论文标题
倾斜表面下吊坠的形状
Shape of pendant droplets under a tilted surface
论文作者
论文摘要
对于吊坠滴的接触线是半径$ r_0 $的圆，我们得出关系$ mg \sinα= {π\ over2}γr_0\，（\cosθ^{\cosθ^{\ rm min} - \cosθ^{\ rm max}） $θ^{\ rm max} $是背面（上坡）的接触角，在前面（下坡），$ m $是滴剂的质量，$γ$是液体的表面张力。债券（或eötvös）的数字为$ bo = mg/（2R_0γ）$。倾斜角$α$可能会从$α= 0 $（无柄滴）增加到$α=π/2 $（固定在垂直壁上）到$α=π$（从天花板上掉落吊坠）。焦点将放在$α=π/2 $和$α=3π/4 $的吊坠上。下降曲线是在相同近似值中精确计算的。将结果与表面Evolver模拟进行比较，显示出良好的一致性约为$ = 1.2 $，例如，与半球形水滴相对应，高达$ 50 \，μ$L。还给出了每个接触角$θ^{\ rm min} $和$θ^{\ rm max} $的明确公式，并将其与几乎精确的表面Evolver值进行比较。
For a pendant drop whose contact line is a circle of radius $r_0$, we derive the relation $mg\sinα={π\over2}γr_0\,(\cosθ^{\rm min}-\cosθ^{\rm max})$ at first order in the Bond number, where $θ^{\rm min}$ and $θ^{\rm max}$ are the contact angles at the back (uphill) and at the front (downhill), $m$ is the mass of the drop and $γ$ the surface tension of the liquid. The Bond (or Eötvös) number is taken as $Bo=mg/(2r_0γ)$. The tilt angle $α$ may increase from $α=0$ (sessile drop) to $α=π/2$ (drop pinned on vertical wall) to $α=π$ (drop pendant from ceiling). The focus will be on pendant drops with $α=π/2$ and $α=3π/4$. The drop profile is computed exactly, in the same approximation. Results are compared with surface evolver simulations, showing good agreement up to about $Bo=1.2$, corresponding for example to hemispherical water droplets of volume up to about $50\,μ$L. An explicit formula for each contact angle $θ^{\rm min}$ and $θ^{\rm max}$ is also given and compared with the almost exact surface evolver values.
