H.C. Mayer and R. Krechetnikov. WALKING WITH COFFEE: WHY DOES IT SPILL?

Abstract

In our busy lives, almost all of us have to walk with a cup of coffee. While often we spill the drink, this familiar phenomenon has never been explored systematically. Here we report on the results of an experimental study of the conditions under which coffee spills for various walking speeds and initial liquid levels in the cup. These observations are analyzed from the dynamical systems and fluid mechanics viewpoints as well as with the help of a model developed here. Particularities of the common cup sizes, the coffee properties, and the biomechanics ofwalking proved to be responsible for the spilling phenomenon. The studied problem represents an example of the interplay between the complex motion of a cup, due to the biomechanics of a walking individual, and the low-viscosity-liquid dynamics in it.

1. Introduction

Abramson, a classic of sloshing dynamics, started his seminal 1967 NASA report with the remark that

it is common everyday knowledge to each of us that any small container filled with liquid must be moved or carried very carefully to avoid spills.

While many of us may have thought about this phenomenon, the question as to why we spill coffee has never been formally studied to the authors’ knowledge. As we have allwalkedwith a cup of coffee or other liquid-filled containers, this is certainly a familiar problem.

The presented study of when and why coffee spills shows that it is a confluence of biomechanics, liquid sloshing engineering, and dynamical systems. While each of these subjects is well developed, the fact that they have different focuses of study does not allow one to penetrate into the posed question immediately. Namely, on one hand, a large body of the biomechanics and medical literature is dedicated to the study of human walking, including quantifying the motion of the

✓ body center of mass;

✓ energy expenditure, and efficiency of process;

✓ gait patterns related to gender, age, health, etc.;

✓ motion of appendages in natural positions;

✓ regularity of walking over long periods of time.

On the other hand, liquid sloshing studies are concerned

✓ with large liquid-filled structures such as rocket fuel tanks, which are subject to considerable accelerations and vibrations;

✓ forces and torques due to liquids experiencing various sloshing motions;

✓ suppression of sloshing necessary for vehicle control.

However, in the walking with coffee problem the motions of the human body, while seemingly regular, are quite complex and are coupled to a coffee cup and liquid therein, which makes it difficult to unravel the precise reasons behind coffee spilling.

Fig. 1. Coffee spill and key liquid motions in an excited cup: (a) Representative image of coffee spilling. (b) Rotational liquid motion in clockwise direction: top left–top right–bottom left–bottom right. (c) Back-and-forth liquid oscillations; (d) Vertical liquid oscillations.

II. Our studies

To address the challenge we set up experiments in which a human subject was asked to walk naturally with different speeds along a straight line path, with acceleration ẍ chosen naturally by the subject. Two different regimes of walking were explored: when the subject was focused - trying not to spill and unfocused - not concerned with spilling. The subject was recorded at 100 fps walking at different average speeds in the range 〈ẋ〉 = 0.5 - 2.5 ^m/_s. The motion was analyzed in the xy plane only, which is justified below, using an image analysis program written in MATLAB, while the instant of a spill was determined with a light-emitting diode signal triggered by a sensor monitoring the coffee level in the cup.

A. Cup excitation vs hydrodynamics in the cup

As known from biomechanics, the key goals during walking are to move the body toward a desired location and at a desired speed, while using the least amount of total energy to achieve the first goal. This means that the body tends to move in as straight a line as possible and with very little up and down movements. The latter is due to the tendency at the freely chosen step rate to consume the least oxygen at any given speed. As follows from the trajectory measurements, an initial acceleration region is followed by a region of an almost constant walking speed, i.e., the maximum acceleration ẍ_max is realized early on. Measurements, shown in Fig. 4(a), demonstrate that ẍ_max chosen by a human subject increases with average speed 〈ẋ〉 and is lower in the focused regime for higher walking speeds. Since the maximum acceleration ẍ_max is a good measure of the initial deflection of the liquid interface, we decided to choose it for further characterization of the data and scale it with respect to the acceleration ẍ_α, defined by the spill angle α for a given coffee level.

Fig. 2. Definition and extraction of the cup dimensions and coordinates in the coffee spill experiments: (a) Walking path as viewed from above: (b) Cup coordinates: xy is the plane of motion, θ the pitching angle, and g gravity; (c) Spill angle α and equivalent acceleration ẍ_α; (d) MATLAB image analysis.

Analyses of all cup excitations-back-and-forth in Fig. 3(a), vertical in Fig. 3(b), and pitching-show the presence of oscillatory-type motions on the order of 1 cm in magnitude superimposed on the average trajectory of the human subject.

Fig. 3. The trajectory of the coffee cup determined from image analysis for (a) back-and-forth and (b) vertical excitation; (c) The decomposition of the signal shown in the shaded region in (b) into smooth (dashed line) and noise (solid line) components; (a) - (c) depict cup excitations in the unfocused regime with walking speed 〈ẋ〉 = 80 ^cm/_s, maximum acceleration 71 ^cm/_s, and coffee level α = 12.1° shaded region corresponds to the regime of constant average velocity, the red (solid vertical) line shows the instant of the first coffee spill, and the dashed vertical lines correspond to step instants.

Vertical excitations are characterized by the maximum of the y velocity synchronized with steps. As for pitching θ motion, it is essentially a combination of arm swing and flexibility of the wrist and is on the order of 2°. Analysis of the dependence of excitation on walking speed in Fig. 4(c) indicates that the standard deviation of the x velocity about the average decreases with increasing walking speed. This is counterintuitive, i.e., the faster one walks the smoother the motion of the cup in the x direction, as opposed to themagnitude of the y excitation in Fig. 4(d), which increases withwalking speed. Fig. 4(b) shows that the coffee cup is excited at a frequency fcup nearly identical to the step frequency f_step, which increases with walking speed and agrees with previous measurements for a variety of individuals. Most importantly, the excitation frequency is in the range of f_cup = 1 - 2.5 Hz.

Fig. 4. (a) and (b) illustrate basic biomechanics for different coffee levels: (a) maximum acceleration ẍ_max in focused and unfocused regimes for different walking speeds and (b) step and cup excitation frequencies and a comparison of our data to the empirical curve. (c) and (d) show representative samples of the dependence of excitation on walking speed, specifically (c) the dependence of back-and-forth excitation 〈ẋ'〉 on walking speed 〈ẋ〉 and (d) the dependence of vertical excitation 〈y'〉 on walking speed 〈ẋ'〉.

Now let us compare the cup excitations to natural frequencies of liquid oscillations in the cup. The natural frequencies of oscillations of a frictionless, vorticity-free, and incompressible liquid in an upright cylindrical container with a free liquid surface arewell known from liquid sloshing engineering:

(1)

where m = 0, 1, 2, ... and n = 1, 2, ... In this equation H is the liquid height, ρ its density, R the cup radius, σ the surface tension, and g the gravity acceleration. The values ∈_mn are the roots of the first derivative (∈) = 0 of the mth-order Bessel function of the first kind; numerical values are ∈₁₁ = 1.841, ∈₁₂ = 5.331, ∈₃₁ = 8.536, ..., ∈_{1 n + 1} = ∈_{1 n} + π. The shape of an antisymmetric mode of the free surface wave is proportional to J₁(^∈_mnr/_R) and for n = 1 is shown in plate 191 of van Dyke’s Album of Fluid Motion. For a typical common size of a coffee cup, R ≈ 3.5 cm and H ≈ 10 cm, which is studied here, the surface tension σ effect is negligible and the angular frequency is ω₁₁ = ω_nat ∼ 23 ^rad/_s, so that the corresponding frequency is ν₁₁ = ν_nat ~ 3.65 Hz and the period of liquid oscillations is T_nat ∼ 0.27 s. In general, coffee cups vary in size roughly from (R, H) = (2.5, 5.7) to (6.7, 8.9) cm for espresso to cappuccino type mugs, which gives natural frequencies in the range 2.6 - 4.3 Hz. Therefore, once compared to the range of excitation frequencies, f_cup = 1 - 2.5 Hz, it is the lowest-frequency antisymmetric mode ωnat that tends to get excited during walking, which exhibits the largest liquidmass participating in the back-and-forth sloshing and thus leads to the largest effect on the dynamics of the system.

From a dynamical systems point of view, the considered system is parametrically excited by the following motions:

(i)   Back-and-forth: this excites the first antisymmetric sloshing mode.

(ii)  Lateral: as known from the measurements of the motion of the body center of gravity during walking, the lateral and forward oscillations are of the same order of amplitude of 20 mm, although the frequency of the lateral motion is almost half that of the forward motion, i.e., ∼ 1.25 Hz. Therefore, its effect is less important as it is further away from the natural frequency of liquid oscillations, ν_nat ~ 3.65 Hz.

(iii) Vertical: Faraday was one of the first to observe the phenomenon of parametric resonance, noting that surface waves in a fluid-filled cylinder under vertical excitation exhibited twice the period of the excitation - the so-called one-half subharmonic response. Since the frequency of vertical excitation in our case is lower than the natural one, one does not expect the Faraday instability to be observed.

(iv) Pitching: Since the pitching frequency is the same as that of back-and-forth motion, it contributes to the excitation of the natural antisymmetric mode.

These basic excitation effects alone, however, do not explain the particularities of coffee spilling. First, let us look at the dependence of the number of steps N before the coffee is spilled on the coffee level α, the walking speed characterized by the maximum acceleration ẍ_max as per Fig. 4(a), and the focused vs unfocused regimes in Fig. 5.

Fig. 5. Experimental measurements of the number of steps N to spill coffee for different coffee levels α. (a) - (c) represent experimental measurements for different coffee levels α for both focused and unfocused regimes of walking, with summarized dependence on the coffee level α and focused vs unfocused regimes in (d); the error bars represent the uncertainty in determining the step time interval.

First, it was determined that most often coffee spills in the range of 4 - 5 m of walking distance. Clearly, N decreases with increasing ^ẍ_max/_ẍα and thuswalking speed, which is consonant with one’s intuition. If ^ẍ_max/_ẍα > 1, a very short spill time is observed as the coffee reaches the critical spill angle α due to a large early acceleration that provides significant initial deflection of the interface; see Fig. 2(c). Also, high coffee levels exhibit fewer N before spilling occurs. Most interestingly, focusing tends to increase N before spilling, which, in part, is due to lower maximum acceleration, which sets a lower initial amplitude for liquid sloshing, and a lower noise level in the excitation for a given walking speed compared to the unfocused regime as explained below.

While walking appears to be a periodic, regular process, closer examination reveals fluctuations in the gait pattern, even under steady conditions. Together with other natural factors - uneven floors, distractions during walking, etc. - this explains why the cup motion during the constant walking speed regime is composed of noise and smooth oscillations of constant amplitude: the smooth signal has a frequency commensurate with that of walking, whereas it is apparent that the noise is a complicated combination of higher-frequency harmonics. Comparison of the focused and unfocused regimes in Fig. 5 suggests that higher levels of noise in the latter regime substantially intensify the spilling.

Mechanical model

To support the above conclusions and explain the experimental findings presented here, we developed a simple mechanical model based on the fully nonlinear spherical pendulum equations with forced oscillations. As a spherical pendulum have characteristics similar to sloshing, several investigators have used this analogy to predict and understand the observed sloshing behavior in other systems.

Since we are mainly interested in the near-planar pendulum motions, it is convenient to choose a coordinate system which has singularities at x = z = 0, y = ±1 and is defined in terms of two angles γ and β:

x = cosβ sinγ, y = sin β, z = − cosβ cos γ	(2)

where − π < γ ⋜ π and − ^π/₂ ⋜ β ⋜ ^π/₂ as shown in Fig. 6.

Fig. 6. Coordinate system for the spherical pendulum model: the pendulum is shown in the shaded plane and driven horizontally in the x direction with time-dependent forcing x₀(t).

Considering a spherical pendulum of length l and mass m, which is forced horizontally in the x direction as x₀(t), the nondimensional kinetic and potential energies are given by

(3)

where we took into account that the horizontal displacements of the pendulum relative to the pivot are lx and ly, respectively. Given the expressions of the kinetic and potential energies, we can form the Lagrangian L = T − V and, expressing x and z in terms of y and γ, derive the Euler - Lagrange equations for γ and y:

The length l of an equivalent spherical pendulum is found by equating the natural frequency of the pendulum to that of the first antisymmetric mode ω₁₁, for liquid sloshing. As the idea was to develop as simple a model as possible, the forcing in the x direction includes the walking frequency ωcup with amplitude A similar to that observed in experiments and a number of higher-frequency harmonics:

(4)

with randomly selected frequencies ωn over a sufficiently large interval, which includes ω_cup, as well as random phase shifts φ_n and amplitudes a_n, so that their total amplitude in the Euclidean norm is a fraction of A to reflect the presence of the noise. The number of harmonics n* is taken sufficiently large so that the results in Fig. 7 become insensitive to further increases in n*.

Fig. 7. Modeling the sloshing of a liquid in a cup as an equivalent spherical pendulum. (a) shows steps to spill N/P versus the normalized initial deflection of the interface ^ẍ_max/_ẍα for three different coffee levels (8° ≲ α ≲ 16°). Each data point shown in the figure corresponds to the average number of steps N of all model runs that yield a spill divided by the fraction of runs P that result in a spill. The error bars suggest the scatter in the number of steps for any given run for any fixed conditions due to the random nature of the noise in the forcing signal; solid lines through data sets are meant to guide the eye. (b) reflects the influence of the noise amplitude |a| on the number of steps to spill for α = 16.5°. Each data point here corresponds to an average as discussed in the previous figure, but the error bars have been left out for clarity. (c) shows the effect of a time-dependent forcing frequency on coffee spilling over short and longer times. (d) shows an example trajectory of a spherical pendulum projected onto the xz plane in the case &alpha = 16.5°. The starting position of the pendulum is denoted by an open circle and is based on the initial angle in the xy plane and a random z component. The critical spill radius in (d) is denoted by the red circle.

The spherical pendulum modelwas integrated from random initial conditions and with randomly selected forcing (4). A considerable number of runs was performed to collect statistical data of spilling events, which occur when the deflection of the pendulum exceeds a critical value corresponding to the spill angle α. Qualitative agreement exists between the outcome of these simulations and the presented experimental observations, namely, that with increasing ^ẍ_max/_ẍα the number of steps to spill, normalized by the fraction of runs P that result in a spill, ^N/_P, decreases significantly up to ^ẍ_max/_ẍα = 1, at which point there is a nearly instantaneous spill. Furthermore, ^N/_P decreases with an increase in the coffee level for any value of ^ẍ_max/_ẍα. All data points in Fig. 7(a), shown in gray, correspond to the low value of the noise amplitude used, ^|a|/_A = 0.1, which is analogous to the focused walking characterized by lower noise levels in Fig. 7(b). The effect of increasing the amplitude |a| of the noise in the forcing signal is to decrease ^N/_P as shown in Fig. 7(b). This is qualitatively similar in nature to the differences observed in experiments between focused and unfocused walking.

Next, since walking starts from rest and is usually nonuniform, the frequency of step changes so that one can expect resonance to occur much earlier than when excitation approaches the natural frequency. This is known as the chirped frequency phenomenon used to trap particles with electromagnetic waves - resonance sets in when the ramped forcing frequency is midway between its initial value and the natural frequency for resonance in the unforced problem. Therefore, since walking is usually nonuniform, the step frequency varies initially from zero to some constant value and over the course of walking, one can expect that resonance will occur even before the step frequency reaches the value of the natural frequency. If this is true, then perhaps the liquid in the cup experiences growth in sloshing at angular frequencies ω lower than the natural frequency ωnat and could lead to spilling. This behavior is exemplified in Fig. 7(c), in which a simple planar pendulum is subject to a ramped forcing frequency ω starting from zero with an arbitrary deflection at t = 0. For the particular case shown in Fig. 7(c), x₀ = 5 and the ramp rate ∈ = 0.003 18 results in a frequency ratio ^ω/_ωnat = 0.6 after 30 cycles. At some point in time, it is observed that the pendulum experiences an increase in the x deflection corresponding to the frequency reaching the jump frequency. If the maximum experienced amplitude is above a critical deflection amplitude, then spilling occurs. This should be contrasted to a simple planar pendulum with a fixed forced frequency corresponding to a fixed frequency ratio 0.6, in which case no substantial increase in the x deflection is achieved. The inset of Fig. 7(c) corresponds to the ramp rate ∈ = 0.0318, but ^ω/_ωnat = 0.6 is reached after three cycles to model the initial stage of walking from rest. It is apparent that the deflection x increases at early times and can reach a critical amplitude versus the motion of an unforced pendulum with the same initial deflection.

A closer look at the liquid motion in the cup suggests that not only back-and-forth but also swirling liquid motions can be excited by walking at typical speeds. While swirl does not contribute much to coffee spilling, it is still an interesting question as to why rotational motion is generated. In addition to the obvious potential reason - movement of the hand in the xz plane - many different experimental studies concerned with back-and-forth sloshing have revealed an interesting type of liquid instability occurring very close to the lowest liquid resonant frequency: the nodal diameter of the antisymmetric wave begins to rotate at an unsteady rate. The essential features of this complex liquid motion can be described qualitatively as an apparent rotation of the liquid about the vertical axis of symmetry of the tank, superimposed on the normal sloshing motion. There is also a somewhat larger frequency range just above the natural frequency in which the nodal diameter can rotate stably at a constant rate. As illustrated in Fig. 7(d), the spherical pendulum does demonstrate, depending on the ratio of the excitation to the natural frequency, instability of planar motion and thus generation of rotational motion.

III. Discussion

As follows from the presented study, we spill coffee either by accelerating too much for a given coffee level or through more complicated dynamical phenomena due to the particular range of sizes of common coffee cups, which is dictated by the convenience of carrying them and the normal consumption of coffee by humans. Namely, first the maximum acceleration occurring early on in the walking sets an initial sloshing amplitude. This interface deflection is then amplified by the back-and-forth and pitching excitations. Vertical excitation does not lead to resonance as it is a subharmonic excitation. The noise component of motion contains higher-frequency harmonics, which make the antisymmetric mode unstable, thus generating a swirl. Time to spill generally depends on whether walking is in a focused or unfocused regime and increases with decreasing maximum acceleration. The difference between the focused and unfocused regimes suggests that walking with coffee is a control problem, since the number of steps to spill is different in these two regimes. As the data analysis shows, this is due to the difference in the level of a noise excitation, while the amplitude of a regular excitation does not change significantly, which is conceivably due to the natural constraints of the biomechanics of walking. However, whether it is a feedback control system - i.e., when a human being identifies the resonant sloshing frequency and then performs a targeted suppression of the resonant mode - or an open loop control system - i.e., when a human being simply becomes more careful about carrying a cup regardless of the natural frequency of the fluid in it - may depend on an individual.Understanding this is beyond the scope of ourwork, as the collected data are too limited to provide conclusive results about such a specific question as to whether it is only the resonant excitation which is suppressed in the focused regime. Another relevant question to study is the influence of the degree of danger of the fluid in a cup, e.g., hot vs cold.

Our simple theoretical model captures the observed phenomena qualitatively - quantitative differences are in part due to differences in the widths of the resonance curves between spherical pendulum and liquid oscillations in a cylindrical container; in the latter case, the response curve is substantially wider, thus explaining the stronger response of the liquid to excitations at frequencies far from resonance. The fact that the studied system demonstrates that in the focused regime, characterized by lower levels of noise, the coffee spill occurs later compared to the unfocused regime makes the phenomenon analogous to that of stochastic resonance, where the periodic forcing is amplified by the presence of noise above a certain threshold.

While the experimental data discussed here are representative of human walking with a cup of coffee, as the involved parameters - excitation amplitudes and frequencies - do not have much variation among individuals, future comparative studies may provide further insights into the variability of the discussed phenomena. For example, gender differences - gait velocity and step length are lower and step frequency is higher for women than for men - may lead to differences in coffee spilling. However, some age effects, such as the changes of the basic gait parameters most frequently seen with advancing age, are a reduction of gait speed and step length, but just small changes of the step frequency are not expected to lead to variations in coffee spilling.

Finally, lessons learned from sloshing engineering may suggest strategies to control spilling, e.g., via using (a) a flexible container to act as a sloshing absorber in suppressing liquid oscillations, or (b) a series of concentric rings arranged around the inner wall of a container. Namely, ring baffles damp the large sloshing mass flow associated with the main resonant frequency of the liquid in the container into several smaller sloshing masses - high-speed turbulent flow around baffles - having considerably higher first-mode resonant frequencies. The baffles can be nonperforated or perforated. Perforation provides better damping and reduces themass of baffles. However, despite the variety of spill control options, the simplicity and convenience of a common coffee cup will likely continue to outweigh the side effect of coffee drinking studied here - occasional spilling.

IV. Acknowlegments

This work was partially supported by DARPA Young Faculty Award Grant No. N66001-11-1-4130. The authors would like to thank UCSB Mechanical Engineering undergraduate Zheqing Liu for the assistance with experiments.

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