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In our treatment the nonlinear equations describing the steady, inviscid,
rotational, axisymmetric flow in the Kerr metric, have been tailored
to form a first-order autonomous dynamical system. The
critical points of the phase trajectories of the flow have been identified
first, following which, a linearized study in the neighbourhood of
these critical points has been carried out. As a consequence of this
exercise, a complete and rigorous mathematical classification scheme
for the nature of the critical points has been derived, and it has
been argued that the critical points can admissibly be only saddle
points and centre-type points for the kind of conserved, axisymmetric
and rotational flow under study here. While all of these are principally
the attributes of the hydrodynamical process itself, the influence of
the black hole (the agent external to the fluid, but driving its flow
nonetheless) has also been
noteworthy to the extent that its intrinsic rotational parameter affects
the character of multitransonicity in general and the properties of an
individual critical point in particular. This is an important result
to have emerged from the study.
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 linearized 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 multi-transonic
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, characterized
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.
Using post-Newtonian formalism, 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 perturbation. The perturbation equation has also
served the dual purpose of enabling and understanding the acoustic geometry for
inviscid and rotational flows.
Next: Ongoing/Future Project
Up: Astrophysical Fluid as an
Previous: Introduction
Tapas Kumar Das
2009-01-17