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Varghese Mathai

Publications and source records attributed to Varghese Mathai.

At least 73 records · Page 4Linked to original sources
arXiv2017

On the impact of a concave nosed axisymmetric body on a free surface

We report on an experimental study of the vertical impact of a concave nosed axisymmetric body on a free surface. Previous studies have shown that bodies with a convex nose, like a sphere, produce a well defined splash with a relatively large cavity behind the model. In contrast, we find that with a concave nose, there is hardly a splash and the cavity extent is greatly reduced. This may be explained by the fact that in the concave nosed case, the initial impact is between a confined air pocket and the free surface unlike in the convex nosed case. From measurements of the unsteady pressure in the concave nose portion, we show that in this case, the maximum pressures are significantly lower than the classically expected "water hammer" pressures and also lower than those generally measured on other geometries. Thus, the presence of an air pocket in the case of a concave nosed body adds an interesting dimension to the classical problem of impact of solid bodies on to a free surface.

physics.flu-dyn↗
arXiv2017

Mass and moment of inertia govern the transition in the dynamics and wakes of freely rising and falling cylinders

In this Letter, we study the motion and wake-patterns of freely rising and falling cylinders in quiescent fluid. We show that the amplitude of oscillation and the overall system-dynamics are intricately linked to two parameters: the particle's mass-density relative to the fluid $m^* \equiv ρ_p/ρ_f$ and its relative moment-of-inertia $I^* \equiv {I}_p/{I}_f$. This supersedes the current understanding that a critical mass density ($m^*\approx$ 0.54) alone triggers the sudden onset of vigorous vibrations. Using over 144 combinations of ${m}^*$ and $I^*$, we comprehensively map out the parameter space covering very heavy ($m^* > 10$) to very buoyant ($m^* < 0.1$) particles. The entire data collapses into two scaling regimes demarcated by a transitional Strouhal number, $St_t \approx 0.17$. $St_t$ separates a mass-dominated regime from a regime dominated by the particle's moment of inertia. A shift from one regime to the other also marks a gradual transition in the wake-shedding pattern: from the classical $2S$~(2-Single) vortex mode to a $2P$~(2-Pairs) vortex mode. Thus, auto-rotation can have a significant influence on the trajectories and wakes of freely rising isotropic bodies.

physics.flu-dyn↗
arXiv2017

Differential topology of semimetals

The subtle interplay between local and global charges for topological semimetals exactly parallels that for singular vector fields. Part of this story is the relationship between cohomological semimetal invariants, Euler structures, and ambiguities in the torsion of manifolds. Dually, a topological semimetal can be represented by Euler chains from which its surface Fermi arc connectivity can be deduced. These dual pictures, and the link to topological invariants of insulators, are organised using geometric exact sequences. We go beyond Dirac-type Hamiltonians and introduce new classes of semimetals whose local charges are subtle Atiyah-Dupont-Thomas invariants globally constrained by the Kervaire semicharacteristic, leading to the prediction of torsion Fermi arcs.

math-ph↗
arXiv2017

Experimental investigation of the turbulence induced by a bubble swarm rising within an incident turbulence

This work reports an experimental characterisation of the flow properties in a homogeneous bubble swarm rising at high-Reynolds-numbers within a homogeneous and isotropic turbulent flow. Both the gas volume fraction α and the velocity fluctuations of the carrier flow before bubble injection are varied, respectively, in the ranges 0% < alpha < 0.93% and 2.3 cm/s < urms < 5.5 cm/s. The so-called bubblance parameter is used to compare the ratio of the kinetic energy generated by the bubbles to the one produced by the incident turbulence, and is varied from 0 to 1.3. Conditional measurements of the velocity field downstream of the bubbles in the vertical direction allow us to disentangle three regions that have specific statistical properties, namely the primary wake, the secondary wake and the far field. While the fluctuations in the primary wake are similar to the one of a single bubble rising in a liquid at rest, the statistics of the velocity fluctuations in the far field follow Gaussian distribution, similar to the one produced by the homogenous and isotropic turbulence at the largest scales. In the secondary wake region, the conditional probability density function (pdf) of the velocity fluctuations is asymmetric and shows an exponential tail for the positive fluctuations and a Gaussian one for the negative fluctuations. The overall agitation thus results from the combination of these three contributions and depends mainly on the bubblance parameter.

physics.flu-dyn↗
arXiv2017

Group dualities, T-dualities, and twisted K-theory

This paper explores further the connection between Langlands duality and T-duality for compact simple Lie groups, which appeared in work of Daenzer-Van Erp and Bunke-Nikolaus. We show that Langlands duality gives rise to isomorphisms of twisted K-groups, but that these K-groups are trivial except in the simplest case of SU(2) and SO(3). Along the way we compute explicitly the map on $H^3$ induced by a covering of compact simple Lie groups, which is either 1 or 2 depending in a complicated way on the type of the groups involved. We also give a new method for computing twisted K-theory using the Segal spectral sequence, giving simpler computations of certain twisted K-theory groups of compact Lie groups relevant for D-brane charges in WZW theories and rank-level dualities. Finally we study a duality for orientifolds based on complex Lie groups with an involution.

hep-th↗
arXiv2016

Geometry of Pseudodifferential algebra bundles and Fourier Integral Operators

We study the geometry and topology of (filtered) algebra-bundles ${\bfΨ}^{\mathbb Z}$ over a smooth manifold $X$ with typical fibre $Ψ^{\mathbb Z}(Z; V)$, the algebra of classical pseudodifferential operators of integral order on the compact manifold $Z$ acting on smooth sections of a vector bundle $V$. First a theorem of Duistermaat and Singer is generalized to the assertion that the group of projective invertible Fourier integral operators ${\rm PGL}({\mathcal F}^\bullet(Z; V))$, is precisely the automorphism group, ${\rm Aut}(Ψ^{\mathbb Z}(Z; V)),$ of the filtered algebra of pseudodifferential operators. We replace some of the arguments in their paper by microlocal ones, thereby removing the topological assumption well as extending their result to sections of a vector bundle. We define a natural class of connections and B-fields the principal bundle to which ${\bfΨ}^{\mathbb Z}$ is associated and obtain a de Rham representative of the Dixmier-Douady class, in terms of the outer derivation on the Lie algebra and the residue trace of Guillemin and Wodzicki; the resulting formula only depends on the formal symbol algebra ${\bfΨ}^{\mathbb Z}/{\bfΨ}^{-\infty}.$ Examples of pseudodifferential algebra bundles are given that are not associated to a finite dimensional fibre bundle over $X.$

math.DG↗
arXiv2016

Microbubbles and microparticles are not faithful tracers of turbulent acceleration

We report on the Lagrangian statistics of acceleration of small (sub-Kolmogorov) bubbles and tracer particles with Stokes number St << 1 in turbulent flow. At decreasing Reynolds number, the bubble accelerations show deviations from that of tracer particles, i.e. they deviate from the Heisenberg-Yaglom prediction and show a quicker decorrelation despite their small size and minute St. Using direct numerical simulations, we show that these effects arise due the drift of these particles through the turbulent flow. We theoretically predict this gravity-driven effect for developed isotropic turbulence, with the ratio of Stokes to Froude number or equivalently the particle drift-velocity governing the enhancement of acceleration variance and the reductions in correlation time and intermittency. Our predictions are in good agreement with experimental and numerical results. The present findings are relevant to a range of scenarios encompassing tiny bubbles and droplets that drift through the turbulent oceans and the atmosphere. They also question the common usage of micro-bubbles and micro-droplets as tracers in turbulence research.

physics.flu-dyn↗
arXiv2016

Translational and rotational dynamics of a large buoyant sphere in turbulence

We report experimental measurements of the translational and rotational dynamics of a large buoyant sphere in isotropic turbulence. We introduce an efficient method to simultaneously determine the position and (absolute) orientation of a spherical body from visual observation. The method employs a minimization algorithm to obtain the orientation from the 2D projection of a specific pattern drawn onto the surface of the sphere. This has the advantages that it does not require a database of reference images, is easily scalable using parallel processing, and enables accurate absolute orientation reference. Analysis of the sphere's translational dynamics reveals clear differences between the streamwise and transverse directions. The translational auto-correlations and PDFs provide evidence for periodicity in the particle's dynamics even under turbulent conditions. The angular autocorrelations show weak periodicity. The angular accelerations exhibit wide tails, however without a directional dependence.

physics.flu-dyn↗
arXiv2016

T-duality simplifies bulk-boundary correspondence

Recently we introduced T-duality in the study of topological insulators. In this paper, we study the bulk-boundary correspondence for three phenomena in condensed matter physics, namely, the quantum Hall effect, the Chern insulator, and time reversal invariant topological insulators. In all of these cases, we show that T-duality trivializes the bulk-boundary correspondence.

hep-th↗
arXiv2016

Spin-structures and proper group actions

We generalise Atiyah and Hirzebruch's vanishing theorem for actions by compact groups on compact Spin-manifolds to possibly noncompact groups acting properly and cocompactly on possibly noncompact Spin-manifolds. As corollaries, we obtain some vanishing results for $\hat A$-type genera.

math.DG↗
arXiv2015

T-duality trivializes bulk-boundary correspondence: the parametrised case

We state a general conjecture that T-duality trivialises a model for the bulk-boundary correspondence in the parametrised context. We give evidence that it is valid by proving it in a special interesting case, which is relevant both to String Theory and to the study of topological insulators with defects in Condensed Matter Physics.

hep-th↗
arXiv2015

Wake-driven dynamics of finite-sized buoyant spheres in turbulence

Particles suspended in turbulent flows are affected by the turbulence and at the same time act back on the flow. The resulting coupling can give rise to rich variability in their dynamics. Here we report experimental results from an investigation of finite-sized buoyant spheres in turbulence. We find that even a marginal reduction in the particle's density from that of the fluid can result in strong modification of its dynamics. In contrast to classical spatial filtering arguments and predictions of particle models, we find that the particle acceleration variance increases with size. We trace this reversed trend back to the growing contribution from wake-induced forces, unaccounted for in current particle models in turbulence. Our findings highlight the need for improved multi-physics based models that account for particle wake effects for a faithful representation of buoyant-sphere dynamics in turbulence.

physics.flu-dyn↗
arXiv2015

T-Duality of Topological Insulators

Topological insulators and D-brane charges in string theory can both be classified by the same family of groups. In this paper, we extend this connection via a geometric transform, giving a novel duality of topological insulators which can be viewed as a condensed matter analog of T-duality in string theory. For 2D Chern insulators, this duality exchanges the rank and Chern number of the valence bands.

hep-th↗
arXiv2015

Quantising proper actions on Spin$^c$-manifolds

Paradan and Vergne generalised the quantisation commutes with reduction principle of Guillemin and Sternberg from symplectic to Spin$^c$-manifolds. We extend their result to noncompact groups and manifolds. This leads to a result for cocompact actions, and a result for non-cocompact actions for reduction at zero. The result for cocompact actions is stated in terms of $K$-theory of group $C^*$-algebras, and the result for non-cocompact actions is an equality of numerical indices. In the non-cocompact case, the result generalises to Spin$^c$-Dirac operators twisted by vector bundles. This yields an index formula for Braverman's analytic index of such operators, in terms of characteristic classes on reduced spaces.

math.DG↗
arXiv2015

Geometric quantization and families of inner products

We formulate a quantization commutes with reduction principle in the setting where the Lie group $G$, the symplectic manifold it acts on, and the orbit space of the action may all be noncompact. It is assumed that the action is proper, and the zero set of a deformation vector field, associated to the momentum map and an equivariant family of inner products on the Lie algebra $\mathfrak{g}$ of $G$, is $G$-cocompact. The central result establishes an asymptotic version of this quantization commutes with reduction principle. Using an equivariant family of inner products on $\mathfrak{g}$ instead of a single one makes it possible to handle both noncompact groups and manifolds, by extending Tian and Zhang's Witten deformation approach to the noncompact case.

math.DG↗
arXiv2015

Twisted chiral de Rham complex, generalized geometry, and T-duality

The chiral de Rham complex of Malikov, Schechtman, and Vaintrob, is a sheaf of differential graded vertex algebras that exists on any smooth manifold $Z$, and contains the ordinary de Rham complex at weight zero. Given a closed 3-form $H$ on $Z$, we construct the twisted chiral de Rham differential $D_H$, which coincides with the ordinary twisted differential in weight zero. We show that its cohomology vanishes in positive weight and coincides with the ordinary twisted cohomology in weight zero. As a consequence, we propose that in a background flux, Ramond-Ramond fields can be interpreted as $D_H$-closed elements of the chiral de Rham complex. Given a T-dual pair of principal circle bundles $Z, \widehat{Z}$ with fluxes $H, \widehat{H}$, we establish a degree-shifting linear isomorphism between a central quotient of the $i \mathbb{R}[t]$-invariant chiral de Rham complexes of $Z$ and $\widehat{Z}$. At weight zero, it restricts to the usual isomorphism of $S^1$-invariant differential forms, and induces the usual isomorphism in twisted cohomology. This is interpreted as T-duality in type II string theory from a loop space perspective. A key ingredient in defining this isomorphism is the language of Courant algebroids, which clarifies the notion of functoriality of the chiral de Rham complex.

math.DG↗
arXiv2015

Exotic twisted equivariant cohomology of loop spaces, twisted Bismut-Chern character and T-duality

We define exotic twisted $S^1$-equivariant cohomology for the loop space $LZ$ of a smooth manifold $Z$ via the invariant differential forms on $LZ$ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Chern character form, a loop space refinement of the twisted Chern character form, which represent classes in the completed periodic exotic twisted $S^1$-equivariant cohomology of $LZ$. We establish a localisation theorem for the completed periodic exotic twisted $S^1$-equivariant cohomology for loop spaces and apply it to establish T-duality in a background flux in type II String Theory from a loop space perspective.

hep-th↗