Waves in Guinness

We describe a simple model of a bubbly two-phase flow which is able to explain why waves propagate downward when a pint of Guinness is poured, and also how the waves are generated. Our theory involves a physically based regularization of the basic equations of the two-phase flow, using interphasic pressure difference and virtual mass terms, together with bulk or eddy viscosity terms. We show that waves can occur through an instability analogous to that which forms roll waves in inclined fluid flows, and we provide a description of the form of these waves, and compare them to observations. Our theory provides a platform for the description of waves in more general bubbly two-phase flows, and the way in which the flow breaks down to form slug flow. (C) 2008 American Institute of Physics

Parameters and values used in numerical simulations.

<p>Parameters and values used in numerical simulations.</p

Bar charts showing the distribution of cases with documented and undocumented (including asymptomatic) dates of symptom onset within the population.

<p>(a) gender of cases, (b) age group of cases in years, (c) education level of cases: No schooling (N), Primary school (P), Secondary school (S), (d) occupation of cases: Student (S), Stay at home (H), Factory worker (W), Construction worker (C), Child (Ch), Vendor (V), Farmer (F).</p

Tornado diagram of univariate sensitivity analysis.

<p>The diagram shows the degree to which a 10% variability in the parameters affects the value of . Each bar is a representation of how uncertainty in that particular parameter affects the estimate of the reproduction number. The baseline scenario is fixed with .</p

Events and rates in the stochastic model.

<p>Events and rates in the stochastic model.</p

A map of Trapeang Roka village, showing all houses for which gps co-ordinates were collected.

<p>The map shows houses with no confirmed infection (unfilled circle), houses with only infections documented by date of onset (black circle), houses with only infections undocumented by date of onset (red circles) and houses which have both cases with documented and undocumented infection onset dates (green circle). The black diagonal line indicates the main road running through the village, about which the houses are clustered.</p

The mean of 1000 stochastic realisations for number of symptomatic cases documented by date of onset (solid black line), symptomatic cases undocumented by date of onset (dotted black line), asymptomatic cases (dashed black line) and the total number of infectious cases (solid blue line) plotted with the epidemic curve (solid red line).

<p>Day 0 corresponds to the start of the epidemic on February 7.</p

The mean of 1000 stochastic realisations for the number of daily symptomatic cases documented by date of onset (solid black line) plotted with the epidemic curve (solid red line).

<p>Also shown is the mean of 1000 realisations of the SEIR model (dashed black line) and an eigendecomposition of the epidemic curve (dashed red line). The grey shaded area shows the 95% confidence interval. Day 0 corresponds to the start of the epidemic on February 7.</p

Epidemic curve showing confirmed chikungunya per day by date of reported onset in the village of Trapeang Roka, Cambodia.

<p>The grey arrow indicates the start of a two-day rain spell.</p