Origem: Wikilivros, livros abertos por um mundo aberto.
Um líquido Newtoniano é um fluido incompressível com viscosidade constante. Muitos líquidos de interesse são líquidos Newtonianos. As equações básicas apresentam uma forma muito simplificada quando comparadas às equações gerais, embora não tanto quanto no caso do líquido ideal.
Portanto, para um líquido Netoniano, teremos
ρ
=
ρ
0
μ
=
μ
0
{\displaystyle \rho \;=\;\rho _{0}\qquad \mu \;=\;\mu _{0}}
A equação de continuidade em forma diferencial
∂
ρ
∂
t
+
(
∇
⋅
(
ρ
v
→
)
)
=
0
{\displaystyle {\frac {\partial \rho }{\partial t}}\;+\;(\nabla \cdot (\rho {\vec {v}}))\;=\;0}
∇
⋅
v
→
=
0
{\displaystyle \nabla \cdot {\vec {v}}\;=\;0}
Conforme visto anteriormente , as equações de Navier-Stokes
(
∂
∂
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)}
∂
v
z
∂
z
+
∂
v
x
∂
x
+
∂
v
y
∂
y
=
0
{\displaystyle {\frac {\partial v_{z}}{\partial z}}\;+\;{\frac {\partial v_{x}}{\partial x}}\;+\;{\frac {\partial v_{y}}{\partial y}}\;=\;0}
∂
∂
z
(
∂
v
z
∂
x
)
=
∂
2
v
z
∂
z
∂
x
=
∂
2
v
z
∂
x
∂
z
=
∂
∂
x
(
∂
v
z
∂
z
)
{\displaystyle {\frac {\partial }{\partial z}}\left({\frac {\partial v_{z}}{\partial x}}\right)\;=\;{\frac {\partial ^{2}v_{z}}{\partial z\partial x}}\;=\;{\frac {\partial ^{2}v_{z}}{\partial x\partial z}}\;=\;{\frac {\partial }{\partial x}}\left({\frac {\partial v_{z}}{\partial z}}\right)}
∂
∂
z
(
∂
v
z
∂
x
)
=
∂
∂
x
(
−
∂
v
y
∂
y
−
∂
v
x
∂
x
)
=
−
∂
2
v
y
∂
x
∂
y
−
∂
2
v
x
∂
x
2
{\displaystyle {\frac {\partial }{\partial z}}\left({\frac {\partial v_{z}}{\partial x}}\right)\;=\;{\frac {\partial }{\partial x}}\left(\;-\;{\frac {\partial v_{y}}{\partial y}}\;-\;{\frac {\partial v_{x}}{\partial x}}\right)\;=\;\;-\;{\frac {\partial ^{2}v_{y}}{\partial x\partial y}}\;-\;{\frac {\partial ^{2}v_{x}}{\partial x^{2}}}}
−
∂
p
∂
x
+
μ
0
(
−
2
3
⋅
0
+
2
∂
2
v
x
∂
x
2
+
∂
2
v
x
∂
y
2
+
∂
2
v
y
∂
y
∂
x
+
∂
2
v
x
∂
z
2
−
∂
2
v
y
∂
x
∂
y
−
∂
2
v
x
∂
x
2
)
=
{\displaystyle -\;{\frac {\partial p}{\partial x}}\;+\;\mu _{0}\left(\;-\;{\frac {2}{3}}\cdot 0\;+\;2\;{\frac {\partial ^{2}v_{x}}{\partial x^{2}}}\;+\;{\frac {\partial ^{2}v_{x}}{\partial y^{2}}}\;+\;{\frac {\partial ^{2}v_{y}}{\partial y\partial x}}\;+\;{\frac {\partial ^{2}v_{x}}{\partial z^{2}}}\;-\;{\frac {\partial ^{2}v_{y}}{\partial x\partial y}}\;-\;{\frac {\partial ^{2}v_{x}}{\partial x^{2}}}\right)\;=\;}
=
ρ
0
(
v
x
∂
v
x
∂
x
+
v
y
∂
v
x
∂
y
+
v
z
∂
v
x
∂
z
+
∂
v
x
∂
t
)
{\displaystyle \;=\;\rho _{0}\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)}
−
∂
p
∂
x
+
μ
0
(
∂
2
v
x
∂
x
2
+
∂
2
v
x
∂
y
2
+
∂
2
v
x
∂
z
2
)
=
ρ
0
(
v
x
∂
v
x
∂
x
+
v
y
∂
v
x
∂
y
+
v
z
∂
v
x
∂
z
+
∂
v
x
∂
t
)
{\displaystyle -\;{\frac {\partial p}{\partial x}}\;+\;\mu _{0}\left({\frac {\partial ^{2}v_{x}}{\partial x^{2}}}\;+\;{\frac {\partial ^{2}v_{x}}{\partial y^{2}}}\;+\;{\frac {\partial ^{2}v_{x}}{\partial z^{2}}}\right)\;=\;\rho _{0}\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)}
−
∂
p
∂
y
+
μ
0
(
∂
2
v
y
∂
x
2
+
∂
2
v
y
∂
y
2
+
∂
2
v
y
∂
z
2
)
=
ρ
0
(
v
y
∂
v
x
∂
x
+
v
y
∂
v
y
∂
y
+
v
z
∂
v
y
∂
z
+
∂
v
x
∂
t
)
{\displaystyle -\;{\frac {\partial p}{\partial y}}\;+\;\mu _{0}\left({\frac {\partial ^{2}v_{y}}{\partial x^{2}}}\;+\;{\frac {\partial ^{2}v_{y}}{\partial y^{2}}}\;+\;{\frac {\partial ^{2}v_{y}}{\partial z^{2}}}\right)\;=\;\rho _{0}\left(v_{y}{\frac {\partial v_{x}}{\partial x}}\;+\;v_{y}{\frac {\partial v_{y}}{\partial y}}\;+\;v_{z}{\frac {\partial v_{y}}{\partial z}}\;+\;{\frac {\partial v_{x}}{\partial t}}\right)}
−
∂
p
∂
z
+
μ
0
(
∂
2
v
z
∂
x
2
+
∂
2
v
z
∂
y
2
+
∂
2
v
z
∂
z
2
)
+
ρ
0
g
=
ρ
0
(
v
x
∂
v
z
∂
x
+
v
y
∂
v
z
∂
y
+
v
z
∂
v
z
∂
z
+
∂
v
z
∂
t
)
{\displaystyle \;-\;\;{\frac {\partial p}{\partial z}}\;+\;\mu _{0}\left({\frac {\partial ^{2}v_{z}}{\partial x^{2}}}\;+\;{\frac {\partial ^{2}v_{z}}{\partial y^{2}}}\;+\;{\frac {\partial ^{2}v_{z}}{\partial z^{2}}}\right)\;+\;\rho _{0}g\;=\;\rho _{0}\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)}
