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}}}
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
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}
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}}}}
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}}}}
(
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}}}}
(
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 }
μ
′
{\displaystyle \mu '}
tenseur de contraintes
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)}
(
∂
ρ
tot
∂
t
v
)
{\displaystyle \left({\frac {\partial \rho _{\text{tot}}}{\partial t}}\mathbf {v} \right)}
La variation par accélération convective
ρ
(
v
⋅
∇
)
v
{\displaystyle \rho \,(\mathbf {v} \cdot \mathbf {\nabla } )\mathbf {v} }
La variation par variation de masse convective
v
(
v
⋅
∇
)
ρ
{\displaystyle \mathbf {v} \,(\mathbf {v} \cdot \mathbf {\nabla } )\mathbf {\rho } }
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)}
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c8fee6e7b8a7d732c12302f671ddd7d9f8400e6)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a222d644aac6aeed4cd7a6fc72f08798bd3bd52)
