Subsections
• Linear Growth
• Exponential Growth
• Explosive Growth
• Volcano Example
• Conclusion

  
Honey on Toast

It's breakfast time, and I am about to spread some honey on my toast. I am using a spoon to dip in the honey jar. A magnified view of the leading edge of the spoon as it scoops up the honey might be like the picture below. This is a rather blunt-edged spoon, but the flow streamlines for a sharp edge would be similar. Actually, I computed these streamlines for flow about the leading edge of a hydrofoil or submarine. In that case the corners need smoothing to make the flow realistic; but that's another story.

 

 
Figure 1: Flow streamlines in a viscous fluid.

Now let's return to breakfast. I've got the honey on my spoon, and am fascinated by its slow-moving viscous behaviour. I lift the spoon high, and turn it upside down, watching for the honey to fall onto my toast. It takes a few seconds before it moves very much at all, but then it falls in a rush, as I now demonstrate.

   
Figure 2: Calculated shape of a drop of viscous fluid.Next I show a video of the same honey-falling flow. This video is not a close-up of the actual demonstration, but is a computer simulation. It was prepared by my PhD student Yvonne Stokes, using a `finite-element' program. This was actually designed to be used with some industrial applications. Yvonne's main work is on the slumping of molten glass into a mould, which is a very important process in the optical industry. Anyway, the computation is clearly seen to be a very good model of actual flow of honey. The final fall occurs so rapidly that we need to split the screen; a compressed image allows more of the final fall to be seen. The previous picture shows a single frame from this video.

For the rest of this talk, I want to illustrate how very simple Year-12 level calculus can throw some light on this sudden-fall phenomenon. The next picture shows a snapshot of the honey-drop during its fall. I am going to be daring enough to ask you to appreciate some algebraic symbols, introducing the symbol $x$ to denote the length of the drop. Also I will use the symbol $t$ in the obvious way to denote the time. The question is, how does length $x$ vary with time $t$?

 

  
Linear Growth

Calculus is about rates of change. For example, if the rate of change of $x$ with respect to $t$ was constant then $x$ would increase steadily (linearly) with time, as in the graph below. Obviously that is not what happens here! Rather than a steady increase, the honey seems hardly to move at all at first, then falls very suddenly.

 

 

  
Exponential Growth

Think of $x$ also as like the amount of money in your savings bank account. Constant rate of change is like simple interest. On the other hand, compound interest is where the rate of change increases as $x$ increases, and your bank balance then increases faster than it would for simple interest. For small compounding intervals this increase is exponential, as below. `Exponential', as a word describing rapid increases in the size of something like population, is in quite common use these days. Its explicit meaning, via the actual exponential function $\exp(x)$, is well known to every Year-12 student of Mathematics, Physics, and especially Biology.

 

 

Population increases are exponential because the more parents there are, the more babies there are; that is, if $x$ measures population size, then the rate of increase of $x$ is proportional to the present value of $x$. Malthus's law of exponential population growth is a famous biological result meaning that the population increases ``for ever''. Of course some other effect like food limitation will eventually intervene to slow down this increase. Similarly, if exponential growth really did apply to honey falling, the length $x$ of the drop would apparently increase for ever. Fortunately the toast eventually gets in the way!

  
Explosive Growth

 

 

But in fact, the honey's fall is not exponential. The increase in length of the honey drop is initially slower but finally faster than exponential growth would suggest. In a certain approximation its growth is `explosive'. Explosive growth is where the rate of increase of $x$ is not just in proportion to the present length $x$, but in proportion to the square of that length, as above.

Every Year-12 mathematics student can (or should be able to!) solve the differential equation for explosive growth. It is actually a bit easier than that for exponential growth. The answer is just that $x$ behaves like the reciprocal of a time difference. The most important consequence is that there is a finite time $t=T$ at which $x$ becomes infinite! This is the time when the explosion actually happens, or the time when the honey really falls. The previous picture shows a graph of $x$ against $t$.

Even for exponential growth $x$ eventually becomes infinite, but only if you wait an infinite time. With explosive growth you don't have to wait an infinite time. The same practical qualification prevents an actual infinite value of $x$ from occurring; namely, the honey hits the toast.

  
Volcano Example

  Explosive growth naturally occurs for real explosions too.
For example, here is a slide of the Mt St Helens explosion of May 1980.

(Photograph by Austin Post, USGS; reproduced from the collection of images available at the Cascades Volcano Observatory.)

  Prior to that explosion, there was a build up over several weeks of a mound of volcanic material, as shown at left. This was like the preliminary drooping of the honey in the spoon, before it really gets moving.

(Photograph by Peter W Lipman, USGS; from the Cascades Volcano Observatory.)

The dome on Mt St Helens, was growing at up to 1.5m per day.

From the technical point of view, there is immense interest in the actual value of the explosion time $T$. If $T$ was known for Mount St Helens, lives could have been saved. Indeed, the earthquake prediction business is precisely about this $T$ estimation. For the honey drop, we can predict $T$ quite well. For example, $T$ is proportional to the viscosity of the honey. Obviously thicker honey takes longer to fall.

  
Conclusion

This type of study is typical of Applied Mathematics.
Although the honey problem may seem rather trivial or silly, I hope you can see the important practical generalisations, some of which require only very elementary mathematics.