Détails des calculs de dérivées particulaires
modifier
d ( ρ v → ) d t = ∂ ( ρ v → ) ∂ t + ( v → ⋅ g r a d → ) ( ρ v → ) = ∂ ( ρ v → ) ∂ t + ( v → ⋅ ∇ → ) ( ρ v → ) {\displaystyle {\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)}
d ( ρ v → ) d t = ρ ∂ v → ∂ t + ∂ ρ ∂ t v → + ρ v x ∂ v → ∂ x + ∂ ρ ∂ x v x v → + ρ v y ∂ v → ∂ y + ∂ ρ ∂ y v y v → + ρ v z ∂ v → ∂ z + ∂ ρ ∂ z v z v → {\displaystyle {\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}}}
d ( ρ v → ) d t = ρ ∂ v → ∂ t + ∂ ρ ∂ t v → + ρ ( v x ∂ v → ∂ x + v y ∂ v → ∂ y + v z ∂ v → ∂ z ) + ( ∂ ρ ∂ x v x + ∂ ρ ∂ y v y + ∂ ρ ∂ z v z ) v → {\displaystyle {\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}}}
d ( ρ v → ) d t = ρ ∂ v → ∂ t + ∂ ρ ∂ t v → + ρ ( v → ⋅ g r a d → ) v → + ( v → ⋅ g r a d → ρ ) v → {\displaystyle {\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}}}
d ( ρ v → ) d t + ρ v → div v → = ρ ( ∂ v → ∂ t + ( v → ⋅ g r a d → ) v → ) + ( ∂ ρ ∂ t + v → ⋅ g r a d → ρ + ρ div v → ) v → {\displaystyle {\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}}}
d ( ρ v → ) d t + ρ v → div v → = ρ ( ∂ v → ∂ t + ( v → ⋅ g r a d → ) v → ) + ( ∂ ρ ∂ t + div ( ρ v → ) ) ⏟ = 0 v → {\displaystyle {\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}}}
d ( ρ v → ) d t + ρ v → div v → = ρ d v → d t {\displaystyle {\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}}}
d ( ρ v → ) d t = ρ d v → d t + d ρ d t v → {\displaystyle {\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}}}
d ( ρ v → ) d t + ρ v → div v → = ρ d v → d t + ( d ρ d t + ρ div v → ) ⏟ = 0 v → {\displaystyle {\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}}}
d ( ρ v → ) d t = ρ ( ∂ v → ∂ t + v x ∂ v → ∂ x + v y ∂ v → ∂ y + v z ∂ v → ∂ z ) + ( ∂ ρ ∂ t + ∂ ρ ∂ x v x + ∂ ρ ∂ y v y + ∂ ρ ∂ z v z ) v → {\displaystyle {\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}}}
d ( ρ v → ) d t = ρ ( ∂ v → ∂ t + ( v → ⋅ ∇ → ) ⋅ v → ) + d ρ d t v → {\displaystyle {\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}}}
d ( ρ v → ) d t + ρ v → div v → = ρ ( ∂ v → ∂ t + ( v → ⋅ ∇ → ) ⋅ v → ) + ρ v → div v → {\displaystyle {\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}}}
d ( ρ v → ) d t + ρ v → div v → = ρ ∂ v → ∂ t + ρ ( v → ⋅ ∇ → ) ⋅ v → {\displaystyle {\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}}}
∇ → ⋅ ( ρ v → ⊗ v → ) = d i v ( ρ v → ) v → + ρ ( v → ⋅ g r a d → ) v → {\displaystyle {\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}}}
∂ ( ρ v → ) ∂ t + ∇ → ⋅ ( ρ v → ⊗ v → ) = − ∇ → p + ∇ → ⋅ τ ¯ ¯ + ρ f → {\displaystyle {\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}}}
∇ → ⋅ ( ρ v → ⊗ v → ) = ∇ → ⋅ ( ( ρ v x ρ v y ρ v z ) × ( v x v y v z ) ) = ∇ → ⋅ ( ρ v x v x ρ v x v y ρ v x v z ρ v y v x ρ v y v y ρ v y v z ρ v z v x ρ v z v y ρ v z v z ) {\displaystyle {\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}}}
∇ → ⋅ ( ρ v → ⊗ v → ) = ( ∂ ( ρ v x v x ) ∂ x + ∂ ( ρ v x v y ) ∂ y + ∂ ( ρ v x v z ) ∂ z ∂ ( ρ v y v x ) ∂ x + ∂ ( ρ v y v y ) ∂ y + ∂ ( ρ v y v z ) ∂ z ∂ ( ρ v z v x ) ∂ x + ∂ ( ρ v z v y ) ∂ y + ∂ ( ρ v z v z ) ∂ z ) {\displaystyle {\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}}}
∇ → ⋅ ( ρ v → ⊗ v → ) = ( ∂ ρ ∂ x v x v x + ρ ∂ v x ∂ x v x + ρ ∂ v x ∂ x v x + ∂ ρ ∂ y v x v y + ρ ∂ v x ∂ y v y + ρ ∂ v y ∂ y v x + ∂ ρ ∂ z v x v z + ρ ∂ v x ∂ z v z + ρ ∂ v z ∂ z v x ∂ ρ ∂ x v y v x + ρ ∂ v y ∂ x v x + ρ ∂ v x ∂ x v y + ∂ ρ ∂ y v y v y + ρ ∂ v y ∂ y v y + ρ ∂ v y ∂ y v y + ∂ ρ ∂ z v y v z + ρ ∂ v y ∂ z v z + ρ ∂ v z ∂ v z v y ∂ ρ ∂ x v z v x + ρ ∂ v z ∂ x v x + ρ ∂ v x ∂ x v z + ∂ ρ ∂ y v z v y + ρ ∂ v z ∂ y v y + ρ ∂ v y ∂ y v z + ∂ ρ ∂ z v z v z + ρ ∂ v z ∂ z v z + ρ ∂ v z ∂ z v z ) {\displaystyle {\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}}}
∇ → ⋅ ( ρ v → ⊗ v → ) = ( ( ρ d i v v → + g r a d → ρ ⋅ v → ) v x + ρ g r a d → v x ⋅ v → ( ρ d i v v → + g r a d → ρ ⋅ v → ) v y + ρ g r a d → v y ⋅ v → ( ρ d i v v → + g r a d → ρ ⋅ v → ) v z + ρ g r a d → v z ⋅ v → ) {\displaystyle {\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}}}
∇ → ⋅ ( ρ v → ⊗ v → ) = ( d i v ( ρ v → ) v x + ρ g r a d → v x ⋅ v → d i v ( ρ v → ) v y + ρ g r a d → v y ⋅ v → d i v ( ρ v → ) v z + ρ g r a d → v z ⋅ v → ) {\displaystyle {\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}}}
∇ → ⋅ ( ρ v → ⊗ v → ) = d i v ( ρ v → ) v → + ρ ( v → ⋅ g r a d → ) v → {\displaystyle {\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}}}
∇ → ⋅ ( ρ v → ⊗ v → ) = − ∂ ρ ∂ t v → + ρ ( v → ⋅ g r a d → ) v → {\displaystyle {\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}}}
∂ ( ρ v → ) ∂ t + ∇ → ⋅ ( ρ v → ⊗ v → ) = ρ ∂ v → ∂ t + ∂ ρ ∂ t v → − ∂ ρ ∂ t v → + ρ ( v → ⋅ g r a d → ) v → {\displaystyle {\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}}}
∂ ( ρ v → ) ∂ t + ∇ → ⋅ ( ρ v → ⊗ v → ) = ρ ∂ v → ∂ t + ρ ( v → ⋅ g r a d → ) v → {\displaystyle {\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}}}
( v → ⋅ ∇ → ) ρ v → = ( ( v → ⋅ ∇ → ) ρ v x ( v → ⋅ ∇ → ) ρ v y ( v → ⋅ ∇ → ) ρ v z ) = ( v x ∂ ρ v x ∂ x + v y ∂ ρ v x ∂ y + v z ∂ ρ v x ∂ z v x ∂ ρ v y ∂ x + v y ∂ ρ v y ∂ y + v z ∂ ρ v y ∂ z v x ∂ ρ v z ∂ x + v y ∂ ρ v z ∂ y + v z ∂ ρ v z ∂ z ) {\displaystyle \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
modifier
Force exercée sur une surface S {\displaystyle S}
modifier
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 :
d F S → = τ ¯ ¯ × d S → = ( σ x x τ x y τ x z τ y x σ y y τ y z τ z x τ z y σ z z ) × ( d S x d S y d S z ) = ( σ x x d S x + τ x y d S y + τ x z d S z τ y x d S x + σ y y d S y + τ y z d S z τ z x d S x + τ z y d S y + σ z z d S z ) {\displaystyle {\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 d V {\displaystyle \mathrm {d} V}
modifier
Les forces surfaciques qui s'appliquent sur les faces d'un élément de volume d V = d x d y d z {\displaystyle \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 d S → = d S x e x → = d y d z e x → {\displaystyle {\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 d F 1 → = σ x x d y d z e x → {\displaystyle {\overrightarrow {\mathrm {d} F_{1}}}=\sigma _{xx}\,\mathrm {d} y\,\mathrm {d} z\,{\overrightarrow {e_{x}}}} ;
deux forces dans le plan de la surface :
d F 2 → = τ y x d y d z e y → {\displaystyle {\overrightarrow {\mathrm {d} F_{2}}}=\tau _{yx}\,\mathrm {d} y\,\mathrm {d} z\,{\overrightarrow {e_{y}}}} dans la direction de l'axe ( O y ) {\displaystyle (Oy)} ;
d F 2 → = τ z x d y d z e z → {\displaystyle {\overrightarrow {\mathrm {d} F_{2}}}=\tau _{zx}\,\mathrm {d} y\,\mathrm {d} z\,{\overrightarrow {e_{z}}}} dans la direction de l'axe ( O y ) {\displaystyle (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 :
d F V → = ∑ i d F i → = ( [ σ x x ( x + d x ) − σ x x ( x ) ] d y d z + [ τ x y ( y + d y ) − τ x y ( y ) ] d x d z + [ τ x z ( z + d z ) − τ x z ( z ) ] d x d y [ τ y x ( x + d x ) − τ y x ( x ) ] d y d z + [ σ y y ( y + d y ) − σ y y ( y ) ] d x d z + [ τ y z ( z + d z ) − τ y z ( z ) ] d x d y [ τ z x ( x + d x ) − τ z x ( x ) ] d y d z + [ τ z y ( y + d y ) − τ z y ( y ) ] d x d z + [ σ z z ( z + d z ) − σ z z ( z ) ] d x d y ) {\displaystyle {\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}}} ,
d F V → = ( ∂ σ x x ∂ x d x d y d z + ∂ τ x y ∂ y d x d y d z + ∂ τ x z ∂ z d x d y d z ∂ τ y x ∂ y d x d y d z + ∂ σ y y ∂ y d x d y d z + ∂ τ y z ∂ y d x d y d z ∂ τ z x ∂ y d x d y d z + ∂ τ z y ∂ y d x d y d z + ∂ σ z z ∂ y d x d y d z ) {\displaystyle {\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}}}
d F V → = d i v → ( σ x x τ x y τ x z τ y x σ y y τ y z τ z x τ z y σ z z ) d V = d i v → τ ¯ ¯ d V {\displaystyle {\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}
σ ¯ ¯ = − P δ ¯ ¯ + τ ¯ ¯ {\displaystyle {\overline {\overline {\sigma }}}=-P\,{\overline {\overline {\delta }}}+{\overline {\overline {\tau }}}}
( σ x x τ x y τ x z τ y x σ y y τ y z τ z x τ z y σ z z ) = − P ( 1 0 0 0 1 0 0 0 1 ) + ( τ x x τ x y τ x z τ y x τ y y τ y z τ z x τ z y τ z z ) {\displaystyle {\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}}}
τ x x + τ y y + τ z z = 0 {\displaystyle \tau _{xx}+\tau _{yy}+\tau _{zz}=0}
τ i j = μ ( ∂ v i ∂ x j + ∂ v j ∂ x i ) + μ ′ d i v v → δ i j {\displaystyle \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}}
μ {\displaystyle \mu } : viscosité dynamique
μ ′ {\displaystyle \mu '} : coefficient de seconde viscosité
tenseur de contraintes
Recherche ː travail des forces surfaciques
modifier
eq de l'énergie ; Tenseur des contraintes
Produit matriciel ; Matrice transposée
div → τ ¯ ¯ = ( ∂ σ x x ∂ x + ∂ τ x y ∂ y + ∂ τ x z ∂ z ∂ τ y x ∂ x + ∂ σ y y ∂ y + ∂ τ y z ∂ z ∂ τ z x ∂ x + ∂ τ z y ∂ y + ∂ σ z z ∂ z ) {\displaystyle {\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
∑ W S y = [ ( τ y x v y ) x + d x − ( τ y x v y ) x ] d y d z + [ ( σ y y v y ) y + d y − ( σ y y v y ) y ] d x d z + [ ( τ y z v y ) z + d z − ( τ y z v y ) z ] d x d y {\displaystyle {\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}}}
∑ W S y = ( ∂ ( τ y x v y ) ∂ x + ∂ ( σ y y v y ) ∂ y + ∂ ( τ y z v y ) ∂ z ) d V {\displaystyle \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}
∑ W S = ( ∂ ( σ x x v x ) ∂ x + ∂ ( τ x y v x ) ∂ y + ∂ ( τ x z v x ) ∂ z + ∂ ( τ y x v y ) ∂ x + ∂ ( σ y y v y ) ∂ y + ∂ ( τ y z v y ) ∂ z + ∂ ( τ z x v z ) ∂ x + ∂ ( τ z y y v z ) ∂ y + ∂ ( σ z z v z ) ∂ z ) d V {\displaystyle {\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}}}
∑ W S = div ( t τ ¯ ¯ × v → ) d V = div ( σ x x v x + τ y x v y + τ z x v z τ x y v x + σ y y v y + τ z y v z τ x z v x + τ y z v y + σ z z v z ) d V {\displaystyle \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}
∑ W S = ( ∂ σ x x ∂ x v x + σ x x ∂ v x ∂ x + ∂ τ x y ∂ y v x + τ x y ∂ v x ∂ y + ∂ τ x z ∂ z v x + τ x z ∂ v x ∂ z + ∂ τ y x ∂ x v y + τ y x ∂ v y ∂ x + ∂ σ y y ∂ y v y + σ y y ∂ v y ∂ y + ∂ τ y z ∂ z v y + τ y z ∂ v y ∂ z + ∂ τ z x ∂ x v z + τ z x ∂ v z ∂ x + ∂ τ z y y ∂ y v z + τ z y y ∂ v z ∂ y + ∂ σ z z ∂ z v z + σ z z ∂ v z ∂ z ) d V {\displaystyle {\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}}}
∑ W S = v → ⋅ div → τ ¯ ¯ + ( σ x x ∂ v x ∂ x + τ x y ∂ v x ∂ y + τ x z ∂ v x ∂ z + τ y x ∂ v y ∂ x + σ y y ∂ v y ∂ y + τ y z ∂ v y ∂ z + τ z x ∂ v z ∂ x + τ z y y ∂ v z ∂ y + σ z z ∂ v z ∂ z ) d V {\displaystyle {\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}}}
En utilisant la dérivée particulaire :
d ( ρ tot v ) d t = ∂ ( ρ tot v ) ∂ t + ( v ⋅ ∇ ) ( ρ tot v ) = ρ tot ∂ v ∂ t + ρ tot ( v ⋅ ∇ ) v + ∂ ρ tot ∂ t v + v ( v ⋅ ∇ ) ρ tot {\displaystyle {\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 ( ρ tot ∂ v ∂ t ) {\displaystyle \left(\rho _{\text{tot}}{\frac {\partial \mathbf {v} }{\partial t}}\right)} et par variation de masse locale ( ∂ ρ tot ∂ t v ) {\displaystyle \left({\frac {\partial \rho _{\text{tot}}}{\partial t}}\mathbf {v} \right)} . Les termes convectifs sont négligés.
La variation par accélération convective ρ ( v ⋅ ∇ ) v {\displaystyle \rho \,(\mathbf {v} \cdot \mathbf {\nabla } )\mathbf {v} } est négligeable.
La variation par variation de masse convective v ( v ⋅ ∇ ) ρ {\displaystyle \mathbf {v} \,(\mathbf {v} \cdot \mathbf {\nabla } )\mathbf {\rho } } est négligeable. La conservation de la quantité de mouvement s'écrit alors :
∂ ( ρ tot v ) ∂ t = − ∇ P tot . ( 3 ) {\displaystyle {\frac {\partial (\rho _{\text{tot}}\mathbf {v} )}{\partial t}}=-\nabla P_{\text{tot}}.\qquad (3)}