The Self-Made Tapestry Phần 5 pps

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highest) to the value in the bulk of the oil. If we think of a model analogy in which the pressure is

equivalent to the height of a hill and the motion of the air bubble is equivalent to the motion of a ball,

the ball accelerates more rapidly down the hill the steeper it isin other words, it is the gradient that

determines the rate of advance. Saffman and Taylor pointed out that the gradient in pressure around a

bulge at the air/oil interface gets steeper as the bulge gets sharper. This sets up a self-amplifying process

in which a small initial bulge begins to move faster than the interface to either side. The sharper and

longer the finger gets, the steeper the pressure gradient at its tip and so the more rapidly it grows (Fig.

5.14).

Fig. 5.14

The Saffman-Taylor instability. As

a bulge develops at the advancing

fluid front, the pressure gradient

at the bulge tip is enhanced and so the

tip advances more rapidly. (Contours of

constant pressureisobars, like those in

weather mapsare shown as dashed lines.)

This amplifies small bulges into sharp fingers.

Compare this to the growth instability in DLA

(Fig. 5.8).

This instability is called the SaffmanTaylor instability. In 1984, Australian physicist Lincoln Paterson

pointed out that the equations that describe it are analogous to those that underlie the DLA instability

described by Witten and Sander. So it is entirely to be expected that viscous fingering and DLA

produce the same kind of fractal branching networks. Both are examples of so-called Laplacian growth,

which can be described by a set of equations derived from the work of the eighteenth-century French

scientist Pierre Laplace. Within these deceptively simple equations are the ingredients for growth

instabilities that lead to branching.

But tenuous fractal patterns directly comparable to those of DLA occur in viscous fingering only under

rather unusual conditions. More commonly one sees a subtlely altered kind of branching structure: the

basic pattern or 'backbone' of the network has a comparable, disorderly form, but the branches

themselves are fat fingers, not wispy tendrils (Fig. 5.15; compare 5.12). And under some conditions the

bubbles cease to have the ragged DLA-like form at all, and instead advance in broad fingers that split at

their tips (Fig. 5.4a). This sort of branching pattern is called the dense-branching morphology, and is

more or less space-filling (twodimensional) rather than fractal. Why then, if the same tip-growth

instability operates in both viscous fingering and DLA, do different patterns result?

All viscous fingering patterns differ from that of DLA in at least one important respectthey have a

characteristic length scale, defined by the average width of the fingers. This length scale is most clearly

apparent at relatively low injection pressures, when the air bubble's boundary advances quite slowly.

Then one sees just a few fat fingers that split as they grow (Fig. 5.16). There is a kind of regularity in

this so-called tip-splitting patternthe fingers seem to define a more or less periodic undulation around

the perimeter of the bubble with a characteristic wavelength. But a length scale is apparent in the widths

of the fingers even for more irregular patterns formed at higher growth rates (for example, Fig. 5.15).

For the self-similar DLA cluster (Fig. 5.7), on the other hand, there is no characteristic sizeit looks the

same on all scales.

Fig. 5.15

Viscous fingering has a characteristic length

scale, which determines the minimum width of the

branches. So the fingers are fatter than the fine filaments

of DLA clusters. (Image: Yves Couder, Ecole Normale

Supérieure, Paris.)

Eshel Ben-Jacob of Tel Aviv University explained the reason for these differences in the mid-1980s:

between the air bubble and the surrounding viscous fluid there is

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an interface with a surface tension. As I explained in Chapter 2, the presence of a surface tension means

that an interface has an energetic cost. Surface tension encourages surfaces to minimize their area.

Clearly, a DLA cluster is highly profligate with surface areathe cluster is about as indiscriminate with

the extent of its perimeter as you can imagine. This is because there is effectively no surface tension

built into the theoretical DLA modelthere is no penalty incurred if new surface is introduced by

sprouting a thin branch. In viscous fingering, on the other hand, there will always be a surface tension

(provided that the two fluids do not mix), and so there would be a crippling cost in energy in forming

the kind of highly crenelated interface found in DLA. The fat fingers represent a compromise between

the Saffman-Taylor instability, which favours the growth of branches on all length scales, and the

smoothing effect of surface tension, which washes out bulges smaller than a certain limit. To a first

approximation, you could say that the characteristic wave-length of viscous fingering is set by the point

at which the advantage in growth rate of ever narrower branches is counterbalanced by their cost in

surface energy.

Fig. 5.16

At low injection pressures, the length scale

of viscous fingering is quite large, and

the advancing bubble front then has a kind of

undulating shape with a well-defined

wavelength.

The relation between DLA and viscous fingering is made very apparent when DLA growth is conducted

in a system where a surface tension is built in. The surface tension has the effect of expanding the

cluster's branches into fat fingers (Fig. 5.17). Ben-Jacob showed that the generic branching pattern in

such cases is the dense-branching morphology. Conversely, a wispy DLA-like 'bubble' can be produced

experimentally in the HeleShaw cell by using fluids whose interface has a very low surface tension.

Fig. 5.17

When surface tension is included in the DLA model, it

generates fat, tip-splitting branches like those in viscous

fingering. Here the bands depict the cluster at different

stages of its growth. (Image: Paul Meakin and Tamás Vicsek.)

Physicists Johann Nittmann and Gene Stanley have shown that, somewhat surprisingly, the fat branches

of viscous fingering can be generated instead of the tenuous DLA morphology even in a system with no

surface tension. They formulated a DLA-type model in which they could vary the amount of

'noise' (that is, of randomizing influences) in the system. In their model the perimeter of the cluster can

grow only after a particle has impinged on it a certain number of times (in pure DLA just one collision

is enough). This reduces the tendency for new branches to sprout at the slightest fluctuation. Nittmann

and Stanley found that, when the noise is very low, the model generates fat branching patterns (Fig.

5.18a), which mutate smoothly to the DLA-type structure as the noise is increased (Fig. 5.18b,c). This

suggests that one way to impose a DLA-like pattern on viscous fingering in a Hele-Shaw cell is to

introduce a randomizing influence (that is, to make the system more 'noisy'). A simple way of doing this

is to score grooves at random into one of the cell plates until it is criss-crossed by a dense network of

disorderly linesthis was how the pattern shown earlier in Fig. 5.4c was obtained. The lesson here is that

noise or randomness can influence a growth pattern in pronounced ways.

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Fig. 5.18

Dense-branching patterns appear in DLA growth even in the absence of surface tension, when the

effect of noise in the system is reduced by reducing the sticking probability of the impinging particles (a).

As the noise is increased (from a to c), the branches contract into the fine tendrils of the DLA-type pattern.

Again, contours denote different stages of the growth process. Note that, despite their differing appearance,

all of the patterns here have a fractal dimension of about 1.7. (Images: Gene Stanley, Boston University.)

The six-petalled flowers