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速度ポテンシャル.
速度ポテンシャル(そくどポテンシャル、英: Velocity potential)[1][2]は、流体力学において、渦なし流れの解析に用いられる。速度ポテンシャルを持つ流れをポテンシャル流と呼ぶ。
速度ポテンシャルΦは次式を満たすようなスカラー場である。
![{\displaystyle {\boldsymbol {u))=\operatorname {grad} \Phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4afb7a443a35d8b83aebf426801da946764f07)
ただし、u は流体の速度であり、渦なし、つまり
![{\displaystyle \operatorname {rot} {\boldsymbol {u))=\nabla \times {\boldsymbol {u))={\boldsymbol {0))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/203436dd239a832fffa45c9065a4021ff2da1a61)
を満たす。これはベクトル解析における
![{\displaystyle \operatorname {rot} (\operatorname {grad} \Phi )={\boldsymbol {0))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32d67b5050a8459a1d8e8c31b21ce9146279edaa)
の性質を用いている(ナブラ#二階微分を参照)。
一般のポテンシャルと異なり、速度ポテンシャルの定義には負号がつかないことに注意。
- 領域が単連結であれば速度ポテンシャルΦは一価関数、多重連結であれば多価関数である。
- 速度u および流線は速度ポテンシャルΦの等値面(等ポテンシャル面)に直交し、速度ポテンシャルの法線方向n の微分が速度を与える:
![{\displaystyle {\frac {\partial \Phi }{\partial n))={\boldsymbol {n))\cdot \operatorname {grad} \Phi ={\boldsymbol {n))\cdot {\boldsymbol {u))=u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd81e9b7eaed94ab767a03487de19d57cd84d43)
- 非圧縮性流体の渦なし流に対しては、速度ポテンシャルΦは調和関数となる。
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