A UCL mathematics student has found that the falling 'curtain' of chocolate in a chocolate fountain surprisingly pulls inwards rather than going straight downwards because of surface tension.
“Chocolate fountains are just cool, aren’t they!” said Adam Townsend (UCL Mathematics), lead author of the paper based on his MSci project. “But it’s also nice that they’re models of some very important aspects of fluid dynamics. We’ve used some serious maths to solve a fun problem – why the chocolate 'curtain' on a chocolate fountain always falls inwards.”
The study, published today (Wednesday 25 November 2015) in European Journal of Physics, used classic work on ‘water bells’ to model the fluid dynamics of the chocolate in the fountain. The physics of the water bell is exactly the same as the falling curtain of chocolate, and by using this model the team discovered that surface tension of the chocolate causes it to fall inwards.
“You can build a water bell really easily in your kitchen,” said Dr Helen Wilson (UCL Mathematics), co-author of the paper and MSci project. “Just fix a pen vertically under a tap with a 10p coin flat on top and you'll see a beautiful bell-shaped fountain of water.”
The researchers studied the flow of chocolate up the pipe to the top of the fountain, over the plastic tiers that form the distinctive chocolate fountain shape and down as a curtain.
“Both the chocolate fountain and water bell experiments are surprisingly simple to perform. However, they allow us to demonstrate several aspects of fluid dynamics, both ‘Newtonian’ which includes everyday fluids such as air, water and syrup, and ‘non-Newtonian’ which is anything with an underlying structure that can be built up or broken down by flow such as biological fluids (e.g. blood) and molten chocolate.” said Dr Wilson.
“Although this was a fun project about chocolate, gaining an understanding of the thin film flows in the chocolate fountain might help with applications as diverse as lava flow on volcanoes, tear films in the eye, and extracting plasmas out of nuclear fusion reactors,” concluded Dr Wilson.
The researchers were also pleased that their work allowed them to engage with the public.
Adam Townsend added, “I've been talking at mathematics enrichment events around London for the last few years. If I can convince just one person that maths is more than Pythagoras' Theorem, I'll have succeeded. Of course, the same mathematics has a wide use in many other important industries – but none of them are quite as tasty as chocolate.”
Abstract: We consider the fluid dynamics of the chocolate fountain. Molten chocolate is a mildly shear-thinning non-Newtonian fluid. Dividing the flow into three main geometries—the pumped flow up the centre, the film flow over each dome, and the freely-falling curtain flow between the domes—we generate a wide-ranging study of Newtonian and non-Newtonian fluid mechanics. The central pumped flow is a benchmark to elucidate the effects of shear-thinning. The dome flow can be modelled as a thin-film flow with the leading-order effects being a simple balance of gravity and viscosity. Finally, the curtain flow is analytically intractable but matches onto the existing theory of water bells (both inviscid and viscous).
In pipe flow, Newtonian fluids exhibit a parabolic velocity profile; shear-thinning makes the profile more blunted. In thin-film flow over the dome, gravitational and viscous effects balance and the dome shape is not important beyond the local slope. We find that the chocolate thins and slows down as it travels down the dome. Finally, in the curtain flow, we predict the shape of the falling sheet for an inviscid fluid, and compare this with the literature to predict the shape for a viscous fluid, having shown that viscous forces are too great to ignore. We also find that the primary effect driving the shape of the curtain (which falls inwards towards the axis of the fountain) is surface tension.
We find that the three domains provide excellent introductions to non-Newtonian mechanics, the important mathematical technique of scaling, and how to manipulate existing data to make our own predictions. We also find that the topic generates interest among the public in our engagement work.