The teapot effect

Prompted by a recent paper in the mathematical literature, here is a much less technical article on the teapot effect (dripping from the spout when tea is poured) and what potters can do to help avoid it.

Pours for thought?

[The teapot effect: theory and practice]

David Jones (Jones the Pots)

Dripping spouts have been annoying teapot users for centuries in spite of potters’ design rules of thumb for avoiding such drops. Here some recent theoretical advances of potential value in designing drip-free spouts are reviewed and the implications for teapotters considered.

On hearing it suggested that they might benefit from reading a very mathematical paper in the Journal of Fluid Mechanics, the eyes of most potters may well glaze over.  So, here I’ll try to summarise its findings in relatively non-mathematical terms, putting them in the context of other research and practice relating to ‘the teapot effect’, and considering the implications for potters.  

Background to the teapot effect

The propensity of teapot spouts to drip was dubbed ‘the teapot effect’ by Markus Reiner in 1956 [1] although related problems had been studied under different names for several decades before that.  The usual explanation at the time for the tea clinging to the teapot spout surface rather than breaking away in a smooth arc towards the target cup was that surface tension (adhesion of the fluid and solid surface) was responsible.  However, in simple experimentation Reiner showed this was not the only underlying mechanism.  Instead, he postulated an explanation in terms of internal vortices (swirls) in the fluid, induced by the fluid being stationary where it meets the solid surface but moving past quite quickly only a short distance away from the surface.   

Shortly thereafter, Joseph Keller [2] concluded that dripping actually resulted from air pressure differences pushing the tea against the lip and further down round against the outside of the spout. Jearl Walker [3] later interpreted his observations as identifying the pressure differences as being induced when a fluid tries to move past a curved surface in a laminar (straight-line) flow. The curvature of the solid surface compresses the flow and hence creates a pressure drop that has the effect of sucking the fluid in to the surface. This is similar to the way a curved aircraft wing produces lift, but in a teapot it can lead to tea wrapping around the end of the spout as it is poured, in some circumstances leading to dripping. 

In 1994 Kistler and Scriven [4] made a substantial start on the mathematical modelling of the flow near the end of a spout.  Then Cyril Duez and colleagues from the University of Lyons [5] showed the importance of a capillary effect leading to a meniscus that keeps the liquid in contact with the surface as it leaves the spout.  They also highlighted the importance of the wettability of the teapot surface by the tea which affects the characteristics of the meniscus. They explored the effect of coating the surface with hydrophobic substances (including carbon black!) which repel water and reduces the friction between the spout and the fluid, and so encourages continuation of the main flow and inhibits drop formation.

All of these papers represent important earlier steps in understanding the teapot effect en route to the latest paper by Bernhard Scheichl and colleagues [6] which develops a more detailed and coherent account of it. 

All of the papers cited include at least diagrams and pictures of the teapot effect. Schiechl et al’s recent paper also has clear videos of the effect (see [7]).  The first 2 describe the phenomena in reasonably lay scientific terms, but the others are mostly more (or very) mathematical.

Meanwhile, in the studio

In parallel with the developments in the scientific literature but for even longer, potters have collectively been accumulating experience about how to avoid drips from their teapot spouts, mostly through informal observation, trial and error, and anecdote rather than well-designed experimentation.  Even Josiah Wedgwood does not appear to have studied spout design systematically in the way that he did glaze colours and pattern designs [8]. In this respect, the age of evidence-based ceramics [9] has yet to arrive in general, though it would be unsurprising to discover that major pottery manufacturers have done (but not openly published) such work. Nonetheless, enough of their teapots still drip to suggest that any solutions are not universally known.

Many studio potters follow rules of thumb about spout design, often derived from their teachers’ advice as well as from their own inevitably somewhat limited observations.   Compilations of guidance are available (see, for example, David Bolton [10], and John Hesselberth [11]).  The latter cites advice from some pottery texts, and of course many other texts and websites also offer advice. Mea Rhee has usefully collected recent blogs [12] which mention teapots. This helps us to see the range of aspects of teapot design believed to be helpful in avoiding drips.   Some of the tenets of best practice often mentioned include longer rather than shorter spouts (to aid development of a fast, smooth flow), a sharp bottom edge to the spout (in keeping with the scientific results above and below), a spout fixed low on the teapot body (to allow a longer spout), and a single large hole in the teapot body where the spout is attached (to facilitate a smooth flow). We’ll consider later the extent to which any of them are supported by experimental evidence. 

Although other potters may agree with some of these recommendations, there will be plenty disagreeing with one or more of them too.  This may be on practical, technical grounds (e.g.  in a ceramic pot a very thin bottom lip may be vulnerable to damage), or aesthetic grounds (e.g. the appearance of spouts of different lengths, varying diameters and so on may be preferred).  Directly contradictory advice or beliefs can of course easily be found; for example, to avoid drips, a bump inside the spout rather than a smooth interior surface has been proposed [13], but so also has a groove [11]! {To be clear, I’m reporting these various recommendations, but not endorsing all of them.}

We’ll return to reconsider these design recommendations from potters later, after consideration of the new research findings outlined below.

How can maths help?

The motion of most fluids – liquid or gas – is governed by the Navier-Stokes equations [14].  These are a set of complex (nonlinear partial differential) equations. To find solutions to these equations some simplifying assumptions need to be made. Unfortunately, the situation of interest here – tea being poured from the spout of a teapot – has several characteristics which don’t correspond well with the (mathematically) most useful simplifications.  We’ll see shortly that although at first sight the title of the recent paper [6] may prompt doubt about its practical relevance, the results are in fact very helpful in explaining the teapot effect. However, while in the longer term a stronger understanding of the underlying mechanisms can be expected to be fruitful in practical terms, sometimes practical improvements can take place in the absence of such understanding (as was the case, for example, in prevention of scurvy by eating citrus fruit before the underlying vitamin-deficiency mechanism was discovered [15]).

How difficult is solution of the teapot effect problem?

Well, it’s a slow process (1956 and counting…).  However, first, some good news: tea is effectively an incompressible liquid and is not very viscous; this makes the maths slightly simpler. Pouring molten chocolate, for example, would pose more of a problem. That said, the whole process of pouring tea from a pot is complex, as the tea in the pot starts from being still (unless recently stirred), then accelerates through the spout as the pot is tipped, but finally decelerates as pouring finishes. At the same time, the orientation of the pot and of the spout in particular changes as the pot is tipped during pouring. This means that the effect of gravity and the arc of the poured liquid change during the pouring process. The geometry of the spout may also pose problems, particularly if there are bends in it or if its diameter varies along its length as these may impact on speed, uniformity and stability of the flow.

So, where and how should we start trying to solve the teapot effect problem?  When poured quickly, the inertia or momentum of the flow means the tea continues to flow after it has passed the end of the spout, so the flow of liquid inevitably separates from the spout.  If drips do form they are observed at the end of the spout, so that some connection with the separation process is plausible a priori.  Separation of a flow from a solid surface is an important phenomenon elsewhere, for example, in aircraft wings and the swing of cricket balls.  Conditions in those contexts differ from those in a teapot spout; some of the underlying mechanisms are the same but other factors differ, and formation of drops is not an issue there. However, as noted in the review above of some of the earlier work on the teapot effect, as well as the inertia there are other mechanisms involved too, variously including capillary effects, surface tension, wettability of surfaces, viscosity and gravity.

Progress perhaps thus understandably appears to be slow, in spite of the international research effort.  Since not just teapots are affected, but also jugs, kettles, wine bottles, and industrial pouring processes (where dripping may be dangerous, if, for example, the liquid is acidic), it’s perhaps less surprising that researchers in Britain, China and Japan are not the only ones interested. 

Teatime in Lyon and Vienna

Research by Cyril Duez and colleagues at the University of Lyon in 2009 established an interplay between the flow of the liquid on a relatively large scale – the main flow from a teapot spout, for example, dominated by inertia/momentum of the flow – and surface capillary effects on a much smaller scale, which may become important when the flow slows down.   

Recently, a team led by Bernard Scheichl based in TU Wien (the Vienna University of Technology) and elsewhere built on Duez et al’s work (and some of the other papers reviewed above) to make progress in solving simplified versions of the governing Navier-Stokes equations relevant to the flow of tea being poured from a teapot, concentrating on the region close to the end of the spout, where the flow separates from the pot surface. This provides a much better understanding of the teapot effect, at the price of a rather complex explanation, and certainly demanding maths, so you may want to skip even the simplified account in the next (starred) section.

It is important first to note, however, that the new work relates only to developed flows – or as we might say in other contexts, midstream flows – and not to the starting and stopping phases of the pouring, so there’s still more relevant research to be done in addition to the progress described below.   

*Poring over the details    [*A little more technically]

Schiechl et al’s paper concentrates on what happens in a region of microscopic scale near the end of the spout when tea is poured from it.  Figure 1 in Schiechl et al’s paper [6] shows the set up, and its complexity. As noted above, they consider only developed flows and not the starting and stopping phases of the pouring which would involve even more mathematical complexity. They make various simplifying (but carefully justified) assumptions about the flow in the various regions shown, including that the Reynolds number is large (meaning that the energy of the flow motion dominates the dissipating effects of viscosity).   In fact, they divide the region near the end of the spout into half-a-dozen regions nested inside each other, as shown in Figure 3 in their paper. Different conditions apply in each region, but a simplified non-mathematical account of this is difficult. However, it is worth explaining that in the innermost region near the end of the spout, the scale on which they are working is so small that curvature of the spout can be ignored, and the lip regarded as flat and unbounded in width!  The inner regions are not directly visible as separate regions. More importantly, the flow in each region influences what happens in neighbouring regions, so that overall the mathematical modelling of the flow is very complex.

In the key regions near the end of the spout, the authors can solve simplified versions of the Navier-Stokes equations. These solutions show that inertia, capillarity, surface tension and viscosity all have roles to play. Perhaps surprisingly, the new work by Schiechl et al indicates that gravity is relatively unimportant in determining whether drops form or not in the teapot effect, although gravity does, of course, affect the direction of the poured arc of fluid and of any falling drops.

If the flow is fast enough, tea flows in an arc after separation from the end of the spout, the inertia of the flow dominating other mechanisms. Drops do not form. At these higher flow rates the so-called boundary layer of fluid closest to the surface of the spout detaches smoothly from the spout surface and so the tea can continue in a smooth arc down towards the tea cup.

However, when the flow is slow enough, a drop can form at end of spout. A capillary effect can dominate or at least compete with the inertial effect, slowing the flow down and pulling some of the tea flow down to form drips which dribble off the spout. Exactly when this switch from a smooth flow to a dribble occurs depends on various characteristics of the tea, the material of the pot, and the pouring process. Importantly, the more hydrophilic (wettable) the material of the teapot is, the later the separation of the liquid flow from the spout will be.  The angle at which the spout is cut back underneath the flow is also important, as this affects how long the liquid clings to the surface before it eventually breaks away – into a smooth stream, or a drip.

Hence, so far 3 factors important in preventing or reducing propensity to form drops in midstream flow and potentially open to manipulation by the maker or user of the teapot have been identified: speed of flow, sharpness of the spout, and wettability of the teapot surface.  

We’ll consider what this means for makers and users of teapots in the next section.

Pourers and potters

Potters can’t prevent dripping from teapots on their own.  They need co-operation from the pourers, and even then drips may occur.  What can potters and pourers do? If we set aside trying to alter the properties of the tea itself, since even making tea ‘strong enough for a spoon to stand up in’ won’t significantly impact on the role of viscosity, for example, the following factors identified in the midstream mathematical modelling are open to intervention:

  • speed of pouring,
  • spout design, and
  • surface wettability.

Speed of pouring.  Mundanely, but nonetheless importantly, the potter can design and make the teapot in such a way as to make pouring as easy as possible, so that, inter alia, the fast midstream flow is easily and quickly achieved by the pourer. A small and light teapot, with an easily used handle, for example, one with enough room for several fingers, and a flattened profile for a firm grip, will help.  So will a relatively long spout. Then the pourer can more easily pour the tea quickly and hold the pot at a steep enough angle to ensure the flow has enough inertia to detach (cleanly) from the spout, at least in the midstream flow period.

Spout design.  A sharply cut away lower lip to the spout will help to prevent dripping, although this needs to be balanced against strength and durability of the lip. The Schiechl paper confirms Duez et al’s suggestion that to avoid dribbling the lip of the spout should be as thin as possible. 

A long spout may enable a fast, stable flow to be established at its end, but this will also depend on how low on the body of the teapot the spout is mounted, and how full of tea the pot is when pouring starts.

What should be recommended re other aspects of the spout design is much less clear and is not addressed in the papers cited above.  Bends, flaring or reduction of the diameter of the spout, ridges inside the spout and grooves on the under edge all have their advocates in the ceramics literature. From a fluid mechanics viewpoint, while some variants of these suggestions could prove advantageous, it is also easy to see how many of them could interrupt establishment of a smooth flow at the end of the spout, perhaps leading to turbulence.

Surface wettability. Duez et al in particular advocated coating the lip with a hydrophobic material which repels water and thus prevents the tea from clinging to the teapot material by reducing the friction between the spout and the fluid. This chimes with the old idea of smearing butter (or latterly, vegan alternatives, presumably) on the spout to stop dribbling.  What about wettability by type of glaze?  Most teapots are glazed with a full gloss rather than a matt glaze, so the here the options are limited. 

Some potters will no doubt respond to the list above by saying that this is what they do anyway – but now, following the latest theoretical work, they could in principle explain why they do so as well.

Pouring: beginning and ending  

The design features above which seek to facilitate pouring may well also be particularly valuable in the starting and stopping phases, about which to date the theory says relatively little. Drips are commonly observed to occur during the stopping phase especially, as the teapot is tipped back to horizontal to stop the flow, and also in the start-up phase before the main flow is fully established.  Not surprisingly, there are recommendations in the ceramics literature relating to prevention of drips, particular when the pouring finishes, but no consensus recommendation emerges. Nonetheless, Hesselberth [11] discusses the pouring situation usefully, citing comments from other texts considering the orientation and slope of spout.   

What’s brewing?   [Next steps]

Although the teapot effect doesn’t for mathematicians quite have the cachet of the Riemann hypothesis [16] we can be sure that it will be investigated further by mathematical methods.  Indeed, Scheichl and colleagues indicated they are already doing so. Although not always straightforward to implement, computational methods of solving the equations governing the flow of fluids could in principle address the starting and stopping phases of pouring, although quite probably not in satisfactory detail to explain what is happening near the end of the spout.  Thus, in due course, more nearly complete mathematical solutions to the teapot effect may, with some difficulty, be developed. 

What can and should potters do while waiting for the maths?  Although it is possible that individual potters may be able to move on from test tiles of glazes to critical evaluation of test teapot designs, a more realistic possibility is for potters to collaborate on an evidence-based ceramics approach. A teapot effect attenuation (TEA) project – a 2nd age of Wedgwood perhaps – would require careful description of all of the key design characteristics of the teapots which each of a number of potters regard as their best in terms of non-dripping potential, followed by experimental comparison of the dripping performance of each teapot using a standard pouring rig set up as, say, part of a postgraduate degree project, perhaps funded by a major tea purveyor.  Anyone for TEA?

Story in a teacup    [Summary] 

In seeking a simple takeaway summary of factors important in the teapot effect, the maxim ‘Everything should be made as simple as possible, but not simpler.’ often attributed to Einstein [17] is appropriate.  The explanation is not all that simple; even in midstream flow inertia/speed of flow, surface tension, capillarity, wettability, and lip wedge shape have parts to play, and there’s more to come re the impact of pouring techniques. The best summary I can give with reasonable accuracy is to say that in midstream pouring the momentum of a fast flow dominates and drops are not a problem, but at lower flow speeds capillary effects come in to play and allow formation of drops from the end of the spout.

Finally, just in case you are asking: Do I make teapots? Yes.  Do I guarantee that they don’t drip? Not as yet – see the picture above!

References

  1. Reiner M   The teapot effect … a problem. Phys Today 1956 9(9) 16-20.
  2. Keller JB Teapot effect. J Appl Phys 1957 28(8) 859-864.
  3. Walker J.  The troublesome teapot effect, or why poured liquid clings to the container. Sci Am 1984 251(4) 144-153.  
  4. Kistler SF and Scriven LE The teapot effect: sheet-forming flows with deflection, wetting and hysteresis. J Fluid Mech 1994 263 19-62.
  5. Duez C, Ybert C, Clanet C, Bocquet L. Wetting controls separation of inertial flows from solid surfaces.  Phys Rev Lett 2010 104  084503.  Draft available at arXiv:0910.3306 
  6. Scheichl B, Bowles RI, Paslas G.  Developed liquid film passing a smoothed and wedge-shaped trailing edge: small-scale analysis and the ‘teapot effect’ at large Reynolds numbers  J Fluid Mech 2021 926 A25
  7. See  https://www.youtube.com/watch?v=jzZ2_Yh8c68 
  8. Wedgwood glaze tests   See  https://collections.vam.ac.uk/item/O1478402/trial/
  9. Worrall J.  Evidence-based everything (but let’s do the basing properly) 2018.  See  https://www.lse.ac.uk/Events/2018/10/20181019t1830vOT/evidence-based-everything
  10. Bolton D  Functional Teapot Options & Rules of Thumb – CLC Ceramics  2018  See http://www.davidwbolton.com/uploads/5/5/1/4/55145091/tpotdiagram07.pdf
  11. Hesselberth J.    How to make drip-free spouts.  Clay Times 1997.  See https://www.claytimes.com/articles/spouts.html
  12. Rhee M   Teapot design.   See  https://community.ceramicartsdaily.org/topic/19430-teapots-that-pour-beautifully/
  13. Kumar D. See  http://news.bbc.co.uk/1/hi/sci/tech/231020.stm
  14. Navier-Stokes equations.  See  https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations
  15. Scurvy. See  https://www.jameslindlibrary.org/articles/james-lind-and-scurvy-1747-to-1795/
  16. Unsolved maths problems.  See  https://www.claymath.org/millennium-problems
  17. Einstein quotes.  See  https://www.nature.com/articles/d41586-018-05004-4

Last amended: 11 May 2022