What is the compressibility module and why is it important in fluid mechanics?

[TeTeC]

Hello everyone !

I'm currently working on a problem in Eugene Hecht's Physics book. It's about static fluid and the "compressibility module". I'm not sure this traduction makes sense, because my book is translated into French and I can't find the correct English expression. For those who have this book within easy reach, just read the exercice number 20 at chapter 11. For others, I'll give you an approximate translation, as good as my English can be...

The compressibility module is defined as B = -(F/A)/(DeltaV/V) where the variation in volume of the body is a result of the application of a uniform F force distributed on the whole surface A. Give for B a more accurate expression taking into account the variation in hydrostatic pressure working on the object. In fluids, we often use the inverse of B, which is the compressibility, K. Explain what is its meaning. Why the sign (-) ?

Ok, I know I have to give what I've already discovered, but unfortunately all I found is nothing, except the fact that F/A = P. This exercice should normally use (as it is classified in the exercices using mathematical tools) something like a derivative, but I can't find the purpose of any derivative. Clearly, I need some help, a start... ;)

Thank a lot !

TeTeC

(Excuse me for the potentially bad English mistakes, French is my mother tongue)

 

[FredGarvin]

You're alos referring to what is called the Bulk Modulus, or Bulk Modulus of Elasticity $$E_v$$. The way I am used to seeing it is in the form of:

$$E_v = - \frac{\delta p}{\delta V / V}$$ This can also be shown in the form of $$E_v = - \frac{\delta p}{\delta \rho / \rho}$$

It simply is a measure of the change in volume of a fluid due to a change in pressure. The negative sign indicates that the volume decreases with an increase in pressure differential. The larger the value, the closer to incompressible a fluid is.

 

[TeTeC]

Ok, the Bulk Modulus explains the "B".

Now that I have the answer, the transformation to apply is easier to find... V = m/rho, and then comes your expression.

I've just been reading that the Bulk Modulus of Elasticity is something like 2.2 x 10^9 N/m². That gives a great reason to say that water is nearly incompressible.

Off Topic : don't you know which software I could download to translate things like MathType objects into Latex ?

Thanks !

TeTeC

 

[FredGarvin]

A good number for how compressible water is is that a about a 1% change in volume happens with 3125 psi. The incompressible assumption on most liquids is a valid one. Gasses on the other hand...no go.

In regards to the software question, I really have no idea on what you can use. Sorry.