Instability of collapsing source under expansion-free condition in einstein Gauss–Bonnet gravity

In this work, we have investigated the dynamical instability of spherically symmetric gravitating object under expansion-free condition in Einstein Gauss–Bonnet gravity. In this context, the field equations and dynamical equations have been established in the Gauss–Bonnet gravity. The linear perturbation scheme has been used on the dynamical equations to construct the collapse equation. The Newtonian, post Newtonian and post Newtonian approximations have been applied to investigate the general dynamical (in)stability equations. It has been observed that the instability range of the collapsing source is independent of adiabatic index Γ (stiffness of the fluid does not play any role). The instability range can be determined by the pressure anisotropy, energy density profile, Gauss–Bonnet parameter α and some constraints at Newtonian, post Newtonian and post Newtonian order.     

Effects of CDTT model on the stability of spherical collapse in Palatini f(R) gravity

The objective of this paper is to study the stability of an adiabatic anisotropic collapsing sphere in the context of Palatini f(R) gravity. In this framework, we construct the collapse equation with the help of the contracted Bianchi identities of the effective as well as the usual energy-momentum tensor. The perturbation scheme is applied on the fluid variables which accordingly cause a perturbation on the Ricci scalar. We explore the instability ranges in the Newtonian and post-Newtonian regimes. It is concluded that the stability of the star is governed by adiabatic index Γ ₁, which depends on the energy density profile, anisotropic pressure and dark source terms of the chosen f(R) model. We also explore our results when f(R)→R.     

Faraday instability with a polymer solution

We present an experimental study of the Faraday instability in which we compare the behavior of a Newtonian fluid (water-glycerine mixture) with that of a semi-dilute non-Newtonian solution of high molecular weight polymer. We show that although the dispersion relation of surface waves, derived for a layer of inviscid fluid, remains valid in that particular non-Newtonian case, the behavior of the instability threshold with frequency strongly differs from the Newtonian case. We explain this effect as a result of a frequency-dependent viscosity. The linear stability analysis of the non-Newtonian case shows a perfect agreement with the experimental results both for the dispersion relation and for the reduction of the instability threshold. We discuss the use of the characteristics of the Faraday experiment as a measurement tool to determine frequency dependent properties of non-Newtonian fluids. Received 5 January 1999     

Stability characteristics of hyper-concentration flow in open channel

The flow instability is related to many engineering problems and belongs to a wide-ranging research field. When the problem on the transition from the laminar to the turbulence caused by the instability of the laminar is studied, the “neutral line” and the critical Reynolds number are always taken as the criterion to judge whether a certain kind of flow is stable, whose corresponding flow medium is the clear water, that is, the single-phase Newtonian fluid. And it is not studied in the traditional instability theory that the hyper-concentration flow widely exists in rivers. This shortage can be covered by this research. Study shows that the instability of non-Newtonian fluid such as hyper-concentration fluid, compared with Newtonian fluid such as clear water, is influenced by not only Reynolds number, the ratio of the inertia force and the viscous force, but also many other factors such as the sediment concentration, the concentration distribution, the grain size, the volumetric weight of the sediment and so on, which make the mechanical principle even more complex. So the results of the research can supply the scientific basis for the explanations of “slurrying river”, the turbulence intensity of the flow carrying sediment and the variance of the turbulence structure.     

Three-dimensional absolute and convective instabilities in mixed convection of a viscoelastic fluid through a porous medium

By using the mathematical formalism of absolute and convective instabilities we study the nature of unstable three-dimensional disturbances of viscoelastic flow convection in a porous medium with horizontal through-flow and vertical temperature gradient. Temporal stability analysis reveals that among three-dimensional (3D) modes the pure down-stream transverse rolls are favored for the onset of convection. In addition, by considering a spatiotemporal stability approach we found that all unstable 3D modes are convectively unstable except the transverse rolls which may experience a transition to absolute instability. The combined influence of through-flow and elastic parameters on the absolute instability threshold, wave number and frequency is then determined, and results are compared to those of a Newtonian fluid.     

Dynamical instability of charged self-gravitating stars in modified gravity

This paper explores the instability limits for the stellar objects in the background of a particular modified gravity theory. In order to accomplish the instability conditions, a spherically symmetric anisotropic charged fluid influenced by the modified gravity is taken under consideration. The modified field equations and the equations of motion are accomplished in background of the Gauss–Bonnet gravity. These equations are perturbed to constitute the collapse equation. The Newtonian and post-Newtonian limits are imposed and found that the dynamical instability of the fluid is explained by the adiabatic index which consists on analytical value depending on static profile of material variables.     

Dynamical instability of shear-free collapsing star in extended teleparallel gravity

We study the spherically symmetric collapsing star in terms of dynamical instability. We take the framework of extended teleparallel gravity with a non-diagonal tetrad, a power-law form of the model presenting torsion and a matter distribution as a non-dissipative anisotropic fluid. The vanishing shear scalar condition is adopted to gain insight in a collapsing star. We apply a first order linear perturbation scheme to the metric, the matter, and f(T) functions. The dynamical equations are formulated under this perturbation scheme to develop collapsing equation for finding dynamical instability limits in two regimes, such as the Newtonian and the post-Newtonian regime. We obtain a constraint-free solution of a perturbed time dependent part with the help of a vanishing shear scalar. The adiabatic index exhibits the instability ranges through the second dynamical equation which depend on physical quantities such as the density, the pressure components, the perturbed parts of the symmetry of the star, etc. We also develop some constraints on the positivity of these quantities and obtain instability ranges to satisfy the dynamical instability condition.     

Symmetries and consistency relations in the large scale structure of the universe

We study the symmetries enjoyed by the Newtonian equations of motion of the non-relativistic dark matter fluid coupled to gravity which give rise to the phenomenon of gravitational instability. We also discuss some consistency relations involving the soft limit of the (n+1)$(n+1)$-correlator functions of matter and galaxy overdensities.     

Instability ranges of collapsing star through adiabatic index in f(T, Θ) theory of gravity

The main purpose of this paper is to study the stability analysis of collapsing star in the scenario of Newtonian and post Newtonian approximations. We consider the cylindrical symmetry of collapsing object which is filled with anisotropic fluid. The f(T, Θ) theory of gravity, where T is the torsion scalar and Θ is the trace of energy-momentum tensor is taken into account. The dynamical field equations are constructed which help to derive collapse equation by applying the perturbation method. The stability behavior of collapsing star is defined in both Newtonian and post Newtonian approximations with the help of an adiabatic index.     

Cyclone–anticyclone vortex asymmetry mechanism and linear Ekman friction

Allowance for the linear Ekman friction has been found to ensure a threshold (in rotation frequency) realization of the linear dissipative–centrifugal instability and the related chiral symmetry breaking in the dynamics of Lagrangian particles, which leads to the cyclone–anticyclone vortex asymmetry. An excess of the fluid rotation rate ω₀ over some threshold value determined by the fluid eigenfrequency ω (i.e., ω₀ > ω) is shown to be a condition for the realization of such an instability. A new generalization of the solution of the Karman problem to determine the steady-state velocity field in a viscous incompressible fluid above a rotating solid disk of large radius, in which the linear Ekman friction was additionally taken into account, has been obtained. A correspondence of this solution and the conditions for the realization of the dissipative–centrifugal instability of a chiral-symmetric vortex state and the corresponding cyclone–anticyclone vortex asymmetry has been shown. A generalization of the well-known spiral velocity distribution in an “Ekman layer” near a solid surface has been established for the case where the fluid rotation frequency far from the disk ω differs from the disk rotation frequency ω₀.     

Subsonic flapping flutter

A mathematical model of two-dimensional flow through a flexible channel is analyzed for its stability characteristics. Linear theory shows that fluid viscosity, modelled by a Darcy friction factor, induces flutter instability when the dimensionsless fluid speed, S, attains a critical flutter speed, S₀. This is in qualitative agreement with experimental results, and it is at variance with previous analytical studies where fluid viscosity was neglected and divergence instability was predicted. The critical flutter speed and the associated critical flutter frequency depend on three other dimensionless parameters: the ratio of fluid to wall damping; the ratio of wall to fluid mass; and the ratio of wall bending resistance to elastance. Non-linear theory predicts stable, finite amplitude flutter for S>S₀, which increases in frequency and amplitude as S increases. Both symmetric and antisymmetric modes of deformation are discussed.     

An experimental study of the effects of pulsating and steady internal fluid flow on an elastic tube subjected to external vibration

The results of an experimental study on both pulsating and steady Newtonian fluid flow in an initially stretched rubber tube subjected to external vibration are reported. A circulating loop system was designed to maintain constant hydrostatic pressure throughout the tests so that the influence of external excitation on the fluid flow could be properly distinguished. The effects of fluid flow velocity and initial stretch rates on the dynamic response and damping of the tube conveying fluid were examined, and it was observed that damping ratios increase with increasing flow velocities, and generally decrease with increasing initial stretch rates for the tube conveying fluid. It was also noted that dynamic responses increase with increasing initial stretch rates, and decrease with increasing flow velocities. The effect of external vibration on fluid flow rates is small in a tube with a thickness-to-radius ratio (D_out−D_in)/D_in=0.617. Fluid pressures vary, in terms of frequency and amplitude, with external vibration as well as Womersley number.     

Thermal convection in Rivlin-Ericksen elastico-viscous fluid in porous medium in hydromagnetics

The thermal instability of a layer of Rivlin-Ericksen elastico-viscous fluid in porous medium acted on by a uniform magnetic field is considered. For stationary convection, Rivlin-Ericksen elastico-viscous fluid behaves like a Newtonian fluid. The magnetic field is found to have stabilizing effect whereas medium permeability has destabilizing effect. The magnetic field introduces oscillatory modes in the system, A sufficient condition for the non-existence of overstability is also obtained.     

Instability analysis of a cylindrical stellar object in Brans–Dicke gravity

This paper investigates instability ranges of a cylindrically symmetric collapsing cosmic filamentary structure in the Brans–Dicke theory of gravity. For this purpose, we use a perturbating approach to the modified field equations as well as dynamic equations and construct a collapse equation. The collapse equation with an adiabatic index (Γ) is used to explore the instability ranges of both isotropic and anisotropic fluid in Newtonian and post-Newtonian approximations. It turns out that the instability ranges depend on the dynamic variables of collapsing filaments. We conclude that the system always remains unstable for 0 < Γ < 1, while Γ > 1 provides instability only in a special case.     

Effects of Hall current on unsteady MHD flows of a second grade fluid

The aim of this present paper is to construct exact solutions corresponding to the motion of magnetohydrodynamic (MHD) fluid in the presence of Hall current, due to cosine and sine oscillations of a rigid plate as well as those induced by an oscillating pressure gradient. A uniform magnetic field is applied transversely to the flow. By using Fourier sine transform steady state and transient solutions are presented. These solutions satisfy the governing equations and all associated initial and boundary conditions. The results for a hydrodynamic second grade fluid can be obtained as a limiting case when B ₀ → 0 and for a Newtonian fluid when α ₁ → 0.     

Thermal instability in Maxwellian and Oldroydian viscoelastic fluids with suspended particles in porous medium

The effect of suspended particles on thermal instability is considered separately in Maxwellian and Oldroydian viscoelastic fluids in a porous medium. The principle of exchange of stabilities is found to hold well under a condition which is the same for Maxwellian as well as Oldroydian fluid. For stationary convection, both the Maxwellian and Oldroydian fluids behave like Newtonian fluid and the medium permeability and the suspended particles have destabilizing effects on the system. The sufficient conditions for the non-existence of overstability for both Maxwellian and Oldroydian viscoelastic fluids are also obtained.     

Newtonian and Post-Newtonian Limits of Relativistic Cosmology

The Newtonian limit of general relativity is by no means as straightforward as is commonly assumed. In particular, the correct limit of the Bianchi identities must be taken to the second (non-linear) order. Furthermore Newtonian cosmology does not have a well-posed initial value formulation, while relativistic cosmology does. We show in this paper that the c _-4 approximation of general relativity, although non-linear, provides a non-standard version of Newtonian theory which is in fact completely equivalent to the Heckmann–Schücking version of Newtonian cosmology. The next approximation (order c _-6), when the limit is taken in a particular way, gives rise to a closed and self-consistent post-Newtonian cosmological theory which has a well-posed initial value problem. This seems to be a suitable, if somewhat complicated, theory for cosmological and astrophysical problems     

Subcritical finite-amplitude solutions for plane Couette flow of viscoelastic fluids

Plane Couette flow of viscoelastic fluids is shown to exhibit a purely elastic subcritical instability at a very small-Reynolds number in spite of being linearly stable. The mechanism of this instability is proposed and the nonlinear stability analysis of plane Couette flow of the Upper-Convected Maxwell fluid is presented. Above a critical Weissenberg number, a small finite-size perturbation is sufficient to create a secondary flow, and the threshold value for the amplitude of the perturbation decreases as the Weissenberg number increases. The results suggest a scenario for weakly turbulent viscoelastic flow which is similar to the one for Newtonian fluids as a function of Reynolds number.