Fw: [討論] 精液算是非牛頓流體嗎
※ [本文轉錄自 sex 看板 #1EehgknU ]
作者: Dola1003 (Road of Major) 看板: sex
標題: Re: [討論] 精液算是非牛頓流體嗎
時間: Sat Oct 22 20:44:26 2011
看到此討論串 想起多年前的一篇文章
就由我大學所學之 抽送現象 Fuck Phenomena
來解其shear stress and velocity profile
已知此cave 為一半徑為R 長度為L 形狀為一 cylinder
Assumption:
1.V=V(r,θ,z)
2.One-Dimensional flow V=V(z) 若假設為2D,則為PDE難解所以假設為1D
3.steady state
4.incompressible Newtonian fluid
5.laminar and fully developed flow, end effect can be neglected
以下為此cavity之 示意圖
╭─╮─────────╮ r
│ │ │ ↑
│ │ │ → z
╰─╯─────────╯
(1) Equation of continuity
dρ/dt+▽‧(ρV)= 0 (註:由於偏微分之partial 符號無法顯示,故用d表示之)
=> d/dz(Vz)= 0
假設 no vortex flow => Vz=Vz(r) only
(2) Shell momentum balance
ΣF= Fgz + Fpz + Fτz =∫∫c.s.ρVz(V‧n) dA + d/dt∫∫∫c.v.ρVz dV
假設此cave為horizontal,則無重力項。
又因fully developed and lamimar flow
其control surface and control volume 之兩項也可以消掉。
但是此cave不為free surface 所以有 pressure drop。
=> P‧r△r△θ|z - P‧r△r△θ|z+△z +τrz r△θ△Z|r - τrz r△θ△Z|r+dr=0
divided by △r△θ△z and take a limit as △r,△θ,△z approach zero
=> -d/dz(P‧r) -d/dr(τrz‧r)=0 (τ為 shear stress)
=> d/dr(τrz‧r)= (-dP/dz)‧r
B.C.1: r=0 τrz=0 => c1=0
∴τrz= 1/2(-dP/dz)‧r = 1/2 (△P/L)‧r
又 Newtonian fluid
τrz= -μdVz/dr (μ為 viscousity)
=> -μdVz/dr= 1/2(-dP/dz)‧r
=> Vz=-1/4μ (-dP/dz)‧r^2 +c2
B.C.2: r=R Vz=0 => c2= R^2/4μ(-dP/dz)
Vz = R^2/4μ(-dP/dz)[1-(r/R)^2]
= R^2/4μ(△P/L)[1-(r/R)^2] ... velocity profile
而此狀態下抽送的次數而導致噴射之關係 需考慮環境因子A 以及人為因素
由於篇幅關係故筆者不在此贅述
了解此原理之後 筆者出幾題課後作業讓西斯版之鄉民回去做個練習
Homework
1.試求出此狀態下之平均速度<v> 以及 體積流率Q
2.試證明此摩擦係數f(Friction factor) = 16/Re (Re為雷諾數 = ρVD/μ)
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