- #1
Adesh
- 735
- 191
I’m currently studying Fluid Mechanics, during an analysis I came across this
We now consider an example that combines centrifugal force and gravity: a liquid in a drum (centrifuge) rotates with constant angular velocity ##\omega## about a vertical axis. The centrifugal force per unit of volume is $$ \mathbf F = F_{r} \hat r ; F_r = \rho r \omega ^2 ; \hat r \text{is a unit vector} \perp \text{to the axis}$$
Its potential is given by $$ U = - \frac{1}{2} \rho r^2 \omega ^2 $$
The total potential of gravity and centrifugal force is (except for a constant) given by
$$ U = - \rho g z - \frac{1}{2} \rho r^2 \omega ^2 = - \gamma \left (
z + \frac{ r^2 \omega ^2}{2 g} \right ) $$ From the equation ## p = -U + constant ## we can write $$ p = \gamma \left (
z + \frac{r^2 \omega^2 }{2g} \right) + constant $$
[we can find the constant by doing few simple things and after getting it we can write] $$ p = \gamma \left(
z-z_0 + \frac{r^2 \omega ^2}{2g} \right)$$ The free surface, characterised by ##p=0##, has the equation $$ z_0 -z = \frac{r^2 \omega ^2}{2g} $$
My problem what does that line “the free surface, characterised by ##p=0##” means? I tried searching Wikipedia and found that free surface means where there is no shear stress but I couldn’t connect it to here. How zero shear stress would make ##p=0## in the above equations?
P.S. : the book from which I have quoted is Sommerfeld’s Lectures on Theoretical Physics Vol II.
We now consider an example that combines centrifugal force and gravity: a liquid in a drum (centrifuge) rotates with constant angular velocity ##\omega## about a vertical axis. The centrifugal force per unit of volume is $$ \mathbf F = F_{r} \hat r ; F_r = \rho r \omega ^2 ; \hat r \text{is a unit vector} \perp \text{to the axis}$$
Its potential is given by $$ U = - \frac{1}{2} \rho r^2 \omega ^2 $$
The total potential of gravity and centrifugal force is (except for a constant) given by
$$ U = - \rho g z - \frac{1}{2} \rho r^2 \omega ^2 = - \gamma \left (
z + \frac{ r^2 \omega ^2}{2 g} \right ) $$ From the equation ## p = -U + constant ## we can write $$ p = \gamma \left (
z + \frac{r^2 \omega^2 }{2g} \right) + constant $$
[we can find the constant by doing few simple things and after getting it we can write] $$ p = \gamma \left(
z-z_0 + \frac{r^2 \omega ^2}{2g} \right)$$ The free surface, characterised by ##p=0##, has the equation $$ z_0 -z = \frac{r^2 \omega ^2}{2g} $$
My problem what does that line “the free surface, characterised by ##p=0##” means? I tried searching Wikipedia and found that free surface means where there is no shear stress but I couldn’t connect it to here. How zero shear stress would make ##p=0## in the above equations?
P.S. : the book from which I have quoted is Sommerfeld’s Lectures on Theoretical Physics Vol II.