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A general theory of thermodynamically consistent biomechanical--biochemical growth in a body, considering mass addition in the bulk and at an incoherent interface, is developed. The incoherency arises due to incompatibility of growth and…
Imbibition phenomena have been widely used experimentally and theoretically to study the kinetic roughening of interfaces. We critically discuss the existing experiments and some associated theoretical approaches on the scaling properties…
An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be formulated as a Laplacian growth model regularized by a 'kinetic undercooling' boundary condition. Using this model we study both the…
The growth of laminar-turbulent band patterns in plane Couette flow is studied in the vicinity of the global stability threshold R_g below which laminar flow ultimately prevails. Appropriately tailored direct numerical simulations are…
Given an orientation-preserving diffeomorphism of the interval [0;1], consider the uniform norm of the differential of its n-th iteration. We get a function of n called the growth sequence. Its asymptotic behaviour is an interesting…
Transport networks, such as vasculature or river networks, provide key functions in organisms and the environment. They usually contain loops whose significance for the stability and robustness of the network is well documented. However,…
I show that the evolution of a two dimensional surface in a Laplacian field can be described by Hamiltonian dynamics. First the growing region is mapped conformally to the interior of the unit circle, creating in the process a set of…
We consider approximations of the Stefan-type condition by imbalances of volume closely around the inner interface and study convergence of the solutions of the corresponding semilinear stochastic moving boundary problems. After a…
Inspired by recent experimental observation of patterning at the membrane of a living cell, we propose a generic model for the dynamics of a fluctuating interface driven by particle-like inclusions which stimulate its growth. We find that…
Using formal asymptotic methods we derive a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour or other biological tissue. The mathematical model takes the form of a closed…
When two surfaces are brought into contact and slide against each other, junctions are formed at the interface. The dynamics of formation, rupture and evolution of these junctions governs the tribological response of the macro-contact.…
This paper establishes the emergence of slowly moving transition layer solutions for the $p$-Laplacian (nonlinear) evolution equation, \[ u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), \qquad x \in (a,b), \; t > 0, \] where $\varepsilon>0$…
We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the half space. We impose a boundary condition that the curves are perpendicular to the boundary for $t>0$, however, the initial curve…
We study the exact non-singular zero-surface tension solutions of the Saffman-Taylor problem for all times. We show that all moving logarithmic singularities a_k(t) in the complex plane \omega = e^{i\phi}, where \phi is the stream function,…
Some physical problems as flame front propagation or Laplacian growth without surface tension have nice analytical solutions which replace its complex integro-differential motion equations by simple differential equations of poles motion in…
This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions…
We study the mean-field version of a model proposed by Leschhorn to describe the depinning transition of interfaces in random media. We show that evolution equations for the distribution of forces felt by the interface sites can be written…
Viscous fingering and wormhole growth are complex nonlinear unstable phenomena. We view both as the result of competition for water in which the capacity of an instability to grow depends on its ability to carry water. We derive empirical…
We define a new model of interface roughening which has the property that the minimum of interface height is conserved locally during the growth. This model corresponds to the limit $q \to \infty$ of the q-color dimer deposition-evaporation…
We study deterministic Loewner evolutions on the complex plane driven by complex-valued functions. This model can be viewed as a generalization of real-driven Loewner evolutions in the upper half-plane, or as the deterministic analogue of…