Equações em forma adimensional
editar
Equação de continuidade
editar
A equação de continuidade em forma diferencial
∂
ρ
∂
t
+
(
∇
⋅
(
ρ
v
→
)
)
=
0
{\displaystyle {\frac {\partial \rho }{\partial t}}\;+\;(\nabla \cdot (\rho {\vec {v}}))\;=\;0}
-1 ] = M·L-3 ·T-1 . Para obter uma equação correspondente adimensional, substituímos a densidade ρ por ρ* ·ρr e a velocidade v por v* ·vr , onde ρr e vr são valores de referência e ρ* e v* são números puros. Aplicamos um procedimento equivalente também aos comprimentos; por exemplo, substituímos x por x* ·Lr , onde Lr é um comprimento de referência. Assim:
∂
ρ
∂
t
=
∂
(
ρ
∗
⋅
ρ
r
)
∂
(
t
∗
⋅
t
r
)
=
ρ
r
t
r
∂
ρ
∗
∂
t
∗
{\displaystyle {\frac {\partial \rho }{\partial t}}\;=\;{\frac {\partial (\rho ^{*}\cdot \rho _{r})}{\partial (t^{*}\cdot t_{r})}}\;=\;{\frac {\rho _{r}}{t_{r}}}\;{\frac {\partial \rho ^{*}}{\partial t^{*}}}}
∇
⋅
(
ρ
v
→
)
=
∂
(
ρ
v
x
)
∂
x
+
∂
(
ρ
v
y
)
∂
y
+
∂
(
ρ
v
z
)
∂
z
=
∂
[
(
ρ
∗
⋅
ρ
r
)
(
v
x
∗
⋅
v
r
)
]
∂
(
x
∗
⋅
L
r
)
+
∂
[
(
ρ
∗
⋅
ρ
r
)
(
v
y
∗
⋅
v
r
)
]
∂
(
y
∗
⋅
L
r
)
+
∂
[
(
ρ
∗
⋅
ρ
r
)
(
v
z
∗
⋅
v
r
)
]
∂
(
z
∗
⋅
L
r
)
{\displaystyle \nabla \cdot (\rho {\vec {v}})\;=\;{\frac {\partial (\rho v_{x})}{\partial x}}\;+\;{\frac {\partial (\rho v_{y})}{\partial y}}\;+\;{\frac {\partial (\rho v_{z})}{\partial z}}\;=\;{\frac {\partial [(\rho ^{*}\cdot \rho _{r})(v_{x}^{*}\cdot v_{r})]}{\partial (x^{*}\cdot L_{r})}}\;+\;{\frac {\partial [(\rho ^{*}\cdot \rho _{r})(v_{y}^{*}\cdot v_{r})]}{\partial (y^{*}\cdot L_{r})}}\;+\;{\frac {\partial [(\rho ^{*}\cdot \rho _{r})(v_{z}^{*}\cdot v_{r})]}{\partial (z^{*}\cdot L_{r})}}}
=
ρ
r
v
r
L
r
(
∂
(
ρ
∗
v
x
∗
)
∂
x
∗
+
∂
(
ρ
∗
v
y
∗
)
∂
y
∗
+
∂
(
ρ
∗
v
z
∗
)
∂
z
∗
)
{\displaystyle \;=\;{\frac {\rho _{r}v_{r}}{L_{r}}}\;\left({\frac {\partial (\rho ^{*}v_{x}^{*})}{\partial x^{*}}}\;+\;{\frac {\partial (\rho ^{*}v_{y}^{*})}{\partial y^{*}}}\;+\;{\frac {\partial (\rho ^{*}v_{z}^{*})}{\partial z^{*}}}\right)}
r tal que
t
r
=
L
r
v
r
{\displaystyle t_{r}\;=\;{\frac {L_{r}}{v_{r}}}}
∂
ρ
∗
∂
t
∗
+
∂
(
ρ
∗
v
x
∗
)
∂
x
∗
+
∂
(
ρ
∗
v
y
∗
)
∂
y
∗
+
∂
(
ρ
∗
v
z
∗
)
∂
z
∗
=
0
{\displaystyle {\frac {\partial \rho ^{*}}{\partial t^{*}}}\;+\;{\frac {\partial (\rho ^{*}v_{x}^{*})}{\partial x^{*}}}\;+\;{\frac {\partial (\rho ^{*}v_{y}^{*})}{\partial y^{*}}}\;+\;{\frac {\partial (\rho ^{*}v_{z}^{*})}{\partial z^{*}}}\;=\;0}
∂
ρ
∗
∂
t
∗
+
(
∇
∗
⋅
(
ρ
∗
v
→
∗
)
)
=
0
{\displaystyle {\frac {\partial \rho ^{*}}{\partial t^{*}}}\;+\;(\nabla \;^{*}\cdot (\rho ^{*}{\vec {v}}\;^{*}))\;=\;0}
∇
∗
⋅
F
=
∂
F
∂
x
∗
+
∂
F
∂
y
∗
+
∂
F
∂
z
∗
{\displaystyle \nabla \;^{*}\cdot F={\dfrac {\partial F}{\partial x^{*}}}+{\dfrac {\partial F}{\partial y^{*}}}+{\dfrac {\partial F}{\partial z^{*}}}}
Equação de Bernoulli
editar
Para escrever a equação de Bernoulli em forma adimensional, a pressão p deve ser substituída por p* ·pr , com p_r = ρvr 2 = ρ* ·ρr ·vr 2 . Assim, se fizermos
v
r
=
g
L
r
{\displaystyle v_{r}\;=\;{\sqrt {gL_{r}}}\;}
p
ρ
+
v
2
2
+
g
z
=
c
o
n
s
t
a
n
t
e
{\displaystyle {\frac {p}{\rho }}\;+\;{\frac {v^{2}}{2}}\;+\;\ gz\;=\;constante}
p
∗
⋅
p
r
ρ
∗
⋅
ρ
r
+
(
v
∗
⋅
v
r
)
2
2
+
g
z
∗
⋅
L
r
=
c
o
n
s
t
a
n
t
e
{\displaystyle {\frac {p^{*}\cdot p_{r}}{\rho ^{*}\cdot \rho _{r}}}\;+\;{\frac {(v^{*}\cdot v_{r})^{2}}{2}}\;+\;\ gz^{*}\cdot L_{r}\;=\;constante}
p
∗
(
ρ
∗
⋅
ρ
r
⋅
v
r
2
)
ρ
∗
⋅
ρ
r
+
v
r
2
(
v
∗
)
2
2
+
v
r
2
⋅
z
∗
=
c
o
n
s
t
a
n
t
e
{\displaystyle {\frac {p^{*}(\rho ^{*}\cdot \rho _{r}\cdot v_{r}^{2})}{\rho ^{*}\cdot \rho _{r}}}\;+\;v_{r}^{2}\;{\frac {(v^{*})^{2}}{2}}\;+\;\ v_{r}^{2}\cdot z^{*}\;=\;constante}
p
∗
+
(
v
∗
)
2
2
+
z
∗
=
c
o
n
s
t
a
n
t
e
{\displaystyle p^{*}\;+\;{\frac {(v^{*})^{2}}{2}}\;+\;\ z^{*}\;=\;constante}
Equações de Navier-Stokes para um líquido Newtoniano
editar
Considerando-se o eixo Z como apontando para cima e desprezando-se um dos eixos horizontais, o que é válido para a maioria das situações, as equações de Navier-Stokes para um líquido Newtoniano (densidade e viscosidade constantes), têm a forma seguinte:
ρ
0
(
∂
v
x
∂
t
+
v
x
∂
v
x
∂
x
+
v
z
∂
v
x
∂
z
)
=
−
∂
p
∂
x
+
μ
0
(
∂
2
v
x
∂
x
2
+
∂
2
v
x
∂
z
2
)
{\displaystyle \rho _{0}\left({\frac {\partial v_{x}}{\partial t}}+v_{x}{\frac {\partial v_{x}}{\partial x}}\;+\;v_{z}{\frac {\partial v_{x}}{\partial z}}\right)\;=\;-\;{\frac {\partial p}{\partial x}}\;+\;\mu _{0}\left({\frac {\partial ^{2}v_{x}}{\partial x^{2}}}\;+\;{\frac {\partial ^{2}v_{x}}{\partial z^{2}}}\right)}
ρ
0
(
∂
v
z
∂
t
+
v
x
∂
v
z
∂
x
+
v
z
∂
v
z
∂
z
)
=
−
ρ
0
g
−
∂
p
∂
z
+
μ
0
(
∂
2
v
z
∂
x
2
+
∂
2
v
z
∂
z
2
)
{\displaystyle \rho _{0}\left({\frac {\partial v_{z}}{\partial t}}+v_{x}{\frac {\partial v_{z}}{\partial x}}\;+\;v_{z}{\frac {\partial v_{z}}{\partial z}}\right)\;=\;-\;\rho _{0}\;g\;-\;{\frac {\partial p}{\partial z}}\;+\;\mu _{0}\left({\frac {\partial ^{2}v_{z}}{\partial x^{2}}}\;+\;{\frac {\partial ^{2}v_{z}}{\partial z^{2}}}\right)}
-1 ] = [[M·L-1 ·T-2 ]·L-1 ] = [M·L-2 ·T-2 ]. Em forma adimensional, teremos
ρ
∗
ρ
r
(
∂
(
v
x
∗
v
r
)
∂
(
t
∗
t
r
)
+
(
v
x
∗
v
r
)
∂
(
v
x
∗
v
r
)
∂
(
x
∗
L
r
)
+
(
v
z
∗
v
r
)
∂
(
v
x
∗
v
r
)
∂
(
z
∗
L
r
)
)
=
−
∂
(
p
∗
p
r
)
∂
(
x
∗
L
r
)
+
μ
∗
μ
r
(
∂
2
(
v
x
∗
v
r
)
∂
(
x
∗
L
r
)
2
+
∂
2
(
v
x
∗
v
r
)
∂
(
z
∗
L
r
)
2
)
{\displaystyle \rho ^{*}\rho _{r}\left({\frac {\partial (v_{x}^{*}v_{r})}{\partial (t^{*}t_{r})}}+(v_{x}^{*}v_{r}){\frac {\partial (v_{x}^{*}v_{r})}{\partial (x^{*}L_{r})}}\;+\;(v_{z}^{*}v_{r}){\frac {\partial (v_{x}^{*}v_{r})}{\partial (z^{*}L_{r})}}\right)\;=\;-\;{\frac {\partial (p^{*}p_{r})}{\partial (x^{*}L_{r})}}\;+\;\mu ^{*}\mu _{r}\left({\frac {\partial ^{2}(v_{x}^{*}v_{r})}{\partial (x^{*}L_{r})^{2}}}\;+\;{\frac {\partial ^{2}(v_{x}^{*}v_{r})}{\partial (z^{*}L_{r})^{2}}}\right)}
ρ
∗
ρ
r
(
∂
(
v
z
∗
v
r
)
∂
(
t
∗
t
r
)
+
(
v
x
∗
v
r
)
∂
(
v
z
∗
v
r
)
∂
(
x
∗
L
r
)
+
(
v
z
∗
v
r
)
∂
(
v
z
∗
v
r
)
∂
(
z
∗
L
r
)
)
=
−
ρ
∗
ρ
r
g
−
∂
(
p
∗
p
r
)
∂
(
z
∗
L
r
)
+
μ
∗
μ
r
(
∂
2
(
v
z
∗
v
r
)
∂
(
x
∗
L
r
)
2
+
∂
2
(
v
z
∗
v
r
)
∂
(
z
∗
L
r
)
2
)
{\displaystyle \rho ^{*}\rho _{r}\left({\frac {\partial (v_{z}^{*}v_{r})}{\partial (t^{*}t_{r})}}+(v_{x}^{*}v_{r}){\frac {\partial (v_{z}^{*}v_{r})}{\partial (x^{*}L_{r})}}\;+\;(v_{z}^{*}v_{r}){\frac {\partial (v_{z}^{*}v_{r})}{\partial (z^{*}L_{r})}}\right)\;=\;-\;\rho ^{*}\rho _{r}g\;-\;{\frac {\partial (p^{*}p_{r})}{\partial (z^{*}L_{r})}}\;+\;\mu ^{*}\mu _{r}\left({\frac {\partial ^{2}(v_{z}^{*}v_{r})}{\partial (x^{*}L_{r})^{2}}}\;+\;{\frac {\partial ^{2}(v_{z}^{*}v_{r})}{\partial (z^{*}L_{r})^{2}}}\right)}
∂
f
(
x
)
∂
g
(
x
)
=
∂
f
(
x
)
∂
x
∂
x
∂
g
(
x
)
=
∂
f
(
x
)
∂
x
∂
g
(
x
)
∂
x
⇒
∂
f
(
x
)
∂
(
k
x
)
=
1
k
∂
f
(
x
)
∂
x
{\displaystyle {\frac {\partial f(x)}{\partial g(x)}}\;=\;{\frac {\partial f(x)}{\partial x}}\;{\frac {\partial x}{\partial g(x)}}\;=\;{\frac {\frac {\partial f(x)}{\partial x}}{\frac {\partial g(x)}{\partial x}}}\Rightarrow \;\;\;{\frac {\partial f(x)}{\partial (kx)}}\;=\;{\frac {1}{k}}\;{\frac {\partial f(x)}{\partial x}}}
∂
2
f
(
x
)
∂
g
(
x
)
2
=
∂
∂
g
(
x
)
(
∂
f
(
x
)
∂
g
(
x
)
)
=
∂
∂
g
(
x
)
(
1
k
∂
f
(
x
)
∂
x
)
=
1
k
∂
∂
(
k
x
)
(
∂
f
(
x
)
∂
x
)
=
1
k
2
∂
∂
(
x
)
(
∂
f
(
x
)
∂
x
)
=
1
k
2
(
∂
2
f
(
x
)
∂
x
2
)
{\displaystyle {\frac {\partial ^{2}f(x)}{\partial g(x)^{2}}}={\frac {\partial }{\partial g(x)}}\left({\frac {\partial f(x)}{\partial g(x)}}\right)={\frac {\partial }{\partial g(x)}}\left({\frac {1}{k}}{\frac {\partial f(x)}{\partial x}}\right)={\frac {1}{k}}{\frac {\partial }{\partial (kx)}}\left({\frac {\partial f(x)}{\partial x}}\right)={\frac {1}{k^{2}}}{\frac {\partial }{\partial (x)}}\left({\frac {\partial f(x)}{\partial x}}\right)={\frac {1}{k^{2}}}\left({\frac {\partial ^{2}f(x)}{\partial x^{2}}}\right)}
ρ
r
v
r
t
r
ρ
∗
∂
v
x
∗
∂
t
∗
+
ρ
r
v
r
2
L
r
ρ
∗
(
v
x
∗
∂
v
x
∗
∂
x
∗
+
v
z
∗
∂
v
x
∗
∂
z
∗
)
=
−
p
r
L
r
∂
p
∗
∂
x
∗
+
μ
r
v
r
L
r
2
μ
∗
(
∂
2
v
x
∗
∂
x
∗
2
+
∂
2
v
x
∗
∂
z
∗
2
)
{\displaystyle {\frac {\rho _{r}v_{r}}{t_{r}}}\rho ^{*}{\frac {\partial v_{x}^{*}}{\partial t^{*}}}+{\frac {\rho _{r}v_{r}^{2}}{L_{r}}}\rho ^{*}\left(v_{x}^{*}{\frac {\partial v_{x}*}{\partial x^{*}}}\;+\;v_{z}^{*}{\frac {\partial v_{x}^{*}}{\partial z^{*}}}\right)\;=\;-\;{\frac {p_{r}}{L_{r}}}\;{\frac {\partial p^{*}}{\partial x^{*}}}\;+\;{\frac {\mu _{r}v_{r}}{L_{r}^{2}}}\mu ^{*}\left({\frac {\partial ^{2}v_{x}^{*}}{\partial x^{*2}}}\;+\;{\frac {\partial ^{2}v_{x}^{*}}{\partial z^{*2}}}\right)}
ρ
r
v
r
t
r
ρ
∗
∂
v
z
∗
∂
t
∗
+
ρ
r
v
r
2
L
r
ρ
∗
(
v
x
∗
∂
v
z
∗
∂
x
∗
+
v
z
∗
∂
v
z
∗
∂
z
∗
)
=
−
ρ
∗
ρ
r
g
−
p
r
L
r
∂
p
∗
∂
z
∗
+
μ
r
v
r
L
r
2
μ
∗
(
∂
2
v
z
∗
∂
x
∗
2
+
∂
2
v
z
∗
∂
z
∗
2
)
{\displaystyle {\frac {\rho _{r}v_{r}}{t_{r}}}\rho ^{*}{\frac {\partial v_{z}^{*}}{\partial t^{*}}}+{\frac {\rho _{r}v_{r}^{2}}{L_{r}}}\rho ^{*}\left(v_{x}^{*}{\frac {\partial v_{z}*}{\partial x^{*}}}\;+\;v_{z}^{*}{\frac {\partial v_{z}^{*}}{\partial z^{*}}}\right)\;=\;-\;\rho ^{*}\rho _{r}g\;-\;{\frac {p_{r}}{L_{r}}}\;{\frac {\partial p^{*}}{\partial z^{*}}}\;+\;{\frac {\mu _{r}v_{r}}{L_{r}^{2}}}\mu ^{*}\left({\frac {\partial ^{2}v_{z}^{*}}{\partial x^{*2}}}\;+\;{\frac {\partial ^{2}v_{z}^{*}}{\partial z^{*2}}}\right)}
(
p
r
=
ρ
∗
ρ
r
v
r
2
e
v
r
=
g
L
r
)
{\displaystyle (p_{r}\;=\;\rho ^{*}\rho _{r}v_{r}^{2}\;\;\;e\;\;\;v_{r}\;=\;{\sqrt {gL_{r}}})}
μ
r
=
ρ
r
L
r
v
r
{\displaystyle \mu _{r}\;=\;\rho _{r}L_{r}v_{r}}
t
r
=
L
r
v
r
{\displaystyle t_{r}={\frac {L_{r}}{v_{r}}}}
ρ
r
v
r
2
L
r
ρ
∗
∂
v
x
∗
∂
t
∗
+
ρ
r
v
r
2
L
r
ρ
∗
(
v
x
∗
∂
v
x
∗
∂
x
∗
+
v
z
∗
∂
v
x
∗
∂
z
∗
)
=
−
ρ
∗
ρ
r
v
r
2
L
r
∂
p
∗
∂
x
∗
+
ρ
r
L
r
v
r
2
L
r
2
μ
∗
(
∂
2
v
x
∗
∂
(
x
∗
)
2
+
∂
2
v
x
∗
∂
(
z
∗
)
2
)
{\displaystyle {\frac {\rho _{r}v_{r}^{2}}{L_{r}}}\rho ^{*}{\frac {\partial v_{x}^{*}}{\partial t^{*}}}+{\frac {\rho _{r}v_{r}^{2}}{L_{r}}}\rho ^{*}\left(v_{x}^{*}{\frac {\partial v_{x}*}{\partial x^{*}}}\;+\;v_{z}^{*}{\frac {\partial v_{x}^{*}}{\partial z^{*}}}\right)\;=\;-\;{\frac {\rho ^{*}\rho _{r}v_{r}^{2}}{L_{r}}}\;{\frac {\partial p^{*}}{\partial x^{*}}}\;+\;{\frac {\rho _{r}L_{r}v_{r}^{2}}{L_{r}^{2}}}\mu ^{*}\left({\frac {\partial ^{2}v_{x}^{*}}{\partial (x^{*})^{2}}}\;+\;{\frac {\partial ^{2}v_{x}^{*}}{\partial (z^{*})^{2}}}\right)}
∂
v
x
∗
∂
t
∗
+
v
x
∗
∂
v
x
∗
∂
x
∗
+
v
z
∗
∂
v
x
∗
∂
z
∗
=
−
∂
p
∗
∂
x
∗
+
μ
∗
ρ
∗
(
∂
2
v
x
∗
∂
(
x
∗
)
2
+
∂
2
v
x
∗
∂
(
z
∗
)
2
)
{\displaystyle {\frac {\partial v_{x}^{*}}{\partial t^{*}}}+v_{x}^{*}{\frac {\partial v_{x}*}{\partial x^{*}}}\;+\;v_{z}^{*}{\frac {\partial v_{x}^{*}}{\partial z^{*}}}\;=\;-\;{\frac {\partial p^{*}}{\partial x^{*}}}\;+\;{\frac {\mu ^{*}}{\rho ^{*}}}\left({\frac {\partial ^{2}v_{x}^{*}}{\partial (x^{*})^{2}}}\;+\;{\frac {\partial ^{2}v_{x}^{*}}{\partial (z^{*})^{2}}}\right)}
∂
v
z
∗
∂
t
∗
+
v
x
∗
∂
v
z
∗
∂
x
∗
+
v
z
∗
∂
v
z
∗
∂
z
∗
=
−
1
−
∂
p
∗
∂
z
∗
+
μ
∗
ρ
∗
(
∂
2
v
z
∗
∂
(
x
∗
)
2
+
∂
2
v
z
∗
∂
(
z
∗
)
2
)
{\displaystyle {\frac {\partial v_{z}^{*}}{\partial t^{*}}}+v_{x}^{*}{\frac {\partial v_{z}*}{\partial x^{*}}}\;+\;v_{z}^{*}{\frac {\partial v_{z}^{*}}{\partial z^{*}}}\;=\;-1\;-\;{\frac {\partial p^{*}}{\partial z^{*}}}\;+\;{\frac {\mu ^{*}}{\rho ^{*}}}\left({\frac {\partial ^{2}v_{z}^{*}}{\partial (x^{*})^{2}}}\;+\;{\frac {\partial ^{2}v_{z}^{*}}{\partial (z^{*})^{2}}}\right)}
r tal que coincida com a velocidade do fluxo na superfície livre. Esse valor é indicado usualmente também como v∞ . Tal substituição implica que a pressão de referência pr e a velocidade de referência sejam relacionadas da mesma forma
(
p
r
=
ρ
∗
ρ
r
v
r
2
)
{\displaystyle (p_{r}\;=\;\rho ^{*}\rho _{r}v_{r}^{2})}
v
r
=
g
L
r
{\displaystyle v_{r}\;=\;{\sqrt {gL_{r}}}}
∂
v
z
∗
∂
t
∗
+
v
x
∗
∂
v
z
∗
∂
x
∗
+
v
z
∗
∂
v
z
∗
∂
z
∗
=
−
g
L
r
v
∞
2
−
∂
p
∗
∂
z
∗
+
μ
∗
ρ
∗
(
∂
2
v
z
∗
∂
(
x
∗
)
2
+
∂
2
v
z
∗
∂
(
z
∗
)
2
)
{\displaystyle {\frac {\partial v_{z}^{*}}{\partial t^{*}}}+v_{x}^{*}{\frac {\partial v_{z}*}{\partial x^{*}}}\;+\;v_{z}^{*}{\frac {\partial v_{z}^{*}}{\partial z^{*}}}\;=\;-\;{\frac {gL_{r}}{v_{\infty }^{2}}}\;-\;{\frac {\partial p^{*}}{\partial z^{*}}}\;+\;{\frac {\mu ^{*}}{\rho ^{*}}}\left({\frac {\partial ^{2}v_{z}^{*}}{\partial (x^{*})^{2}}}\;+\;{\frac {\partial ^{2}v_{z}^{*}}{\partial (z^{*})^{2}}}\right)}
![{\displaystyle \nabla \cdot (\rho {\vec {v}})\;=\;{\frac {\partial (\rho v_{x})}{\partial x}}\;+\;{\frac {\partial (\rho v_{y})}{\partial y}}\;+\;{\frac {\partial (\rho v_{z})}{\partial z}}\;=\;{\frac {\partial [(\rho ^{*}\cdot \rho _{r})(v_{x}^{*}\cdot v_{r})]}{\partial (x^{*}\cdot L_{r})}}\;+\;{\frac {\partial [(\rho ^{*}\cdot \rho _{r})(v_{y}^{*}\cdot v_{r})]}{\partial (y^{*}\cdot L_{r})}}\;+\;{\frac {\partial [(\rho ^{*}\cdot \rho _{r})(v_{z}^{*}\cdot v_{r})]}{\partial (z^{*}\cdot L_{r})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84d0b4414037b00e2f33197022d714942526a5a8)
