Originally posted by joe shmo
Ok, i'll show how I arrived at the equation
Start with a fluid element submerged in a liquid at a depth "h"
The element has a differential thickness "dh"
The force on the elements surface at depth "h" is:
F_h = -P*A
The force on the elements surface at depth h + dh is:
F_h+dh = ( P + dP)*A
The only other force acting on the element is ...[text shortened]...
[b]dP = dp*g*h + p*dg*h + p*g*dh
Can anyone see what may not be valid about that part?[/b]
I understand your development of dF_w based on F_w. I'm not sure why your F_h=-P*A is not similarly developed as dF_h=-A*dP. Instead you have
F_{h+dh} = (P+dP)*A,
but isn't that missing a negative sign? Shouldn't it be
F_{h+dh} = -(P+dP)*A …?
Then your equation
(P + dP)*A - P*A - A(dp*g*h + p*dg*h + p*g*dh) = 0
would become
-(P + dP)*A - P*A - A(dp*g*h + p*dg*h + p*g*dh) = 0,……(Eq1)
which still seems wrong. It's been a long time since I've dabbled seriously in physics and its handling of so-called "differentials," but it seems to me that non-differential quantities such as P*A are incommensurate with differential quantities. They're being mixed together here. In your version of the equation the P*A terms just happen to cancel out, and in my version they don't. But anyway, if we just replace -(P + dP)*A - P*A with dF_h=-A*dP to get
-A*dP - A(dp*g*h + p*dg*h + p*g*dh) = 0,
we seem to get crap:
dP = -(dp*g*h + p*dg*h + p*g*dh).
That negative sign just isn't going to help! But maybe (Eq1) is salvageable?
But what's the meaning of F_w = -p*V*g = -p*A*h*g? You're writing V=A*h, but are you saying the volume of the fluid element is its cross-sectional area times its depth? Is this truly a submerged fluid element, or the column of fluid above the element? Are you maybe mixing up the calculation of P at depth h with the calculation of F_w?
Finally, how can F_h and F_w cancel out, given they are two force vectors that point downward? In fact, how can F_h=-P*A be a vector at all, considering that pressure and area are scalar quantities? Are we dealing with magnitudes instead? But the magnitudes can't cancel out either.
I'm discombooberated and conflusterflated. 😉