9 research outputs found
Conformal mapping methods for interfacial dynamics
The article provides a pedagogical review aimed at graduate students in
materials science, physics, and applied mathematics, focusing on recent
developments in the subject. Following a brief summary of concepts from complex
analysis, the article begins with an overview of continuous conformal-map
dynamics. This includes problems of interfacial motion driven by harmonic
fields (such as viscous fingering and void electromigration), bi-harmonic
fields (such as viscous sintering and elastic pore evolution), and
non-harmonic, conformally invariant fields (such as growth by
advection-diffusion and electro-deposition). The second part of the article is
devoted to iterated conformal maps for analogous problems in stochastic
interfacial dynamics (such as diffusion-limited aggregation, dielectric
breakdown, brittle fracture, and advection-diffusion-limited aggregation). The
third part notes that all of these models can be extended to curved surfaces by
an auxilliary conformal mapping from the complex plane, such as stereographic
projection to a sphere. The article concludes with an outlook for further
research.Comment: 37 pages, 12 (mostly color) figure
Three-dimensional stability of Burgers vortices
Burgers vortices are explicit stationary solutions of the Navier-Stokes
equations which are often used to describe the vortex tubes observed in
numerical simulations of three-dimensional turbulence. In this model, the
velocity field is a two-dimensional perturbation of a linear straining flow
with axial symmetry. The only free parameter is the Reynolds number , where is the total circulation of the vortex and is
the kinematic viscosity. The purpose of this paper is to show that Burgers
vortex is asymptotically stable with respect to general three-dimensional
perturbations, for all values of the Reynolds number. This definitive result
subsumes earlier studies by various authors, which were either restricted to
small Reynolds numbers or to two-dimensional perturbations. Our proof relies on
the crucial observation that the linearized operator at Burgers vortex has a
simple and very specific dependence upon the axial variable. This allows to
reduce the full linearized equations to a vectorial two-dimensional problem,
which can be treated using an extension of the techniques developped in earlier
works. Although Burgers vortices are found to be stable for all Reynolds
numbers, the proof indicates that perturbations may undergo an important
transient amplification if is large, a phenomenon that was indeed observed
in numerical simulations.Comment: 31 pages, no figur
Generalized Contour Dynamics: A Review
Contour dynamics is a computational technique to solve for the motion of vortices in incompressible inviscid flow. It is a Lagrangian technique in which the motion of contours is followed, and the velocity field moving the contours can be computed as integrals along the contours. Its best-known examples are in two dimensions, for which the vorticity between contours is taken to be constant and the vortices are vortex patches, and in axisymmetric flow for which the vorticity varies linearly with distance from the axis of symmetry. This review discusses generalizations that incorporate additional physics, in particular, buoyancy effects and magnetic fields, that take specific forms inside the vortices and preserve the contour dynamics structure. The extra physics can lead to time-dependent vortex sheets on the boundaries, whose evolution must be computed as part of the problem. The non-Boussinesq case, in which density differences can be important, leads to a coupled system for the evolution of both mean interfacial velocity and vortex sheet strength. Helical geometry is also discussed, in which two quantities are materially conserved and whose evolution governs the flow
Preface to special issue on the dynamics of thin liquid films
A detailed understanding of flows in thin liquid films is important for a wide range of modern engineering processes. This is particularly so in chemical and process engineering, where thin liquid films are encountered in heat-and-mass-transfer devices (e.g. distillation columns and spinning-disk reactors), and in coating processes (e.g. spin coating, blade coating, spray painting and rotational moulding). In order to design these processes for safe and efficient operation it is important to build mathematical models that can predict their performance, to have confidence in the predictions of the models, and to be able to use the models to optimise the design and operation of the devices involved. Thin liquid films also occur in a variety of biological contexts, including the thin liquid linings of the airways in the lung and the thin tear films that coat the eyes, and greater understanding of these films should lead to improved treatment of diseases in these organs