Pressure difference between two points in a tapered pipe

[I_Try_Math]

Homework Statement

The pipe in the figure is transporting oil (density 850 kg/m3). The velocity at point 1
is 0.5m/s, but at point 2 it is 1.0m/s. Calculate the difference in height in the two open thin tubes

Relevant Equations

Bernoulli's equation

Trying to find an equation for the pressure difference between point 1 and 2. Not sure if my overall reasoning is incorrect, or I introduced a sign error somehow. The equation I come up with implies that the difference is a negative number, but that can't be true if the area of pipe at point 1 is larger than it is at point 2, as it in the given diagram?

Here's my work:

$l_1$ and $l_2$ are intended to mean the distance from points 1 and 2 to where the fluid meets the air.

$P_1 + \rho g l_1 = P_{atm}$

$P_2 + \rho g l_2 = P_{atm}$

$P_1 + \rho g l_1 = P_2 + \rho g l_2$

$P_1 - P_2 + \rho g l_1 = \rho g l_2$

$P_1 - P_2 = \rho g l_2 - \rho g l_1$

$P_1 - P_2 = \rho g(l_2 - l_1)$

$P_1 - P_2 = \rho g(-h)$

 

[Lnewqban]

The height difference should not have any sign.
The lesser height reached by its liquid column is the effect caused by the lower internal static pressure in cross-section 2.

 

[haruspex]

The height difference should not have any sign.

You are missing the point. There is an error in the algebra.

$P_1 + \rho g l_1 = P_{atm}$

Think about that again.

 

[I_Try_Math]

You are missing the point. There is an error in the algebra.

Think about that again.

I still can't tell what is incorrect about the equation relating $l_1$, $P_1$, and $P_{atm}$. Any hints are appreciated.

 

[hutchphd]

Any hints are appreciated.

Look at some limit cases.....what if atmospheric pressure is nil......does your equation for P1 seem correct? What if it gets bigger than 0?

 

[Lnewqban]

I still can't tell what is incorrect about the equation relating $l_1$, $P_1$, and $P_{atm}$. Any hints are appreciated.

If you are normally swimming, your ears feel atmospheric pressure only.
If you then are diving at a depth of 10 meter under the surface of the sea, your ears are feeling the pressure exerted by the column of water above you plus the pressure exerted by the atmosphere.

 

[Chestermiller]

Are you saying that there is no effect of the fluid velocities at the two points?