Nonbarotropic Flows Research Articles

Integral invariants for non-barotropic flows in a four dimensional space time manifold

Properties of non-barotropic flows are described using Lie derivatives of differential forms in a Euclidean four dimensional space-time manifold. Vanishing of the Lie derivative implies that the corresponding physical quantity remains invariant along the integral curves of the flow. Integral invariants of non-barotropic perfect and viscous flows are studied using the concepts of relative and absolute invariance of forms. The four dimensional expressions for the rate of change of the generalized circulation, generalized vorticity flux, generalized helicity and generalized parity in the case of ideal and viscous non-barotropic flows are thereby obtained.

Singularity formation of the compressible non-barotropic flows with zero heat conductivity

We study a criterion for the breakdown of the strong solutions to a three-dimensional Cauchy problem of compressible non-barotropic magnetohydrodynamic equations without heat conductivity. It is shown that the strong solution exists globally if the norms of ‖(ρ,P)‖L∞(0,T;L∞) and ‖H‖L∞(0,T;Ls) are bounded for some s ∈ [6, +∞]. In particular, a direct application to the compressible Navier-Stokes equations implies that the concentration of the density or the pressure will be responsible for the breakdown of strong solutions.

Vortex Steady Planar Entropic Flows of Ideal Gases

We find all solutions to the submodel of vortex, steady, planar, barotropic, entropic flows of an ideal gas and show that possible motions are exhausted by rectilinear motions under a constant pressure and motions along concentric circles. We present a group classification of the model of planar, vortex, entropic, nonbarotropic flows, examine invariant submodels, and propose a physical interpretation of certain solutions.

Asymptotic behavior for cylindrically symmetric nonbarotropic flows in exterior domains with large data

We study the initial–boundary value problem for the compressible Navier–Stokes equations describing the cylindrically symmetric motion of a viscous nonbarotropic fluid in the domain exterior to a ball in R3. The global solution is proved to exist uniquely and be asymptotically stable as time tends to infinity for large initial data. Moreover, the density and temperature are shown to be bounded from above and below uniformly in both time and space. Our analysis is based on nonlinear energy methods.

On formation of singularity of spherically symmetric nonbarotropic flows

We study an initial boundary value problem on a ball for the heat-conductive system of compressible Navier–Stokes–Fourier equations, in particular, a criterion for the breakdown of the classical solution. For smooth initial data away from vacuum, we prove that the classical solution which is spherically symmetric loses its regularity in a finite time if and only if the density concentrates or vanishes or the velocity becomes unbounded around the center. One possible situation is that a vacuum ball appears around the center and the density may concentrate on the boundary of the vacuum ball simultaneously.

Corotational instability of inertial-acoustic modes in black hole accretion discs: non-barotropic flows

We study the effect of corotation resonance on the inertial-acoustic oscillations (p-modes) of black-hole accretion discs. Previous works have shown that for barotropic flows (where the pressure depends only on the density), wave absorption at the corotation resonance can lead to mode growth when the disc vortensity, $\zeta=\kappa^2/(2\Omega\Sigma)$ (where $\Omega, \kappa, \Sigma$ are the rotation rate, radial epicyclic frequency and surface density of the disc, respectively), has a positive gradient at the corotation radius. Here we generalize the analysis of the corotation resonance effect to non-barotropic fluids. We show that the mode instability criterion is modified by the finite radial Brunt-V\"as\"al\"a frequency of the disc. We derive an analytic expression for the reflectivity when a density wave impinges upon the corotation barrier, and calculate the frequencies and growth rates of global p-modes for disc models with various $\alpha$-viscosity parameterizations. We find that for disc fluids with constant adiabatic index $\Gamma$, super-reflection and mode growth depend on the gradient of the effective vortensity, $\zeta_{\rm eff} = \zeta/S^{2/\Gamma}$ (where $S \equiv P/\Sigma^{\Gamma}$ measures the entropy): when $d\zeta_{\rm eff}/dr > 0$ at the corotation radius, wave absorption leads to amplification of the p-mode. Our calculations show that the lowest-order p-modes with azimuthal wave number $m=2, 3, 4,...$ have the largest growth rates, with the frequencies approximately in (but distinct from) the $2:3:4...$ commensurate ratios. We discuss the implications of our results for the high-frequency quasi-periodic oscillations observed in accreting black-hole systems.

Symmetric Nonbarotropic Flows with Large Data and Forces

We prove the global existence of weak solutions of the Navier-Stokes equations of compressible, nonbarotropic flow in three space dimensions with initial data and external forces which are large, discontinuous, and spherically or cylindrically symmetric. The analysis allows for the possibility that a vacuum state emerges at the origin or axis of symmetry, and the equations hold in the sense of distributions in the set where the density is positive. In addition, the mass and momentum equations hold weakly in the entire space-time domain, but with a nonstandard interpretation of the viscosity terms as distributions. Solutions are obtained as limits of solutions in annular regions between two balls or cylinders, and the analysis allows for the possibility that energy is absorbed into the origin or axis, and is lost in the limit as the inner radius goes to zero.

Variational principle and conservation laws for nonbarotropic flows

A variational formulation for nonbarotropic perfect fluid flow is presented. By introducing suitable field variables, Euler’s equations of motion from the variational principles are obtained. The conservation laws of energy, impulse, and angular momenta follow from application of Noether’s theorem.

The dynamics of isocirculational flows

Abstract Barotropic flows are described in terms of mass preserving velocity fields of the vortex lines and non-barotropic flows by mass preserving velocity fields that move with the entropy and the potential vorticity. In these representations, all locally conserved quantities of inviscid flows are a priori given functions of the positions of the elements of fluid. The Hamiltonian action for these “isocirculational” flows is also given and is suitable for studying motions around given steady states. Stability conditions of stationary flows with respect to perturbations that are not necessarily small but that do not change the values of the conserved quantities including the circulations are obtained by minimizing the potential terms of the Lagrangian functional.

Some vorticity theorems and conservation laws for non-barotropic fluids

Some theorems concerning the vorticity in barotropic flows of perfect fluids are generalized for non-barotropic flows. The generalization involves replacing the velocity in certain parts of the equations by a time-dependent quantity which is a function of the velocity and thermodynamic properties of the fluid. Results which are generalized include Kelvin's circulation theorem and conservation laws for potential vorticity and helicity. It is shown how the results can be further generalized to include dissipative effects. The possibility of using some of the results in deriving a complete set of Lagrangian conservation laws for perfect fluids is discussed.