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Appendix A

Vector and Tensor Algebra

A.1 Vector Space. Basis

A real vector space is an algebraic structure consisting of a set V endowed withtwo operations. One of them is an internal operation denoted by + and satisfyingthe following properties:

i) Associativity.a + (b + c) = (a + b) + c ∀ a, b, c.

ii) Existence of a neutral element 0 such that

a + 0 = 0 + a = a ∀ a.

iii) Existence of a symmetric element : for each a there exists b such that

a + b = b + a = 0.

iv) Commutativity.a + b = b + a ∀ a, b.

Thus, V with the + operation is a commutative group.

The second operation is an external one, the product by real numbers, satis-fying the following properties:

v) λ(a + b) = λa + λb.

vi) λ(µa) = (λµ)a.

vii) (λ + µ)a = λa + µa.

162 Appendix A. Vector and Tensor Algebra

viii) 1a = a.

Elements of V are called vectors.

A set of vectors eiNi=1 is said to be

• linearly independent if

N∑i=1

λiei = 0 ⇒ λi = 0 ∀i ∈ 1, . . . , N, (A.1)

• a system of generators if

∀v ∈ V ∃λi, i = 1, . . . , N, such that v =N∑

i=1

λiei, (A.2)

• a basis if it is both linearly independent and a system of generators.

If B = e1, . . . , eN is a basis of V , then each v ∈ V can be written as alinear combination,

v =N∑

i=1

viei,

in a unique way. The vi ∈ R, i = 1, . . . , N, are called coordinates of v with respectto basis B.

Let us consider two bases in V , eiNi=1 and EiN

i=1. In particular, vectorsEi can be written as linear combinations of vectors eiN

i=1, namely,

El =N∑

i=1

Cliei. (A.3)

Now, let v be any vector in V . Let us denote by vei and vE

i , i = 1, . . . , N , thecoordinates of v with respect to basis eiN

i=1 and EiNi=1, respectively. If we

denote by ve the column vector of the former and by vE the column vector of thelatter, it is easy to see that

vE = [C]ve, (A.4)

where [C] is the matrix [C]ij = Cij with Cij given in (A.3).

A.2. Inner Product 163

A.2 Inner Product

A mapping ϕ : V × V −→ R is called an inner product if the following propertieshold:

1) ϕ is bilinear :

1.1) ϕ(λ1v1 + λ2v2,w) = λ1ϕ(v1,w) + λ2ϕ(v2,w),

1.2) ϕ(v, λ1w1 + λ2w2,w) = λ1ϕ(v,w1) + λ2ϕ(v,w2).

2) ϕ is symmetric:

ϕ(v,w) = ϕ(w,v).

3) ϕ is positive definite:

ϕ(v,v) > 0 ∀ v = 0.

Finite dimensional vector spaces with an inner product are called Euclideanvector spaces. For them we can define a norm by

|v| = ϕ(v,v)12 .

In what follows we write v · w instead of ϕ(v,w).

Two vectors v and w are said to be orthogonal if v ·w = 0.

A basis B = e1, . . . , eN is said to be orthogonal if ei · ej = 0 ∀ i = j andis said to be orthonormal if

ei · ej = δij =

1 if i = j,0 if i = j.

If B is orthonormal, then the i-th coordinate of a vector v is given by

vi = v · ei.

Moreover, the matrix to change coordinates with respect to two orthonormal basisis orthogonal, namely,

[C][C]t = [I], (A.5)

where [I] is the identity matrix.


A.3 Tensors

A tensor of order p is any p-linear mapping

T :← p →

V × ... × V→ R.

The set of tensors of order p is denoted by Lp(V). When endowed with the twoalgebraic operations:

• Sum(f + g)(v) = f(v) + g(v) ∀v ∈ V ,

• Product by scalars

(λf)(v) = λf(v), ∀λ ∈ R ∀v ∈ V ,

it is a vector space.Given a basis B = e1, ..., eN of V , we can associate to any f ∈ Lp(V) the

p-dimensional array

fi1,...,ip := f(ei1 , ..., eip), 1 ≤ i1, ..., ip ≤ N. (A.6)

These numbers are called coordinates of f with respect to basis B. A tensor iscalled isotropic if its coordinates do not depend on the basis.

For p = 1, the space L1(V) of first order tensors is nothing but the dual spaceof V which is isomorphic to V . Each f ∈ L1(V) is identified to the unique v ∈ Vsuch that

f(w) = v · w ∀w ∈ V .

For any given orthonormal basis, B, one can show that the coordinates of tensorf , defined by A.6 with p = 1, coincides with the coordinates of vector v. Moreover,the only isotropic first order tensor is f ≡ 0.

For p = 2, the space L2(V) is isomorphic to Lin, the space of endomorphismsof V . A tensor f ∈ L2(V) is identified to the unique endomorphism S ∈ Lin suchthat

f(v,w) = Sw · v ∀v,w ∈ V .

It is easy to see that, given a basis, the entries of the matrix associated with theendomorphism S are the coordinates of f given by (A.6) with p = 2. Moreover,one can show that f is isotropic if and only if S = QSQt ∀Q ∈ Orth+.

For p = 4, the space L4(V) is isomorphic to L(Lin, Lin), the space of en-domorphisms of Lin. Indeed, each f ∈ L4(V) can be identified to the uniqueendomorphism of Lin, l, such that

f(v1,v2,v3,v4) = l(v3 ⊗ v4) · (v1 ⊗ v2) ∀vi ∈ V , i = 1, . . . , 4.

In what follows, due to the above identifications, the elements of Lin will becalled (second order) tensors and the endomorphisms of Lin will be called fourthorder tensors.

A.3. Tensors 165

Second Order Tensors

Let a and b be two vectors in V . The tensor

(a ⊗ b)v := v · b a

is called the tensor product of a and b.If e is a unit vector, then

(e⊗ e)v = (v · e) e

is the orthogonal projection of v on the straight line generated by e. Moreover,

(I − e⊗ e)v = v − (v · e)e, (A.7)

is the projection of v on the orthogonal plane to e.

If B = e1, . . . , eN, the set of tensors ei ⊗ ej : 1 ≤ i, j ≤ N is a basisof Lin. More precisely, we have

S =N∑

i,j=1

Sij ei ⊗ ej ,

where

Sij = S ej · ei

are the coordinates of S with respect to basis B.

We notice that the coordinates of a tensor S can be arranged as a matrix,

[S] =

⎛⎜⎝ S11 . . . S1N

.... . .

...SN1 . . . SNN

⎞⎟⎠ .

For example, the matrix of coordinates of tensor a ⊗ b is the matrix product⎛⎜⎜⎜⎝a1

a2

...aN

⎞⎟⎟⎟⎠ (b1 b2 · · · bN), (A.8)

where a =N∑

i=1

aiei and b =N∑

i=1

biei.

Let us see how tensor coordinates change when we change the basis in vectorspace V . Let S ∈ Lin be a tensor. Let us denote by Se

ij and SEij , i, j = 1, . . . , N ,


its coordinates with respect to basis eiNi=1 and EiN

i=1, respectively. Then it isnot difficult to see that the following relation holds:

[SE] = [C][Se][C]t. (A.9)

In vector space Lin we can introduce another internal operation which is themapping composition: given two tensors S and T , ST is the tensor defined by

(ST )v = S(Tv).

It is easy to see that the matrix of coordinates of ST is the product of thosecorresponding to S and T , i.e.,

[ST ] = [S][T ]. (A.10)

The transpose tensor of S is the unique tensor St satisfying

Sa · b = a · Stb ∀ a,b ∈ V .

A tensor is called symmetric if S = St and skew if S = −St. The subspace ofsymmetric tensors will be denoted by Sym and that of skew tensors by Skw.

We can also define the inner product of tensors by

S · T = tr(StT ),

where tr denotes the trace operator which is the unique linear operator from Lininto R satisfying

tr(a ⊗ b) = a · b.

We have

tr(S) = tr

⎛⎝ N∑i,j=1

Sijei ⊗ ej

⎞⎠ =N∑

i,j=1

Sijei · ej =N∑

i,j=1

Sijδij =N∑

i=1

Sii,

and hence

S · T =N∑

i,j=1

SijTij .

If T ∈ Sym and W ∈ Skw then it easy to show that T · W = 0.

A.3. Tensors 167

Any tensor S is the sum of one symmetric tensor E and one skew tensor W .More precisely, E and W are defined by

E =12(S + St

), (A.11)

W =12(S − St

). (A.12)

Tensor E is called the symmetric part of S while W is called the skew part of S.Furthermore, this decomposition is unique. More precisely,

Lin = Sym

⊥⊕Skw.

The following equalities can be easily proved:

(a ⊗ b)(c ⊗ d) = (b · c)a ⊗ d. (A.13)

R · (ST ) = (StR) · T = (RT t) · S ∀R, S, T ∈ Lin. (A.14)

1 · S = tr(S). (A.15)

S · (a ⊗ b) = a · Sb. (A.16)

(a ⊗ b)(c ⊗ d) = (a · c)(b · d). (A.17)

Let N = 3 and e1, e2, e3 be a positively oriented orthonormal basis in V .We define the cross product of vectors a and b by

a × b =3∑

i,j,k=1

Eijk ajbk ei.

where

Eijk =

⎧⎨⎩1 if ijk is an even permutation of (1 2 3),−1 if ijk is an odd permutation of (1 2 3),0 otherwise .

One can prove the following equality:

a × (b × c) = (a · c)b − (a · b)c (A.18)

showing that the cross product is not associative.

There is an isomorphism between the vector space V and the vector space ofskew tensors. It is defined by

V −→ Skww −→ W


with Wv := w × v ∀v ∈ V . Vector w is called the axial vector associated to W .A tensor S is invertible if there exists another tensor S−1, called the inverse

of S, such thatSS−1 = S−1S = I. (A.19)

A tensor Q is orthogonal if it preserves the inner product, that is,

Qu · Qv = u · v ∀u,v ∈ V . (A.20)

A necessary and sufficient condition for Q to be orthogonal is that

QQt = QtQ = I, (A.21)

which is equivalent toQt = Q−1. (A.22)

In particular, detQ must be 1 or −1.A rotation is an orthogonal tensor with positive determinant. The set of

rotations is a subspace of Lin which will be denoted by Orth+.One can show that a tensor S is isotropic if and only if it commutes with any

rotation, that is,S = QSQt ∀Q ∈ Orth+.

Furthermore (see, for instance, [5]), this condition is satisfied if and only if S isspherical, i.e., S = ωI for some scalar ω.

A tensor S is called deviatoric if it is symmetric and traceless. We denoteby Dev the subspace of deviatoric tensors. Any symmetric tensor S ∈ Sym canbe uniquely decomposed as the sum of a spherical tensor and a deviatoric tensor.Indeed, the tensor

SD = S − 1N

tr(S)I

is a deviatoric tensor, and we can rewrite this equality in the form

S =1N

tr(S)I + SD.

Moreover the subspace of spherical tensors is orthogonal to the subspace of devi-atoric tensors because I · S = tr(S) = 0 for any S ∈ Dev. Hence, we have theorthogonal direct sum decomposition

Sym = Sph

⊥⊕Dev,

where Sph denotes the subspace of spherical tensors.A tensor S is positive definite if

v · Sv > 0 ∀v = 0. (A.23)

A.3. Tensors 169

The principal invariants of a tensor S are the three numbers

ı1(S) = tr(S), (A.24)

ı2(S) =12[( tr(S))2 − tr(S2)], (A.25)

ı3(S) = det(S) =16[( tr(S))3 + 2( tr(S))2 − 3 tr(S2) tr(S)]. (A.26)

The set of principal invariants of S will be denoted by IS .

Fourth Order Tensors

Let T, S ∈ Lin. The fourth order tensor T ⊗ S ∈ L(Lin, Lin) defined by

(T ⊗ S)R = S · RT ∀R ∈ Lin,

is called the tensor product of T and S.For the basis B of V , the set

(ei ⊗ ej) ⊗ (ek ⊗ el), 1 ≤ i, j, k, l ≤ N

is a basis of the space of fourth order tensors. In fact, for any fourth order tensorl we have

l =N∑

i,j,k,l=1

lijkl (ei ⊗ ej) ⊗ (ek ⊗ el),

wherelijkl = l(ek ⊗ el) · (ei ⊗ ej) (A.27)

are the coordinates of l with respect to basis B.One can show that l is isotropic (i.e., its coordinates are independent of the

basis) if and only if

l(QSQt) = Ql(S)Qt ∀Q ∈ Orth+ ∀S ∈ Lin. (A.28)

The following result has important applications in continuum mechanics:

Proposition 1.3.1. Let l ∈ L(Lin, Lin) be a fourth order tensor. Then l is isotropicif and only if there exist three scalars α, β and γ such that

l(S) = α tr(S)I + βS + γSt. (A.29)

Proof. If l is of the form (A.29) then it is isotropic because (A.28) holds. Indeed,let Q ∈ Orth+. We have

l(QSQt) = α tr(QSQt)I + βQSQt + γ(QSQt)t (A.30)

= Q(α tr(QSQt)I + βS + γS

)Qt = Ql(S)Qt, (A.31)


because tr(QSQt) = tr(S) ∀Q ∈ Orth. The proof of this follows immediately byusing (A.14):

tr(QSQt) = I · QSQt = Qt · SQt = QQt · S = I · S = tr(S).

The proof of the converse result is much more difficult and will be omittedhere (see [4]).

Corollary 1.3.2. Let l ∈ L(Lin, Lin) be a fourth order tensor. Then, if l is isotropic,there exist two scalars λ and µ such that

l(E) = λ tr(E)I + 2µE ∀E ∈ Sym.

Proof. It is enough to take λ = α and µ = β+γ2 in (A.29).

Corollary 1.3.3. The isotropic fourth order tensor l given by (A.29) satisfies l(S) ∈Sym ∀S ∈ Lin if and only if β = γ in which case

l(S) = l

(S + St

2

)∀S ∈ Lin.

Proof. It follows from the fact that tr(S) = tr(St) ∀S ∈ Lin.

It is immediate to see that the coordinates of l given by (A.29) are, in anyorthonormal basis,

lijkm = αδijδkm + βδikδmj + γδimδjk.

Indeed, according to (A.27),

lijkl = l(ek ⊗ el) · (ei ⊗ ej) = (α tr(ek ⊗ em)I + βek ⊗ em + γem ⊗ ek) · ei ⊗ ej

= αek · emei · ej + βek · eiem · ej + γem · eiek · ej , (A.32)

and the result follows.

A.4 The Affine Space

An affine space is a triple (E ,V , +) consisting of a set E , a vector space V and amapping

E × V +−→ E(p,v) −→ p + v

satisfying the following property:

For any p, q ∈ E there exists a unique v ∈ V such that p + v = q.

This vector v is usually denoted by −→pq or q − p.

A.4. The Affine Space 171

If V is a Euclidean vector space, then the corresponding affine space is calledEuclidean affine space.

In a Euclidean affine space we can define a distance by

E × E d−→ R

(p, q) −→ d(p, q) = |−→pq|.

Thus (E , d) is a metric space and hence a topological space.

In a Euclidean affine space, a cartesian coordinate frame is a pair(o, ei, i = 1, . . . , N) where o is any fixed point in E called the origin and ei, i =1, . . . , N is an orthonormal basis of the Euclidean vector space V .

For any point p ∈ E its coordinates with respect to the above cartesian frameare the numbers pi, i = 1, . . . , N, such that

−→op =N∑

i=1

piei.

Appendix B

Vector and Tensor Analysis

A field is any mapping defined in a subset of E and valued in R, V or Lin, in whichcase it is called a scalar, vector or tensor field, respectively.

B.1 Differential Operators

Let W be a normed vector space (in practice W = R, V , Lin). A mappingf : R ⊂ E → W defined in an open set R of E is said to be differentiable at pointx ∈ R if there exists a linear mapping X : V −→ W such that

f(x + h) − f(x) −X (h) = o(h) as h → 0,

which meanslimh→0

‖f(x + h) − f(x) −X (h)‖‖h‖ = 0.

If f is differentiable at x, then linear mapping X is called the differential off at x. It is denoted by Df(x).

Particular cases:

1) Let us assume W = R, i.e., f = ϕ is a scalar field differentiable at x. Then

Dϕ(x) : V −→ R

is a linear mapping and the unique vector ∇ϕ(x) such that

Dϕ(x)a = ∇ϕ(x) · a ∀a ∈ V

is called the gradient of ϕ at x.

If ϕ is differentiable at all points in R we can define the vector field

174 Appendix B. Vector and Tensor Analysis

R ⊂ E −→ Vx −→ ∇ϕ(x),

which is called the gradient of ϕ.

2) Let us consider the case where W = V , i.e., f = u is a vector field. ThenDu(x) : V −→ V , that is, Du(x) ∈ Lin.

If u is differentiable at each point in R, we can define the tensor field

R ⊂ E −→ Linx −→ Du(x),

which is called the gradient of u.

In order to denote this differential operator we will use ∇u or gradu,instead of Du,

Now we introduce other differential operators. Given a differentiable vectorfield u, the scalar field

div u : R ⊂ E −→ R

x −→ div u(x) =: tr(∇u(x))

is called the divergence of u.

Let S : R ⊂ E −→ Lin be a smooth tensor field. The divergence of S atpoint x ∈ R is defined as the unique vector, div S(x), such that

div S(x) · a = div(Sta)(x) ∀ a ∈ V .

Let u : R ⊂ E −→ V be a smooth vector field. The curl of u at x is theunique vector, curl u(x), such that,

curl u(x) × a = (∇u(x) −∇ut(x))a ∀ a ∈ V .

Let S : R ⊂ E −→ Lin be a smooth tensor field. We define the curl of S atpoint x as the unique tensor curlS(x) such that

curlS(x)a = curl(Sta)(x) ∀ a ∈ V .

For a scalar field ϕ : R ⊂ E −→ R, its Laplacian is the scalar field

∆ϕ(x) = div(∇ϕ)(x).

B.2. Curves and Curvilinear Integrals 175

For a vector field u : R ⊂ E −→ V, its Laplacian is the vector field

∆u(x) = div(∇u)(x).

The following equalities hold:

∇(ϕv) = ϕ∇v + v ⊗∇ϕ,

div(ϕv) = ϕ div v + v · ∇ϕ,

∇(v · w) = (∇w)tv + (∇v)tw,

div(v ⊗ w) = v div w + (∇v)w,

div(Stv) = S · ∇v + v · div S,

div(ϕS) = ϕ div S + S∇ϕ,

div(v × w) = w · curl v − v · curlw,

div∇vt = ∇(div v),

curl(ϕv) = ∇ϕ × v + ϕ curl v,

curl(v × w) = ∇vw − w div v + v div w −∇wv,

div curl v = 0,

curl∇ϕ = 0,

∆v = ∇(div v) − curl(curl v),

∆(∇ϕ) = ∇(∆ϕ),

div(∆v) = ∆(div v),

curl(∆v) = ∆(curl v),

∆(ϕψ) = ϕ∆ψ + ψ∆ϕ + 2∇ϕ · ∇ψ.

B.2 Curves and Curvilinear Integrals

A (regular) curve c in R is a smooth map

c : [0, 1] −→ R

such that c(σ) = 0 ∀σ ∈ [0, 1].

The curve is closed if c(0) = c(1).

The length of c is the number


length(c) =∫ 1

0

|c(σ)| dσ.

Let v : R ⊂ E −→ V be a continuous vector field. The curvilinear integral ofv along c is the scalar defined by∫

c

v(x) · dx =∫ 1

0

v(c(σ)) · c(σ) dσ.

Similarly, let S : R ⊂ E −→ Lin be a continuous tensor field. The curvilinearintegral of S along c is the vector defined by∫

c

S(x)dx =∫ 1

0

S(c(σ))c(σ) dσ.

If v is the gradient of a scalar field ϕ, then

∫c

v(x) · dx =∫

c

∇ϕ(x) · dx =∫ 1

0

∇ϕ(c(σ)) · c(σ)dσ

=∫ 1

0

d

dσ(ϕ c)(σ) dσ = ϕ(c(1)) − ϕ(c(0)). (B.1)

In particular∫c∇ϕ(x) · dx = 0 whenever c is closed.

A subset R ⊂ E is simply connected if any closed curve in R can be contin-uously deformed to a point without leaving R.

An open region is any connected subset of E . The closure of an open regionwill be called a closed region. We designate by regular region any closed regionwith smooth boundary ∂R.

Let R be a closed region and Φ a field defined in R. We say that Φ is C1(R) ifit is continuously differentiable in the interior of R, and Φ and ∇Φ have continuousextensions to all of R.

We end this section on vector calculus by recalling some fundamental results.

Theorem 2.2.1. Potential Theorem. Let v be a smooth vector field on an open orclosed simply connected region R and assume that,

curl v = 0.

Then there is a C2 scalar field ϕ : R −→ R such that,

v = ∇φ.

B.3. Gauss’ and Green’s Formulas. Stokes’ Theorem 177

B.3 Gauss’ and Green’s Formulas. Stokes’ Theorem

Theorem 2.3.1. Divergence Theorem. Let R be a bounded regular region and letϕ : R −→ R, v : R −→ V and S : R −→ Lin be smooth fields. Then,

1.∫

∂Rϕn dAx =

∫R∇ϕ dVx,

2.∫

∂Rv ⊗ n dAx =

∫R∇v dVx,

3.∫

∂Rv · n dAx =

∫R

div v dVx,

4.∫

∂RSn dAx =

∫R

div S dVx.

Theorem 2.3.2. Green’s formulas. Let R be a bounded regular region and let ϕ, ψ :R −→ R, v, w : R −→ V and S : R −→ Lin be smooth fields. Then,

1.∫

∂R

∂ϕ

∂nψ dAx =

∫R∇ϕ · ∇ψ dVx +

∫R

∆ϕψ dVx,

2.∫

∂RSn · v dAx =

∫R

S · ∇v dVx +∫R

v · div S dVx,

3.∫

∂R(Sn) ⊗ v dAx =

∫R

S∇vt dVx +∫R

div S ⊗ v dVx,

4.∫

∂Rv · nψ dAx =

∫R

div v ψ dVx +∫R

v · ∇ψ dVx,

5.∫

∂R(w · n)v dAx =

∫R

(div w) v dVx +∫R

(∇v)w dVx,

6.∫

∂R(v × w) · n dAx =

∫R

w · curl v dVx −∫R

v · curlw dVx.

Theorem 2.3.3. Stokes’ Theorem. Let v (respectively, S) be a smooth vector (re-spectively, tensor) field on an open set R ⊂ E. Let S be a smooth surface in Rbounded by c, supposed to be a closed curve. At each point of S we choose a unitnormal vector n consistent with the direction of traversing c in that it causes aright-handed screw to advance along n. Then,∫

Scurl v · n dAx =

∫c

v · dlx,

∫S(curlS)tn dAx =

∫c

S dlx.


B.4 Change of Variable in Integrals

Theorem 2.4.1. Let f be a deformation of a body B and ϕ : f(B) → R a smoothscalar field. Let c (resp. S) be a smooth curve (resp. surface) in B and P a part ofB. Then we have

1.∫f(c)

ϕt dlx =∫

c

ϕ f F kdlp,

2.∫f(S)

ϕn dAx =∫S

ϕ f detF F−tm dAp,

3.∫f(c)

ϕdlx =∫

c

ϕ f ‖Fk‖ dlp,

4.∫f(S)

ϕdAx =∫S

ϕ f detF‖F−tm‖ dAp,

5.∫f(P)

ϕdVx =∫P

ϕ f detF dVp,

where k (resp. t) denotes a unit tangent vector to the curve c (resp. f(c)) and m(resp. n) denotes a unit normal vector to surface S (resp. f(S)).

B.5 Transport Theorems

Theorem 2.5.1. Let X be a motion of a body B. Let c, S and P be a curve, asurface and a part of B. Let ϕ : T → R and w : T → V be a scalar field and avector field, respectively. Then we have

1.ddt

∫ct

ϕt dlx =∫

ct

(ϕt + ϕLt) dlx,

2.ddt

∫St

ϕn dAx =∫St

(ϕn + ϕdivvn − ϕLtn

)dAx,

3.ddt

∫Pt

ϕdVx =∫Pt

(ϕ + ϕdivv) dVx =∫Pt

(ϕ′ + div(ϕv)) dVx

=∫Pt

ϕ′dVx +∫

∂Pt

ϕv · n dAx,

4.ddt

∫ct

w ⊗ t dlx =∫

ct

(w ⊗ t + w ⊗ Lt) dlx,

5.ddt

∫ct

w · t dlx =∫

ct

(w · t + Ltw · t

)dlx,

6.ddt

∫St

w ⊗ n dAx =∫St

(w ⊗ n + divvw ⊗ n− w ⊗ Ltn

)dAx,

B.6. Localization Theorem 179

7.ddt

∫St

w · n dAx =∫St

(w · n + divvw · n −∇vw · n) dAx

=∫St

(w′ · n + divwv · n + curl(w × v) · n) dAx,

8.ddt

∫Pt

wdVx =∫Pt

(w + divvw) dVx =∫Pt

(w′ + div(w ⊗ v)) dVx

=∫Pt

w′dVx +∫

∂Pt

wv · n dAx,

where t denotes a unit tangent vector to ct and n is a unit normal vector to St.

B.6 Localization Theorem

Theorem 2.6.1. Let Φ be a continuous scalar or vector valued mapping defined inan open set R ⊂ E. If ∫

B

Φ dVx ≥ 0

for every closed ball B ⊂ R, then Φ ≥ 0 in R. Hence, if∫B

Φ dVx = 0

for every closed ball B ⊂ R, then Φ ≡ 0 in R.

B.7 Differential Operators in Coordinates

B.7.1 Cartesian Coordinates

Definition

f : Ω = R3 → R

3 (B.2)f(u1, u2, u3) = (u1, u2, u3). (B.3)

Basis

• Contravariantg1 = e1, g2 = e2, g3 = e3. (B.4)

• Covariantg1 = e1, g2 = e2, g3 = e3. (B.5)

• Physicale1, e2, e3. (B.6)


Christoffel symbols of the second kind

Γ1jk =

⎛⎝ 0 0 00 0 00 0 0

⎞⎠ , (B.7)

Γ2jk =

⎛⎝ 0 0 00 0 00 0 0

⎞⎠ , (B.8)

Γ3jk =

⎛⎝ 0 0 00 0 00 0 0

⎞⎠ . (B.9)

Differential Operators in the Physical Basis

• Gradient of scalar field

∇φ =∂φ

∂x1e1 +

∂φ

∂x2e2 +

∂φ

∂x3e3. (B.10)

• Gradient of a vector field

∇w =∂w1

∂x1e1 ⊗ e1 +

∂w1

∂x2e1 ⊗ e2 +

∂w1

∂x3e1 ⊗ e3

+∂w2

∂x1e2 ⊗ e1 +

∂w2

∂x2e2 ⊗ e2 +

∂w2

∂x3e2 ⊗ e3

+∂w3

∂x1e3 ⊗ e1 +

∂w3

∂x2e3 ⊗ e2 +

∂w3

∂x3e3 ⊗ e3. (B.11)

• Convective term

(∇w)w =

w1∂w1

∂x1+ w2

∂w1

∂x2+ w3

∂w1

∂x3

e1

+

w1∂w2

∂x1+ w2

∂w2

∂x2+ w3

∂w2

∂x3

e2

+

w1∂w3

∂x1+ w2

∂w3

∂x2+ w3

∂w3

∂x3

e3. (B.12)

• Curl of a vector field

curlw =(

∂w3

∂x2− ∂w2

∂x3

)e1

+(

∂w1

∂x3− ∂w3

∂x1

)e2

+(

∂w2

∂x1− ∂w1

∂x2

)e3. (B.13)

B.7. Differential Operators in Coordinates 181

• Divergence of a vector field

div w =∂w1

∂x1+

∂w2

∂x2+

∂w3

∂x3. (B.14)

• Divergence of a tensor field

div S =[∂S11

∂x1+

∂S12

∂x2+

∂S13

∂x3

]e1

+[∂S21

∂x1+

∂S22

∂x2+

∂S23

∂x3

]e2

+[∂S31

∂x1+

∂S32

∂x2+

∂S33

∂x3

]e3. (B.15)

• Laplacian of a scalar field

∆φ =∂2φ

∂x21

+∂2φ

∂x22

+∂2φ

∂x23

. (B.16)

• Laplacian of a vector field

∆w = ∆w1e1 + ∆w2e2 + ∆w3e3. (B.17)

Measure elements for integration

• Line

dx1, dx2, dx3. (B.18)

• Surface

dx2 δx3, dx1dx3, dx1dx2. (B.19)

• Volume

dx1 dx2 dx3. (B.20)


B.7.2 Cylindrical Coordinates

Definition

f : Ω = (0,∞) × (0, 2π) × (−∞,∞) → R3 (B.21)

f(r, θ, z) = (r cos θ, r sin θ, z). (B.22)

Basis

• Contravariant ⎧⎨⎩g1(r, θ, z) = cos θe1 + sin θe2,g2(r, θ, z) = −r sin θe1 + r cos θe2,g3(r, θ, z) = e3.

(B.23)

• Covariant ⎧⎨⎩g1(r, θ, z) = cos θe1 + sin θe2,g2(r, θ, z) = − 1

r sin θe1 + 1r cos θe2,

g3(r, θ, z) = e3.(B.24)

• Physical ⎧⎨⎩er(r, θ, z) = cos θe1 + sin θe2,eθ(r, θ, z) = − sin θe1 + cos θe2,ez(r, θ, z) = e3.

(B.25)


Γ1jk =

⎛⎝ 0 0 00 −r 00 0 0

⎞⎠ , (B.26)

Γ2jk =

⎛⎝ 0 1r 0

1r 0 00 0 0

⎞⎠ , (B.27)

Γ3jk =

⎛⎝ 0 0 00 0 00 0 0

⎞⎠ . (B.28)

Differential operators in the physical basis

• Gradient of scalar field

∇φ =∂φ

∂rer +

1r

∂φ

∂θeθ +

∂φ

∂zez. (B.29)



∇w =∂wr

∂rer ⊗ er +

[1r

∂wr

∂θ− 1

rwθ

]er ⊗ eθ +

∂wr

∂zer ⊗ ez

+∂wθ

∂reθ ⊗ er +

[1r

∂wθ

∂θ+

1rwr

]eθ ⊗ eθ +

∂wθ

∂zeθ ⊗ ez

+∂wz

∂rez ⊗ er +

1r

∂wz

∂θez ⊗ eθ +

∂wz

∂zez ⊗ ez. (B.30)

• Convective term

∇ww =

wr∂wr

∂r+

1rwθ

∂wr

∂θ+ wz

∂wr

∂z− (wθ)2

r

er

+

wr∂wθ

∂r+

1rwθ

∂wθ

∂θ+ wz

∂wθ

∂z+

wrwθ

r

eθ

+

wr∂wz

∂r+

1rwθ

∂wz

∂θ+ wz

∂wz

∂z

ez. (B.31)


curlw =(

1r

∂wz

∂θ− ∂wθ

∂z

)er

+(−∂wz

∂r+

∂wr

∂z

)eθ

+(

1r

∂

∂r(rwθ) −

1r

∂wr

∂θ

)ez. (B.32)


div w =1r

∂

∂r(rwr) +

1r

∂wθ

∂θ+

∂wz

∂z. (B.33)


div S =1r

[∂

∂r(rSrr) +

∂Srθ

∂θ+ r

∂Srz

∂z− 1

rSθθ

]er

+[∂Sθr

∂r+

1r

∂Sθθ

∂θ+

∂Sθz

∂z+

1rSrθ +

1rSθr

]eθ

+1r

[∂

∂r(rSzr) +

∂Szθ

∂θ+ r

∂Szz

∂z

]ez. (B.34)



∆φ =1r

[∂

∂r

(r∂φ

∂r

)+

∂

∂θ

(1r

∂φ

∂θ

)+

∂

∂z

(r∂φ

∂z

)]=

1r

∂

∂r

(r∂φ

∂r

)+

1r2

∂2φ

∂θ2+

∂2φ

∂z2. (B.35)


∆w =[∆wr −

1r2

wr −2r2

∂wθ

∂θ

]er +

[∆wθ −

1r2

wθ +2r2

∂wr

∂θ

]eθ + ∆wzez.

(B.36)


• Line

dr, r dθ, dz. (B.37)

• Surface

r dθdz, drdz, r drdθ. (B.38)

• Volume

r drdθdz. (B.39)

B.7.3 Spherical Coordinates

Definitionf : Ω = (0,∞) × (0, π) × (0, 2π) → R

3,f(r, θ, ϕ) = (r sin θ cosϕ, r sin θ sin ϕ, r cos θ). (B.40)

Basis

• Contravariant⎧⎨⎩g1(r, ϕ, θ) = sin θ cosϕe1 + sin θ sinϕe2 + cos θe3,g2(r, ϕ, θ) = r cos θ cosϕe1 + r cos θ sin ϕe2 − r sin θe3,g3(r, ϕ, θ) = −r sin θ sin ϕe1 + r sin θ cosϕe2.

(B.41)

• Covariant⎧⎨⎩g1(r, ϕ, θ) = sin θ cosϕe1 + sin θ sin ϕe2 + cos θe3,g2(r, ϕ, θ) = 1

r cos θ cosϕe1 + 1r cos θ sin ϕe2 − 1

r sin θe3,g3(r, ϕ, θ) = − 1

r sin θ sinϕe1 + 1r sin θ cosϕe2.

(B.42)


• Physical⎧⎨⎩er(r, ϕ, θ) = sin θ cosϕe1 + sin θ sin ϕe2 + cos θe3,eθ(r, ϕ, θ) = cos θ cosϕe1 + cos θ sinϕe2 − sin θe3,eϕ(r, ϕ, θ) = − sinϕe1 + cosϕe2.

(B.43)


Γ1jk =

⎛⎝ 0 0 00 −r 00 0 −r sin2 θ

⎞⎠ , (B.44)

Γ2jk =

⎛⎝ 0 1r 0

1r 0 00 0 − 1

2 sin 2θ

⎞⎠ , (B.45)

Γ3jk =

⎛⎝ 0 0 1r

0 0 cot θ1r cot θ 0

⎞⎠ . (B.46)

Differential operators

• Gradient of a scalar field

∇φ =∂φ

∂rg1 +

∂φ

∂θg2 +

∂φ

∂ϕg3 =

∂φ

∂rer +

1r

∂φ

∂θeθ +

1r sin θ

∂φ

∂ϕeϕ. (B.47)


∇w=∂wr

∂rer ⊗ er+

[1r

∂wr

∂θ− 1

rwθ

]er ⊗ eθ+

[1

r sin θ

∂wr

∂ϕ− 1

rwϕ

]er ⊗ eϕ

+∂wθ

∂reθ ⊗ er+

[1r

∂wθ

∂θ+

1rwr

]eθ ⊗ eθ+

[1

r sin θ

∂wθ

∂ϕ− 1

rcot θ wϕ

]eθ ⊗ eϕ

+∂wϕ

∂reϕ ⊗ er+

1r

∂wϕ

∂θeϕ ⊗ eθ+

[1

r sin θ

∂wϕ

∂ϕ+

1rwr +

1r

cot θwθ

]eϕ ⊗ eϕ.

(B.48)

• Convective term

(∇w)w =

wr

∂wr

∂r+

1rwθ

∂wr

∂θ+

wϕ

r sin θ

∂wr

∂ϕ−

w2θ + w2

ϕ

r

er

+

wr

∂wθ

∂r+

1rwθ

∂wθ

∂θ+

wϕ

r sin θ

∂wθ

∂ϕ+

wrwθ

r−

w2ϕ cot θ

r

eθ

+

wr∂wϕ

∂r+

1rwθ

∂wϕ

∂θ+

wϕ

r sin θ

∂wϕ

∂ϕ+

wrwϕ

r+

wθwϕ cot θ

r

eϕ.

(B.49)



curlw =1

r sin θ

(∂

∂θ(wϕ sin θ) − ∂wθ

∂ϕ

)er

+(

1r sin θ

∂wr

∂ϕ− 1

r

∂

∂r(rwϕ)

)eθ

+1r

(∂

∂r(rwθ) −

∂wr

∂θ

)eϕ. (B.50)


div w =1r2

∂

∂r(r2wr) +

1r sin θ

∂

∂θ(sin θwθ) +

1r sin θ

∂wϕ

∂ϕ. (B.51)


div S

=[∂Srr

∂r+

1r

∂Srθ

∂θ+

1r sin θ

∂Sϕr

∂ϕ+

1r

(2Srr − Sθθ − Sϕϕ + Srθ cot θ)]er

+[∂Srθ

∂r+

1r

∂Sθθ

∂θ+

1r sin θ

∂Sθϕ

∂ϕ+

1r

((Sθθ − Sϕϕ) cot θ + 3Srθ)]eθ

+[∂Sϕr

∂r+

1r

∂Sθϕ

∂θ+

1r sin θ

∂Sϕϕ

∂ϕ

]eϕ. (B.52)


∆ϕ =1

r2 sin θ

[∂

∂r

(r2 sin θ

∂φ

∂r

)+

∂

∂θ

(sin θ

∂φ

∂θ

)+

∂

∂ϕ

(1

sin θ

∂φ

∂ϕ

)]=

1r2

∂

∂r(r2 ∂φ

∂r) +

1r2 sin θ

∂

∂θ(sin θ

∂φ

∂θ) +

1r2 sin2 θ

∂2φ

∂ϕ2. (B.53)


∆w =[∆wr −

2r2

wr −2r2

∂wθ

∂θ− 2

r2wθ cot θ − 2

r2 sin θ

∂wϕ

∂ϕ

]er

+[∆wθ +

2r2

∂wr

∂θ− 1

r2 sin2 θwθ − 2

r2

cos θ

sin2 θ

∂wϕ

∂ϕ

]eθ

+[∆wϕ − 1

r2 sin2 θwϕ +

2r2 sin θ

∂wr

∂ϕ+

2r2

cos θ

sin2 θ

∂wθ

∂ϕ

]eϕ (B.54)

where the Laplacian operators in the right-hand side are given by (B.53).



• Line

dr, r dθ, r sin θ dϕ. (B.55)

• Surface

r2 sin θ dθdϕ, r sin θ drdϕ, r drdθ. (B.56)

• Volume

r2 sin θ drdθdϕ. (B.57)

Appendix C

Some Equations of ContinuumMechanics in CurvilinearCoordinates

In this section we write some partial differential equations of continuum mechanicsin cylindrical and spherical coordinates.

C.1 Mass Conservation Equation

It is given by∂ρ

∂t+ div(ρv) = 0, (C.1)

where ρ is density and v is velocity.

• Cartesian coordinates

∂ρ

∂t+

∂(ρv1)∂x1

+∂(ρv2)∂x2

+∂(ρv3)∂x3

= 0. (C.2)

• Cylindrical coordinates

∂ρ

∂t+

1r

∂

∂r(rρvr) +

1r

∂(ρvθ)∂θ

+∂(ρvz)

∂z= 0. (C.3)

• Spherical coordinates

∂ρ

∂t+

1r2

∂

∂r(r2ρvr) +

1r sin θ

∂

∂θ(sin θρvθ) +

1r sin θ

∂(ρvϕ)∂ϕ

= 0. (C.4)

190 Appendix C. Some Equations in Curvilinear Coordinates

C.2 Motion Equation

The general motion equation has the form

ρ(∂v∂t

+ (gradv)v) = div T + b, (C.5)

where T is the Cauchy stress tensor and b is the body force.


ρ

∂v1

∂t+ v1

∂v1

∂x1+ v2

∂v1

∂x2+ v3

∂v1

∂x3

=

[∂T11

∂x1+

∂T12

∂x2+

∂T13

∂x3

]+ b1, (C.6)

ρ

∂v2

∂t+ v1

∂v2

∂x1+ v2

∂v2

∂x2+ v3

∂v2

∂x3

=

[∂T21

∂x1+

∂T22

∂x2+

∂T23

∂x3

]+ b2, (C.7)

ρ

∂v3

∂t+ v1

∂v3

∂x1+ v2

∂v3

∂x2+ v3

∂v3

∂x3

=

[∂T31

∂x1+

∂T32

∂x2+

∂T33

∂x3

]+ b3. (C.8)


ρ

∂vr

∂t+ vr

∂vr

∂r+

vθ

r

∂vr

∂θ+ vz

∂vr

∂z− v2

θ

r

=

1r

[∂

∂r(rTrr) +

∂Trθ

∂θ+ r

∂Trz

∂z− 1

rTθθ

]+ br, (C.9)

ρ

∂vθ

∂t+ vr

∂vθ

∂r+

vθ

r

∂vθ

∂θ+ vz

∂vθ

∂z+

vrvθ

r

=

[∂Tθr

∂r+

1r

∂Tθθ

∂θ+

∂Tθz

∂z+

1rTrθ +

1rTθr

]+ bθ, (C.10)

ρ

∂vz

∂t+ vr

∂vz

∂r+

vθ

r

∂vz

∂vθ+ vz

∂vz

∂z

=

1r

[∂

∂r(rTzr) +

∂Tzθ

∂θ+ r

∂Tzz

∂z

]+ bz. (C.11)

C.3. Constitutive Law for Newtonian Viscous Fluids in Cooordinates 191


ρ

∂vr

∂t+ vr

∂vr

∂r+

1rvθ

∂vr

∂θ+

vϕ

r sin θ

∂vr

∂ϕ−

v2θ + v2

ϕ

r

=[∂Trr

∂r+

1r

∂Trθ

∂θ+

1r sin θ

∂Tϕr

∂ϕ+

1r

(2Trr − Tθθ − Tϕϕ + Trθ cot θ)]

+br,

(C.12)

ρ

∂vθ

∂t+ vr

∂vθ

∂r+

1rvθ

∂vθ

∂θ+

vϕ

r sin θ

∂vθ

∂ϕ+

vrvθ

r−

v2ϕ cot θ

r

=[∂Trθ

∂r+

1r

∂Tθθ

∂θ+

1r sin θ

∂Tθϕ

∂ϕ+

1r

((Tθθ − Tϕϕ) cot θ + 3Trθ)]

+ bθ,

(C.13)

ρ

∂vϕ

∂t+ vr

∂vϕ

∂r+

1rvθ

∂vϕ

∂θ+

vϕ

r sin θ

∂vϕ

∂ϕ+

vrvϕ

r+

vθvϕ cot θ

r

=[∂Tϕr

∂r+

1r

∂Tθϕ

∂θ+

1r sin θ

∂Tϕϕ

∂ϕ

]+ bϕ.

(C.14)

C.3 Constitutive Law for Newtonian Viscous Fluids inCooordinates

In terms of the stretching D, it is given by

T = 2ηD + ξ div vI, (C.15)

with

D =12(gradv + gradvt). (C.16)

192 Appendix C. Some Equations in Curvilinear Coordinates


T11 = ξ div v + 2η∂v1

∂x1, (C.17)

T22 = ξ div v + 2η∂v2

∂x2, (C.18)

T33 = ξ div v + 2η∂v3

∂x3, (C.19)

T12 = T21 = η

(∂v1

∂x2+

∂v2

∂x1

), (C.20)

T13 = T31 = η

(∂v1

∂x3+

∂v3

∂x1

), (C.21)

T23 = T32 = η

(∂v2

∂x3+

∂v3

∂x2

), (C.22)

with

div v =∂v1

∂x1+

∂v2

∂x2+

∂v3

∂x3. (C.23)


Trr = ξ div v + 2η∂vr

∂r, (C.24)

Tθθ = ξ div v + 2η

(1r

∂vθ

∂θ+

vr

r

), (C.25)

Tzz = ξ div v + 2η∂vz

∂z, (C.26)

Trθ = Tθr = η

[1r

∂vr

∂θ+ r

∂

∂r

(vθ

r

)], (C.27)

Trz = Tzr = η

(∂vr

∂z+

∂vz

∂r

), (C.28)

Tθz = Tzθ = η

(∂vθ

∂z+

1r

∂vz

∂θ

), (C.29)

with

div v =1r

∂

∂r(rvr) +

1r

∂vθ

∂θ+

∂vz

∂z. (C.30)

C.3. Constitutive Law for Newtonian Viscous Fluids in Cooordinates 193


Trr = ξ div v + 2η∂vr

∂r, (C.31)

Tθθ = ξ div v + 2η

(1r

∂vθ

∂θ+

vr

r

), (C.32)

Tϕϕ = ξ div v + 2η

(1

r sin θ

∂vϕ

∂ϕ+

vr

r+

vθ cot θ

r

), (C.33)

Trθ = Tθr = η

[1r

∂vr

∂θ+ r

∂

∂r

(vθ

r

)], (C.34)

Trϕ = Tϕr = η

[r

∂

∂r

(vϕ

r

)+

1r sin θ

∂vr

∂ϕ

], (C.35)

Tθϕ = Tϕθ = η

[1

r sin θ

∂vθ

∂ϕ+

sin θ

r

∂

∂θ

( vϕ

sin θ

)], (C.36)

with

div v =1r2

∂

∂r(r2vr) +

1r sin θ

∂

∂θ(sin θvθ) +

1r sin θ

∂vϕ

∂ϕ. (C.37)

Appendix D

Arbitrary Lagrangian-Eulerian(ALE) Formulations of theConservation Equations

In some fluid-structure interaction problems, the domain where the motion of thefluid is taking place changes with time because the structure interacting withthe fluid also moves. This is the case, for instance, of aeroelasticity problems.Moreover, there are free boundary flows for which the sets Bt, i.e., the positionsoccupied by the body, change with time and are a priori unknown. In these twosituations, numerical methods based on so-called Arbitrary Lagrangian-Eulerian(ALE) formulations can be very useful.

The main goal of this chapter is to write several ALE formulations of theconservation laws.

D.1 ALE Configuration

The situation is summarized in Figure D.1. Recall that B is the reference or La-grangian configuration, Bt is the Eulerian configuration and Ω denotes the ALEconfiguration. Y(·, t) is a motion transforming Ω into Bt at each time t and

Z(·, t) = [Y(·, t)]−1 X(·, t).

Hence,Y(Z(p, t), t) = X(p, t). (D.1)

Fields can be written in terms of Eulerian (spatial), Lagrangian (material) or ALEconfiguration. More precisely, if Φ is an Eulerian field, we can define its Lagrangian

196 Appendix D. ALE Formulations of the Conservation Equations

Z( , )t Y( , )t

X( , )t

BBt

Figure D.1: Lagrangian, Eulerian and ALE configurations

and ALE descriptions by

Φm(p, t) = Φ(X(p, t), t),Φa(z, t) = Φ(Y(z, t), t),

respectively. We recall that, in particular, vm = X.Similarly, for a Lagrangian field Ψ we can define its Eulerian and ALE coun-

terparts by

Ψs(x, t) = Ψ(P(x, t), t),Ψa(z, t) = Ψ(PZ(z, t), t),

where P is the reference map for motion X, i.e., P(·, t) = [X(·, t)]−1, and PZ(z, t) =[Z(·, t)]−1. Hence,

PZ(z, t) = P(Y(z, t), t).

In particular v = Xs.We will use the notations grada, diva and ∂

∂t =′ for the corresponding dif-ferential operators applied to fields in ALE configuration. Let us denote w thespatial field defined by wa = ∂Y

∂t . Field wa is called the ALE velocity. We alsointroduce d :=

(∂Z∂t

)a.

By taking the time derivative of (D.1) we obtain

∂X∂t

(p, t) =∂Y∂t

(Z(p, t), t) + gradaY(Z(p, t), t)∂Z∂t

(p, t). (D.2)

Let p = PZ(z, t). Then z = Z(p, t) and (D.2) yields

va(z, t) = wa(z, t) + G(z, t)d(z, t), (D.3)

D.2. Conservative ALE Form of Conservation Equations 197

where G denotes the tensor field gradaY.For a part P of B, let us denote by Pa

t the subset of Ω given by

Pat = Z(P , t).

Then, from the Reynolds Transport Theorem we get

Theorem 4.1.1. Let ϕ and Φ be an ALE scalar field and an ALE vector field,respectively. Then

d

dt

∫Pa

t

ϕdVz =∫Pa

t

ϕ′ dVz +∫Pa

t

diva(ϕd) dVz , (D.4)

d

dt

∫Pa

t

Φ dVz =∫Pa

t

Φ′ dVz +∫Pa

t

diva(Φ⊗ d) dVz . (D.5)

Corollary 4.1.2. We have

1. d

dt

∫Pa

t

ϕdVz =∫Pa

t

ϕ′ dVz +∫Pa

t

diva

[ϕG−1(va − wa)

]dVz, (D.6)

2. d

dt

∫Pa

t

Φ dVz =∫Pa

t

Φ′ dVz +∫Pa

t

diva

[Φ⊗ (va − wa)] G−t

dVz . (D.7)

Proof. It is an immediate consequence of (D.3). Indeed, we get d = G−1(va −wa)and then

diva(Φ ⊗ d) = diva

[Φ ⊗ G−1(va − wa)

]= diva

[Φ⊗ (va − wa)] G−t

. (D.8)

D.2 Conservative ALE Form of Conservation Equations

Now we are in a position to obtain the ALE formulations of the conservation equa-tions in conservative form. For this purpose we recall the conservation principlesin integral form and Eulerian coordinates, namely,

• Massd

dt

∫Pt

ρ dVx = 0. (D.9)

• Linear Momentum

d

dt

∫Pt

ρv dVx =∫

∂Pt

Tn dAx +∫Pt

b dVx. (D.10)


• Energy

d

dt

∫Pt

ρE dVx =∫

∂Pt

Tn ·vdAx +∫Pt

b ·v dVx −∫

∂Pt

q ·n dAx +∫Pt

f dVx.

(D.11)

The corresponding ALE formulations are obtained by making, first, the change ofvariable x = Y(z, t) in the integrals. Let J(z, t) = detG(z, t) and Pt = Y(Pa

t , t).We have

• Massd

dt

∫Pa

t

ρaJ dVz = 0. (D.12)

By using (D.11) for ϕ = ρaJ we obtain∫Pa

t

(ρaJ)′ + diva

[ρaJG−1(va − wa)

]dVz = 0, (D.13)

and hence, from the Localization Theorem,

(ρaJ)′ + diva

[ρaJG−1(va − wa)

]= 0. (D.14)

• Linear momentum

d

dt

∫Pa

t

ρaJ dVz =∫

∂Pat

JTaG−tm dAz +∫Pa

t

Jba dVz . (D.15)

We transform this equality by using (D.7) for Φ = ρaJva and Gauss’ Theo-rem. We obtain∫

Pat

[(ρavaJ)′ + diva

ρaJ [va ⊗ (va − wa)]G−t

]dVz (D.16)

=∫Pa

t

diva

[JTaG

−t]dVz +

∫Pa

t

Jba dVz, (D.17)

and then, the Localization Theorem yields

(ρavaJ)′ + diva

J [ρava ⊗ (va − wa) − Ta]G−t

= Jba. (D.18)

• Energy

d

dt

∫Pa

t

ρaEaJ dVz =∫

∂Pat

JTava · G−tm dAz (D.19)

+∫Pa

t

Jba · va dVz −∫

∂Pat

Jqa · G−tm dAz +∫Pa

t

Jfa dVz . (D.20)

D.2. Conservative ALE Form of Conservation Equations 199

By using (D.11) for ϕ = ρaEaJ and Gauss’ Theorem, this equality becomes∫Pa

t

(ρaEaJ)′ + diva

[ρaEaJG−1(va − wa)

]dVz

=∫Pa

t

diva

(JG−1Tava

)dVz +

∫Pa

t

Jba · va dVz

−∫Pa

t

diva

(JG−1qa

)dVz +

∫Pa

t

Jfa dVz, (D.21)

and, from the Localization Theorem,

(ρaEaJ)′ + diva

JG−1 [ρaEa(va − wa) − Tava + qa]

= Jba · va + Jfa. (D.22)

Similar computations allow us to deduce the ALE formulation of the energy equa-tion in terms of the internal energy, e = E − 1

2 |v|2, namely,

(ρaeaJ)′ + diva

JG−1 [ρaea(va − wa) + qa]

= JTaG

−t · gradava + Jfa. (D.23)

We notice that ALE partial differential equations (D.14), (D.18), (D.22) and (D.23)hold in the fixed domain Ω.

D.2.1 Mixed Conservative ALE Form of the ConservationEquations

Now, we are going to deduce another conservative form to be called mixed formbecause it includes fields in both ALE and spatial configuration while only partialdifferential operators in the spatial configuration are involved. For this purpose wewill make extensive use of the following results:

Lemma 4.2.1. Let g be a vector spatial field and S a tensor spatial field. We have

diva

[JG−1ga

]= G−t · gradaga = J (divg)a , (D.24)

diva

[JSaG−t

]= J (div S)a . (D.25)

Proof. We use the following equalities:

diva(Rty) = R · graday + y · diva R, (D.26)

which holds for any smooth ALE tensor field R and vector field y, and

diva(JG−t) = 0 (Piola’s identity). (D.27)


Firstly, from (D.26) and (D.27) we get

diva

[JG−1ga

]= diva

[JG−t

]· ga + JG−t · gradaga = JG−t · gradaga. (D.28)

Moreover, by the chain rule,

gradaga = ( gradg)a G (D.29)

and hence

( gradg)a = gradagaG−1. (D.30)

From this equality we deduce

(div g)a = tr ( gradg)a = tr(gradagaG−1

)= G−t · gradaga. (D.31)

Then, equality (D.24) follows by using this equality in (D.28).In order to prove (D.25) we recall that, from the definition of the divergence

operator of a tensor field, we have,

diva(JSaG−t) · e = diva

[JG−1St

ae]

= diva

[JG−t

]· St

ae

+JG−t · grada(Stae) = J

(div(Ste)

)a

= J (div S)a · e, (D.32)

for all vectors e, where we have used (D.26), (D.27), and (D.24) for y = Ste andR = JG−1.

From this Lemma, and (D.14), (D.18) and (D.22) we easily get

(ρaJ)′ + J (div [ρ(v − w)])a = 0, (D.33)(ρavaJ)′ + J (div [ρv ⊗ (v − w) − T ])a = Jba, (D.34)

(ρaEaJ)′ + J (div [ρE(v − w) − Tv + q])a = Jba · va + Jfa, (D.35)

respectively.

D.3 Mixed Nonconservative Form of ALE ConservationEquations

Finally, we want to obtain mixed nonconservative ALE formulations of the con-servation equations.

• Mass. Firstly we have,

(ρaJ)′ = ρ′aJ + ρaJ ′ = ρ′aJ + ρaJ (div w)a (D.36)

D.3. Mixed Nonconservative Form of ALE Conservation Equations 201

and

J (div [ρ(v − w)])a = J [( gradρ)a · (va − wa) + ρa (div(v − w))a]

= JGt( gradρ)a · G−1(va − wa) + Jρa (div(v − w))a

= J gradaρa · d + Jρa (div(v − w))a . (D.37)

Replacing these expressions in (D.33) we get

J (ρ′a + gradaρa · d) + Jρa (div v)a = 0 (D.38)

and finallyρa + ρa (div v)a = 0, (D.39)

where ρa denotes the material time derivative of ALE field ρa with respectto motion Z.

• Momentum. Firstly, we have

(ρaJva)′ = ρaJv′a + (ρaJ)′va (D.40)

and

J (div [ρv ⊗ (v − w)])a = Jva (div [ρ(v − w)])a

+Jρa ( gradv)a (va − wa) . (D.41)

We add these equalities and then subtract the mass conservation equation(D.33) multiplied by va. We get

(ρaJva)′ + J (div [ρv ⊗ (v − w)])a = Jρav′a + Jρa ( gradv)a (va − wa)

= Jρav′a + Jρa ( gradv)a GG−1 (va − wa) = Jρav′

a + Jρa gradavad

= Jρava.(D.42)

By replacing this equality in (D.34) we finally obtain

ρava − (div T )a = ba. (D.43)

• Energy. Firstly we have

(ρaJEa)′ = ρaJE′a + (ρaJ)′Ea (D.44)

and

J (div [ρE(v − w)])a = JEa (div [ρ(v − w)])a

+ Jρa ( gradE)a · (va − wa) . (D.45)


We add these equalities and then subtract the mass conservation equation(D.33) multiplied by Ea. We get

(ρaJEa)′ + J (div [ρE(v − w)])a = JρaE′a + Jρa ( gradE)a · (va − wa)

= JρaE′a + Jρa ( gradE)a · GG−1 (va − wa) = JρaE′

a + Jρa gradaEa · d= JρaEa.

(D.46)

By replacing this equality in (D.35) we finally obtain

ρaEa − (div [Tv])a + (div q)a = ba · va + fa. (D.47)

Bibliography

[1] A. Bermudez, Obtaining the linear equations for the small perturbations of aflow. Mat. Notae XLI (2001/02), 123–138.

[2] P. G. Ciarlet, Mathematical Elasticity. North Holland, 1988.

[3] B. D. Coleman, W. Noll, Thermodynamics of viscosity, heat conduction andelasticity. Arch. for Rational Mech. and Anal., 13 (1963), 167–178. Also in TheFoundations of Mechanics and Thermodynamics. Springer. Berlin, 1974.

[4] Z. H. Guo, The representation theorem for isotropic, linear asymmetric stress-strain relations. J. Elasticity 13 (1983), 121–124.

[5] M. E. Gurtin, An Introduction to Continuum Mechanics. Academic Press, NewYork, 1981.

[6] K. K. Kuo, Principles of Combustion. John Wiley and Sons, Hoboken, N.J.,2005.

[7] B. Mohammadi, O. Pironneau, Analysis of the K-Epsilon Turbulence Model.John Wiley and Sons, 1994.

[8] W. Noll, Representations of certain isotropic tensor functions, Arch. Math. 21(1970), 87–90.

[9] W. R. Smith, R. W. Missen, Chemical Reaction Equilibrium Analysis: Theoryand Algorithms. John Wiley and Sons, 1982.

Index

β-function, 156

absolute temperature, 11acceleration, 3, 65accumulated thermal expansion, 59acoustic intensity, 91activation energy, 120activation temperature, 120affine space, 170ALE configuration

energy equation, 198linear momentum equation, 198mass equation, 198velocity, 196

Arbitrary Lagrangian-Eulerian (ALE)configuration, 195

Arrhenius law, 119associativity, 161atoms, 122axial vector, 168

balance of linear and angular momen-tum, 5

basis, 162bilinear, 163body, 1body force, 5, 137body heat, 8Boussinesq hypothesis, 106Boussinesq model, 79bulk viscosity, 64

calorically perfect, 95cartesian coordinate frame, 171

cartesian coordinates, 179constitutive law for Newtonian

viscous fluids, 192mass conservation equation, 189motion equation, 190

Cauchy’s hypothesis, 7Cauchy’s theorem, 8Cauchy-Green strain tensors, 2change of specific free energy, 128,

129, 131characteristic temperature of vibra-

tion, 95chemical affinity, 128, 134chemical equilibrium, 125chemical potential, 115Clausius-Duhem inequality, 11closure models, 106coefficient

of isothermal compressibility, 68of linear thermal expansion at

constant stress, 55of volumetric thermal expansion

at constant pressure, 67Coleman-Noll material, 18commutative group, 161commutativity, 161compatible, 132compressible Euler equations, 98compressible Navier-Stokes

equations, 98concentration

of species Ei, 111of the mixture, 112

conductive heat flux, 137coordinates, 162, 165, 169

206 Index

Crocco’s theorem, 66cross product, 167curl, 174curve, 175

closed, 175length, 175

curvilinear integral, 176cylindrical coordinates, 182

constitutive law for Newtonianviscous fluids, 192

mass conservation equation, 189motion equation, 190

Dalton’s law, 111deformation, 1density, 150

in the motion, 4of the mixture, 110reference, 4

deviatoric, 168differentiable, 173differential, 173diffusion velocity, 110, 135displacement, 2dissipation inequality, 19dissipation rate, 24dissipative acoustics, 81distance, 171distribution function, 154divergence, 174divergence theorem, 177double delta function, 156dynamic viscosity, 64, 103

eddy dynamic viscosity, 106, 107elastic fluid, 96elasticity tensor, 50element mass fractions, 150endomorphisms, 1energy, 126energy equation

conservative, 137linearized for thermoelasticity, 54non-conservative, 138

enthalpy, 64, 138, 142, 150of formation, 139of the mixture, 110standard, 124

entropy, 11of the mixture, 110

equilibrium constant, 129–131based on concentrations, 129based on partial pressures, 128

Euclidean affine space, 171Euclidean vector space, 163Eulerian field, 3Eulerian fluctuation, 90Eulerian fluctuation of pressure, 79expectation, 154, 155extended symmetry group, 37extent, 127, 130

Favre averaged, 155Fick’s law, 136, 137field, 173filter, 105first principle of thermodynamics, 8flow

adiabatic, 149fluid, 61

energy conservation law, 64energy equation, 70motion equation, 64

Fourier’s law, 42free energy, 12, 129frequency factor, 120full stoichiometric matrix, 133

generalized Hooke’s law, 52Gibbs free energy, 77, 114, 127, 130,

141, 142of formation, 97, 115of the mixture, 111, 114standard, 97

gradient, 173of the motion, 2

gravity force, 87Green’s formulas, 177

Index 207

Green-Lagrange strain tensor, 2Green-Saint Venant strain tensor, 2,

41

heat flux, 8heat rate, 8Helmholtz free energy, 12

of the mixture, 111hyperelastic material

with heat conduction and viscos-ity, 18

ideal fluid, 103ideal gases, 96incompressible Euler equations, 104incompressible Navier-Stokes

equations, 106incompressible Newtonian fluid, 101incremental methods, 57infinitesimal strain tensor, 2, 48inlets, 146inner product, 163, 166internal energy, 9

of formation, 123of the mixture, 110, 121standard, 123

invertible, 168isobaric process, 149isotropic, 164

k-atoms, 122, 140, 146k-moles, 112kernel, 132

Lagrangian coordinates, 13Lagrangian fields, 3Lame’s coefficients, 55Laplacian, 174least action principle, 125Lewis number, 138linearly independent, 162local equilibrium chemistry, 149local equilibrium problem, 141localization theorem, 179

Mach number, 142, 149mass, 126, 141, 142, 150

of the mixture, 120mass action law, 119mass conservation equation, 135

conservative, 136mass diffusion, 136mass diffusion coefficient, 138mass distribution, 4mass fraction, 110, 140, 146material body, 18

isotropic, 39material fields, 3material frame-indifference principle,

28material points, 1material time derivative, 6Mayer equation, 94mean value, 154mechanical energy, 91mechanical equilibrium, 77mixture fractions, 148, 150, 155molar fraction, 112molecular mass, 93

of the mixture, 112motion, 1motion equation

conservative, 5, 137linear approximation of the , 52non-conservative, 5, 137

natural convection, 78neutral element, 161non-dissipative acoustics, 85norm, 163

observer change, 27origin, 171orthogonal, 163, 168orthonormal, 163outlet, 146

passive scalars, 141Pekeris equation, 87

208 Index

perfect gas, 93coefficient of isothermal

compressibility, 94coefficient of thermal expansion,

94enthalpy, 93entropy, 94internal energy, 93sound speed, 94specific heat at constant

pressure, 93positive definite, 56, 163positivity, 126, 142, 143, 150potential theorem, 176power stress, 6pre-exponential exponent, 120pressure, 102

Lagrangian fluctuation, 86of the mixture, 111standard , 129

principal invariants, 169probability density functions (PDF),

153, 154probability space, 153

random variable, 153continuous, 154independent, 154

random vector, 154(jointly) continuous, 154distribution function, 154

rank, 132rate of total mass per unit volume,

146ratio of specific heats, 94real vector space, 161reference configuration, 1reference map, 3region

closed, 176open, 176regular, 176

response mappings, 18rotation, 168

Saint Venant-Kirchhoff material, 41Schmidt numbers, 156second principle of thermodynamics,

11second viscosity coefficient, 64simply connected, 176skew, 166skew part, 167Smagorinsky’s model, 106small deformations, 57sound speed, 72source of mass, 135sources, 146spatial fields, 3, 135species, 109

density, 109enthalpy, 109entropy, 109Gibbs free energy, 109Helmholtz free energy, 109internal energy, 109molecular mass, 109pressure, 109specific heat at constant

pressure, 110specific heat at constant volume,

109velocity, 109

specific heatat constant deformation, 34at constant pressure, 69

of the mixture, 124at constant volume, 67, 121

of the mixture, 111, 121spherical, 168spherical coordinates, 184

constitutive law for Newtonianviscous fluids, 193

mass conservation equation, 189motion equation, 191

stagnation enthalpy, 65, 98state law, 125, 126, 142, 150stirred tank, 119, 125, 140stoichiometric coefficients, 119

Index 209

stoichiometric method, 132Stokes’ theorem, 177streams, 148stress tensor

Boussinesq, 13Cauchy, 5First Piola-Kirchhoff, 13Piola-Lagrange, 13Reynolds, 106Second Piola-Kirchhoff, 13

surface force, 5surface heat, 8symmetric, 163, 166symmetric element, 161symmetric part, 167symmetry transformation, 37system of forces, 5system of generators, 162system of heat, 8, 120

tensor, 164positive definite, 168

tensor of thermal expansion at cons-tant stress, 51

tensor product, 165, 169the response of a material body is in-

dependent of the observer,27

theorem of power expended, 6thermal conductivity, 42, 138thermal enthalpy, 139thermodynamic pressure, 63thermodynamic process, 17

adiabatic, 25Eulerian, 25homentropic, 66isentropic, 24isochoric, 101steady, 66

thermomechanical equilibrium, 78total energy, 8, 98trace, 166trajectory, 2transpose, 166

turbulence models, 105turbulent dissipation rate, 107turbulent kinetic energy, 107turbulent Prandtl, 156

Vaisala-Bruntfrequency, 89tensor, 88

variance, 154, 155vectors, 162velocity, 3, 119

of the mixture, 110, 135

wall, 146wave equation, 87

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