HW6

steady

ρdw

dt= ρg + μ

d2w

dx2

ρg + μd2w

dx2= 0,

d2w

dx2= −

𝜌𝑔

𝜇

dw

dx= −

𝜌𝑔

𝜇𝑥 + 𝐶1, w(x) = −

𝜌𝑔

2𝜇𝑥2 + 𝐶1𝑥 + 𝐶2

Boundary condition 𝑤(−ℎ) = 0 → w(−h) = 0 = −𝜌𝑔

2𝜇ℎ2 − 𝐶1ℎ + 𝐶2

𝑤(ℎ) = 0 → w(h) = 0 = −𝜌𝑔

2𝜇ℎ2 + 𝐶1ℎ + 𝐶2

∴ 𝐶1 = 0 , 𝐶2 =𝜌𝑔

2𝜇ℎ2

▶ w(x) = −𝜌𝑔

2𝜇𝑥2 +

𝜌𝑔

2𝜇ℎ2

w(x) =𝜌𝑔

2𝜇(ℎ2 − 𝑥2) Ans.