REYNOLDS LUBRICATION EQUATION

[RuchiJay]

Please help me to integrate this eqn. with respect to r

1/r * d/dr(rh^3 dp/dr) = 6nu dh/dr + 12nv

I need to find dp/dr

 

[soroban]

Hello, RuchiJay!

This is a strange problem. .Where did it come from?

\(\displaystyle \text{Integrate with respect to }r.\)

. . \(\displaystyle \frac{1}{r}\cdot \frac{d}{dr}\!\left(rh^3\frac{dp}{dr}\right) \:=\: 6nu\frac{dh}{dr} + 12nv\)

\(\displaystyle \text{I need to find }\frac{dp}{dr}\)

\(\displaystyle \text{I will assume that only }p\text{ and }r\text{ are variables.}\)


\(\displaystyle \text{Multiply by }r\,dr\!:\;\;d\left(rh^3\frac{dp}{dr}\right) \;=\;6nur\,dh + 12nvr\,dr\)

\(\displaystyle \text{Integrate: }\:\int\! d\left(rh^3\frac{dp}{dr}\right) \;=\;6nur\!\!\int\! dh + 12nv\!\! \int\! r\,dr\)

. . . . . . . . . . . . . \(\displaystyle rh^3\frac{dp}{dr} \;=\;6nur\,h + 6nv\,r^2 + C\)

. . . . . . . . . . . . . \(\displaystyle rh^3\frac{dp}{dr} \;=\;6nr(uh + vr) + C\)

. . . . . . . . . . . . . . . \(\displaystyle \frac{dp}{dr} \;=\;\frac{6n(uh + vr) + C}{h^3}\)