Pressure and the Young-Laplace Equation

[Chestermiller]

$$-\sigma\left(\frac{\sin\phi}{R}+\frac{d\phi}{dS}\right)+s_0^2\rho g Z(S)+\sigma\left(\frac{\sin\phi}{R}+\frac{d\phi}{dS}\right)_{S=0}=s_0^3\frac{\rho \omega^2R^2(S)}{2}$$
Is this what you had in mind?

Yes. Now divide the equation by $\rho \omega^2 s^3_0$. What do you get?

 

[member 428835]

Yes. Now divide the equation by $\rho \omega^2 s^3_0$. What do you get?

$$\frac{\sigma}{\rho \omega^2 s^3_0}\left[\left(\frac{\sin\phi}{R}+\frac{d\phi}{dS}\right)_{S=0}-\left(\frac{\sin\phi}{R}+\frac{d\phi}{dS}\right)\right]+\frac{g }{\omega^2 s_0}Z(S)=\frac{R^2(S)}{2}$$ Starting to look like the "McCraney number".

 

[Chestermiller]

$$\frac{\sigma}{\rho \omega^2 s^3_0}\left[\left(\frac{\sin\phi}{R}+\frac{d\phi}{dS}\right)_{S=0}-\left(\frac{\sin\phi}{R}+\frac{d\phi}{dS}\right)\right]+\frac{g }{\omega^2 s_0}Z(S)=\frac{R^2(S)}{2}$$ Starting to look like the "McCraney number".

Good. Now, there are a couple of ways to proceed further with the dimensional analysis. One way is to now set $s_0=g/\omega^2$, so that the coefficient of Z is unity. What does this give you?

 

[member 428835]

Good. Now, there are a couple of ways to proceed further with the dimensional analysis. One way is to now set $s_0=g/\omega^2$, so that the coefficient of Z is unity. What does this give you?

Awesome, this is perfect. Thanks so much!
$$M\left[\left(\frac{\sin\phi}{R}+\frac{d\phi}{dS}\right)_{S=0}-\left(\frac{\sin\phi}{R}+\frac{d\phi}{dS}\right)\right]+Z(S)=\frac{R^2(S)}{2}$$

 

[Chestermiller]

Awesome, this is perfect. Thanks so much!
$$M\left[\left(\frac{\sin\phi}{R}+\frac{d\phi}{dS}\right)_{S=0}-\left(\frac{\sin\phi}{R}+\frac{d\phi}{dS}\right)\right]+Z(S)=\frac{R^2(S)}{2}$$

The important thing is to assimilate this kind of methodology for reducing the equations for a system to dimensionless form.

 

[member 428835]

Good. Now, there are a couple of ways to proceed further with the dimensional analysis. One way is to now set $s_0=g/\omega^2$, so that the coefficient of Z is unity. What does this give you?

The important thing is to assimilate this kind of methodology for reducing the equations for a system to dimensionless form.

I couldn't agree more. How did you know to scale $Z$ with $R^2$, which is to say what to set as $O(1)$?

 

[Chestermiller]

I couldn't agree more. How did you know to scale $Z$ with $R^2$, which is to say what to set as $O(1)$?

I wanted the terms for the case of zero surface tension to be preserved in the limit of zero surface tension.