Looking for use of a modified form for the mathematical definition of work

I apologize in advance for my peculiar and nebulous question.

Work is defined as the integral of pressure and an infinitesimal change in volume

$$W = \int p\ \mathrm{d}V$$

Is there any analysis or measurement where the term

$$W^\prime := \int p\ \frac{\mathrm{d}V}{V}$$

pops up and is of importance or has some significance? The reason I ask is that maybe this term can be some measure of pumping effort when the volume of the (incompressible) fluid within the boundary is changing? But apart from this, does this particular combination of $$p$$ and $$V$$ ever show up in any other analysis? I am just looking for anything that uses or spits out this exact term during the course of analysis.

• How can the volume of an incompressible fluid change? – Jake Rose Apr 3 at 23:33
• @JakeRose There is flow through the boundary (or when the boundary does not advect with fluid velocity) – shk92 Apr 3 at 23:38
• @shk92 For an incompressible flow the net flow through the boundary is zero, unless the boundary is moving and the volume inside it changes. Whatever your integral might be it isn't "work" or "energy" because it has the wrong dimensions! – alephzero Apr 3 at 23:57
• What makes you even think that there is such an example? – Chet Miller Apr 4 at 0:25
• @alephzero yes you're right the units are not work... I'm interpreting it as work done per unit volume – shk92 Apr 4 at 2:33

I do not know of any specific reference which shows this integral in the context of any work done, however one might come across this integral when dealing with ideal gases.

Let's say we have an ideal gas whose volume is $$V$$, pressure is $$p$$, temperature is $$T$$ and it consists of $$n$$ moles. Thus by the ideal gas law,

$$pV=nRT$$

Now let's assume it undegoes a process in which it's temperature and amount (moles) stay constant, which implies,

$$pV=nRT=\text{constant}$$

Thus,

\begin{align} \mathrm d (pV) & =0\\ \tag{1} p\mathrm d V + V \mathrm d p & =0\\ -V \mathrm d p&= p \mathrm d V\\ -\mathrm d p&= p \frac{\mathrm d V}{V}\\ -\int \mathrm d p&= \int p \frac{\mathrm d V}{V}\\ -\int \mathrm d p&= W' \end{align}

Though we have derived this integral, however in my experience, I have never seen this kind of integral, neither have I used it to solve anything. However the differential equation $$(1)$$ comes up pretty often while relating pressure and volume when the temperature is held constant.