Black hole interactions at large D: brane blobology

In the large dimension (D) limit, Einstein’s equation reduces to an effective theory on the horizon surface, drastically simplifying the black hole analysis. Especially, the effective theory on the black brane has been successful in describing the non-linear dynamics not only of black branes, but also of compact black objects which are encoded as solitary Gaussian-shaped lumps, blobs. For a rigidly rotating ansatz, in addition to axisymmetric deformed branches, various non-axisymmetric solutions have been found, such as black bars, which only stay stationary in the large D limit.In this article, we demonstrate the blob approximation has a wider range of applicability by formulating the interaction between blobs and subsequent dynamics. We identify that this interaction occurs via thin necks connecting blobs. Especially, black strings are well captured in this approximation sufficiently away from the perturbative regime. Highly deformed black dumbbells and ripples are also found to be tractable in the approximation. By defining the local quantities, the effective force acting on distant blobs are evaluated as well. These results reveal that the large D effective theory is capable of describing not only individual black holes but also the gravitational interactions between them, as a full dynamical theory of interactive blobs, which we call brane blobology.

On the numerical solutions for a parabolic system with blow-up

<abstract><p>We study the finite difference approximation for axisymmetric solutions of a parabolic system with blow-up. A scheme with adaptive temporal increments is commonly used to compute an approximate blow-up time. There are, however, some limitations to reproduce the blow-up behaviors for such schemes. We thus use an algorithm, in which uniform temporal grids are used, for the computation of the blow-up time and blow-up behaviors. In addition to the convergence of the numerical blow-up time, we also study various blow-up behaviors numerically, including the blow-up set, blow-up rate and blow-up in $ L^\sigma $-norm. Moreover, the relation between blow-up of the exact solution and that of the numerical solution is also analyzed and discussed.</p></abstract>

Steady states of thin film droplets on chemically heterogeneous substrates

We study steady-state thin films on chemically heterogeneous substrates of finite size, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the 1D steady-state solutions that exist on such substrates into six different branches and develop asymptotic estimates for the steady states on each branch. Using perturbation expansions, we show that leading-order solutions provide good predictions of the steady-state thin films on stepwise-patterned substrates. We show how the analysis in one dimension can be extended to axisymmetric solutions. We also examine the influence of the wettability contrast of the substrate pattern on the linear stability of droplets and the time evolution for dewetting on small domains. Results are also applied to describe 2D droplets on hydrophilic square patches and striped regions used in microfluidic applications.

The mechanism of vortex instability in electromagnetically driven flow in an annular thin layer of electrolyte

A circumferential flow of a conducting fluid in an annular channel can be created by the action of a Lorentz force arising as a result of the interaction between an applied vertical magnetic field and a radial electric current flowing through the electrolyte. Quite unexpectedly, experiments revealed that a robust vortex system appears near the outer cylindrical wall in such flows. McCloughan and Suslov (J. Fluid Mech. 887:A23, 2020) (McCS) reported comprehensive linear stability results of such a flow for variable Lorentz forcing. Here we complement that study by investigating the flow structure as a function of the channel aspect ratio. Remarkably, despite the completely different physical nature of parametric dependences, dynamic in McCS and purely geometric here, we show that in both scenarios vortices appear on a background of a steady axisymmetric flow at the boundary between two counter-rotating toroidal structures and have a similar energy distributions. The two studies demonstrate the robustness of the mechanism responsible for the vortex formation: Rayleigh's inviscid centrifugal instability aided by radial shear in the boundary layer near the outer cylindrical wall. References P. A. Davidson. An introduction to magnetohydrodynamics. Cambridge University Press, 2nd edition, 2017. doi:10.1017/CBO9780511626333. J. McCloughan and S. A. Suslov. Linear stability and saddle–node bifurcation of electromagnetically driven electrolyte flow in an annular layer. J. Fluid Mech., 887:A23.1–30, 2020. doi:10.1017/jfm.2020.29. J. Perez-Barrera, J. E. Perez-Espinoza, A. Ortiz, E. Ramos, and S. Cuevas. Instability of electrolyte flow driven by an azimuthal Lorentz force. Magnetohydrodynamics, 51(2):203–213, 2015. http://mhd.sal.lv/contents/2015/2/MG.51.2.4.R.html. S. A. Suslov, J. Perez-Barrera, and S. Cuevas. Electromagnetically driven flow of electrolyte in a thin annular layer: Axisymmetric solutions. J. Fluid Mech., 828: 573–600, 2017. doi:10.1017/jfm.2017.551.

Extension Rules of Newman–Janis Algorithm for Rotation Metrics in General Relativity

Aims: The aim of this study is to extend the formula of Newman–Janis algorithm (NJA) and introduce the rules of the complexifying seed metric. The extension of NJA can help determine more generalized axisymmetric solutions in general relativity.Methodology: We perform the extended NJA in two parts: the tensor structure and the seed metric function. Regarding the tensor structure, there are two prescriptions, the Newman–Penrose null tetrad and the Giampieri prescription. Both are mathematically equivalent; however, the latter is more concise. Regarding the seed metric function, we propose the extended rules of a complex transformation by r2/Σ and combine the mass, charge, and cosmologic constant into a polynomial function of r. Results: We obtain a family of axisymmetric exact solutions to Einstein’s field equations, including the Kerr metric, Kerr–Newman metric, rotating–de Sitter, rotating Hayward metric, Kerr–de Sitter metric and Kerr–Newman–de Sitter metric. All the above solutions are embedded in ellipsoid- symmetric spacetime, and the energy-momentum tensors of all the above metrics satisfy the energy conservation equations. Conclusion: The extension rules of the NJA in this research avoid ambiguity during complexifying the transformation and successfully generate a family of axisymmetric exact solutions to Einsteins field equations in general relativity, which deserves further study.

An explicit representation for the axisymmetric solutions of the free Maxwell equations

Garay-Avendaño and Zamboni-Rached defined two classes of axisymmetric solutions of the free Maxwell equations. We prove that the linear combinations of these two classes of solutions cover all totally propagating time-harmonic axisymmetric free Maxwell fields – and hence, by summation on frequencies, all propagating axisymmetric free Maxwell fields. It provides an explicit representation for these fields. This will be important, e.g., to have the interstellar radiation field in a disc galaxy modeled as an exact solution of the free Maxwell equations.