ir e ction ν b elo n ging to the i nterse ction of the c ones C i , i = 1 , 2 define d in the or em 5.1. W e not e that Γ p k s defined b y (5.8) are alw a ys zero if the expansion o f La- grangian Λ in T a ylor series at the vicinity of equilibrium state do es not con tain linear terms with resp ect to ∂ j k ∂ z . Sometimes, w e are able to obtain a “global” stabilit y . F o r example, let us consider a particular case of bubbly liquids, f o r N /ρ = const, defined in Section 3 L = 1 2 ρ | u | 2 + a 2 dρ dt ! 2 − ε ( ρ ) where ε ( ρ ) is a conv ex function of the den sity and a is a positive function of ρ . Then, L = 1 2 | j | 2 l ∗ j − l ∗ j ! + a 2 ∂ ∂ z ( l ∗ j ) j l ∗ j ! 2 − ε ( l ∗ j ) Since the gov erning equations are in v aria n t with respect to the Galilean trans- formation t ′ = t, x ′ = x + U t, u ′ = u + U w e can alw ay s assume that u e = 0 (it is sufficien t to consider the g o v erning system in the reference frame moving with the velocity U = u e ). Hence, j e = ρ e l , where l ∗ = (1 , 0 , 0 , 0). Omitting index ” e ”, we can calculate t he matrix G defined b y (5.11) G =      − ∂ 2 ε ∂ ρ 2 + aβ 2 1 0 ∗ 0 I ρ      Hence, Adj ( G ) =      1 ρ 3 0 ∗ 0  aβ 2 1 − ∂ 2 ε ∂ ρ 2  I ρ 2      16 and disp ersion relation (5.10) reads β 2 1 ρ 3 + aβ 2 1 − ∂ 2 ε ∂ ρ 2 ρ 2  β 2 2 + β 2 3 + β 2 4  = 0 (5 . 12) Let ν b e the direction of time in the t ime-space, then ν = l and (5.1 2) is equiv alen t to ω 2 ∂ 2 ε ∂ ρ 2 − aω 2 = ρ | β σ | 2 (5 . 13) The graph o f the dispersion relation ( 5.13) for p ositiv e v a lues of ω is presen ted in F igure 1. W e ha v e denoted by ω ∗ = s ∂ 2 ε ∂ ρ 2 1 a the eig e nfr e quency of bubbles and by c 0 = s ρ ∂ 2 ε ∂ ρ 2 the e quilibrium sound sp e e d of bubbly liquid (see [8]). Fig. 1. Disp ersion relation for bu b bly liquid s 6 Go v erning equations for mixtures Let us consider homogeneous binary mixtures. The mixture is describ ed b y the v elo cities u α , the av erage densities ρ α and the sp ecific en tropies s α for each comp onen t ( α = 1 , 2). W e introduce tw o r eference frames asso ciated with eac h comp onen t in the fo rm      t = g α ( λ α , X α ) x = φ α ( λ α , X α ) (6 . 1) 17 and the in v erse mappings      λ α = h α ( t , x ) X α = Ψ α ( t , x ) (6 . 2) The corresp onding families of virtual motions generated by (6.1) ar e defined b y      t = G α ( λ α , X α , ε α ) x = Φ α ( λ α , X α , ε α ) with      G α ( λ α , X α , 0) = g α ( λ α , X α ) Φ α ( λ α , X α , 0) = φ α ( λ α , X α ) W e define the t w o Eulerian displacemen t ζ α = ( τ α , ξ α ) where τ α = ∂ G α ∂ ε α ( λ α , X α , 0) , ξ α = ∂ Φ α ∂ ε α ( λ α , X α , 0) As in Section 3 w e define tensor quantities asso ciated with the tw o virtual motions and v aria tions ∧ δ α and ∼ δ α . In the general case, we ha v e tw o four- dimensional solenoidal v ectors j 0( α ) = ρ α v α corresp onding to the α th comp o- nen t, where ρ α is the densit y and v ∗ α = ( 1 , u ∗ α ) is the four-dimensional v elo c- it y v ector. As in Section 3, w e introduce additional ph ysical quan tities a k ( α ) asso ciated with the four-dimensional v ectors j k ( α ) = a k ( α ) j 0( α ) , α = 1 , 2, k ( α ) = 1 , . . . , m ( α ) submitted t o the constrain ts Div j k ( α ) = 0 (6 . 3) F or Hamilton’s action in the for m a = Z Ω Λ  j k ( α ) , ∂ j k ( α ) ∂ z , . . . , ∂ n j k ( α ) ∂ z n , z  d Ω the Hamilton principle reads: for e ach field of virtual displac ements z ∈ Ω − → ζ α such that ζ α and its derivatives ar e zer o on ∂ Ω, δ α Z Ω Λ d Ω = 0 Here δ α a is the deriv a tiv e o f the Hamilto n action with resp ect to ε α , and ζ α are the virtual displacemen ts expressed in Eulerian co or dina t es z by means of eq uations (6.2 ). T he metho d dev elop ed in Section 3 yield s the equations of motion in the form m ( α ) X k ( α )=0 j ∗ k ( α ) ∂ K k ( α ) ∂ z −  ∂ K k ( α ) ∂ z  ∗ ! = 0 , K ∗ k ( α ) = δ Λ δ j k ( α ) (6 . 4) 18 Div j k ( α ) = 0 , α = 1 , 2 and k ( α ) = 0 , . . . , m ( α ) As in Theorem 4.1, for the case Λ = Λ  j k ( α ) , ∂ j k ( α ) ∂ z , z  w e can obtain the follo wing iden tit y Div   2 X α =1 m ( α ) X k ( α )=0 K ∗ k ( α ) j k ( α ) I − j k ( α ) K ∗ k ( α ) + A ∗ k ( α ) ∂ j k ( α ) ∂ z ! − Λ I   + ∂ Λ ∂ z + 2 X α =1 m ( α ) X k ( α )=0 K ∗ k ( α ) Div j k ( α ) − 2 X α =1 m ( α ) X k ( α )=0 j ∗ k ( α ) R k ( α ) ≡ 0 where A ∗ k ( α ) = ∂ Λ ∂ ∂ j k ( α ) ∂ z ! and R k ( α ) = ∂ K k ( α ) ∂ z −  ∂ K k ( α ) ∂ z  ∗ Hence, the gov erning equations (6.4) admit the conserv ation law s Div   2 X α =1 m ( α ) X k ( α )=0 K ∗ k ( α ) j k ( α ) I − j k ( α ) K ∗ k ( α ) + A ∗ k ( α ) ∂ j k ( α ) ∂ z ! − Λ I   + ∂ Λ ∂ z = 0 Div j k ( α ) = 0 , α = 1 , 2 and k ( α ) = 0 , . . . , m ( α ) W e no t ice tha t for a one-v elo cit y mo del, the n umber of scalar conserv ation la ws is m + d + 1 where d is the dimension of the time-space. The n um b er of unkno wn v ariables is m + d . Due to Theorem 4.3, the m + d + 1 conserv ation la ws are connected by the ”Gibbs iden tity ”. In the case of mixtures, w e o btain X α m ( α ) + d + 1 conserv at io n la ws f o r X α m ( α ) + 2 d unkno wn v ariables. In the g eneral case, the classical approac h based on conserv ation laws, do es no t allo w to obtain Rankine-Hugoniot conditions for this sys tem. Nev ertheless, in [15-17] w e ha v e o bta ined the jump conditions. The Hamilton principle provides these conditions without any am biguit y . 7 Linear stability of mixtures. W e consider the Lagrangian in the form [15- 19] L = 1 2 ρ 1 | u 1 | 2 + 1 2 ρ 2 | u 2 | 2 − W ( ρ 1 , ρ 2 , η 1 , η 2 , w ) (7 . 1) where w = u 2 − u 1 and η α is the entrop y p er unit v olume of the α th com- p onen t. A generalisation of (7.1) for thermo capillary fluids w as also prop osed 19 in [20]. If W is a n isotro pic function of w , Lagrangian (7.1) can b e rewritten as follows Λ = 1 2 2 X α = 1  | j 0( α ) | 2 l ∗ j 0( α ) − l ∗ j 0( α )  − W  l ∗ j 0(1) , l ∗ j 0(2) , l ∗ j 1(1) , l ∗ j 1(2) , µ  (7 . 2) where µ = 1 2      j 0(2) l ∗ j 0(2) − j 0(1) l ∗ j 0(1)      2 (7 . 3) Here w e shall also restrict our study t o the case s α = const . T o av o id double indices, w e will denote j 0(1) , j 0(2) b y j 1 , j 2 . Hence, ( 7.2) and ( 7 .3) in v olve Λ( j 1 , j 2 ) = 1 2 2 X α =1 | j α | 2 l ∗ j α − l ∗ j α ! − ∼ W ( l ∗ j 1 , l ∗ j 2 , µ ) (7 . 4) where ∼ W ( ρ 1 , ρ 2 , µ ) = W ( ρ 1 , ρ 2 , ρ 1 s 1 , ρ 2 s 2 , µ ). Omitting the tilde sym b ol, w e consider the p o ten tial motio ns        K ∗ α ≡ ∂ Λ ∂ j α = ∂ ϕ α ∂ z Div j α = 0 , α = 1 , 2 (7 . 5) Let us consider the Legendre transformation of Λ( j 1 , j 2 ) ∆( K 1 , K 2 ) = 2 X α =1 j ∗ α K α − Λ (7 . 6) If the ma t r ices Λ ( αβ ) ≡ ∂ ∂ j α ∂ Λ ∂ j β ! ∗ ! = Λ ∗ ( β α ) (7 . 7) are non-degenerate, relation (7.6 ) in v olv es j ∗ α = ∂ ∆ ∂ K α (7 . 8) Substituting (7 .8) in D iv j α = 0 , w e g et tr ∂ j α ∂ z ! = tr ∂ ∂ z ∂ ∆ ∂ K α ! ∗ ! = tr   2 X γ =1 ∂ ∂ K γ ∂ ∆ ∂ K α ! ∗ ∂ K γ ∂ z   = tr   2 X γ =1 ∆ ( γ α ) ϕ ′′ γ   = 0 with ∆ ( γ α ) ≡ ∂ ∂ K γ ∂ ∆ ∂ K α ! ∗ ! = ∆ ∗ ( αγ ) (7 . 9) 20 Hence, the equations of p o ten tial motio ns are tr   2 X γ =1 ∆ ( γ α ) ϕ ′′ γ   = 0 (7 . 10) In the equilibrium state ∆ ( γ α ) = const. Substituting ϕ γ b y i Φ γ e i β ∗ z in (7.10), w e obtain the disp ersion relation in the symmetric form det  β ∗ ∆ (11) β β ∗ ∆ (21) β β ∗ ∆ (12) β β ∗ ∆ (22) β  = 0 whic h is the generalisation of the dispersion relation (5.5). How eve r, to cal- culate ∆ ( γ α ) in terms of Λ ( γ α ) defined by (7.7), (7.9) ) , w e hav e to solve the follo wing system of matr ix equations 2 X γ =1 ∆ ( γ α ) Λ ( β γ ) = I δ αβ , δ αβ =  1 , α = β 0 , α 6 = β A simpler metho d is to consider the linearised system generated from (7.5) 2 X γ =1 Λ ( γ α ) j γ − ∂ ϕ α ∂ z ! ∗ = 0 , Div j α = 0 where Λ ( γ α ) are taken at the equilibrium state j k e . Substituting j γ b y J γ e i β ∗ z and ϕ α b y i Φ α e i β ∗ z , we obtain the disp ersion relation in the symmetric form det      Λ (11) Λ (21) β 0 Λ (12) Λ (22) 0 β β ∗ 0 ∗ 0 0 0 ∗ β ∗ 0 0      = 0 (7 . 11) Equation (7.11) is presen ted in terms of t he matrices Λ ( γ α ) calculated directly from the L a grangian. Let us consider Lag rangian ( 7 .4). Supp ose tha t the ve - lo cities of eac h comp o nent ar e the same a t equilibrium ( u 1 e = u 2 e ). Due to the in v ariance of the gov erning equations with resp ect to the Ga lilean trans- formation w e assume, without loss of the generalit y , that u 1 e = u 2 e = 0 . Suppressing the index ” e ” t o a v oid double indices, w e g et                      Λ (11) =  ρ 1 − ∂ W ∂ µ  I − ll ∗ ρ 2 1 − ∂ 2 W ∂ ρ 2 1 ll ∗ Λ (21) = Λ (12) = 1 ρ 1 ρ 2 ∂ W ∂ µ ( I − ll ∗ ) − ∂ 2 W ∂ ρ 1 ∂ ρ 2 ll ∗ Λ (22) =  ρ 2 − ∂ W ∂ µ  I − ll ∗ ρ 2 2 − ∂ 2 W ∂ ρ 2 2 ll ∗ (7 . 12) 21 All matrices Λ ( αβ ) are diagonal. No w w e ar e able t o calculate the disp ersion relation (7.1 1) whic h is the determinant of a square matrix o f dimension 10. Let us denote a = − ∂ W ∂ µ , w 11 = ∂ 2 W ∂ ρ 2 1 , w 12 = ∂ 2 W ∂ ρ 1 ∂ ρ 2 , w 22 = ∂ 2 W ∂ ρ 2 2 (7 . 13) Supp ose tha t a > 0 , w 11 > 0 , w 22 > 0 , w 11 w 22 − w 2 12 > 0 (7 . 14) T aking into accoun t (7.12) - (7.13 ) w e obtain b y straightforw a rd calculations of the determinan t (7.11 )  a ( ρ 1 + ρ 2 ) + ρ 1 ρ 2  β 4 1 −  a ( ρ 2 1 w 11 + 2 ρ 1 ρ 2 w 12 + ρ 2 2 w 22 ) + ρ 1 ρ 2 ( ρ 2 w 22 + ρ 1 w 11 )  × ( β 2 2 + β 2 3 + β 2 4 ) β 2 1 + ( β 2 2 + β 2 3 + β 2 4 ) 2 ( w 22 w 11 − w 2 12 ) ρ 2 1 ρ 2 2 = 0 (7 . 15) Let ν = l , β = ω ν + β σ , λ = ω | β σ | . The disp ersion relation (7.15) tak es the form  a ( ρ 1 + ρ 2 ) + ρ 1 ρ 2  λ 4 −  a ( ρ 2 1 w 11 + 2 ρ 1 ρ 2 w 12 + ρ 2 2 w 22 ) + ρ 1 ρ 2 ( ρ 2 w 22 + ρ 1 w 11 )  λ 2 + ( w 22 w 11 − w 2 12 ) ρ 2 1 ρ 2 2 = 0 (7 . 16) The following result is prov ed in [21]: Theorem 7.1 If p otential W ( ρ 1 , ρ 2 , µ ) satisfies c on d itions (7.14) , al l the r o ots λ of the p o l ynom e (7.16) ar e r e a l . Conditions (7.14) mean that internal energy U = W − w ∂ W ∂ w , where w = | w | , is con v ex [15-1 6 , 21]. Hence, if the in ternal energy is a con v ex function and the relat ive v elo cit y w is small enough, the equilibrium state is stable in time direction in the t ime-space. It is in teresting to note that the Lagra ng ian o f a t w o-fluid bubbly liquid with incompressible liquid phase (heterogeneous case) has the same form as (7.1). Indeed, in [18,19] the p otential W for a bubbly liquid is propo sed in the form W = ρ 2 ε 20 ( ρ 20 ) − ρ 10 2 m ( c ) | w | 2 (7 . 17) where ρ 2 = c ρ 20 , c is the v olume concen tration of gas pha se and the index ”0” means the real density of gas and liquid phase. Concen tration c is expressed 22 b y c = 4 3 π R 3 N whe re R is the av erage radius of the bubbles, N is the n um b er of bubbles p er unit v olume of the mixture. The in ternal energy p er unit mass ε 20 of the gas phase and the virtual mass co efficien t m are kno wn functions of ρ 20 and c . By in tro ducing t he av erage density of the liquid phase ρ 1 = ρ 10 (1 − c ) with ρ 10 = const w e can rewrite the p oten tial (7.1 7) as W = ρ 2 ε 20  ρ 2 ρ 10 ρ 10 − ρ 1  − ρ 10 2 m  ρ 10 − ρ 1 ρ 10  | w | 2 The increase of the v olume fraction c pro ducing the in teraction b et w een gas bubbles changes not only the co efficien t m ( c ) but also the in terfacial energy ε in ( c ) so that p otential W can b e g eneralized to the form W = ρ 2 ε 20 ( ρ 20 ) − ρ 10 2 m ( c ) | w | 2 + ε in ( c ) (7 . 18) Quan tity ε in ( c ) pro duces an additional pressure term due to the in terfacial effect [22]. Therefore, Lagrangia n (7.1) describ es not only binary molecular mixtures but also partial cases of suspensions. Case (7.1 7) is actually degen- erated: w 11 w 22 − w 2 12 ≡ 0. The in tro duction o f the in terfacial term in (7.18) in v olv es the conv exity conditio n w 11 w 22 − w 2 12 > 0 provid ed ∂ 2 ε in ∂ c 2 > 0. A APPE NDIX F or any function ◦ f ( λ , X ) (see definitions (2 .9)-(2.10)) and its image f ( t , X ) in the ( t , X )-co ordinates, w e obtain the followin g relations ∂ ◦ f ∂ λ ∂ g ∂ λ ! − 1 = ∂ f ∂ t , ∂ ◦ f ∂ X − ∂ g ∂ X ∂ ◦ f ∂ λ ∂ g ∂ λ ! − 1 = ∂ f ∂ X ( A. 1) A.1 V ariation of the velo city The definition (2.6) of the v elo cit y u yields ◦ u = ∂ φ ∂ λ ∂ g ∂ λ ! − 1 23 In Lagrang ia n co ordinates, p erturbation of u is represen ted b y the formula ∼ u ( λ , X , ε ) = ∂ Φ ∂ λ ( λ , X , ε ) ∂ G ∂ λ ( λ , X , ε ) ! − 1 T aking the deriv ative of ∼ u with respect to ε at ε = 0 and using (A.1), w e get ∼ δ u = ∂ ◦ ξ ∂ λ ∂ g ∂ λ ! − 1 − ◦ u ∂ ◦ τ ∂ λ ∂ g ∂ λ ! − 1 = ∂ ξ ∂ t − u ∂ τ ∂ t = d ξ dt − u dτ dt where d dt = ∂ ∂ t + u ∗ ▽ ∗ is the material deriv ative. W e obtain then ∼ δ v = ( I − vl ∗ ) ∂ ζ ∂ z v ( A. 2) where v ∗ = ( 1 , u ∗ ), I is the iden tit y tensor and l ∗ = ( 1 , 0 , 0 , 0). A.2 V ariation of the deformation gr a dient In La g rangian co ordinates ( λ, X ), the p erturbation of the deformation gradi- en t is g iv en by (2.6) ∼ F ( λ , X , ε ) = ∂ Φ ∂ X − ∂ Φ ∂ λ ∂ G ∂ X ∂ G ∂ λ ! − 1 Com bination of the deriv ative of ∼ F with respect to ε at ε = 0 and relation (A.1) giv es ∼ δ F = ∂ ◦ ξ ∂ X − ∂ ◦ ξ ∂ λ ∂ g ∂ X ∂ g ∂ λ ! − 1 − ∂ Φ ∂ λ ∂ ◦ τ ∂ X ∂ g ∂ λ ! − 1 + ∂ φ ∂ λ ∂ g ∂ X ∂ g ∂ λ ! − 2 ∂ ◦ τ ∂ λ = ∂ ξ ∂ X − u ∂ τ ∂ X =  ∂ ξ ∂ x − u ∂ τ ∂ x  F ( A. 3) A.3 V ariation of the density The mass conserv ation is represen ted by the form ula ρ det F = ρ 0 ( X ) 24 The p erturbat io n of ρ is in the form ∼ ρ ( λ , X , ε ) det ∼ F ( λ , X , ε ) = ρ 0 ( X ) and consequen tly ∼ δ ρ det ◦ F + ◦ ρ ∼ δ (det F ) = 0 ( A. 4) By using the Euler-Jacobi identit y , δ (det F ) = det F tr ( F − 1 ∼ δ F ) relations (A.3) and (A.4) lead to ∼ δ ρ = − tr  ρ ( I − vl ∗ ) ∂ ζ ∂ z  ( A. 5) A.4 V ariation of sp e cific quantities Along eac h tra jectory the sp ecific quan tities a k are constan t. Conseque ntly , a k = a 0 k ( X ). The p erturbatio n o f a k is suc h that ∼ a k ( λ , X , ε ) = a 0 k ( X ). Hence, ∼ δ a k = 0 ( A. 6) B APPE NDIX F orm ulae (A.2) and (A.5) yield dir ectly the v ariation of j 0 ∼ δ j 0 = tr ( vl ∗ − I ) ∂ ζ ∂ z ! I − ( vl ∗ − I ) ∂ ζ ∂ z ! j 0 = ∂ ζ ∂ z − ( D iv ζ ) I ! j 0 ( B . 1) Let us consider the v ariatio ns ∼ δ j k , k = 1 , . . . , m with j k = a k j 0 where the scalar fields a k are suc h that a k = a k 0 ( X ). Due to (A.6) ∼ δ a k = 0 and ∼ δ j k = a k ∼ δ j 0 . Consequen tly ∼ δ j k =  ∂ ζ ∂ z − (D iv ζ ) I  j k , k = 0 , . . . , m ( B . 2) Hence, (2.12) in v olv es ∧ δ j k =  ∂ ζ ∂ z − (Div ζ ) I  j k − ∂ j k ∂ z ζ , k = 0 , . . . , m ( B . 3) 25 C APPEN DIX Let A , B b e t w o linear tra nsformations dep ending on z . L et us define the lin- ear form tr  A ∂ B ∂ z  suc h that f o r any constan t v ector field a , tr  A ∂ B ∂ z  a ≡ tr  A ∂ B a ∂ z  . As a conseque nce, w e get Div ( A B ) = Div ( A ) B + tr  A ∂ B ∂ z  ( C . 1) Indeed, Div ( A B a ) = D iv ( A ) B a + tr  A ∂ B a ∂ z  = (Div A ) B + tr A ∂ B ∂ z !! a Let us also recall the follo wing useful f orm ulae for a n y v ector fields b ( z ), c ( z ) Div ( b c ∗ ) = c ∗ Div b + b ∗  ∂ c ∂ z  ∗ ( C . 2) Div ( b ∗ c I ) = ∂ ∂ z ( b ∗ c ) = b ∗ ∂ c ∂ z + c ∗ ∂ b ∂ z ( C . 3) By using for mulae (C.1) - (C.3), w e ha v e the following iden tit ies Div ( K ∗ k j k I ) ≡ K ∗ k ∂ j k ∂ z + j ∗ k ∂ K k ∂ z ( C . 4) Div ( j k K ∗ k ) ≡ − K ∗ k Div j k − j ∗ k  ∂ K k ∂ z  ∗ ( C . 5) Div  A ∗ k ∂ j k ∂ z  ≡ Div ( A ∗ k ) ∂ j k ∂ z + tr A ∗ k ∂ ∂ z  ∂ j k ∂ z  ! 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