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If you have a glass of water, say, three quarters full and you pour it at an angle of say, $45^{\circ}$ with respect to the the table, the water comes out of the glass and goes directly down towards the floor.

However, when the glass is more full, or even three quarters full and the 'angle of pouring' is far less with respect the table, when the water comes out of the glass, rather than going straight down it kind of stays stuck to the glass and travels down the outside of it.

Why does this happen? (I'm sure many of you have observed it yourself by accident and ended up making a mess).

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Related: physics.stackexchange.com/q/23354/2451 and links therein. –  Qmechanic Oct 19 at 18:22

2 Answers 2

up vote 8 down vote accepted

This is due to surface tension. Water wants to stick to hard surfaces as this is a lower energy arrangement. Component of gravity perpendicular to glass pulls water away from glass wall, and surface tension pulls water to glass wall. When the angle between glass wall and vertical direction is small, component of gravity perpendicular to glass wall is small and surface tension prevails.

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You would think that's an easy question, but it's not! Actually many things involving fluid mechanics are far harder than they seem. Anyhow a team of scientists at the University of Lyons in France have been working on this. See http://arxiv.org/abs/0910.3306 for their paper or http://www.telegraph.co.uk/science/science-news/6454568/How-to-stop-a-teapot-dribbling.html for a more user friendly version.

Water has a tendancy to stick to glass, so the water in your glass (or in the example above your teapot) has to detach from the glass at the lip to avoid dribbling. If you pour fast the momentum of the water will pull it free of the glass and it will pour cleanly. If you pour slowly enough it is more energetically favourable for the water to remain stuck to the glass and it will flow over the lip and down the outside of the glass.

In the paper above the scientists recommend controlling the wetting properties of the teapot to reduce the adhesion of the water to the spout. You could try dipping your glass in fabric conditioner as this will hydrophobe the surface. Also a sharp edge means the water has to change direction fast to dribble, so it will reduce the flow rate at which dribbling starts.

Response to Olly's comment: to do this with any degree of accuracy you need to reach for your finite element software and numerically solve the Navier-Stokes equations. But let's have a go using a very simplified model. NB I'm making this up as I go along, so you should probably check it before sending it to the Nobel committee.

Consider water flowing off an edge; like a river flowing over the edge of a cliff, but on a teapot sized scale. The diagram below shows the geometry.

Dribbling

The water is flowing at a velocity $v$ through a channel (i.e. spout) of width $l$ and depth $d$.

Where the water leaves the channel and flows out into the air you are exchanging a water/teapot interface for a water/air interface. let's call the water/teapot interfacial tension $\gamma_{wt}$ and the water/air interfacial tension (aka the surface tension) $\gamma_{wa}$. The units of $\gamma$ are force per unit length i.e. if you draw a line of some length $\ell$ the force normal to that line is $\gamma\ell$.

If you look at the point where I've drawn the force $F$ acting,i.e. where the water leaves the edge, then the force $F$ acting in the direction I've drawn (opposite to the velocity) is:

$$F = (\gamma_{wa} - \gamma_{wt})\ell $$

where the length $\ell = l + 2d$. A positive value of this force means the water is being pulled back into the channel. The force will normally be positive because the water/air interfacial tension is greater than the water/teapot interfacial tension. That's why water droplets on the surface of the teapot tend to spread out instead of rolling up.

So far so good. Now, my model is that if this force is great enough to bring the water stream to a stop the water will dribble down the edge, while if the force smaller than this the water will flow cleanly off. This is obviously an approximation because it's possible that the bottom of the water stream may slow and dribble while to top flows cleanly, but let's go with this and see where it gets us.

We'll use the fact that force is equal to rate of change of momentum. The momentum of the water flowing off the edge in one second is simply:

$$p = \rho vA \times v$$

because $A = ld$ is the area of the channel, and the velocity $v$ is the length that flows in one second so $vA$ is the volume and therefore $\rho vA$ the mass. So, if the water comes to a stop at the edge the rate of change of momentum is $p$, and therefore we will get dribbling when:

$$ (\gamma_{wa} - \gamma_{wt})\ell > \rho Av^2 $$

or since the velocity is the only thing we can easily vary, we get dribbling when:

$$ v < \sqrt{\frac{(\gamma_{wa} - \gamma_{wt})\ell}{\rho A} }$$

You can now immediately see why the water/teapot interfacial tension matters. Remember $\gamma_{wt}$ is normally less than $\gamma_{wa}$, but if we hydrophobe the teapot surface we make $\gamma_{wt}$ bigger. If we can make it big enough to equal the water/air interfacial tension, so $\gamma_{wa} - \gamma_{wt} = 0$ then our equation becomes:

$$ v < \sqrt{\frac{zero \times\ell}{\rho A} }$$

so we will never get dribbling.

Note also that the dribbling velocity depends on the ratio $\ell/A$ i.e. the ratio of the channel perimeter to it's area. This means a semi-circular channel would be less likely to dribble than a wide shallow channel.

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