Harmonic moment dynamics in Laplacian growth

Harmonic moments are integrals of integer powers of $z=x+iy$ over a domain. Here, the domain is an exterior of a bubble of air growing in an oil layer between two horizontal closely spaced plates. Harmonic moments are a natural basis for such Laplacian growth phenomena because, unlike other representations, these moments linearize the zero surface tension problem [S. Richardson, J. Fluid Mech. 56, 609 (1972)], so that all moments except the lowest one (the area of the bubble) are conserved in time. In our experiments, we directly determine the harmonic moments and show that for nonzero surface tension, all moments (except the lowest one) decay in time rather than exhibiting the divergences of other representations. Further, we derive an expression that relates the derivative of the $k^{th}$ harmonic moment $M_k$ to measurable quantities (surface tension, viscosity, the distance between the plates, and a line integral over the contour encompassing the growing bubble). The laboratory observations are in good accord with the expression we derive for $d{M}_k/dt$, which is proportional to the surface tension; thus in the zero surface tension limit, the moments (above $k=0$) are all conserved, in accord with Richardson’s theory. In addition, from the measurements of the time evolution of the harmonic moments we obtain a value for the surface tension that is within 20% of the accepted value. In conclusion, our analysis and laboratory observations demonstrate that an interface dynamics description in terms of harmonic moments is physically realizable and robust.

Figure 1

Viscous fingering patterns of air in oil contained between two plates separated by $125\mu \text{m}$ (image size is $14\text{cm}\times 14\text{cm}$). (a) Bubble grown at a constant pumping rate of 0.52 mL/min. (b), (c), (d) Bubbles grown with varying pumping rates, as described in Sec. 4B. The time development of bubble (b) is shown in Fig. 3.

Figure 2

In order to grow an approximately $n$-fold symmetric bubble, the pumping rate is adjusted to follow Eq. (17), as illustrated here by curves for different $n$. The sevenfold bubble in Fig. 1b was grown in this way, as indicated by the bold line. We measure the area of a growing bubble in real time from the video images, and compute from that area a radius for a circle with that area. Then, given a target $n$-fold symmetry, the pumping rate is adjusted (in small steps) to stay on the desired $n$-fold curve of the graph. These curves apply to a cell with gap thickness of $125\mu \text{m}$.

Figure 3

(a) An air bubble growing in oil in the Hele-Shaw cell (plate separation, $125\mu \text{m}$). Adjacent interfaces are separated by 56 s, and the maximum span of the bubble is 8 cm. The time-dependent pumping rate selects the sevenfold symmetry (cf. Figure 2). (b) Amplitudes of moments 6, 7, 8, 10, and 14, computed using Eq. (6); note that different moments have different units. All moments decay in time, in contrast to the zero surface tension case where the moments are constant.

Figure 4

An air bubble growing without a clean $n$-fold symmetry. Adjacent interfaces are separated by 20 s.

Figure 5

The ratio of the surface tension value deduced from theory for the data in Fig. 4 to the value independently measured by a traditional method (see text). The upper points $(\times )$ are without a wetting correction, and the lower points $(•)$ are with the wetting correction in Eq. (16) [the term with the factor 3.8]. The error bars show the standard deviation for each group of points. The value of $t_2-t_1$ in Eq. (16) was 10 s. The horizontal axis corresponds to $t_1$.

Figure 6

Spectra of the nondimensional moment amplitudes $\left|M_k\right|/{\left|M_0\right|}^{(2-k)/2}$ [Eq. (15)] for two bubbles: (a) A bubble with approximately sevenfold symmetry [Fig. 3a], where the main moments are 7, 14, and 21. The moments were calculated at the 112th second of the experiment. (b) A bubble without $n$-fold symmetry (Fig. 4), where there is a broad spectrum with no dominant moment. The moments were calculated at the 64th second of the experiment.

Figure 7

The distribution of the ratios of surface tension deduced from theory, ${\sigma }_{measured}$, to ${\sigma }_{reference}$, determined by the Wilhelmy Plate method. The solid line represents computations that include the wetting term, while the dotted line neglects the wetting correction. The bin size is 0.05. A few outlying points are outside of the abscissa’s range.