Notes de cours de l'ISIMA, troisième année http://www.isima.fr/leborgne
Objectivity in classical mechanics (continuum mechanics)
Motions, Eulerian and Lagrangian functions. Deformation gradient. Lie derivatives.
Velocity-addition formula, Coriolis. Objectivity.
Gilles Leborgne January 31, 2022
In classical mechanics, there are two objectivities: 1- The covariant objectivity which concerns the general laws of physics and requires that these laws be observer independent: It deals with qualitative aspects in Mechanics. This is the main subject of this manuscript. 2- The isometric objectivity which concerns the constitutive laws of materials (frame invariance principle).
To describe the covariant objectivity, we need motions, the associated Eulerian functions, and the velocity addition formula. We also introduce the Lie derivative for vectors which might meet some needs of engineers, and which is covariant objective. (Cauchy would certainly have used it if it had existed during his lifetime; In fact, to get a stress, Cauchy had to compare two vectors, whereas one vector suces when using the derivative of Lie.)
Thus we follow Maxwell's needs, [13]: [Preliminary (on the measurement of quantities)] (...) 2. (...) The formula at which we arrive must be such that a person of any nation, by substituting for the dierent symbols the numerical value of the quantities as measured by his own national units, would arrive at a true result. (...) 10. (...) The introduction of coordinate axes into geometry by Des Cartes was one of the greatest steps in mathematical progress, for it reduced the methods of geometry to calculations performed on numerical quantities. The position of a point is made to depend on the length of three lines which are always drawn in determinate directions (...) But for many purposes in physical reasoning, as distinguished from calculation, it is desirable to avoid explicitly introducing the Cartesian coordinates, and to x the mind at once on a point of space instead of its three coordinates, and on the magnitude and direction of a force instead of its three components. This mode of contemplating geometrical and physical quantities is more primitive and more natural than the other,...
Or see the (short) historical note given in the introduction of Abraham and Marsden book Foun- dations of Mechanics [1], about qualitative versus quantitative theory: Mechanics begins with a long tradition of qualitative investigation culminating with Kepler and Galileo. Following this is the period of quantitative theory (1687-1889) characterized by concomitant developments in mechanics, mathemat- ics, and the philosophy of science that are epitomized by the works of Newton, Euler, Lagrange, Laplace, Hamilton, and Jacobi. (...) For celestial mechanics (...) resolution we owe to the genius of Poincaré, who resurrected the qualitative point of view (...) One advantage of this model is that by suppressing unnecessary coordinates the full generality of the theory becomes evident.
In this manuscript, we examine simple applications of qualitative methods to continuum mechanics;
No dierential geometry knowledge is required, except for the tangent space at a point of our ane space, and the tangent bundle, which shed light on the subject. We start with denitions and characterizations (qualitative approach), before quantifying with bases and/or inner dot products.
A fairly long appendix (half of the manuscript) gives standard denitions (qualitative), propositions and proofs, and notations used for calculations (quantication). Discussions with colleagues are a great help.
2 CONTENTS
Contents
I Motions, Eulerian and Lagrangian descriptions, ows 11
1 Motions 11
1.1 Referential . . . 11
1.2 Einstein's convention (duality notation) . . . 11
1.3 Motion of an object . . . 12
1.4 Congurations and spatial (Eulerian) variables . . . 13
1.5 Eulerian and Lagrangian variables . . . 13
1.6 Trajectories . . . 13
1.7 Virtual and real motion . . . 13
1.8 Tangent space,~Rnt, ber, bundle . . . 13
2 Eulerian description (spatial description at actual timet) 14 2.1 Eulerian function . . . 14
2.2 Eulerian velocity (spatial velocity) and speed . . . 15
2.3 Spatial derivative of the Eulerian velocity . . . 15
2.4 The convective objective termdf.~v, written(~v. ~grad)f in a basis... . . 16
2.5 ... and the subjective gradf~ (depends on a Euclidean dot product) . . . 16
2.6 Streamline (current line) . . . 17
2.7 Material time derivative (dérivées particulaires) . . . 18
2.7.1 Usual denition . . . 18
2.7.2 Bis: Space-time denition . . . 19
2.7.3 The material time derivative is a derivation . . . 19
2.7.4 Commutativity issue . . . 20
2.8 Eulerian acceleration . . . 20
2.9 Taylor expansion ofΦe . . . 21
3 Motion on an initial conguration 21 3.1 Denition . . . 21
3.2 Dieomorphism between congurations . . . 22
3.3 Trajectories . . . 22
3.4 Streaklines (lignes d'émission) . . . 23
4 Lagrangian description 23 4.1 Lagrangian function . . . 23
4.1.1 Denition . . . 23
4.1.2 A Lagrangian function is a two point tensor . . . 24
4.2 Lagrangian function associated with a Eulerian function . . . 24
4.2.1 Associated Lagrangian function . . . 24
4.2.2 Remarks . . . 24
4.3 Lagrangian velocity . . . 24
4.3.1 Denition . . . 24
4.3.2 Lagrangian velocity versus Eulerian velocity . . . 25
4.3.3 Relation between dierentials . . . 25
4.3.4 Computation ofL=d~vfrom Lagrangian variables . . . 25
4.4 Lagrangian acceleration . . . 26
4.5 Time Taylor expansion ofΦt0 . . . 27
4.6 A vector eld which let itself be deformed by a ow . . . 27
5 Deformation gradient F 27 5.1 Denitions . . . 27
5.1.1 F:=dΦ . . . 27
5.1.2 Its values: Push-forward . . . 28
5.1.3 A full denition ofF: A two point tensors . . . 29
5.2 The unfortunate notationd~x=F.d ~X . . . 29
3 CONTENTS
5.2.3 Vector approach... . . 30
5.2.4 ... and dierential approach . . . 30
5.2.5 The ambiguous notationd~•x = • F .d ~X . . . 31
5.3 Quantication with bases . . . 31
5.4 Remark: Tensorial notations . . . 32
5.5 Change of coordinate system attforF . . . 33
5.6 Spatial Taylor expansion of Φtt0 andFtt0 . . . 33
5.7 Time Taylor expansion ofFpt0t 0 . . . 33
6 Flow 34 6.1 Introduction: Motion versus ow . . . 34
6.2 Denition . . . 34
6.3 CauchyLipschitz theorem . . . 35
6.4 Examples . . . 36
6.5 Composition of ows . . . 36
6.5.1 Law of composition of ows . . . 37
6.5.2 Stationnary case . . . 37
6.6 Velocity on the trajectory traveled in the opposite direction . . . 38
6.7 Variation of the ow as a function of the initial time . . . 38
6.7.1 Ambiguous and non ambiguous notations . . . 38
6.7.2 Variation of the ow as a function of the initial time . . . 39
7 Decomposition of d~v 39 7.1 Rate of deformation tensor and spin tensor . . . 39
7.2 Quantication with a basis . . . 40
8 Interpretation of the rate of deformation tensor 40 9 Rigid body motions and the spin tensor 41 9.1 Ane motions and rigid body motions . . . 41
9.1.1 Ane motions . . . 41
9.1.2 Rigid body motion . . . 42
9.1.3 Rigid body motion: d~v+d~vT = 0. . . 43
9.2 Representation of the spin tensorΩ: vector, and pseudo vector . . . 43
9.2.1 Reminder . . . 43
9.2.2 Denition of the vector product (cross product) . . . 44
9.2.3 Antisymmetric endomorphism represented by a vector . . . 45
9.2.4 Curl . . . 46
9.2.5 Pseudo-cross product and pseudo-vector (column matrix) . . . 47
9.2.6 Antisymmetric matrix represented by a pseudo-vector . . . 47
9.2.7 Antisymmetric endomorphism and its pseudo-vectors representations . . . 47
9.3 Examples . . . 48
9.3.1 Rectilinear motion . . . 48
9.3.2 Circular motion . . . 48
9.3.3 Motion of a planet (centripetal acceleration) . . . 49
II Push-forward 53
10 Push-forward 53 10.1 Denition . . . 5310.2 Push-forward and pull-back of points . . . 53
10.3 Push-forward and pull-back of scalar functions . . . 54
10.3.1 Denitions . . . 54
10.3.2 Interpretation: Why is it useful? . . . 55
10.4 Push-forward and pull-back of curves . . . 55
10.5 Push-forward and pull-back of vector elds . . . 56
10.5.1 Approximate description: Transport of a small bipoint vector . . . 56
10.5.2 Denition of the push-forward of a vector eld . . . 57
10.5.3 Interpretation: Essential to continuum mechanics . . . 57
4 CONTENTS
10.5.4 Pull-back of a vector eld . . . 58
10.6 Quantication with bases . . . 58
11 Homogeneous and isotropic material 60 12 The inverse of the deformation gradient 61 12.1 Denition ofH =F−1 . . . 61
12.2 Time derivatives ofH . . . 61
13 Push-forward and pull-back of dierential forms 62 13.1 Denition . . . 62
13.2 Incompatibility: Riesz representation and push-forward . . . 63
14 Push-forward and pull-back of tensors 64 14.1 Push-forward and pull-back of order 1 tensors . . . 64
14.2 Push-forward and pull-back of order 2 tensors . . . 65
14.3 Push-forward and pull-back of endomorphisms . . . 66
14.4 Application to derivatives of vector elds . . . 66
14.5 Application to derivative of dierential forms . . . 66
14.6 Ψ∗(d ~w)versusd(Ψ∗w)~ : No commutativity . . . 67
14.7 Ψ∗(dα)versusd(Ψ∗α): No commutativity . . . 67
III Lie derivative 68
15 Lie derivative 68 15.1 Introduction . . . 6815.2 Ubiquity gift not required . . . 68
15.3 Denition rewritten . . . 69
15.4 Lie derivative of a scalar function . . . 70
15.5 Lie derivative of a vector eld . . . 71
15.6 Examples and interpretations . . . 72
15.6.1 Flow resistance measurement . . . 72
15.6.2 Lie Derivative of a vector eld along itself . . . 72
15.6.3 Lie derivative along a uniform ow . . . 72
15.6.4 Lie derivative of a uniform vector eld . . . 72
15.6.5 Uniaxial stretch of an elastic material . . . 73
15.6.6 Simple shear of an elastic material . . . 73
15.6.7 Shear ow . . . 74
15.6.8 Spin . . . 75
15.6.9 Second order Lie derivative . . . 75
15.7 Lie derivative of a dierential form . . . 75
15.8 Incompatibility with the representation vector . . . 77
15.9 Lie derivative of a tensor . . . 77
15.9.1 Formula . . . 77
15.9.2 Lie derivative of a mixed tensor . . . 77
15.9.3 For a non mixed tensor . . . 78
15.9.4 Lie derivative of a up-tensor . . . 78
15.9.5 Lie derivative of a down-tensor . . . 78
IV Velocity-addition formula and Objectivity 79
16 Change of referential 79 16.1 Introduction and problem . . . 7916.2 Framework . . . 79
16.3 The translatorΘtfor positions . . . 80
16.3.1 Dention . . . 80
5 CONTENTS
16.5 The Θ-velocity = the drive velocity . . . 82
16.6 The velocity-addition formula . . . 83
16.7 The Acceleration-addition formula . . . 83
16.8 Inter-referential change of basis formula . . . 84
16.9 A summary: Commutative diagrams . . . 85
16.9.1 Motions and translator, Eulerian . . . 85
16.9.2 Motions and translator, Lagrangian . . . 85
16.9.3 Dierentials . . . 86
17 Coriolis force 86 17.1 Fundamental principal: In a Galilean referential . . . 86
17.2 Inertial and Coriolis forces, and Fundamental Principle . . . 86
18 Objectivities 87 18.1 Covariant objectivity of a scalar function . . . 87
18.2 Covariant objectivity of a vector eld . . . 88
18.3 Isometric objectivity and Frame Invariance Principle . . . 88
18.4 Objectivity of dierential forms . . . 89
18.5 Objectivity of tensors . . . 89
18.6 Non objectivity of the velocities . . . 90
18.6.1 Eulerian velocities . . . 90
18.6.2 Lagrangian velocities . . . 90
18.6.3 d~v is not objective . . . 90
18.6.4 d~v is not isometric objective . . . 90
18.6.5 d~v+d~vT is isometric objective . . . 91
18.7 The Lie derivative are covariant objective . . . 91
18.7.1 Scalar functions . . . 92
18.7.2 Vector elds . . . 92
18.7.3 Tensors . . . 93
18.8 Taylor expansions and ubiquity gift . . . 93
18.8.1 InRn with ubiquity . . . 93
18.8.2 General case . . . 94
V Appendix 95
A Classical and duality notations 95 A.1 Contravariant vector and basis . . . 95A.1.1 Contravariant vector . . . 95
A.1.2 Basis . . . 95
A.1.3 Canonical basis . . . 95
A.1.4 Cartesian basis . . . 95
A.2 Representation of a vector relative to a basis . . . 95
A.3 Bilinear forms . . . 96
A.3.1 Denition . . . 96
A.3.2 Inner dot product, and metric . . . 96
A.3.3 Quantication: Matrices[gij] . . . 97
A.4 Linear maps . . . 98
A.4.1 Denition . . . 98
A.4.2 Quantication: Matrices[Lij] = [Lij]. . . 98
A.5 Transposed matrix . . . 99
A.6 The transposed endomorphisms of an endomorphism . . . 99
A.6.1 Denition (requires an inner dot product) . . . 99
A.6.2 Quantication with bases . . . 99
A.6.3 Symmetric endomorphism . . . 100
A.7 The transposed of a linear map . . . 101
A.7.1 Denition (needs two inner dot products) . . . 101
A.7.2 Quantication with bases . . . 101
A.7.3 Deformation gradient symmetric: Absurd . . . 101
A.7.4 Isometry . . . 102
6 CONTENTS
A.8 Dual basis . . . 102
A.8.1 Linear forms: Covariant vectors . . . 102
A.8.2 Covariant dual basis (the functions which give the components of a vector) . . . . 103
A.8.3 Example: aeronautical units . . . 103
A.8.4 Matrix representation of a linear map . . . 104
A.8.5 Example: Thermodynamic . . . 104
A.9 Tensorial product and tensorial notations . . . 105
A.9.1 Denition . . . 105
A.9.2 Application to bilinear forms . . . 105
A.9.3 Bidual basis (and contravariance) . . . 105
A.9.4 Tensorial representation of a linear map . . . 106
A.10 Einstein convention . . . 107
A.10.1 Denition . . . 107
A.10.2 Do not mistake yourself . . . 107
A.11 Change of basis formulas . . . 107
A.11.1 Change of basis endomorphism and transition matrix . . . 107
A.11.2 Inverse of the transition matrix . . . 108
A.11.3 Change of dual basis . . . 109
A.11.4 Change of coordinate system for vectors and linear forms . . . 109
A.11.5 Notations for transitions matrices for linear maps and bilinear forms . . . 109
A.11.6 Change of coordinate system for bilinear forms . . . 110
A.11.7 Change of coordinate system for linear maps . . . 110
A.12 The vectorial dual bases of one basis . . . 111
A.12.1 An inner dot product and the associated vectorial dual basis . . . 111
A.12.2 An interpretation: (·,·)g-Riesz representatives . . . 111
A.12.3~eig is a contravariant vector . . . 111
A.12.4 Components of~ejg relative to (~ei) . . . 112
A.12.5 Notation problem . . . 113
A.12.6 (Huge) dierences between the (covariant) dual basis and a dual vectorial basis . 113 A.12.7 About the notationgij . . . 113
A.13 The adjoint of a linear map . . . 114
B Euclidean Frameworks 115 B.1 Euclidean basis . . . 115
B.2 Euclidean dot product . . . 115
B.3 Change of Euclidean basis . . . 116
B.3.1 Two Euclidean dot products are proportional . . . 116
B.3.2 Counterexample : non existence of a Euclidean dot product . . . 117
B.4 Euclidean transposed of the deformation gradient . . . 117
B.5 The Euclidean transposed for endomorphisms . . . 117
C Riesz representation theorem 118 C.1 The Riesz representation theorem . . . 118
C.1.1 Framework . . . 118
C.1.2 Riesz representation theorem . . . 118
C.1.3 Riesz representation mapping . . . 119
C.1.4 Change of Riesz representation vector: Euclidean case . . . 119
C.1.5 Quantication with a basis . . . 120
C.1.6 A Riesz representation vector is contravariant . . . 120
C.1.7 Change of Riesz representation vector, general case . . . 121
C.2 Question: What is a vector versus a(·,·)g-vector? . . . 121
C.3 Problems due to a Euclidean framework . . . 121
D Determinants 122 D.1 Alternating multilinear form . . . 122
D.2 Leibniz formula . . . 122
D.3 Determinant of vectors . . . 123
D.4 Determinant of a matrix . . . 124
7 CONTENTS
D.6 Determinant of an endomorphism . . . 125
D.6.1 Denition and basic properties . . . 125
D.6.2 The determinant of an endomorphism is objective . . . 126
D.7 Determinant of a linear map . . . 126
D.7.1 Denition and rst properties . . . 126
D.7.2 Jacobian of a motion, and dilatation . . . 127
D.7.3 Determinant of the transposed . . . 127
D.8 Dilatation rate . . . 127
D.8.1 ∂J∂tt0(t, P) =Jt0(t, P) div~v(t, pt). . . 128
D.8.2 Leibniz formula . . . 128
D.9 ∂J/∂F=J F−T . . . 129
D.9.1 Meaning of ∂L∂ij for linear maps? . . . 129
D.9.2 Meaning of∂J/∂F? . . . 129
E CauchyGreen deformation tensor C 130 E.1 Introduction and remarks . . . 130
E.2 Transposed FT . . . 131
E.2.1 Framework . . . 131
E.2.2 Denition ofFT: Inner dot products required . . . 131
E.2.3 Quantication with bases (matrix representation) . . . 131
E.2.4 Remark: F∗ . . . 132
E.3 CauchyGreen deformation tensor . . . 132
E.3.1 Denition . . . 132
E.3.2 Quantication with bases . . . 133
E.4 Applications . . . 134
E.4.1 Stretch . . . 134
E.4.2 Change of angle . . . 134
E.4.3 Spherical and deviatoric tensors . . . 134
E.4.4 Rigid motion . . . 134
E.4.5 Diagonalization ofC . . . 134
E.4.6 Mohr circle . . . 135
E.5 C[ and pull-backg∗ . . . 136
E.5.1 The at[notation (endomorphismL and its(·,·)g-associated 0 2 tensorL[g) . . . 136
E.5.2 Two inner dot products andC[ . . . 136
E.5.3 The pulled-back metricg∗ . . . 137
E.5.4 CGg[ =g∗ . . . 137
E.6 Time Taylor expansion forC . . . 137
E.6.1 First and second order . . . 137
E.6.2 Associated results and interpretation problems . . . 138
F Prospect: Elasticity and objectivity? 138 F.1 Introduction: Remarks . . . 138
F.2 Polar decompositions of F . . . 138
F.2.1 F=R.U (right polar decomposition) . . . 139
F.2.2 F=V.R(left polar decomposition) . . . 140
F.3 Elasticity: A Classical tensorial approach . . . 141
F.3.1 Classical approach and issue . . . 141
F.3.2 A functional (tensorial) formulation? . . . 141
F.3.3 Second functional formulation: With the the Finger tensor . . . 143
F.4 Elasticity: An objective approach? . . . 144
G Finger tensor (left CauchyGreen tensor) 145 G.1 Denition . . . 145
G.2 b−1 . . . 146
G.3 Time derivatives ofb−1 . . . 146
H GreenLagrange deformation tensor 147 H.1 Denition . . . 147
H.2 Time Taylor expansion ofE . . . 147
8 CONTENTS
I EulerAlmansi tensor 148
I.1 Denition . . . 148
I.2 Time Taylor expansion fora. . . 148
J Innitesimal strain tensor ε 148 J.1 Small displacement hypothesis . . . 148
J.2 Denition ofε. . . 149
J.3 A second mathematical denition (EulerAlmansi) . . . 149
K Displacement 150 K.1 The displacement vector U~ . . . 150
K.2 The dierential of the displacement vector . . . 150
K.3 Deformation tensor (matrix) ε, bis . . . 151
K.4 Small displacement hypothesis, bis . . . 151
K.5 Displacement vector with dierential geometry . . . 151
K.5.1 The shifter . . . 151
K.5.2 The displacement vector . . . 152
L Transport of volumes and areas 152 L.1 Transformed parallelepiped . . . 153
L.2 Transformed volumes . . . 153
L.3 Transformed parallelogram . . . 153
L.4 Transformed surface . . . 154
L.4.1 Deformation of a surface . . . 154
L.4.2 Euclidean dot product and unit normal vectors . . . 154
L.4.3 Relations between surfaces . . . 155
L.5 Piola identity . . . 155
L.6 Piola transformation . . . 156
M Work and power 156 M.1 Introduction . . . 156
M.1.1 Work for a 1-D material . . . 156
M.1.2 Power density for a 1-D material . . . 157
M.2 Denitions for an-D material . . . 157
M.2.1 Power to work . . . 157
M.2.2 Work to power . . . 158
M.2.3 Objective internal power . . . 158
M.2.4 Power and initial conguration . . . 159
M.3 PiolaKirchho tensors . . . 159
M.3.1 The rst PiolaKirchho tensor . . . 159
M.3.2 The second PiolaKirchho tensor . . . 160
M.4 Classical hyper-elasticity: ∂W/∂F . . . 161
M.4.1 Framework: A scalar function acting on linear maps . . . 161
M.4.2 Expression with bases: The∂W/∂Lij . . . 161
M.4.3 Motions andω-lemma . . . 162
M.4.4 Application to classical hyper-elasticity: PK=∂W/∂F . . . 162
M.4.5 Corollary (hyper-elasticity): SK=∂W/∂C. . . 163
M.5 Hyper-elasticity and Lie derivative . . . 164
N Conservation of mass 166 O Balance of momentum 167 O.1 Framework . . . 167
O.2 Master balance law . . . 167
O.3 Cauchy theoremT~ =σ.~n(stress tensorσ) . . . 168
O.4 Toward an objective formulation . . . 169
P Balance of moment of momentum 169
9 CONTENTS
Q Uniform tensors in Lrs(E) 170
Q.1 Tensorial product and multilinear form . . . 170
Q.1.1 Tensorial product of functions . . . 170
Q.1.2 Tensorial product of linear forms: multilinear forms . . . 170
Q.2 Uniform tensors inL0s(E) . . . 170
Q.2.1 Denition of type0suniform tensors . . . 170
Q.2.2 Example: Type 01 uniform tensor . . . 171
Q.2.3 Example: Type 02 uniform tensor . . . 171
Q.2.4 Example: Determinant . . . 171
Q.3 Uniform tensors inLrs(E) . . . 171
Q.3.1 Denition of typer suniform tensors . . . 171
Q.3.2 Example: Type 10 uniform tensor: Identied with a vector . . . 172
Q.3.3 Example: Type 11 uniform tensor . . . 172
Q.3.4 Example: Type 12 uniform tensor . . . 172
Q.4 Exterior tensorial products . . . 173
Q.5 Contractions . . . 173
Q.5.1 Objective contraction of a linear form with a vector . . . 173
Q.5.2 Objective contraction of an endomorphism and a vector . . . 173
Q.5.3 Objective contractions of uniform tensors . . . 174
Q.5.4 Objective double contractions of uniform tensors . . . 175
Q.5.5 Non objective double contraction: Double matrix contraction . . . 176
Q.6 Endomorphism and tensorial notation . . . 176
Q.6.1 Endomorphism identied to a 1 1 uniform tensor . . . 176
Q.6.2 Simple and double objective contractions of endomorphisms . . . 177
Q.6.3 Double matrix contraction (not objective) . . . 177
Q.7 Kronecker contraction tensor, trace . . . 177
R Tensors inTsr(U) 178 R.1 Introduction, module, derivation . . . 178
R.2 Functions and vector elds . . . 179
R.2.1 Framework . . . 179
R.2.2 Field of functions . . . 179
R.2.3 Vector elds . . . 179
R.3 Dierential forms, covariance and contravariance . . . 180
R.3.1 Dierential forms . . . 180
R.3.2 Covariance and contravariance . . . 180
R.4 Denition of tensors . . . 180
R.5 Example: Type 01tensor = dierential forms . . . 181
R.6 Example: Type 10tensor = identied to a vector eld . . . 181
R.7 Example: A metric is a type 02tensor . . . 181
R.8 Example: Type 11tensor... . . 181
R.9 ... and identication with elds of endomorphisms . . . 182
R.10 Example: Type 20tensor... . . 182
R.11 Unstationary tensor . . . 182
S A dierential, its eventual gradients, divergence 182 S.1 Denitions . . . 182
S.2 Quantication and thej-th partial derivative . . . 183
S.3 Example: Quantication for the dierential of a scalar valued function . . . 183
S.4 Possible gradient associated with a dierential . . . 185
S.5 Example: Quantication for the dierential of a vector valued function . . . 186
S.6 Trace of an endomorphism . . . 186
S.6.1 Denition . . . 186
S.6.2 Alternative denition: With one-one tensors . . . 186
S.7 Divergence of a vector eld: invariant . . . 187
S.8 Unit normal vector, unit normal form, integration . . . 188
S.8.1 Framework . . . 188
S.8.2 Unit normal vector . . . 188
S.8.3 Unit normal form . . . 188
10 CONTENTS
S.8.4 Integration by parts . . . 189
S.9 Objective divergence for 1 1 tensors or endomorphisms . . . 190
S.9.1 Dierential of a 1 1 tensor or of an endomorphism . . . 190
S.9.2 Denition: Objective divergence . . . 190
S.9.3 Objective divergences of a 2 0 tensor . . . 192
S.9.4 Non existence of an objective divergence of a 0 2 tensor . . . 192
S.10 Euclidean framework and classic divergence of a tensor (subjective) . . . 192
S.10.1 Classic divergence of a 1 1 tensor or of an endomorphism . . . 192
S.10.2 Classic divergence for 2 0 and 0 2 tensors . . . 193
T Natural canonical isomorphisms 193 T.1 The adjoint of a linear map . . . 193
T.2 An isomorphism E'E∗ is never natural . . . 194
T.2.1 Denition . . . 194
T.2.2 Question . . . 194
T.2.3 The Theorem . . . 194
T.2.4 Illustrations (two fundamental examples) . . . 194
T.3 Natural canonical isomorphismE'E∗∗ . . . 195
T.3.1 Framework and denition . . . 195
T.3.2 The Theorem . . . 195
T.4 Natural canonical isomorphisms L(E;F)' L(F∗, E;R)' L(E∗;F∗) . . . 196
U Distribution in brief: A covariant concept 196 U.1 Denitions . . . 197
U.2 Derivation of a distribution . . . 198
U.3 Hilbert spaceH1(Ω) . . . 198
U.3.1 Motivation . . . 198
U.3.2 Denition ofH1(Ω) . . . 198
U.3.3 SubspaceH01(Ω)and its dual spaceH−1(Ω) . . . 199
11
Part I
Motions, Eulerian and Lagrangian descriptions, ows
A quantityf being given, the notationg :=f means: g is dened by g =f. To dene Eulerian and Lagrangian functions, we rst need to dene a motion of an object. The framework is classical mechanics, time being decoupled from space.
1 Motions
1.1 Referential
LetR3be the classical geometric ane space (space of points), and let (~
R3,+, .) =noted ~
R3 be the usual associated vector space of bipoint vectors. And we also considerR and R2 as subspaces of R3. So we considerRn,n= 1,2,3, and the associated vector space~Rn.
Origin: An observer chooses an originO ∈ Rn. Thus a pointp∈Rn can be located by the observer thanks to the bipoint vector−→
Op=~x∈R~n, so thatp=O+~x.
Another observer chooses an origin O0 ∈Rn. Thus a pointp∈Rn can be located by this observer thanks to the bipoint vector−−→
O0p=~x0∈~
Rn, so thatp=O0+~x0. And we have~x0=−−→
OO0+~x.
Cartesian coordinate system: A Cartesian coordinate system in the ane spaceRn is a setRc = (O,(~ei)i=1,...,n) chosen by an observer, whereO is a point called the origin, and(~ei) := (~ei)i=1,...,n is a basis in~Rn. Then, quantication of the location of a pointp∈Rnby the observer who denedRc: there existsx1, ..., xn∈Rs.t.
p=O+~x=O+
n
X
i=1
xi~ei, and [−→
Op]|~e= [~x]|~e=
x1
...
xn
(1.1)
is the column matrix containing the components of−→
Op=~xin the basis(~ei).
Quantication by another observer with his Cartesian referential R0c = (O0,(~ei0)i=1,...,n): p=O0+~x0 =O0+
n
X
i=1
xi0~ei0. (1.2)
Chronology: A chronology (or temporal coordinate system) is a set Rt = (t0,(∆t)) chosen by an observer, wheret0∈Ris a point called the time origin, and(∆t)is called the time unit (a basis inR).~ Referentiel: A referentialRis the set
R= (Rt,Rc) = (t0,(∆t),O,(~ei)i=1,...,n)= (chronologie,Cartesian coordinate system) (1.3) chosen by an observer, made of a chronology and a Cartesian coordinate system.
In the following (framework of classical mechanics), to simplify the writings, the same implicit chronol- ogy is used by all observers, and a referentialR= (Rt,Rc)will simply be notedR=Rc = (O,(~ei)).
1.2 Einstein's convention (duality notation)
We will also use Einstein's convention (duality notation), see A.10: The components xi of ~xin (1.1) are also namedxi=xi with Einstein's convention:
~x=
n
X
i=1
xi~ei
| {z }
=
n
X
i=1
xi~ei
| {z }
, so [~x]|~e=
x1
...
xn
=
x1
...
xn
. (1.4)
12 1.3. Motion of an object
Moreover Einstein's convention uses the notation Pn
i=1xi~ei=notedxi~ei, i.e. the sum signPn
i=1 can be omitted when an index is used twice, once up and once down. However this omission will not be made in this manuscript: The LaTeX program makes it easy to printPn
i=1.
Example 1.1 The height of a child is represented on a wall by a vertical bipoint vector~xstarting from the ground up to a pencil line. The vector~xis objective = qualitative: It is the same for any observer.
Question: What is the size of the child ? (Quantitative = subjective.)
Answer: It depends... on the observer. E.g., an English observer chooses a basis vector ~a1 which length is one English foot (ft). So he writes ~x =x1~a1, and for him the size of the child (size of ~x) is x1 in foot. A French observer chooses a basis vector~b1 which length is one meter (m). So he writes
~
x=y1~b1, and for him the size of the child (size of~x) is y1 meter. E.g., ifx1 = 4then y1 '1.22, since 1 ft = 0.3048m: The child (the vector~x) is both 4 ft and 1.22 m tall.
With Einstein duality notation: ~x=x1~a1=y1~b1, and ifx1= 4theny1'1.22.
This manuscript deals with covariant objectivity, thus an English engineer (and his foot) and a French engineer (and his meter) will be able to work together. And they will be able to use the results of Galileo, Descartes, Newton, Euler... who used their own unit of length (and knew nothing about scalar products invented in the 19th century).
1.3 Motion of an object
LetObj be a real object, or material object, made of particles (e.g., the Moon: Exists independently of an observer).
Denition 1.2 The motion ofObj in Rn is the map
Φ :e
[t1, t2]×Obj →Rn (t, PObj)
| {z }
particle
→ p
position at|{z} t
=Φ(t, Pe Obj) =position ofPObj att inRn, (1.5)
which describes the motion of the particlesPObj ∈Obj in the ane spaceRn. Andtis the time variable, pis the space variable, and(t, p)∈R×Rn is the time-space variable.
An observer can also choose an originOand use the bi-point motion vectorϕ(t, Pe~ Obj) :=−−−−−−−−→
OΦ(t, Pe Obj) instead of the pointΦ(t, Pe Obj):
e~ ϕ:
[t1, t2]×Obj →R~n
(t, PObj) →~x=ϕ(t, Pe~ Obj) =−−−−−−−−→
OΦ(t, Pe Obj).
(1.6)
But then, two observers with two dierent originsOandO0have two dierent bi-point vectors~xand~x0. Therefore, in the following we won't useϕe~: We will exclusively useΦe, cf. (1.5). Moreover, in a non-planar surface considered on its own (a manifold), the notion of bi-point vector is meaningless (it goes through the surface: The only available vectors are tangent vectors).
Quantication: An observer chooses a Cartesian referentialR= (O,(~ei))to describe the motionΦe: p=Φ(t, Pe Obj) =O+~x=O+
n
X
i=1
xi~ei =position ofPObj attin R. (1.7) Remark 1.3 Hypothesis of both Newtonian mechanics (Galileo relativity) and general relativity (Ein- stein): 1- You can describe a phenomenon only at the actual timetand from the location pt you are at (you have neither time or space ubiquity gift); 2- You don't know the future; 3- You can use your memory (use the past), or someone else memory if you can communicate objectively.
Remark 1.4 The motion of an object Obj (e.g. a planet) has been described before the invention of groups, rings, vector spaces, algebra (19th century) (Copernicus 1473-1543, Descartes 1596-1650).
13 1.4. Congurations and spatial (Eulerian) variables
1.4 Congurations and spatial (Eulerian) variables
LetΦe be a motion, cf. (1.5). Let t∈[t1, t2]be xed, and dene
Φet:
( Obj →Rn
PObj 7→p=Φet(PObj) :=Φ(t, Pe Obj). (1.8) Denition 1.5 The conguration att of Obj is the subset ofRn (ane space) dened by
Ωt=Φet(Obj) = Im(eΦt) =the range (or image) ofΦet
:={p∈Rn :∃PObj ∈Obj s.t.p=Φet(PObj)}. (1.9) Andp=Φet(PObj)∈Ωtis the spatial variable (att), or Eulerian variable, relative toPObj at t. And if a Cartesian referentialR= (O,(~ei))has been chosen, then−−→
Opt=Pn
i=1xi~eiis called a vectorial spatial variable, or vectorial Eulerian variable, relative toPObj att and relative to the referentialR. Hypothesis: At any timet, the mapΦet is assumed to be one-to-one (= injective): Obj does not crash onto itself. AndΩtis supposed to be a a smooth domain inRn, that is, the closure of an open set inR3, or of a 2-D dierentiable surface inR3, or of a 1-D dierentiable curve inR3 (continuum mechanics).
1.5 Eulerian and Lagrangian variables
Denition 1.6
• Ift is the actual time, then Ωt is called the actual conguration or current conguration, and the spatial variablept∈Ωt=Φet(PObj)is called the Eulerian variable (location ofPObj at actual time).
• Ift0 is a time in the past, thenΩt0 is called the initial conguration, or reference conguration, and the spatial variablept0 ∈Ωt0 =Φet0(PObj)is called the Lagrangian variable relative tot0.
1.6 Trajectories
LetΦe be a motion of Obj, cf. (1.5). LetPObj ∈Obj be a particle (e.g., a particle in the Moon).
Denition 1.7 The (parametric) trajectory ofPObj betweent1andt2is the function
ΦePObj :
([t1, t2] →Rn,
t 7→p(t) =ΦePObj(t) :=Φ(t, Pe Obj) (position ofPObj at t). (1.10) And its rangeIm(eΦPObj) =ΦePObj([t1, t2])is the (geometric) trajectory of PObj.
1.7 Virtual and real motion
Denition 1.8 A virtual (or possible) motion ofObj is a functionΦe regular enough for the calculations to be meaningful: In the following, the parametric trajectoriesΦePObj are at leastC2 for velocities and accelerations to exist. Among all the virtual motions, the observed motion is called the real motion.
1.8 Tangent space, R ~
nt, ber, bundle
Rn is the ane space of points, the same at all time (classical mechanics), associated with the vector spaceR~n (made of bipoint vectors). However, to deal with surfaces (manifolds), a vector is considered to be a vector tangent to the surface at a point. E.g., on the surface of a sphereS (e.g. Earth, Moon...) a tangent vector at a pointpcannot be a tangent vector at some other point (a sphere is not at).
In Rn, let m ∈[1, n]N and letS be a regularm-surface (a m-dierentiable manifold in Rn). That is, inRn =R3, a3-surfaceS is an open setΩinR3, a2-surfaceS is a usual surface, a1-surfaceS is a usual curve.
Denition 1.9
The tangent space atpnoted= TpS :={tangent vectorsw~p atS atp}. (1.11) Particular case: IfS= Ωis an open set in n, then T S=T Ω =~n is independent ofp.
14 2.1. Eulerian function
Denition 1.10
The ber atp:={p} ×TpS={couple(p, ~wp)
| {z }
:= pointed vector
∈ {p} ×TpS}, (1.12) that is, is the set of pointed vectors atp (a vector equipped with a base point to which it is attached).
If the context is clear, a pointed vector(p, ~wp)is simply notedw~p.
Particular case: IfS= Ωis an open set inRn, then the ber atpisTpΩ ={p} ×~Rn. Denition 1.11
The tangent bundle:= [
p∈S
({p} ×TpS)noted= T S, (1.13) that is, is the union of the bers.
Particular case: IfS= Ωis an open set inRn, then T S=TΩ =S
p∈S({p} ×~Rn).
2 Eulerian description (spatial description at actual time t )
2.1 Eulerian function
LetΦe be a motion ofObj, cf. (1.5), andΩt=Φet(Obj)⊂Rn be the conguration att, cf. (1.9). Let
C
be the set of congurations, that is the subset in the Cartesian time-space R×Rn dened byC
:= [t∈[t1,t2]
({t} ×Ωt) (⊂R×Rn)
={(t, p)∈R×Rn :∃(t, PObj)∈[t1, t2]×Obj, p=Φ(t, Pe Obj)},
(2.1)
Question: Why don't we simply useS
t∈[t1,t2]Ωtinstead of
C
=St∈[t1,t2]({t} ×Ωt)?
Answer:
C
gives the lm of the life of Obj = the succession of the photosΩt taken at eacht. (And Ωt is obtained fromC
thanks to the pause feature att.) WhereasSt∈[t1,t2]Ωt is just one photo = the superposition of all the photos on an unique photo: The lm is superimposed on one photo... and we do not distinguish the past from the present.
Denition 2.1 In short, and with (2.1) (relative toObj) andm∈N∗, a Eulerian function is a function Eul:
(
C
→R~m(or more generally a suitable set of tensors)(t, p) → Eul(t, p). (2.2)
The spatial variablepis the Eulerian variable.
Example 2.2 Eul(t, p) =θ(t, p)∈R=temperature of the particlePObj which is attatp=Φ(t, Pe Obj); Example 2.3 Eul(t, p) =~u(t, p)∈R~n =force applied on the particle which is attat p.
Denition 2.4 In details, a functionEulbeing given as in (2.2), the associated Eulerian functionEulc is the function dened by
Ecul:
(
C
→C
×R~m(orC
×some suitable set of tensors)(t, p) →Eul(t, p) = ((t, p);c Eul(t, p)), (2.3) and is called a eld of functions; SoEul(t, p)c is the pointed function at(t, p)(in time-space).
So, the rangeIm(Ecul) =Eul(c
C
)of an Eulerian functionEculis the graph ofEul. (Recall: The graph of a functionf :x∈A→f(x)∈B is the subset{(x, f(x))∈A×B} ⊂A×B: gives the drawing off).AndEulc is writtenEul for short, if there is no ambiguity.
Question: Why introduceEulc? Isn'tEulsucient?
Answer: With p+ = (t, p)∈R×R3, a valuey =Eul(p+) =Eul(t, p) is drawn on they-axis, when the pointed value Eul(pc +) = (p+, y) = (p+,Eul(p+))is drawn on the graph ofEul.
E.g., a vector~v(p+) =−−→ AB∈ ~
R3(bipoint vector) can be drawn at any point, while the pointed vector
~v(p ) = (p ;~v(p ))is−−→
AB drawn atp .
15 2.2. Eulerian velocity (spatial velocity) and speed
Example 2.5 1- bθ(t, p) = ((t, p);θ(t, p)) = temperature of the particle PObj which is at t at p = Φ(t, Pe Obj) ∈ R3. Usually represented by a color at (t, p) (on the graph of θ): On the photo at t, the colors gives the dierent temperatures at dierentp.
2-~bv(t, p) = ((t, p);~v(t, p)) =a force on the particle PObj which is attat p. Usually represented by a arrow at(t, p): On the graph of~v. So on the photo att, you see the dierent vectors at dierentp.
Att, withEult(p) :=Eul(t, p), the Eulerian eld attis
Eulct:
(Ωt →Ωt× Lrs(~Rn)
p →Eulct(p) := (p,Eult(p)). (2.4) Remark 2.6 E.g., the initial framework of Cauchy (for his description of forces) is Eulerian: The Cauchy stress vector~t=σ.~nis considered at the actual timet at a pointpt∈Ωt. (It is not Lagrangian.)
2.2 Eulerian velocity (spatial velocity) and speed
Consider a particlePObj and its (regular) trajectoryΦePObj :t→p(t) =ΦePObj(t), cf. (1.10).
Denition 2.7 In short, the Eulerian velocity of the particlePObj which is at tat p=Φ(t, Pe Obj)is the vectorial valued map dened on
C
=St∈[t1,t2]({t} ×Ωt)by
~v(t, p) :=ΦePObj0(t) = deΦPObj
dt (t) (= lim
h→0
ΦePObj(t+h)−ΦePObj(t)
h = lim
h→0
−−−−−−−−−−−−−→
ΦePObj(t)eΦPObj(t+h)
h ), (2.5)
i.e.~v(t, p) is the tangent vector at t at p=ΦePObj(t) to the trajectory ΦePObj. (It depends on the chosen unit of time, e.g. per second, or per hour...) Also written
~v(t, p) = ∂Φe
∂t(t, PObj). (2.6)
In details, cf. (2.3), the Eulerian velocity is the function dened with (2.5) by b~
v(t, p) = ((t, p), ~v(t, p)) (2.7)
(pointed vector), and it is represented by the vector~v(t, p)drawn at(t, p)(on the graph of~v).
Remark 2.8 dΦedtPObj(t) =~v(t,ΦePObj(t)), cf. (2.5), is often written dp
dt(t) =~v(t, p(t)), or d~x
dt(t) =~v(t, ~x(t)), (2.8) wherep(t) :=ΦePObj(t), the last equality with a chosen originO and~x(t) =−−−→
Op(t) =ϕ(t, Pe~ Obj), cf. (1.6).
Such an equation is the prototype of an ODE (ordinary dierential equation) solved with the Cauchy Lipschitz theorem, see 6 and remark 1.3. (A Lagrangian velocity does not produce an ODE, see (4.9).) Denition 2.9 If an observer chooses a Euclidean dot product (·,·)g (e.g. built with the foot or the meter cf. B.1), the associated norm being||.||g, then the length||~v(t, p)||g is named the speed ofPObj, or scalar velocity ofPObj (e.g. given in ft/s or in m/s).
And the context must remove the ambiguities: the velocity is either the vector velocity ~v(t, p) = ΦePObj0(t)(depends on the time unit), or the speed||~v(t, p)||g (also depends on the length unit).
2.3 Spatial derivative of the Eulerian velocity
Lett∈[t1, t2]and Eult(p) :=Eul(t, p). HereEult: Ωt→R~m is supposed to be regular inΩt.
Denition 2.10 The space derivative dEul of the Eulerian function Eul is the dierential dEult of the functionEult, that is,dEulis dened attat p∈Ωtby, for all w~ ∈R~nt vector atp,
dEul(t, p). ~w:=dEult(p). ~w= lim
h→0
Eult(p+h ~w)− Eult(p)
h (= lim
h→0
Eul(t, p+h ~w)− Eul(t, p)
h ). (2.9)
ThusdEul(t, p)gives inΩt(the photo att) the spatial rate of variations ofEulat p.
E.g., the space derivative d~v of the Eulerian velocity eld is, attat p∈Ωt, for allw~ ∈R~nt, d~v(t, p). ~w= lim ~v(t, p+h ~w)−~v(t, p)
(= lim ~vt(p+h ~w)−~vt(p)
). (2.10)
16 2.4. The convective objective termdf.~v, written(~v. ~grad)f in a basis...
2.4 The convective objective term df.~ v , written (~ v. ~ grad)f in a basis...
Recall: IfΩis an open set in Rn and if f : Ω→Ris dierentiable atp, then its dierential atpis the linear mapdf(p) :R~n →Rdened by, for all~u∈~Rn (vector atp),
df(p).~u= lim
h→0
f(p+h~u)−f(p)
h (2.11)
Quantication: Let(~ei)be a Cartesian basis inR~n, and let (usual denition)
∂f
∂xi(p) :=df(p).~ei, and [df(p)]|~e= (∂x∂f1(p) ... ∂x∂fn(p) ) (line matrix). (2.12) That is,(ei)being the dual basis of(~ei), cf. (A.33), we havedf(p) =Pn
i=1
∂f
∂xi(p)ei, and the linear form df(p)is represented by a line matrix. And we get, with~u=Pn
i=1ui~ei, df(p).~u= [df(p)]|~e.[~u]|~e=
n
X
i=1
∂f
∂xi(p)ui=
n
X
i=1
ui∂f
∂xi(p)noted= (~u. ~grad)f(p). (2.13) We have thus dened the operator (the linear map) relative to a basis(~ei):
~
u. ~grad =
n
X
i=1
ui ∂
∂xi :
C1(Ω;R) →C0(Ω;R) f →(~u. ~grad)(f) =
n
X
i=1
ui∂f
∂xi (=df.~u). (2.14) For vector valued functionsf~: Ω→R~m, the above steps apply to the components off~in a basis(~bi) inR~m: Iff~=Pm
i=1fi~bi, then d ~f .~u=
m
X
i=1
(dfi.~u)~bi=
m
X
i=1
((~u. ~grad)fi)~bi, and [d ~f]|~e,~b= [∂fi
∂xj] (the Jacobian matrix). (2.15) Application: Consider a motionΦePObj of a particlePObj ∈Obj, cf. (1.10), lett∈R, letp=ΦePObj(t), let
~v(t, p) =ΦePObj0(t)(the Eulerian velocity att at p). And consider a dierentiable Eulerian function Eul, cf. (2.2), and letEul(t, p) =notedEult(p). Then, withf =Eultand~u=~v(t, p)in (2.11) we dene:
Denition 2.11 The convective derivative of the Eulerian functionEulis dened attatptby (derivative along the trajectory)
(dEult.~vt)(p) =dEul(t, p).~v(t, p) :=dEult(p).~vt(p) (= lim
h→0
Eult(p+h~vt(p))− Eult(p)
h ). (2.16)
Quantication: Let(~ei)be a Cartesian basis in ~
Rn. Then (2.13) gives dEul.~v= (~v. ~grad)Eult=
n
X
i=1
vi∂Eul
∂xi =the convective derivative in a basis. (2.17)
2.5 ... and the subjective gradf ~ (depends on a Euclidean dot product)
An observer chooses a distance unit (foot, meter...) and uses the associated Euclidean dot product(·,·)g
inR~nt, cf. B.2 (the following results will depend on (·,·)g, i.e. on the observer).
LetΩbe an open set inRn andf ∈C1(Ω;R)(scalar valued function, e.g.f =Eult∈C1(Ωt;R)). Let p∈Ω.
Denition 2.12 The(·,·)g-Riesz representation vectorgrad~ gf(p)∈~Rn of the dierential formdf(p)is dened by, cf. (C.1),
~n ~ written ~ (2.18)
17 2.6. Streamline (current line)
Quantication with a basis (~ei): Let (~ei)be a Cartesian basis inRn, and let ∂x∂fi :=df.~ei, cf. (2.12).
Then (2.18) gives[df(p)]|~e.[~u]|~e= [grad~ gf(p)]T|~e.[g]|~e.[~u]|~e,for all~u∈~
Rnt, thus (with[g]|~esymmetric) [grad~ gf(p)]|~e= [g]|~e.[df(p)]|~eT (column matrix). (2.19) That isgrad~ gf =Pn
i=1ai~ei whereai=Pn
j=1gij∂x∂fj for alli.
Case (~ei) is a (·,·)g-orthonormal basis: Then [grad~ gf]|~e = [df]|~eT (since [g]|~e = I) and grad~ gf = Pn
i=1
∂f
∂xi~ei.
Be careful: The gradient grad~ gf depends on (·,·)g, cf. (2.18)-(2.19), while (~u. ~grad)f does not, cf. (2.14): It only depends on the choice of a basis for the denition of the ∂x∂fi.
For vector valued functionsf~: Ω→R~m, the above steps apply to the componentsfi off~relative to a basis(~bi)inR~m... But there is a notation problem...:
1- For dierentiald ~f there is just one Jacobian matrix (relative to a given basis), cf. (2.15), sometimes also called the gradient matrix (although no Euclidean dot product is required).
2- But the gradient of f~ ... depends on the authors: It could mean the Jacobian matrix or its transposed, and the use of some Euclidean dot product (which one?) may be required... or not...
3- In the objective setting of this manuscript, we will never talk about the gradient of a vectorial functionf~: only the dierential (objective) and the Jacobian off~will be used (after a choice of a basis).
Exercice 2.13 A Euclidean setting being chosen, prove (~v. ~grad)~v= 1
2
grad(||~~ v||2) +rot~~ v∧~v.
Answer. Euclidean basis(E~i), Euclidean dot product(·,·)g=noted(·,·), associated norm||.||g=noted||.||. Thus
~v=Pn
i=1viE~i gives ||~v||2 =X
i
(vi)2, thus ∂||~v||2
∂xk =X
i
2vi∂vi
∂xk, for anyk = 1,2,3. And, the rst component of rot~~ v is (rot~~ v)1 = ∂v3
∂x2 −∂v2
∂x3, idem for (rot~~ v)2 and (rot~~ v)3 (circular permutation). Thus (rst component) (rot~~ v∧~v)1 = (∂v1
∂x3−∂v3
∂x1)v3−(∂v2
∂x1−∂v1
∂x2)v2, idem for(rot~~ v∧~v)2and(rot~~ v∧~v)2. Thus(12grad(||~~ v||2)+rot~~ v∧~v)1= v1∂v∂x11 +v2∂v∂x21 +v3∂v∂x31 +∂v∂x13v3−∂v∂x31v3−∂v∂x21v2+∂x∂v12v2=v1∂v∂x11 +v2∂x∂v12 +v3∂v∂x13 = (~v. ~grad)v1. Idem for the other components.
2.6 Streamline (current line)
Lett∈R. Consider the photoΩt=Φet(Obj). Letpt∈Ωt,ε >0, and consider a spatial curve inΩtatpt:
cpt :
(]−ε, ε[ →Ωt
s →q(s) =cpt(s) )
, s.t. cpt(0) =pt. (2.20) Sosis a (spatial) curvilinear abscissa (dimension of a length), andcpt(]−ε, ε[) = Im(cpt)is drawn inΩt
(drawn in the photo att), andεis small enough for cpt(]−ε, ε[)to be inΩt.
Denition 2.14 ~v being the Eulerian velocity eld of a motionΦe, a (parametric) streamline throughpt
is a curvecpt solution of the dierential equation dcpt
ds (s) =~vt(cpt(s)) with cpt(0) =pt. (2.21) AndIm(cpt) =cpt(]−ε, ε[)(inΩt) is the geometric associated streamline. And (2.21) is also written
dq
ds(s) =~vt(q(s)) with q(0) =pt, or d~x
ds(s) =~vt(~x(s)) with ~x(0) =−−→
Opt (2.22) once an originO has been chosen inRn.
18 2.7. Material time derivative (dérivées particulaires)
NB: (2.8) cannot be confused with (2.21): the problem (2.8) is time dependent, while, at each t, the problem (2.21) is time independent (the variable is the spatial variables).
Usual notation: A Cartesian coordinate system R= (O,(~ei)) is chosen att, and~x(s) := −−−−−→
Ocpt(s) =
−−−→Oq(s) = Pn
i=1xi(s)~ei, thus d~dsx(s) = Pn i=1
dxi
ds(s)~ei. Thus (2.22) reads as the dierential system of equation in ~
Rn
∀i= 1, ..., n, dxi
ds(s) =vi(t, x1(s), ..., xn(s)) with xi(0) = (−−→
Opt)i (2.23) (thexi:s→xi(s)are thensolutions of the nequations). Also written
dx1
v1
=...= dxn
vn
=ds. (2.24)
Whatever the notation, it is the dierential system ofnequations (2.23) which must be solved.
(With duality notations, dxdsi(s) =vi(t, x1(s), ..., xn(s)).)
2.7 Material time derivative (dérivées particulaires)
2.7.1 Usual denition
Consider a regular motionΦe , cf. (1.5), and the associated Eulerian velocity eld~v, cf. (2.5).
Goal: to measure the variations of a Eulerian function Eul along the trajectory of a particle PObj. E.g.,PObj being is attatp(t) =Φ(t, Pe Obj), we look for the variations of the temperatureθ(t, p(t))ofPObj
through time. Thus consider a Eulerian functionEuland
gPObj(t) :=Eul(t,ΦePObj(t)) =Eul(t, p(t)) when p(t) :=ΦePObj(t). (2.25) Denition 2.15 The Material time derivative ofEul at(t, p(t))is
DEul
Dt (t, p(t)) :=gPObj0(t) = the derivative of gat t (= lim
h→0
Eul(t+h, p(t+h))− Eul(t, p(t))
h ). (2.26)
With (2.25) andΦe0P
Obj(t) =~v(t, p(t))(Eulerian velocity), we get (gPObj0(t) =) DEul
Dt (t, p(t)) = ∂Eul
∂t (t, p(t)) +dEul(t, p(t)).~v(t, p(t)), (2.27) that is, in
C
=St∈[t1,t2]({t} ×Ωt),
DEul
Dt := ∂Eul
∂t +dEul.~v . (2.28)
Remark 2.16 •The notation dtd (lowercase letters) concerns a function of one variable, e.g. dgdtPObj(t) :=
gPObj0(t) := limh→0gPObj(t+h))−gh PObj(t);
• The notation ∂t∂ concerns a function of more than one variable, e.g. ∂Eul∂t (t, p) = limh→0Eul(t+h,p)−Eul(t,p)
h ;
• The notation DtD (capital letters) concerns a Eulerian function when the variables t and p are made dependent thanks to a motion, that is DEulDt (t, p(t)) := limh→0Eul(t+h,p(t+h))−Eul(t,p(t))
h whenp(τ) =
ΦePObj(τ).
Other notations (often practical, but might be ambiguous if composed functions are considered):
DEul
Dt (t, p(t))noted= dEul(t, p(t))
dt , and DEul
Dt (t0, p(t0))noted= dEul(t, p(t)) dt |t=t0
. (2.29)
Quantication. Let(~e1, ..., ~en)be a Cartesian basis and(dxi)its dual basis (we use duality notations:
use classic notations if you prefer). Then DEul= ∂Eul
dt+dEul=∂Eul dt+
n
X∂Eul
dxi, (2.30)
