Laplacian networks: Growth, local symmetry, and shape optimization

Goldilocks Fluctuations: Dynamic Constraints on Loop Formation in Scale-Free Transport Networks

Adaptive transport networks are known to contain loops when subject to hydrodynamic fluctuations. However, fluctuations are no guarantee that a loop will form, as shown by loop-free networks driven by oscillating flows. We provide a complete stability analysis of the dynamical behavior of any loop formed by fluctuating flows. We find a threshold for loop stability that involves an interplay of geometric constraints and hydrodynamic forcing mapped to constant and fluctuating components. Loops require fluctuation in the relative size of the flux between nodes, not just a temporal variation in the flux at a given node. Hence, there is both a minimum and a maximum amount of fluctuation relative to the constant-flux component where loops are supported.

Long-term day-by-day tracking of microvascular networks sprouting in fibrin gels: From detailed morphological analyses to general growth rules

Understanding and controlling of the evolution of sprouting vascular networks remains one of the basic challenges in tissue engineering. Previous studies on the vascularization dynamics have typically focused only on the phase of intense growth and often lacked spatial control over the initial cell arrangement. Here, we perform long-term day-by-day analysis of tens of isolated microvasculatures sprouting from endothelial cell-coated spherical beads embedded in an external fibrin gel. We systematically study the topological evolution of the sprouting networks over their whole lifespan, i.e., for at least 14 days. We develop a custom image analysis toolkit and quantify (i) the overall length and area of the sprouts, (ii) the distributions of segment lengths and branching angles, and (iii) the average number of branch generations-a measure of network complexity. We show that higher concentrations of vascular endothelial growth factor (VEGF) lead to earlier sprouting and more branched networks, yet without significantly affecting the speed of growth of individual sprouts. We find that the mean branching angle is weakly dependent on VEGF and typically in the range of 60°-75°, suggesting that, by comparison with the available diffusion-limited growth models, the bifurcating tips tend to follow local VEGF gradients. At high VEGF concentrations, we observe exponential distributions of segment lengths, which signify purely stochastic branching. Our results-due to their high statistical relevance-may serve as a benchmark for predictive models, while our new image analysis toolkit, offering unique features and high speed of operation, could be exploited in future angiogenic drug tests.

Geological records of transient fluid drainage into the shallow mantle wedge

Pore fluid pressure on subduction zone megathrusts is lowered by fluid drainage into the overlying plate, affecting subduction zone seismicity. However, the spatial and temporal scales of fluid flow through suprasubduction zones are poorly understood. We constrain the duration and velocity of fluid flow through a shallow mantle wedge based on the analyses of vein networks consisting of high-temperature serpentine in hydrated ultramafic rocks from the Oman ophiolite. On the basis of a diffusion model and the time-integrated fluid flux, we show that the channelized fluid flow was short-lived (2.1 × 10^-1 to 1.1 × 10¹ years) and had a high fluid velocity (2.7 × 10^-3 to 4.9 × 10^-2 meters second^-1), which is close to the propagation velocities of seismic events in present-day subduction zones. Our results suggest that the drainage of fluid into the overlying plate occurs as episodic pulses, which may influence the recurrence of megathrust earthquakes.

Through history to growth dynamics: deciphering the evolution of spatial networks

Many ramified, network-like patterns in nature, such as river networks or blood vessels, form as a result of unstable growth of moving boundaries in an external diffusive field. Here, we pose the inverse problem for the network growth-can the growth dynamics be inferred from the analysis of the final pattern? We show that by evolving the network backward in time one can not only reconstruct the growth rules but also get an insight into the conditions under which branch splitting occurs. Determining the growth rules from a single snapshot in time is particularly important for growth processes so slow that they cannot be directly observed, such as growth of river networks and deltas or cave passages. We apply this approach to analyze the growth of a real river network in Vermont, USA. We determine its growth rule and argue that branch splitting events are triggered by an increase in the tip growth velocity.

A Two-Dimensional Phase-Field Investigation on Unidirectionally Solidified Tip-Splitting Microstructures

The onset of morphological instabilities along a solidifying interface has a tendency to influence the microstructural characteristics of cast alloys. In the present study, the initiation as well as the mechanism of microstructural pattern formation is investigated by a quantitative phase-field approach. For energetically isotropic interfaces, we show that the presence of grain boundary grooves promotes the initiation of morphological instabilities, and with progressive solidification, they subsequently amplify into tip-splitting microstructures. We also demonstrate that the grain boundary groove shape influences the amplification of the ridge-shaped instability near the pit region. The structural transition of tip splitting to dendritic microstructures is showcased through the effect of interfacial anisotropy. In addition, the prediction of the tip-splitting position is discussed through an analytical criterion, wherein the sign of the surface Laplacian of interfacial curvature dictates the formation of crest and trough positions in a tip-splitting pattern. In complete agreement with the sharp-interface theory, our phase-field simulations validate the analytically obtained tip-splitting position and suggest that the two tips evolve symmetrically on either side of the hindered concave region. Furthermore, the role of lattice anisotropy on the tip-splitting phenomenon is also discussed in detail.

Geodesic Loewner paths with varying boundary conditions

Equations of the Loewner class subject to non-constant boundary conditions along the real axis are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian growth in which the slits represent thin fingers growing in a diffusion field. A single finger follows a curved path determined by the forcing function appearing in Loewner's equation. This function is found by solving an ordinary differential equation whose terms depend on curvature properties of the streamlines of the diffusive field in the conformally mapped 'mathematical' plane. The effect of boundary conditions specifying either piecewise constant values of the field variable along the real axis, or a dipole placed on the real axis, reveal a range of behaviours for the growing slit. These include regions along the real axis from which no slit growth is possible, regions where paths grow to infinity, or regions where paths curve back toward the real axis terminating in finite time. Symmetric pairs of paths subject to the piecewise constant boundary condition along the real axis are also computed, demonstrating that paths which grow to infinity evolve asymptotically toward an angle of bifurcation of π/5.

Application of the Schwarz-Christoffel map to the Laplacian growth of needles and fingers

A numerical procedure based on the Schwarz-Christoffel map suitable for the study of the Laplacian growth of thin two-dimensional protrusions is presented. The protrusions take the form of either straight needles or curved fingers satisfying Loewner's equation, and are represented by slits in the complex plane. Particular use is made of Driscoll's numerical procedure, the SC Toolbox, for computing the Schwarz-Christoffel map from a half plane to a slit half plane. Since the Schwarz-Christoffel map applies only to polygonal regions, the growth of curved fingers is approximated by an increasing number of short straight line segments. The growth rate is given by a fixed power η of the harmonic measure at the finger or needle tips and so includes the possibility of "screening" as the needles of fingers interact with themselves and with boundaries. The method is illustrated with examples of multiple needle and finger growth in half-plane and channel geometries. The effect of η on the trajectories of asymmetric bifurcating fingers is also studied.

Symmetric rearrangement of groundwater-fed streams

Streams shape landscapes through headward growth and lateral migration. When these streams are primarily fed by groundwater, recent work suggests that their tips advance to maximize the symmetry of the local Laplacian field associated with groundwater flow. We explore the extent to which such forcing is responsible for the lateral migration of streams by studying two features of groundwater-fed streams in Bristol, Florida: their confluence angle near junctions and their curvature. First, we find that, while streams asymptotically form a 72° angle near their tips, they simultaneously exhibit a wide 120° confluence angle within approximately 10 m of their junctions. We show that this wide angle maximizes the symmetry of the groundwater field near the junction. Second, we argue that streams migrate laterally within valleys and present a new spectral analysis method to relate planform curvature to the surrounding groundwater field. Our results suggest that streams migrate laterally in response to fluxes from the surrounding groundwater table, providing evidence of a new mechanism that complements Laplacian growth at their tips.