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Ellande

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Détails des calculs de dérivées particulaires

Méthode 1

${\frac {\mathrm {d} (\rho {\overrightarrow {v}})}{\mathrm {d} t}}={\frac {\partial (\rho {\overrightarrow {v}})}{\partial t}}+({\vec {v}}\cdot {\overrightarrow {\mathrm {grad} }})(\rho {\overrightarrow {v}})={\frac {\partial (\rho {\overrightarrow {v}})}{\partial t}}+({\vec {v}}\cdot {\overrightarrow {\nabla }})\left(\rho {\overrightarrow {v}}\right)$

${\frac {\mathrm {d} (\rho {\overrightarrow {v}})}{\mathrm {d} t}}=\rho {\frac {\partial {\overrightarrow {v}}}{\partial t}}+{\frac {\partial \rho }{\partial t}}{\overrightarrow {v}}+\rho \,v_{x}{\frac {\partial {\overrightarrow {v}}}{\partial x}}+{\frac {\partial \rho }{\partial x}}v_{x}{\overrightarrow {v}}+\rho \,v_{y}{\frac {\partial {\overrightarrow {v}}}{\partial y}}+{\frac {\partial \rho }{\partial y}}v_{y}{\overrightarrow {v}}+\rho \,v_{z}{\frac {\partial {\overrightarrow {v}}}{\partial z}}+{\frac {\partial \rho }{\partial z}}v_{z}{\overrightarrow {v}}$

${\frac {\mathrm {d} (\rho {\overrightarrow {v}})}{\mathrm {d} t}}=\rho {\frac {\partial {\overrightarrow {v}}}{\partial t}}+{\frac {\partial \rho }{\partial t}}{\overrightarrow {v}}+\rho \left(v_{x}{\frac {\partial {\overrightarrow {v}}}{\partial x}}+v_{y}{\frac {\partial {\overrightarrow {v}}}{\partial y}}+v_{z}{\frac {\partial {\overrightarrow {v}}}{\partial z}}\right)+\left({\frac {\partial \rho }{\partial x}}v_{x}+{\frac {\partial \rho }{\partial y}}v_{y}+{\frac {\partial \rho }{\partial z}}v_{z}\right){\overrightarrow {v}}$

${\frac {\mathrm {d} (\rho {\overrightarrow {v}})}{\mathrm {d} t}}=\rho {\frac {\partial {\overrightarrow {v}}}{\partial t}}+{\frac {\partial \rho }{\partial t}}{\overrightarrow {v}}+\rho \left({\overrightarrow {v}}\cdot {\overrightarrow {\mathrm {grad} }}\right){\overrightarrow {v}}+\left({\overrightarrow {v}}\cdot {\overrightarrow {\mathrm {grad} }}\,\rho \right){\overrightarrow {v}}$

${\frac {\mathrm {d} (\rho {\overrightarrow {v}})}{\mathrm {d} t}}+\rho \,{\overrightarrow {v}}\,{\hbox{div}}\ {\overrightarrow {v}}=\rho \left({\frac {\partial {\overrightarrow {v}}}{\partial t}}+\left({\overrightarrow {v}}\cdot {\overrightarrow {\mathrm {grad} }}\right){\overrightarrow {v}}\right)+\left({\frac {\partial \rho }{\partial t}}+{\overrightarrow {v}}\cdot {\overrightarrow {\mathrm {grad} }}\,\rho +\rho \,{\hbox{div}}\,{\overrightarrow {v}}\right){\overrightarrow {v}}$

${\frac {\mathrm {d} (\rho {\overrightarrow {v}})}{\mathrm {d} t}}+\rho \,{\overrightarrow {v}}\,{\hbox{div}}\ {\overrightarrow {v}}=\rho \left({\frac {\partial {\overrightarrow {v}}}{\partial t}}+\left({\overrightarrow {v}}\cdot {\overrightarrow {\mathrm {grad} }}\right){\overrightarrow {v}}\right)+\underbrace {\left({\frac {\partial \rho }{\partial t}}+{\hbox{div}}\left(\rho \,{\overrightarrow {v}}\right)\right)} _{=0}{\overrightarrow {v}}$

${\frac {\mathrm {d} \left(\rho {\overrightarrow {v}}\right)}{\mathrm {d} t}}+\rho \,{\overrightarrow {v}}\,{\hbox{div}}\ {\overrightarrow {v}}=\rho {\frac {\mathrm {d} {\overrightarrow {v}}}{\mathrm {d} t}}$

Méthode 2

${\frac {\mathrm {d} \left(\rho {\overrightarrow {v}}\right)}{\mathrm {d} t}}=\rho {\frac {\mathrm {d} {\overrightarrow {v}}}{\mathrm {d} t}}+{\frac {\mathrm {d} \rho }{\mathrm {d} t}}{\overrightarrow {v}}$

${\frac {\mathrm {d} \left(\rho {\overrightarrow {v}}\right)}{\mathrm {d} t}}+\rho \,{\overrightarrow {v}}\,{\hbox{div}}\ {\overrightarrow {v}}=\rho {\frac {\mathrm {d} {\overrightarrow {v}}}{\mathrm {d} t}}+\underbrace {\left({\frac {\mathrm {d} \rho }{\mathrm {d} t}}+\rho \ {\hbox{div}}\,{\overrightarrow {v}}\right)} _{=0}{\overrightarrow {v}}$

Méthode 3

${\frac {\mathrm {d} \left(\rho {\overrightarrow {v}}\right)}{\mathrm {d} t}}=\rho \left({\frac {\partial {\overrightarrow {v}}}{\partial t}}+v_{x}{\frac {\partial {\overrightarrow {v}}}{\partial x}}+v_{y}{\frac {\partial {\overrightarrow {v}}}{\partial y}}+v_{z}{\frac {\partial {\overrightarrow {v}}}{\partial z}}\right)+\left({\frac {\partial \rho }{\partial t}}+{\frac {\partial \rho }{\partial x}}v_{x}+{\frac {\partial \rho }{\partial y}}v_{y}+{\frac {\partial \rho }{\partial z}}v_{z}\right){\overrightarrow {v}}$

${\frac {\mathrm {d} \left(\rho {\overrightarrow {v}}\right)}{\mathrm {d} t}}=\rho \left({\frac {\partial {\overrightarrow {v}}}{\partial t}}+({\overrightarrow {v}}\cdot {\overrightarrow {\nabla }})\cdot {\overrightarrow {v}}\right)+{\frac {\mathrm {d} \rho }{\mathrm {d} t}}{\overrightarrow {v}}$

${\frac {\mathrm {d} \left(\rho {\overrightarrow {v}}\right)}{\mathrm {d} t}}+\rho \,{\overrightarrow {v}}\,{\hbox{div}}\ {\overrightarrow {v}}=\rho \left({\frac {\partial {\overrightarrow {v}}}{\partial t}}+({\overrightarrow {v}}\cdot {\overrightarrow {\nabla }})\cdot {\overrightarrow {v}}\right)+\rho \,{\overrightarrow {v}}\,{\hbox{div}}\ {\overrightarrow {v}}$

${\frac {\mathrm {d} \left(\rho {\overrightarrow {v}}\right)}{\mathrm {d} t}}+\rho \,{\overrightarrow {v}}\,{\hbox{div}}\ {\overrightarrow {v}}=\rho {\frac {\partial {\overrightarrow {v}}}{\partial t}}+\rho \,({\overrightarrow {v}}\cdot {\overrightarrow {\nabla }})\cdot {\overrightarrow {v}}$

Autre

${\overrightarrow {\nabla }}\cdot \left(\rho {\overrightarrow {v}}\otimes {\overrightarrow {v}}\right)=\mathrm {div} \,(\rho {\overrightarrow {v}})\,{\overrightarrow {v}}+\rho ({\overrightarrow {v}}\cdot {\overrightarrow {\mathrm {grad} }})\,{\overrightarrow {v}}$

${\frac {\partial \left(\rho {\vec {v}}\right)}{\partial t}}+{\vec {\nabla }}\cdot \left(\rho {\vec {v}}\otimes {\vec {v}}\right)=-{\vec {\nabla }}p+{\vec {\nabla }}\cdot {\overline {\overline {\tau }}}+\rho {\vec {f}}$

${\vec {\nabla }}\cdot \left(\rho {\vec {v}}\otimes {\vec {v}}\right)={\vec {\nabla }}\cdot \left({\begin{pmatrix}\rho v_{x}\\\rho v_{y}\\\rho v_{z}\end{pmatrix}}\times {\begin{pmatrix}v_{x}&v_{y}&v_{z}\end{pmatrix}}\right)={\vec {\nabla }}\cdot {\begin{pmatrix}\rho v_{x}v_{x}&\rho v_{x}v_{y}&\rho v_{x}v_{z}\\\rho v_{y}v_{x}&\rho v_{y}v_{y}&\rho v_{y}v_{z}\\\rho v_{z}v_{x}&\rho v_{z}v_{y}&\rho v_{z}v_{z}\end{pmatrix}}$

${\vec {\nabla }}\cdot \left(\rho {\vec {v}}\otimes {\vec {v}}\right)={\begin{pmatrix}{\frac {\partial (\rho v_{x}v_{x})}{\partial x}}+{\frac {\partial (\rho v_{x}v_{y})}{\partial y}}+{\frac {\partial (\rho v_{x}v_{z})}{\partial z}}\\{\frac {\partial (\rho v_{y}v_{x})}{\partial x}}+{\frac {\partial (\rho v_{y}v_{y})}{\partial y}}+{\frac {\partial (\rho v_{y}v_{z})}{\partial z}}\\{\frac {\partial (\rho v_{z}v_{x})}{\partial x}}+{\frac {\partial (\rho v_{z}v_{y})}{\partial y}}+{\frac {\partial (\rho v_{z}v_{z})}{\partial z}}\end{pmatrix}}$

${\vec {\nabla }}\cdot \left(\rho {\vec {v}}\otimes {\vec {v}}\right)={\begin{pmatrix}{\frac {\partial \rho }{\partial x}}v_{x}v_{x}+\rho {\frac {\partial v_{x}}{\partial x}}v_{x}+\rho {\frac {\partial v_{x}}{\partial x}}v_{x}+{\frac {\partial \rho }{\partial y}}v_{x}v_{y}+\rho {\frac {\partial v_{x}}{\partial y}}v_{y}+\rho {\frac {\partial v_{y}}{\partial y}}v_{x}+{\frac {\partial \rho }{\partial z}}v_{x}v_{z}+\rho {\frac {\partial v_{x}}{\partial z}}v_{z}+\rho {\frac {\partial v_{z}}{\partial z}}v_{x}\\{\frac {\partial \rho }{\partial x}}v_{y}v_{x}+\rho {\frac {\partial v_{y}}{\partial x}}v_{x}+\rho {\frac {\partial v_{x}}{\partial x}}v_{y}+{\frac {\partial \rho }{\partial y}}v_{y}v_{y}+\rho {\frac {\partial v_{y}}{\partial y}}v_{y}+\rho {\frac {\partial v_{y}}{\partial y}}v_{y}+{\frac {\partial \rho }{\partial z}}v_{y}v_{z}+\rho {\frac {\partial v_{y}}{\partial z}}v_{z}+\rho {\frac {\partial v_{z}}{\partial v_{z}}}v_{y}\\{\frac {\partial \rho }{\partial x}}v_{z}v_{x}+\rho {\frac {\partial v_{z}}{\partial x}}v_{x}+\rho {\frac {\partial v_{x}}{\partial x}}v_{z}+{\frac {\partial \rho }{\partial y}}v_{z}v_{y}+\rho {\frac {\partial v_{z}}{\partial y}}v_{y}+\rho {\frac {\partial v_{y}}{\partial y}}v_{z}+{\frac {\partial \rho }{\partial z}}v_{z}v_{z}+\rho {\frac {\partial v_{z}}{\partial z}}v_{z}+\rho {\frac {\partial v_{z}}{\partial z}}v_{z}\end{pmatrix}}$

${\vec {\nabla }}\cdot \left(\rho {\vec {v}}\otimes {\vec {v}}\right)={\begin{pmatrix}(\rho \,\mathrm {div} \,{\vec {v}}+{\overrightarrow {\mathrm {grad} }}\,\rho \cdot {\vec {v}})v_{x}+\rho \,{\overrightarrow {\mathrm {grad} }}\,v_{x}\cdot {\vec {v}}\\(\rho \,\mathrm {div} \,{\vec {v}}+{\overrightarrow {\mathrm {grad} }}\,\rho \cdot {\vec {v}})v_{y}+\rho \,{\overrightarrow {\mathrm {grad} }}\,v_{y}\cdot {\vec {v}}\\(\rho \,\mathrm {div} \,{\vec {v}}+{\overrightarrow {\mathrm {grad} }}\,\rho \cdot {\vec {v}})v_{z}+\rho \,{\overrightarrow {\mathrm {grad} }}\,v_{z}\cdot {\vec {v}}\end{pmatrix}}$

${\vec {\nabla }}\cdot \left(\rho {\vec {v}}\otimes {\vec {v}}\right)={\begin{pmatrix}\mathrm {div} \,(\rho {\vec {v}})\,v_{x}+\rho \,{\overrightarrow {\mathrm {grad} }}\,v_{x}\cdot {\vec {v}}\\\mathrm {div} \,(\rho {\vec {v}})\,v_{y}+\rho \,{\overrightarrow {\mathrm {grad} }}\,v_{y}\cdot {\vec {v}}\\\mathrm {div} \,(\rho {\vec {v}})\,v_{z}+\rho \,{\overrightarrow {\mathrm {grad} }}\,v_{z}\cdot {\vec {v}}\end{pmatrix}}$

${\vec {\nabla }}\cdot \left(\rho {\vec {v}}\otimes {\vec {v}}\right)=\mathrm {div} \,(\rho {\vec {v}})\,{\vec {v}}+\rho ({\vec {v}}\cdot {\overrightarrow {\mathrm {grad} }})\,{\vec {v}}$

${\vec {\nabla }}\cdot \left(\rho {\vec {v}}\otimes {\vec {v}}\right)=-{\frac {\partial \rho }{\partial t}}\,{\vec {v}}+\rho ({\vec {v}}\cdot {\overrightarrow {\mathrm {grad} }})\,{\vec {v}}$

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${\frac {\partial \left(\rho {\vec {v}}\right)}{\partial t}}+{\vec {\nabla }}\cdot \left(\rho {\vec {v}}\otimes {\vec {v}}\right)=\rho {\frac {\partial {\vec {v}}}{\partial t}}+{\frac {\partial \rho }{\partial t}}{\vec {v}}-{\frac {\partial \rho }{\partial t}}\,{\vec {v}}+\rho ({\vec {v}}\cdot {\overrightarrow {\mathrm {grad} }})\,{\vec {v}}$

${\frac {\partial \left(\rho {\vec {v}}\right)}{\partial t}}+{\vec {\nabla }}\cdot \left(\rho {\vec {v}}\otimes {\vec {v}}\right)=\rho \,{\frac {\partial {\vec {v}}}{\partial t}}+\rho \,({\vec {v}}\cdot {\overrightarrow {\mathrm {grad} }})\,{\vec {v}}$

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$\left({\overrightarrow {v}}\cdot {\overrightarrow {\nabla }}\right)\rho {\overrightarrow {v}}={\begin{pmatrix}{\left({\overrightarrow {v}}\cdot {\overrightarrow {\nabla }}\right)\rho \,v_{x}}\\{\left({\overrightarrow {v}}\cdot {\overrightarrow {\nabla }}\right)\rho \,v_{y}}\\{\left({\overrightarrow {v}}\cdot {\overrightarrow {\nabla }}\right)\rho \,v_{z}}\end{pmatrix}}={\begin{pmatrix}{v_{x}{\frac {\partial \rho \,v_{x}}{\partial x}}+v_{y}{\frac {\partial \rho \,v_{x}}{\partial y}}+v_{z}{\frac {\partial \rho \,v_{x}}{\partial z}}}\\{v_{x}{\frac {\partial \rho \,v_{y}}{\partial x}}+v_{y}{\frac {\partial \rho \,v_{y}}{\partial y}}+v_{z}{\frac {\partial \rho \,v_{y}}{\partial z}}}\\{v_{x}{\frac {\partial \rho \,v_{z}}{\partial x}}+v_{y}{\frac {\partial \rho \,v_{z}}{\partial y}}+v_{z}{\frac {\partial \rho \,v_{z}}{\partial z}}}\end{pmatrix}}$

Tenseur des contraintes

Force exercée sur une surface $S$

En ajoutant les forces orientées dans la même direction, la résultante de l'ensemble des forces sur une surface élémentaire d'orientation quelconque s'exprime :

${\overrightarrow {\mathrm {d} F_{S}}}={\overline {\overline {\tau }}}\times {\overrightarrow {\mathrm {d} S}}={\begin{pmatrix}\sigma _{xx}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{yy}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{zz}\\\end{pmatrix}}\times {\begin{pmatrix}\mathrm {d} S_{x}\\\mathrm {d} S_{y}\\\mathrm {d} S_{z}\end{pmatrix}}={\begin{pmatrix}\sigma _{xx}\,\mathrm {d} S_{x}+\tau _{xy}\,\mathrm {d} S_{y}+\tau _{xz}\,\mathrm {d} S_{z}\\\tau _{yx}\,\mathrm {d} S_{x}+\sigma _{yy}\,\mathrm {d} S_{y}+\tau _{yz}\,\mathrm {d} S_{z}\\\tau _{zx}\,\mathrm {d} S_{x}+\tau _{zy}\,\mathrm {d} S_{y}+\sigma _{zz}\,\mathrm {d} S_{z}\\\end{pmatrix}}$.

Illustration

Force exercée sur un élément de volume $\mathrm {d} V$

Les forces surfaciques qui s'appliquent sur les faces d'un élément de volume $\mathrm {d} V=\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z$ sont modélisées par le tenseur des contraintes. L'illustration ci-contre permet de comprendre comment se décomposent ces forces.

Sur chaque face, par convention, le vecteur surface est orienté vers l'extérieur du volume. Par exemple, la face la plus proche de nous sur l'illustration a un vecteur surface ${\overrightarrow {\mathrm {d} S}}=\mathrm {d} S_{x}\,{\overrightarrow {e_{x}}}=\mathrm {d} y\,\mathrm {d} z\,{\overrightarrow {e_{x}}}$. La force qui s'exerce sur cette face peut être décomposée en 3 forces :

• une force normale à la surface ${\overrightarrow {\mathrm {d} F_{1}}}=\sigma _{xx}\,\mathrm {d} y\,\mathrm {d} z\,{\overrightarrow {e_{x}}}$ ;
• deux forces dans le plan de la surface :
  • ${\overrightarrow {\mathrm {d} F_{2}}}=\tau _{yx}\,\mathrm {d} y\,\mathrm {d} z\,{\overrightarrow {e_{y}}}$ dans la direction de l'axe $(Oy)$ ;
  • ${\overrightarrow {\mathrm {d} F_{2}}}=\tau _{zx}\,\mathrm {d} y\,\mathrm {d} z\,{\overrightarrow {e_{z}}}$ dans la direction de l'axe $(Oy)$.

Les indices associés à chaque contrainte indique, dans l'ordre, la direction de la force et la face sur laquelle la force s'applique. Les contraintes situées sur la diagonale correspondent à des forces de pression ce qui justifie que l'on leur affecte un nom différent. Les autres correspondent à des contraintes de cisaillement dues à la viscosité dans le cas de la mécanique des fluides.

La résultante des forces qui s'exerce sur l'élément de volume peut s'écrire :

${\overrightarrow {\mathrm {d} F_{V}}}=\sum _{i}{\overrightarrow {\mathrm {d} F_{i}}}={\begin{pmatrix}\left[\sigma _{xx}(x+\mathrm {d} x)-\sigma _{xx}(x)\right]\mathrm {d} y\,\mathrm {d} z+\left[\tau _{xy}(y+\mathrm {d} y)-\tau _{xy}(y)\right]\mathrm {d} x\,\mathrm {d} z+\left[\tau _{xz}(z+\mathrm {d} z)-\tau _{xz}(z)\right]\mathrm {d} x\,\mathrm {d} y\\\left[\tau _{yx}(x+\mathrm {d} x)-\tau _{yx}(x)\right]\mathrm {d} y\,\mathrm {d} z+\left[\sigma _{yy}(y+\mathrm {d} y)-\sigma _{yy}(y)\right]\mathrm {d} x\,\mathrm {d} z+\left[\tau _{yz}(z+\mathrm {d} z)-\tau _{yz}(z)\right]\mathrm {d} x\,\mathrm {d} y\\\left[\tau _{zx}(x+\mathrm {d} x)-\tau _{zx}(x)\right]\mathrm {d} y\,\mathrm {d} z+\left[\tau _{zy}(y+\mathrm {d} y)-\tau _{zy}(y)\right]\mathrm {d} x\,\mathrm {d} z+\left[\sigma _{zz}(z+\mathrm {d} z)-\sigma _{zz}(z)\right]\mathrm {d} x\,\mathrm {d} y\\\end{pmatrix}}$,

${\overrightarrow {\mathrm {d} F_{V}}}={\begin{pmatrix}{\frac {\partial \sigma _{xx}}{\partial x}}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z+{\frac {\partial \tau _{xy}}{\partial y}}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z+{\frac {\partial \tau _{xz}}{\partial z}}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z\\{\frac {\partial \tau _{yx}}{\partial y}}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z+{\frac {\partial \sigma _{yy}}{\partial y}}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z+{\frac {\partial \tau _{yz}}{\partial y}}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z\\{\frac {\partial \tau _{zx}}{\partial y}}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z+{\frac {\partial \tau _{zy}}{\partial y}}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z+{\frac {\partial \sigma _{zz}}{\partial y}}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z\\\end{pmatrix}}$

${\overrightarrow {\mathrm {d} F_{V}}}={\overrightarrow {\mathrm {div} }}{\begin{pmatrix}\sigma _{xx}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{yy}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{zz}\\\end{pmatrix}}\,\mathrm {d} V={\overrightarrow {\mathrm {div} }}\,{\overline {\overline {\tau }}}\,\mathrm {d} V$

Décomposition

${\overline {\overline {\sigma }}}=-P\,{\overline {\overline {\delta }}}+{\overline {\overline {\tau }}}$

${\begin{pmatrix}\sigma _{xx}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{yy}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{zz}\\\end{pmatrix}}=-P\,{\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\\end{pmatrix}}+{\begin{pmatrix}\tau _{xx}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\tau _{yy}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\tau _{zz}\\\end{pmatrix}}$

$\tau _{xx}+\tau _{yy}+\tau _{zz}=0$

$\tau _{ij}=\mu \left({\partial v_{i} \over \partial x_{j}}+{\partial v_{j} \over \partial x_{i}}\right)+\mu '\,\mathrm {div} \,{\overrightarrow {v}}\,\delta _{ij}$

$\mu$ : viscosité dynamique

$\mu '$ : coefficient de seconde viscosité

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tenseur de contraintes

Recherche ː travail des forces surfaciques

eq de l'énergie ; Tenseur des contraintes

Produit matriciel ; Matrice transposée

${\overrightarrow {\hbox{div}}}\ {\overline {\overline {\mathrm {\tau } }}}={\begin{pmatrix}{\frac {\partial \sigma _{xx}}{\partial x}}+{\frac {\partial \tau _{xy}}{\partial y}}+{\frac {\partial \tau _{xz}}{\partial z}}\\{\frac {\partial \tau _{yx}}{\partial x}}+{\frac {\partial \sigma _{yy}}{\partial y}}+{\frac {\partial \tau _{yz}}{\partial z}}\\{\frac {\partial \tau _{zx}}{\partial x}}+{\frac {\partial \tau _{zy}}{\partial y}}+{\frac {\partial \sigma _{zz}}{\partial z}}\\\end{pmatrix}}$

Travail des forces de surface

forces dont la direction est selon l'axe y

${\begin{alignedat}{2}\sum W_{S_{y}}=&\left[(\tau _{yx}\,v_{y})_{x+dx}-(\tau _{yx}\,v_{y})_{x}\right]\mathrm {d} y\,\mathrm {d} z\\&+\left[(\sigma _{yy}\,v_{y})_{y+dy}-(\sigma _{yy}\,v_{y})_{y}\right]\mathrm {d} x\,\mathrm {d} z\\&+\left[(\tau _{yz}\,v_{y})_{z+dz}-(\tau _{yz}\,v_{y})_{z}\right]\mathrm {d} x\,\mathrm {d} y\end{alignedat}}$

$\sum W_{S_{y}}=\left({\frac {\partial (\tau _{yx}\,v_{y})}{\partial x}}+{\frac {\partial (\sigma _{yy}\,v_{y})}{\partial y}}+{\frac {\partial (\tau _{yz}\,v_{y})}{\partial z}}\right)\ \mathrm {d} V$

${\begin{aligned}\sum W_{S}=&{\biggl (}{\frac {\partial (\sigma _{xx}\,v_{x})}{\partial x}}+{\frac {\partial (\tau _{xy}\,v_{x})}{\partial y}}+{\frac {\partial (\tau _{xz}\,v_{x})}{\partial z}}\\&+{\frac {\partial (\tau _{yx}\,v_{y})}{\partial x}}+{\frac {\partial (\sigma _{yy}\,v_{y})}{\partial y}}+{\frac {\partial (\tau _{yz}\,v_{y})}{\partial z}}\\&+{\frac {\partial (\tau _{zx}\,v_{z})}{\partial x}}+{\frac {\partial (\tau _{zyy}\,v_{z})}{\partial y}}+{\frac {\partial (\sigma _{zz}\,v_{z})}{\partial z}}{\biggr )}\mathrm {d} V\end{aligned}}$

$\sum W_{S}={\hbox{div}}\left({}^{t}{\overline {\overline {\tau }}}\times {\overrightarrow {v}}\right)\mathrm {d} V={\hbox{div}}{\begin{pmatrix}\sigma _{xx}\,v_{x}+\tau _{yx}\,v_{y}+\tau _{zx}\,v_{z}\\\tau _{xy}\,v_{x}+\sigma _{yy}\,v_{y}+\tau _{zy}\,v_{z}\\\tau _{xz}\,v_{x}+\tau _{yz}\,v_{y}+\sigma _{zz}\,v_{z}\end{pmatrix}}\mathrm {d} V$

${\begin{aligned}\sum W_{S}=&{\biggl (}{\frac {\partial \sigma _{xx}}{\partial x}}\,v_{x}+\sigma _{xx}\,{\frac {\partial v_{x}}{\partial x}}+{\frac {\partial \tau _{xy}}{\partial y}}\,v_{x}+\tau _{xy}\,{\frac {\partial v_{x}}{\partial y}}+{\frac {\partial \tau _{xz}}{\partial z}}\,v_{x}+\tau _{xz}\,{\frac {\partial v_{x}}{\partial z}}\\&+{\frac {\partial \tau _{yx}}{\partial x}}\,v_{y}+\tau _{yx}\,{\frac {\partial v_{y}}{\partial x}}+{\frac {\partial \sigma _{yy}}{\partial y}}\,v_{y}+\sigma _{yy}\,{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial \tau _{yz}}{\partial z}}\,v_{y}+\tau _{yz}\,{\frac {\partial v_{y}}{\partial z}}\\&+{\frac {\partial \tau _{zx}}{\partial x}}\,v_{z}+\tau _{zx}\,{\frac {\partial v_{z}}{\partial x}}+{\frac {\partial \tau _{zyy}}{\partial y}}\,v_{z}+\tau _{zyy}\,{\frac {\partial v_{z}}{\partial y}}+{\frac {\partial \sigma _{zz}}{\partial z}}\,v_{z}+\sigma _{zz}\,{\frac {\partial v_{z}}{\partial z}}{\biggr )}\mathrm {d} V\end{aligned}}$

${\begin{aligned}\sum W_{S}={\overrightarrow {v}}\cdot {\overrightarrow {\hbox{div}}}\ {\overline {\overline {\mathbf {\tau } }}}+&{\biggl (}\sigma _{xx}\,{\frac {\partial v_{x}}{\partial x}}+\tau _{xy}\,{\frac {\partial v_{x}}{\partial y}}+\tau _{xz}\,{\frac {\partial v_{x}}{\partial z}}\\&+\tau _{yx}\,{\frac {\partial v_{y}}{\partial x}}+\sigma _{yy}\,{\frac {\partial v_{y}}{\partial y}}+\tau _{yz}\,{\frac {\partial v_{y}}{\partial z}}\\&+\tau _{zx}\,{\frac {\partial v_{z}}{\partial x}}+\tau _{zyy}\,{\frac {\partial v_{z}}{\partial y}}+\sigma _{zz}\,{\frac {\partial v_{z}}{\partial z}}{\biggr )}\mathrm {d} V\end{aligned}}$

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Euler

En utilisant la dérivée particulaire :

${\frac {\mathrm {d} (\rho _{\text{tot}}\mathbf {v} )}{\mathrm {d} t}}={\frac {\partial (\rho _{\text{tot}}\mathbf {v} )}{\partial t}}+(\mathbf {v} \cdot \mathbf {\nabla } )(\rho _{\text{tot}}\mathbf {v} )=\rho _{\text{tot}}{\frac {\partial \mathbf {v} }{\partial t}}+\rho _{\text{tot}}\,(\mathbf {v} \cdot \mathbf {\nabla } )\mathbf {v} +{\frac {\partial \rho _{\text{tot}}}{\partial t}}\mathbf {v} +\mathbf {v} \,(\mathbf {v} \cdot \mathbf {\nabla } )\rho _{\text{tot}}$.

On ne tient compte que des deux premiers termes qui correspondent à la variation de quantité de mouvement par accélération locale $\left(\rho _{\text{tot}}{\frac {\partial \mathbf {v} }{\partial t}}\right)$ et par variation de masse locale $\left({\frac {\partial \rho _{\text{tot}}}{\partial t}}\mathbf {v} \right)$. Les termes convectifs sont négligés.

• La variation par accélération convective $\rho \,(\mathbf {v} \cdot \mathbf {\nabla } )\mathbf {v}$ est négligeable.
• La variation par variation de masse convective $\mathbf {v} \,(\mathbf {v} \cdot \mathbf {\nabla } )\mathbf {\rho }$ est négligeable.

La conservation de la quantité de mouvement s'écrit alors :

${\frac {\partial (\rho _{\text{tot}}\mathbf {v} )}{\partial t}}=-\nabla P_{\text{tot}}.\qquad (3)$