# I Bubble cavitation - what does a dash mean?

1. Mar 2, 2016

### rwooduk

For a bubble in water in equilibrium the supporting forces due to pressure of the gas Pg and vapour Pv in the bubble must equal the crushing forces i.e. the hydrostatic forces Ph and those due to surface tension 2sigma/R0. So we have

$P_{v}+P_{g}=P_{h}+ \frac{2\sigma }{R_{0}}$

Now we apply ultrasound so the pressure in the liquid is now Ph + Pa where:

$P_{a}=P_{A} sin 2\pi ft$

BUT the ultrasound causes cavitation of the bubble! And when the bubble is largest we have:

$P_{V}>(P_{h}-P_{a}) + \frac{2\sigma }{R_{0}} - P_{g}$

Now my question, he writes the next equation as:

$P_{V}-\frac{2\sigma }{R_{0}}=P_{V}^{'}>(P_{h}-P_{a})$

BUT he doesnt define $P_{V}^{'}$ so I'm a little lost.

Could someone tell me what a dash would generally mean?

It's from p.61 of this paper:

http://gendocs.ru/docs/37/36518/conv_1/file1.pdf

Thanks in advance for any help!

2. Mar 2, 2016

### Serena

Are you sure that the link for your paper is correct?

3. Mar 2, 2016

4. Mar 2, 2016

### Serena

I am sorry, gendocs.ru doesn't provide a direct link to the PDF from Scholar. Do you know the DOI of this article?

5. Mar 2, 2016

### rwooduk

hmm sorry I tried to upload it here:

http://www.filedropper.com/12_7 [Broken]

Last edited by a moderator: May 7, 2017
6. Mar 2, 2016

### Serena

I think it's just a conventional way to express the threshold value for the coupled vapor pressure and surface tension effects on cavitation.

7. Mar 2, 2016

### nasu

Do you mean the word "dash" or a dash like this "-" (a punctuation mark).
I cannot find it on page 61 of that document.

8. Mar 3, 2016

### rwooduk

Many thanks for your help! The thing that confuses me is that the PV term for vapour pressure is in the same equation as the P':

$P_{V}-\frac{2\sigma }{R_{0}}=P_{V}^{'}>(P_{h}-P_{a})$

If the $P_{V}^{'}$ term expresses in part the vapour pressure then why is there a PV at the other side of the equation?

Thanks again for your help.

I mean a dash at the top of the symbol, in the paper P becomes P'