Shaking Sauce From a Bottle

Have you ever noticed that sometimes it takes more effort to shake a few drops of sauce or liquid out of a bottle than other times?

I began pondering this when using Tabasco sauce one day. When the bottle was full, it seemed difficult to get the liquid out, and it appeared less of a problem after having used it for some time. What was behind this?

It turns out, it's a balance between the volume of air in the bottle available to expand, and the weight of the fluid that serves to lowers the pressure in that volume.

Here is how it works:

When the bottle is inverted, the weight of the fluid counteracts a portion of the atmospheric pressure, reducing the pressure in the empty volume in the top of the inverted bottle. The greater the height of the fluid column, the lower the pressure of the empty volume. As the pressure decreases, the volume expands until the sum of the air pressure in the empty volume and the hydraulic pressure of the fluid column equal atmospheric pressure. This change in volume of the air in the bottle is what allows some of the fluid to leave the bottle (e.g the volume of tabasco sauce you manage to put onto your burrito).

The greater the initial air gap, the larger the change in volume will be. However, there is only a finite volume in a bottle, so as the air volume increases, the fluid volume decreases. This means that although a decrease in fluid volume means there is a greater volume of air available to expand, there is less hydraulic pressure available to help reduce the pressure in that volume (as illustrated below in cases A, B, C), limiting the effectiveness of the larger volume.

The question now becomes:

What is the relationship between fluid fill level, and the resulting change in volume of the air within the bottle (and thereby the volume of fluid that leaves the bottle) when the bottle is inverted?

For this problem we'll make some very basic assumptions to simplify the calculations:

Bubbles don't travel from the mouth of the bottle to the unfilled volume:
This is usually true for bottles with a small opening such as the Tabasco sauce bottle in question.
Surface tension effects are ignored:
Because of surface tension, A droplet would normally cling to the opening, aiding in supporting the weight of the fluid column. Such behavior would result in a slightly higher pressure in the unfilled space, and therefore a smaller change in volume.
The bottle remains rigid:
A flexible bottle would deform due to the lowered pressure, allowing more fluid to escape than if the bottle were rigid.
Viscous effects are ignored:
Since this is a hydrostatic problem (and we're ignoring dynamic cases), viscous effects are not expected to play a role.
Uniform bottle diameter:
A non uniform bottle diameter would result in a non-linear change in volume with change in fluid fill level. Although it is not terribly difficult to account for this, it requires knowing the exact relationship (equation to model the curves of the bottle), and this won't really teach us anything more than if we just simply the shape of the bottle to a cylinder.

These assumptions mean we won't get an exact answer for how much of any specific fluid leaves any specific container, but the underlying trend would still remain accurate.

Here is a chart that shows the amount of fluid leaving a bottle after it has been inverted, as a function of initial fill level:

It is immediately obvious that you can get the most out of a bottle when it is exactly 50% full. This is quite an elegant result, and agrees with our experience. The same equations used to generate the above chart can be rearranged to maintain a fixed change in volume, and determine what amount of acceleration (in addition to gravitational acceleration) is needed to extract that volume (one drop for example) of fluid from the bottle.

The following questions are then posed:

What acceleration is needed to extract 1 mL of fluid from the bottle
At what fill level will I need to shake the bottle to get the fluid out?

The result here is also quite pleasing, as it too agrees with our own experiences. When the bottle is nearly full or nearly empty, you have to shake it a bit for the sauce to come out. When the bottle is 20-80% full, all you have to do is tip it upsidedown and some sauce will spill out on its own. Since the red line represents earth's gravitational acceleration, we see that if we were on the Moon (.17 G) or Mars (.38 G), we will be shaking bottles to get sauce out no matter how full they are, not to mention those poor people in the space station! Fortunately it's not too difficult to generate a few G's by shaking the bottle.

Attachments:

MATLAB script used to generate the plots above: Draining_Bottle.m.
Copy of work used to derive the equations: Inverted_bottle_work.JPG