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Conforme visto anteriormente , a equação de conservação do momento linear em forma diferencial, para coordenadas cartesianas, pode ser escrita
(
∂
σ
x
x
∂
x
+
∂
τ
y
x
∂
y
+
∂
τ
z
x
∂
z
)
u
→
x
+
(
∂
τ
x
y
∂
x
+
∂
σ
y
y
∂
y
+
∂
τ
z
y
∂
z
)
u
→
y
+
(
∂
τ
x
z
∂
x
+
∂
τ
y
z
∂
y
+
∂
σ
z
z
∂
z
−
ρ
g
)
u
→
z
=
ρ
(
v
→
⋅
∇
v
→
+
∂
v
→
∂
t
)
{\displaystyle \left({\frac {\partial \sigma _{xx}}{\partial x}}\;+\;{\frac {\partial \tau _{yx}}{\partial y}}\;+\;{\frac {\partial \tau _{zx}}{\partial z}}\right)\;{\vec {u}}_{x}\;+\;\left({\frac {\partial \tau _{xy}}{\partial x}}\;+\;{\frac {\partial \sigma _{yy}}{\partial y}}\;+\;{\frac {\partial \tau _{zy}}{\partial z}}\right)\;{\vec {u}}_{y}\;+\;\left({\frac {\partial \tau _{xz}}{\partial x}}\;+\;{\frac {\partial \tau _{yz}}{\partial y}}\;+\;{\frac {\partial \sigma _{zz}}{\partial z}}\;-\;\rho g\right)\;{\vec {u}}_{z}\;=\;\rho \left({\vec {v}}\cdot \nabla {\vec {v}}\;+\;{\frac {\partial {\vec {v}}}{\partial t}}\right)}
análise cinemática anterior , as tensões de cisalhamento estão relacionadas com a velocidade da seguinte maneira
∂
v
x
∂
y
+
∂
v
y
∂
x
=
1
μ
τ
y
x
=
1
μ
τ
x
y
{\displaystyle {\frac {\partial v_{x}}{\partial y}}\;+\;{\frac {\partial v_{y}}{\partial x}}\;=\;{\frac {1}{\mu }}\;\tau _{yx}\;=\;{\frac {1}{\mu }}\;\tau _{xy}}
∂
v
x
∂
z
+
∂
v
z
∂
x
=
1
μ
τ
z
x
=
1
μ
τ
x
z
{\displaystyle {\frac {\partial v_{x}}{\partial z}}\;+\;{\frac {\partial v_{z}}{\partial x}}\;=\;{\frac {1}{\mu }}\;\tau _{zx}\;=\;{\frac {1}{\mu }}\;\tau _{xz}}
∂
v
z
∂
y
+
∂
v
y
∂
z
=
1
μ
τ
y
z
=
1
μ
τ
z
y
{\displaystyle {\frac {\partial v_{z}}{\partial y}}\;+\;{\frac {\partial v_{y}}{\partial z}}\;=\;{\frac {1}{\mu }}\;\tau _{yz}\;=\;{\frac {1}{\mu }}\;\tau _{zy}}
análise estática anterior
σ
x
x
=
−
p
−
2
3
μ
(
∂
v
x
∂
x
+
∂
v
y
∂
y
+
∂
v
z
∂
z
)
+
2
μ
∂
v
x
∂
x
{\displaystyle \sigma _{xx}\;=\;-\;p\;-\;{\frac {2}{3}}\mu \left({\frac {\partial v_{x}}{\partial x}}\;+\;{\frac {\partial v_{y}}{\partial y}}\;+\;{\frac {\partial v_{z}}{\partial z}}\right)\;+\;2\mu {\frac {\partial v_{x}}{\partial x}}}
σ
y
y
=
−
p
−
2
3
μ
(
∂
v
x
∂
x
+
∂
v
y
∂
y
+
∂
v
z
∂
z
)
+
2
μ
∂
v
y
∂
y
{\displaystyle \sigma _{yy}\;=\;-\;p\;-\;{\frac {2}{3}}\mu \left({\frac {\partial v_{x}}{\partial x}}\;+\;{\frac {\partial v_{y}}{\partial y}}\;+\;{\frac {\partial v_{z}}{\partial z}}\right)\;+\;2\mu {\frac {\partial v_{y}}{\partial y}}}
σ
z
z
=
−
p
−
2
3
μ
(
∂
v
x
∂
x
+
∂
v
y
∂
y
+
∂
v
z
∂
z
)
+
2
μ
∂
v
z
∂
z
{\displaystyle \sigma _{zz}\;=\;-\;p\;-\;{\frac {2}{3}}\mu \left({\frac {\partial v_{x}}{\partial x}}\;+\;{\frac {\partial v_{y}}{\partial y}}\;+\;{\frac {\partial v_{z}}{\partial z}}\right)\;+\;2\mu {\frac {\partial v_{z}}{\partial z}}}
equações de Navier-Stokes .
Na direção do eixo X, teremos:
(
∂
∂
x
(
−
p
−
2
3
μ
(
∂
v
x
∂
x
+
∂
v
y
∂
y
+
∂
v
z
∂
z
)
+
2
μ
∂
v
x
∂
x
)
+
∂
∂
y
(
μ
∂
v
x
∂
y
+
μ
∂
v
y
∂
x
)
+
∂
∂
z
(
μ
∂
v
x
∂
z
+
μ
∂
v
z
∂
x
)
)
=
{\displaystyle \left({\frac {\partial }{\partial x}}\left(-\;p\;-\;{\frac {2}{3}}\mu \left({\frac {\partial v_{x}}{\partial x}}\;+\;{\frac {\partial v_{y}}{\partial y}}\;+\;{\frac {\partial v_{z}}{\partial z}}\right)\;+\;2\mu {\frac {\partial v_{x}}{\partial x}}\right)\;+\;{\frac {\partial }{\partial y}}\left(\mu {\frac {\partial v_{x}}{\partial y}}\;+\;\mu {\frac {\partial v_{y}}{\partial x}}\right)\;+\;{\frac {\partial }{\partial z}}\left(\mu {\frac {\partial v_{x}}{\partial z}}\;+\;\mu {\frac {\partial v_{z}}{\partial x}}\right)\right)\;=\;}
=
ρ
(
v
x
∂
v
x
∂
x
+
v
y
∂
v
x
∂
y
+
v
z
∂
v
x
∂
z
+
∂
v
x
∂
t
)
{\displaystyle \;=\;\rho \left(v_{x}{\frac {\partial v_{x}}{\partial x}}\;+\;v_{y}{\frac {\partial v_{x}}{\partial y}}\;+\;v_{z}{\frac {\partial v_{x}}{\partial z}}\;+\;{\frac {\partial v_{x}}{\partial t}}\right)}
(
∂
∂
y
(
−
p
−
2
3
μ
(
∂
v
x
∂
x
+
∂
v
y
∂
y
+
∂
v
z
∂
z
)
+
2
μ
∂
v
y
∂
y
)
+
∂
∂
x
(
μ
∂
v
x
∂
y
+
μ
∂
v
y
∂
x
)
+
∂
∂
z
(
μ
∂
v
z
∂
y
+
μ
∂
v
y
∂
z
)
)
=
{\displaystyle \left({\frac {\partial }{\partial y}}\left(-\;p\;-\;{\frac {2}{3}}\mu \left({\frac {\partial v_{x}}{\partial x}}\;+\;{\frac {\partial v_{y}}{\partial y}}\;+\;{\frac {\partial v_{z}}{\partial z}}\right)\;+\;2\mu {\frac {\partial v_{y}}{\partial y}}\right)\;+\;{\frac {\partial }{\partial x}}\left(\mu {\frac {\partial v_{x}}{\partial y}}\;+\;\mu {\frac {\partial v_{y}}{\partial x}}\right)\;+\;{\frac {\partial }{\partial z}}\left(\mu {\frac {\partial v_{z}}{\partial y}}\;+\;\mu {\frac {\partial v_{y}}{\partial z}}\right)\right)\;=\;}
=
ρ
(
v
x
∂
v
y
∂
x
+
v
y
∂
v
y
∂
y
+
v
z
∂
v
y
∂
z
+
∂
v
y
∂
t
)
{\displaystyle \;=\;\rho \left(v_{x}{\frac {\partial v_{y}}{\partial x}}\;+\;v_{y}{\frac {\partial v_{y}}{\partial y}}\;+\;v_{z}{\frac {\partial v_{y}}{\partial z}}\;+\;{\frac {\partial v_{y}}{\partial t}}\right)}
(
∂
∂
z
(
−
p
−
2
3
μ
(
∂
v
x
∂
x
+
∂
v
y
∂
y
+
∂
v
z
∂
z
)
+
2
μ
∂
v
z
∂
z
)
+
∂
∂
x
(
μ
∂
v
x
∂
z
+
μ
∂
v
z
∂
x
)
+
∂
∂
y
(
μ
∂
v
z
∂
y
+
μ
∂
v
y
∂
z
)
)
−
ρ
g
=
{\displaystyle \left({\frac {\partial }{\partial z}}\left(-\;p\;-\;{\frac {2}{3}}\mu \left({\frac {\partial v_{x}}{\partial x}}\;+\;{\frac {\partial v_{y}}{\partial y}}\;+\;{\frac {\partial v_{z}}{\partial z}}\right)\;+\;2\mu {\frac {\partial v_{z}}{\partial z}}\right)\;+\;{\frac {\partial }{\partial x}}\left(\mu {\frac {\partial v_{x}}{\partial z}}\;+\;\mu {\frac {\partial v_{z}}{\partial x}}\right)\;+\;{\frac {\partial }{\partial y}}\left(\mu {\frac {\partial v_{z}}{\partial y}}\;+\;\mu {\frac {\partial v_{y}}{\partial z}}\right)\right)\;-\;\rho g\;=\;}
=
ρ
(
v
x
∂
v
z
∂
x
+
v
y
∂
v
z
∂
y
+
v
z
∂
v
z
∂
z
+
∂
v
z
∂
t
)
{\displaystyle \;=\;\rho \left(v_{x}{\frac {\partial v_{z}}{\partial x}}\;+\;v_{y}{\frac {\partial v_{z}}{\partial y}}\;+\;v_{z}{\frac {\partial v_{z}}{\partial z}}\;+\;{\frac {\partial v_{z}}{\partial t}}\right)}
