Fluid Pressure Question (not a homework problem)

[Geezer]

It's spring break right now, so I thought I'd take the time to brush up on stat mech and thermo before classes resume next week...

My question is this (Problem #1096 in "Problems and Solutions on Thermodynamics and Statistical Mechanics" Edited by Yung-Kuo Lim):

A flask of conical shape contains raw milk. The pressure is measured inside the flask at the bottom. After a sufficiently long time, the cream rises to the top and the milk settles to the bottom (the total volume of the liquid remains the same). Does the pressure increase, decrease, or remain the same? Explain.

Instinctively, I wanted to respond that the pressure remains the same, but the book says it doesn't. The final "solution," as presented in this book, is that the pressure decreases. Does that seem right to you?

Here's the link to the Google Book preview so that you can see the full solution yourself: http://books.google.com/books?id=dQGC0ifkE34C&pg=PA94&lpg=PA94&dq=flask+of+conical+shape+contains+raw+milk&source=bl&ots=Zh3L3i65hi&sig=PabAlSKz6pDQGkFIxsxfePkb32k&hl=en&sa=X&ei=KMVzT-m9EKm5iwKpl_2uCw&ved=0CCAQ6AEwAA#v=onepage&q=flask%20of%20conical%20shape%20contains%20raw%20milk&f=false

 

[K^2]

It does decrease, and it's due to the shape of the flask. Let's simplify it. Imagine that a flask comes to a point (basically, a hollowed-out cone with circular base) and is completely filled with an equal parts mixture of two fluids with densities of 1 and 2. Take the column of fluid at the center of the flask. What is the average density of that column? It's 1.5, obviously. Now, imagine that the fluids separate. What is the average density of the narrow column in the center now?

 

[jim hardy]

Now THAT's a good one.

I guess pressure is really $\int\rho g$ dh and I'm not accustomed to $\rho$ being $f$(h) .