arXiv subjects

Arnab K. Ray

Publications and source records attributed to Arnab K. Ray.

35 records · Page 2Linked to original sources

arXiv2008

Analytical modelling of terminal properties in industrial growth

In this pedagogical study, carried out by adopting standard mathematical methods of nonlinear dynamics, we have presented some simple analytical models to understand terminal behaviour in industrial growth. This issue has also been addressed from a dynamical systems perspective, with especial emphasis on the concept of the Balanced Scorecard. Our study enables us to make the general claim that although the fortunes of an industrial organization can rise with exponential rapidity on relatively short time scales, its growth will ultimately and inevitably be saturated on long time scales by various factors which are nonlinear in character. We have mathematically demonstrated the likely occurrence of this feature under various possible circumstances, including the Red Ocean and the Blue Ocean. Finally and most importantly, our arguments and their associated mathematical modelling have received remarkable support from the growth pattern indicated by empirical data gathered from a well-recognized global company like IBM.

q-fin.GN↗

arXiv2007

Standing and travelling waves in the shallow-water circular hydraulic jump

A wave equation for a time-dependent perturbation about the steady shallow-water solution emulates the metric an acoustic white hole, even upon the incorporation of nonlinearity in the lowest order. A standing wave in the sub-critical region of the flow is stabilised by viscosity, and the resulting time scale for the amplitude decay helps in providing a scaling argument for the formation of the hydraulic jump. A standing wave in the super-critical region, on the other hand, displays an unstable character, which, although somewhat mitigated by viscosity, needs nonlinear effects to be saturated. A travelling wave moving upstream from the sub-critical region, destabilises the flow in the vicinity of the jump, for which experimental support has been given.

cond-mat.other↗

arXiv2007

Secular instability in quasi-viscous disc accretion

A first-order correction in the $α$-viscosity parameter of Shakura and Sunyaev has been introduced in the standard inviscid and thin accretion disc. A linearised time-dependent perturbative study of the stationary solutions of this "quasi-viscous" disc leads to the development of a secular instability on large spatial scales. This qualitative feature is equally manifest for two different types of perturbative treatment -- a standing wave on subsonic scales, as well as a radially propagating wave. Stability of the flow is restored when viscosity disappears.

astro-ph↗

arXiv2007

Perturbations on steady spherical accretion in Schwarzschild geometry

The stationary background flow in the spherically symmetric infall of a compressible fluid, coupled to the space-time defined by the static Schwarzschild metric, has been subjected to linearized perturbations. The perturbative procedure is based on the continuity condition and it shows that the coupling of the flow with the geometry of space-time brings about greater stability for the flow, to the extent that the amplitude of the perturbation, treated as a standing wave, decays in time, as opposed to the amplitude remaining constant in the Newtonian limit. In qualitative terms this situation simulates the effect of a dissipative mechanism in the classical Bondi accretion flow, defined in the Newtonian construct of space and time. As a result of this approach it becomes impossible to define an acoustic metric for a conserved spherically symmetric flow, described within the framework of Schwarzschild geometry. In keeping with this view, the perturbation, considered separately as a high-frequency travelling wave, also has its amplitude reduced.

astro-ph↗

arXiv2007

Critical properties of spherically symmetric accretion in a fractal medium

Spherically symmetric transonic accretion of a fractal medium has been studied in both the stationary and the dynamic regimes. The stationary transonic solution is greatly sensitive to infinitesimal deviations in the outer boundary condition, but the flow becomes transonic and stable, when its evolution is followed through time. The evolution towards transonicity is more pronounced for a fractal medium than what is it for a continuum. The dynamic approach also shows that there is a remarkable closeness between an equation of motion for a perturbation in the flow, and the metric of an analogue acoustic black hole. The stationary inflow solutions of a fractal medium are as much stable under the influence of linearised perturbations, as they are for the fluid continuum.

astro-ph↗

arXiv2007

Static and dynamic aspects of transonicity in Bondi accretion

Transonicity in a spherically symmetric accreting system has been considered in both the stationary and the dynamic regimes. The stationary flow, set up as a dynamical system, has been shown to be greatly unstable to even the minutest possible deviation in the boundary condition for transonicity. With the help of a simple analytical model, and some numerical modelling, it has then been argued that the flow indeed becomes transonic and stable, when the evolution of the flow is followed through time. The time-dependent approach also shows that there is a remarkable closeness between an equation of motion for a perturbation in the flow, and the metric of an analog acoustic black hole.

astro-ph↗

arXiv2007

Evolution of transonicity in an accretion disc

For inviscid, rotational accretion flows driven by a general pseudo-Newtonian potential on to a Schwarzschild black hole, the only possible fixed points are saddle points and centre-type points. For the specific choice of the Newtonian potential, the flow has only two critical points, of which the outer one is a saddle point while the inner one is a centre-type point. A restrictive upper bound is imposed on the admissible range of values of the angular momentum of sub-Keplerian flows through a saddle point. These flows are very unstable to any deviation from a necessarily precise boundary condition. The difficulties against the physical realisability of a solution passing through the saddle point have been addressed through a temporal evolution of the flow, which gives a non-perturbative mechanism for selecting a transonic solution passing through the saddle point. An equation of motion for a real-time perturbation about the stationary flows reveals a very close correspondence with the metric of an acoustic black hole, which is also an indication of the primacy of transonicity.

astro-ph↗

arXiv2007

Axisymmetric black hole accretion in the Kerr metric as an autonomous dynamical system

In a stationary, general relativistic, axisymmetric, inviscid and rotational accretion flow, described within the Kerr geometric framework, transonicity has been examined by setting up the governing equations of the flow as a first-order autonomous dynamical system. The consequent linearised analysis of the critical points of the flow leads to a comprehensive mathematical prescription for classifying these points, showing that the only possibilities are saddle points and centre-type points for all ranges of values of the fixed flow parameters. The spin parameter of the black hole influences the multitransonic character of the flow, as well as some of its specific critical properties. The special case of a flow in the space-time of a non-rotating black hole, characterised by the Schwarzschild metric, has also been studied for comparison and the conclusions are compatible with what has been seen for the Kerr geometric case.

astro-ph↗

arXiv2007

Critical properties of spherically symmetric black hole accretion in Schwarzschild geometry

The stationary spherically symmetric accretion flow in the Schwarzschild metric has been set up as an autonomous first-order dynamical system, and it has been studied completely analytically. Of the three possible critical points in the flow, the one that is physically realistic behaves like the saddle point of the standard Bondi accretion problem. One of the two remaining critical points exhibits the strange mathematical behaviour of being either a saddle point or a centre-type point, depending on the values of the flow parameters. The third critical point is always unphysical and behaves like a centre-type point. The treatment has been extended to pseudo-Schwarzschild flows for comparison with the general relativistic analysis.

astro-ph↗

arXiv2006

Critical properties and stability of stationary solutions in multi-transonic pseudo-Schwarzschild accretion

For inviscid, rotational accretion flows, both isothermal and polytropic, a simple dynamical systems analysis of the critical points has given a very accurate mathematical scheme to understand the nature of these points, for {\em any} pseudo-potential by which the flow may be driven on to a Schwarzschild black hole. This allows for a complete classification of the critical points for a wide range of flow parameters, and shows that the only possible critical points for this kind of flow are saddle points and centre-type points. A restrictive upper bound on the angular momentum of critical solutions has been established. A time-dependent perturbative study reveals that the form of the perturbation equation, for both isothermal and polytropic flows, is invariant under the choice of any particular pseudo-potential. Under generically true outer boundary conditions, the inviscid flow has been shown to be stable under an adiabatic and radially propagating perturbtion. The perturbation equation has also served the dual purpose of enabling an understanding of the acoustic geometry for inviscid and rotational flows.

astro-ph↗

arXiv2005

A dynamical systems approach to a thin accretion disc and its time-dependent behaviour on large length scales

Modelling the flow in a thin accretion disc like a dynamical system, we analyse the nature of the critical points of the steady solutions of the flow. For the simple inviscid disc there are two critical points, with the outer one being a saddle point and the inner one a centre type point. For the weakly viscous disc, there are four possible critical points, of which the outermost is a saddle point, while the next inner one is a spiral. Coupling the nature of the critical points with the outer boundary condition of the flow, gives us a complete understanding of all the important physical features of the flow solutions in the subsonic regions of the disc. In the inviscid disc, the physical realisability of the transonic solution passing through the saddle point is addressed by considering a temporal evolution of the flow, which is a very likely non-perturbative mechanism for selecting the transonic inflow solution from among a host of other possible stable solutions. For the weakly viscous disc, while a linearised time-dependent perturbation imposed on the steady mass inflow rate causes instability, the same perturbative analysis reveals that for the inviscid disc, there is a very close correspondence between the equation for the propagation of the perturbation and the metric of an acoustic black hole. Compatible with the transport of angular momentum to the outer regions of the disc, a viscosity-limited length scale is defined for the spatial extent of the inward rotational drift of matter.

astro-ph↗

arXiv2005

A Dynamical Systems Approach to an Inviscid and Thin Accretion Disc

The inviscid and thin accretion disc is a simple and well understood model system in accretion studies. In this work, modelling such a disc like a dynamical system, we analyse the nature of the fixed points of the stationary solutions of the flow. We show that of the two fixed points, one is a saddle and the other is a centre type point. We then demonstrate, using a simple but analogous mathematical model, that a temporal evolution of the flow is a very likely non-perturbative mechanism for the selection of an inflow solution that passes through the saddle type critical point.

astro-ph↗

arXiv2005

Linearized perturbation on stationary inflow solutions in an inviscid and thin accretion disc

The influence of a linearized perturbation on stationary inflow solutions in an inviscid and thin accretion disc, has been studied here, and it has been argued, that a perturbative technique would indicate that all possible classes of inflow solutions would be stable. The choice of the driving potential, Newtonian or pseudo-Newtonian, would not particularly affect the arguments which establish the stability of solutions. It has then been surmised that in the matter of the selection of a particular solution, adoption of a non-perturbative technique, based on a more physical criterion, as in the case of the selection of the transonic solution in spherically symmetric accretion, would give a more conclusive indication about the choice of a particular branch of the flow.

astro-ph↗

arXiv2005

Large scale properties in turbulent spherically symmetric accretion

The role of turbulence in a spherically symmetric accreting system has been studied on very large spatial scales of the system. This is also a highly subsonic flow region and here the accreting fluid has been treated as nearly incompressible. It has been shown here that the coupling of the mean flow and the turbulent fluctuations, gives rise to a scaling relation for an effective "turbulent viscosity". This in turn leads to a dynamic scaling for sound propagation in the accretion process. As a consequence of this scaling, the sonic horizon of the transonic inflow solution is shifted inwards, in comparison with the inviscid flow.

astro-ph↗

arXiv2005

Hydraulic Jump in One-dimensional Flow

In the presence of viscosity the hydraulic jump in one dimension is seen to be a first-order transition. A scaling relation for the position of the jump has been determined by applying an averaging technique on the stationary hydrodynamic equations. This gives a linear height profile before the jump, as well as a clear dependence of the magnitude of the jump on the outer boundary condition. The importance of viscosity in the jump formation has been convincingly established, and its physical basis has been understood by a time-dependent analysis of the flow equations. In doing so, a very close correspondence has been revealed between a perturbation equation for the flow rate and the metric of an acoustic white hole. We finally provide experimental support for our heuristically developed theory.

cond-mat.other↗

arXiv2004

Viscosity in spherically symmetric accretion

The influence of viscosity on the flow behaviour in spherically symmetric accretion, has been studied here. The governing equation chosen has been the Navier-Stokes equation. It has been found that at least for the transonic solution, viscosity acts as a mechanism that detracts from the effectiveness of gravity. This has been conjectured to set up a limiting scale of length for gravity to bring about accretion, and the physical interpretation of such a length-scale has been compared with the conventional understanding of the so-called "accretion radius" for spherically symmetric accretion. For a perturbative presence of viscosity, it has also been pointed out that the critical points for inflows and outflows are not identical, which is a consequence of the fact that under the Navier-Stokes prescription, there is a breakdown of the invariance of the stationary inflow and outflow solutions -- an invariance that holds good under inviscid conditions. For inflows, the critical point gets shifted deeper within the gravitational potential well. Finally, a linear stability analysis of the stationary inflow solutions, under the influence of a perturbation that is in the nature of a standing wave, has indicated that the presence of viscosity induces greater stability in the system, than has been seen for the case of inviscid spherically symmetric inflows.

astro-ph↗

arXiv2004

Realizability of stationary spherically symmetric transonic accretion

The spherically symmetric stationary transonic (Bondi) flow is considered a classic example of an accretion flow. This flow, however, is along a separatrix, which is usually not physically realizable. We demonstrate, using a pedagogical example, that it is the dynamics which selects the transonic flow.

astro-ph↗