# 迪恩數

## 定義

${\displaystyle {\mathit {De}}={\frac {\rho V\!d}{\mu }}\left({\frac {d/2}{R}}\right)^{1/2}}$
• ${\displaystyle \rho }$ 為流體密度
• ${\displaystyle \mu }$ 為流體的粘度
• ${\displaystyle V}$ 是軸向的速度值
• ${\displaystyle d}$ 為彎管直徑（若截面不是圓形，可以用等效直徑，請參考雷諾數
• ${\displaystyle R}$ 是彎管的曲率半徑

## 迪恩方程

${\displaystyle D\left({\frac {\mathrm {D} u_{x}}{\mathrm {D} t}}+u_{z}^{2}\right)=-D{\frac {\partial p}{\partial x}}+\nabla ^{2}u_{x}}$
${\displaystyle D{\frac {\mathrm {D} u_{y}}{\mathrm {D} t}}=-D{\frac {\partial p}{\partial y}}+\nabla ^{2}u_{y}}$
${\displaystyle D{\frac {\mathrm {D} u_{z}}{\mathrm {D} t}}=1+\nabla ^{2}u_{z}}$
${\displaystyle {\frac {\partial u_{x}}{\partial x}}+{\frac {\partial u_{y}}{\partial y}}=0}$

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}=u_{x}{\frac {\partial }{\partial x}}+u_{y}{\frac {\partial }{\partial y}}}$實質導數

## 參考資料

1. ^ Berger, S. A.; Talbot, L.; Yao, L. S. Flow in Curved Pipes. Ann. Rev. Fluid Mech. 1983, 15: 461–512. Bibcode:1983AnRFM..15..461B. doi:10.1146/annurev.fl.15.010183.002333.
2. ^ Chapter5 Geometry and Flow p.3 互联网档案馆存檔，存档日期2016-03-04.
3. ^ Mestel, J. Flow in curved pipes: The Dean equations页面存档备份，存于互联网档案馆）, Lecture Handout for Course M4A33, Imperial College.
4. ^ Dennis, C. R.; Ng, M. Dual solutions for steady laminar-flow through a curved tube. Q. J. Mech. Appl. Math. 1982, 35: 305. doi:10.1093/qjmam/35.3.305.