Origem: Wikilivros, livros abertos por um mundo aberto.
A velocidade de um fluido incompressível em um canal que se estreita continuamente é dada pela equação
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{\displaystyle {\vec {v}}\;=\;v_{1}\left[1\;+\;{\frac {x}{L}}\right]\;{\vec {u}}_{x}}
Aqui é preciso usar a forma diferencial das equações. A partir da velocidade, podemos encontrar a aceleração calculando a derivada direcional de v na direção do eixo X:
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{\displaystyle {\vec {a}}_{x}\;=\;\left({\frac {D}{Dt}}{\vec {v}}\right)\cdot {\vec {u}}_{x}\;=\;\left({\frac {\partial {\vec {v}}_{x}}{\partial t}}\;+\;v_{x}\;{\frac {\partial v_{x}}{\partial x}}\;+\;v_{y}\;{\frac {\partial v_{x}}{\partial y}}\;+\;v_{z}\;{\frac {\partial v_{x}}{\partial z}}\right)\cdot {\vec {u}}_{x}}
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{\displaystyle {\vec {a}}_{x}\;=\;\left(0\;+\;{\frac {v_{1}}{L}}\cdot v_{1}\left[1\;+\;{\frac {x}{L}}\right]\;+\;0\;+\;0\right)\cdot {\vec {u}}_{x}\;=\;{\frac {v_{1}^{2}}{L}}\left[1\;+\;{\frac {x}{L}}\right]\;{\vec {u}}_{x}}
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{\displaystyle v_{x}\;=\;{\vec {v}}\cdot {\vec {u}}_{x}\Rightarrow \;\;\;{\frac {dx_{P}}{dt}}\;=\;v_{1}\left[1\;+\;{\frac {x_{P}}{L}}\right]}
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{\displaystyle {\frac {dx_{P}}{1\;+\;{\frac {x_{P}}{L}}}}\;=\;v_{1}dt\Rightarrow \;\;\;\int _{0}^{x}{\frac {dx_{P}}{1\;+\;{\frac {x_{P}}{L}}}}\;=\;\int _{0}^{t}v_{1}dt}
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{\displaystyle \left.L\ln \left(1\;+\;{\frac {x_{P}}{L}}\right)\right|_{0}^{x}\;=\;\left.v_{1}t\right|_{0}^{t}\Rightarrow \;\;\;L\ln {\frac {1\;+\;{\frac {x}{L}}}{1\;+\;{\frac {0}{L}}}}\;=\;v_{1}(t\;-\;0)}
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{\displaystyle \ln \left(1\;+\;{\frac {x}{L}}\right)\;=\;{\frac {v_{1}}{L}}t\Rightarrow \;\;\;\left(1\;+\;{\frac {x}{L}}\right)\;=\;e^{{\frac {v_{1}}{L}}t}}
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{\displaystyle x\;=\;L\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)}
A aceleração poderia ser calculada também derivando-se a função de posição
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{\displaystyle {\vec {a}}\;=\;{\frac {\partial ^{2}x}{\partial t^{2}}}\;{\vec {u}}_{x}\;+\;{\frac {\partial ^{2}y}{\partial t^{2}}}\;{\vec {u}}_{y}\;+\;{\frac {\partial ^{2}z}{\partial t^{2}}}\;{\vec {u}}_{z}\;=\;{\frac {\partial ^{2}}{\partial t^{2}}}\left[L\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)\right]\;{\vec {u}}_{x}}
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{\displaystyle {\vec {a}}\;=\;{\frac {\partial }{\partial t}}\left[L{\frac {v_{1}}{L}}\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)\right]\;{\vec {u}}_{x}\;=\;\left[v_{1}{\frac {v_{1}}{L}}\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)\right]\;{\vec {u}}_{x}}
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{\displaystyle {\vec {a}}\;=\;\left[{\frac {v_{1}^{2}}{L}}\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)\right]\;{\vec {u}}_{x}}
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{\displaystyle {\vec {a}}_{x}\;=\;{\frac {v_{1}^{2}}{L}}\left[1\;+\;{\frac {x}{L}}\right]\;{\vec {u}}_{x}\;=\;{\frac {v_{1}^{2}}{L}}\left[1\;+\;{\frac {L\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)}{L}}\right]\;{\vec {u}}_{x}\;=\;{\frac {v_{1}^{2}}{L}}\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)\;{\vec {u}}_{x}}
![{\displaystyle {\vec {v}}\;=\;v_{1}\left[1\;+\;{\frac {x}{L}}\right]\;{\vec {u}}_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d549731e4eaaa3f8af89c60b1562825f882981)
![{\displaystyle {\vec {a}}_{x}\;=\;\left(0\;+\;{\frac {v_{1}}{L}}\cdot v_{1}\left[1\;+\;{\frac {x}{L}}\right]\;+\;0\;+\;0\right)\cdot {\vec {u}}_{x}\;=\;{\frac {v_{1}^{2}}{L}}\left[1\;+\;{\frac {x}{L}}\right]\;{\vec {u}}_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc33ca67fe4695b436374440c1c432af34756d2)
![{\displaystyle v_{x}\;=\;{\vec {v}}\cdot {\vec {u}}_{x}\Rightarrow \;\;\;{\frac {dx_{P}}{dt}}\;=\;v_{1}\left[1\;+\;{\frac {x_{P}}{L}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80160d859ddf76e87bdfe2d49a38c884868050da)
![{\displaystyle {\vec {a}}\;=\;{\frac {\partial ^{2}x}{\partial t^{2}}}\;{\vec {u}}_{x}\;+\;{\frac {\partial ^{2}y}{\partial t^{2}}}\;{\vec {u}}_{y}\;+\;{\frac {\partial ^{2}z}{\partial t^{2}}}\;{\vec {u}}_{z}\;=\;{\frac {\partial ^{2}}{\partial t^{2}}}\left[L\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)\right]\;{\vec {u}}_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/664e10169c6a5843f0a436ae2db5f8098a6dab53)
![{\displaystyle {\vec {a}}\;=\;{\frac {\partial }{\partial t}}\left[L{\frac {v_{1}}{L}}\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)\right]\;{\vec {u}}_{x}\;=\;\left[v_{1}{\frac {v_{1}}{L}}\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)\right]\;{\vec {u}}_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e96c9a7cf289c06c716d88fa70ab8e2b6a9b80)
![{\displaystyle {\vec {a}}\;=\;\left[{\frac {v_{1}^{2}}{L}}\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)\right]\;{\vec {u}}_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac89023e571a28532be8bfcce7d6e2d6aafeb60)
![{\displaystyle {\vec {a}}_{x}\;=\;{\frac {v_{1}^{2}}{L}}\left[1\;+\;{\frac {x}{L}}\right]\;{\vec {u}}_{x}\;=\;{\frac {v_{1}^{2}}{L}}\left[1\;+\;{\frac {L\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)}{L}}\right]\;{\vec {u}}_{x}\;=\;{\frac {v_{1}^{2}}{L}}\left(e^{{\frac {v_{1}}{L}}t}\;-\;1\right)\;{\vec {u}}_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b52fdf9405b19743934fb8f6fc2da4a63c794972)