Evolution and breakup of viscous rotating drops

We study the evolution of a viscous fluid drop rotating about a fixed axis at constant angular velocity $Omega$ or constant angular momentum L surrounded by another viscous fluid. The problem is considered in the limit of large Ekman number and small Reynolds number. The analysis is carried out by combining asymptotic analysis and full numerical simulation by means of the boundary element method. We pay special attention to the stability/instability of equilibrium shapes and the possible formation of singularities representing a change in the topology of the fluid domain. When the evolution is at constant $Omega$, depending on its value, drops can take the form of a flat film whose thickness goes to zero in finite time or an elongated filament that extends indefinitely. When evolution takes place at constant L and axial symmetry is imposed, thin films surrounded by a toroidal rim can develop, but the film thickness does not vanish in finite time. When axial symmetry is not imposed and L is sufficiently large, drops break axial symmetry and, depending on the value of L, reach an equilibrium configuration with a 2-fold symmetry or break up into several drops with a 2- or 3-fold symmetry. The mechanism of breakup is also described

Geometric characteristics of conics in Bézier form

Derive coordinate-free expressions for geometric characteristics of conics written in Bézier form in terms of their control points and weights.

Convective transport and stability in films of binary mixtures

Thin polymer films are increasingly used in advanced technological applications. The use of these films as coatings is often limited by their lack of stability due to their wettability properties on the substrates

A second order in time modified Lagrange-Galerkin finite element method for the incompressible Navier-Stokes equations

We introduce a second order in time modified Lagrange--Galerkin (MLG) method for the time dependent incompressible Navier--Stokes equations. The main ingredient of the new method is the scheme proposed to calculate in a more efficient manner the Galerkin projection of the functions transported along the characteristic curves of the transport operator. We present error estimates for velocity and pressure in the framework of mixed finite elements when either the mini-element or the $P2/P1$ Taylor--Hood element are used.

Lagrangian skeleton on the Gulf of Mexico oil spill

The understanding of the circulation of ocean currents, the exchange of CO2 between atmosphere and oceans, and the influence of the oceans on the distribution of heat on a global scale is key to our ability to predict and assess the future evolution of climate [1, 2]. Global climate change is affecting sea breathing through mechanisms not yet understood.

A Semi-Lagrangian Particle Level Set Finite Element Method for Interface Problems

We present a quasi-monotone semi-Lagrangian particle level set (QMSL-PLS) method for moving interfaces. The QMSL method is a blend of first order monotone and second order semi-Lagrangian methods. The QMSL-PLS method is easy to implement, efficient, and well adapted for unstructured, either simplicial or hexahedral, meshes. We prove that it is unconditionally stable in the maximum discrete norm, � · �h,∞, and the error analysis shows that when the level set solution u(t) is in the Sobolev space Wr+1,∞(D), r ≥ 0, the convergence in the maximum norm is of the form (KT/Δt)min(1,Δt � v �h,∞ /h)((1 − α)hp + hq), p = min(2, r + 1), and q = min(3, r + 1),where v is a velocity. This means that at high CFL numbers, that is, when Δt > h, the error is O( (1−α)hp+hq) Δt ), whereas at CFL numbers less than 1, the error is O((1 − α)hp−1 + hq−1)). We have tested our method with satisfactory results in benchmark problems such as the Zalesak’s slotted disk, the single vortex flow, and the rising bubble.

On a two species chemotaxis model with slow chemical diffucion.

In this paper we consider a system of three parabolic equations modeling the behavior of two biological species moving attracted by a chemical factor. The chemical substance verifies a parabolic equation with slow diffusion. The system contains second order terms in the first two equations modeling the chemotactic effects. We apply an iterative method to obtain the global existence of solutions using that the total mass of the biological species is conserved. The stability of the homogeneous steady states is studied by using an energy method. A final example is presented to illustrate the theoretical results.

Lagrangian tools and the assessment of the predictive capacity of geophysical data sets

We examine, with recently developed Lagrangian tools, altimeter data and numerical simulations obtained from the HYCOM model in the Gulf of Mexico. Our data correspond to the months just after the Deepwater Horizon oil spill in the year 2010. Our Lagrangian analysis provides a skeleton that allows the interpretation of transport routes over the ocean surface. The transport routes are further verified by the simultaneous study of the evolution of several drifters launched during those months in the Gulf of Mexico. We find that there exist Lagrangian structures that justify the dynamics of the drifters, although the agreement depends on the quality of the data. We discuss the impact of the Lagrangian tools on the assessment of the predictive capacity of these data sets.

Asymptotic analysis of vertical geothermal boreholes in the limit of slowly varyng heat injection rates

Theoretical models for the thermal response of vertical geothermal boreholes often assume that the characteristic time of variation of the heat injection rate is much larger than the characteristic diffusion time across the borehole. In this case, heat transfer inside the borehole and in its immediate surroundings is quasi-steady in the first approximation, while unsteady effects enter only in the far field. Previous studies have exploited this disparity of time scales, incorporating approximate matching conditions to couple the near-borehole region with the outer unsteady temperatura field. In the present work matched asymptotic expansion techniques are used to analyze the heat transfer problem, delivering a rigorous derivation of the true matching condition between the two regions and of the correct definition of the network of thermal resistances that represents the quasi-steady solution near the borehole. Additionally, an apparent temperature due to the unsteady far field is identified that needs to be taken into account by the near-borehole region for the correct computation of the heat injection rate. This temperature differs from the usual mean borehole temperature employed in the literatura.