EFFECTIVE Q CRITERION FOR DISK STABILITY IN AN EXTERNAL POTENTIAL

standard Q criterion (with Q > 1) describes the stability against local, axisymmetric perturbations in a disk supported by rotation and random motion. Most astrophysical disks, however, are under the influence of an external gravitational potential, which can significantly affect their stability. A typical example is a galactic disk embedded in a dark matter halo. Here, we do a linear perturbation analysis for a disk in an external potential and obtain a generalized dispersion relation and the effective stability criterion. An external potential, such as that due to the dark matter halo concentric with the disk, contributes to the unperturbed rotational field and significantly increases its stability. We obtain the values for the effective Q parameter for the Milky Way and for a low surface brightness galaxy, UGC 7321. We find that in each case the stellar disk by itself is barely stable and it is the dark matter halo that stabilizes the disk against local, axisymmetric gravitational instabilities. Thus, the dark matter halo is necessary to ensure local disk stability. This result has been largely missed so far because in practice the Q parameter for a galactic disk is obtained using the observed rotational field that already includes the effect of the halo.

Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams

In this paper, the governing equations for free vibration of a non-homogeneous rotating Timoshenko beam, having uniform cross-section, is studied using an inverse problem approach, for both cantilever and pinned-free boundary conditions. The bending displacement and the rotation due to bending are assumed to be simple polynomials which satisfy all four boundary conditions. It is found that for certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, the assumed polynomials serve as simple closed form solutions to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of analytical polynomial functions possible for material mass density, shear modulus and elastic modulus distributions, which share the same frequency and mode shape for a particular mode. The derived results are intended to serve as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of rotating non-homogeneous Timoshenko beams.

Modal tailoring and closed-form solutions for rotating non-uniform Euler-Bernoulli beams

In this paper, the free vibration of a rotating Euler-Bernoulli beam is studied using an inverse problem approach. We assume a polynomial mode shape function for a particular mode, which satisfies all the four boundary conditions of a rotating beam, along with the internal nodes. Using this assumed mode shape function, we determine the linear mass and fifth order stiffness variations of the beam which are typical of helicopter blades. Thus, it is found that an infinite number of such beams exist whose fourth order governing differential equation possess a closed form solution for certain polynomial variations of the mass and stiffness, for both cantilever and pinned-free boundary conditions corresponding to hingeless and articulated rotors, respectively. A detailed study is conducted for the first, second and third modes of a rotating cantilever beam and the first and second elastic modes of a rotating pinned-free beam, and on how to pre-select the internal nodes such that the closed-form solutions exist for these cases. The derived results can be used as benchmark solutions for the validation of rotating beam numerical methods and may also guide nodal tailoring. (C) 2014 Elsevier Ltd. All rights reserved.

Analysis of Entropy Generation During Mixed Convective Heat Transfer of Nanofluids Past a Rotating Circular Cylinder

The entropy generation due to mixed convective heat transfer of nanofluids past a rotating circular cylinder placed in a uniform cross stream is investigated via streamline upwind Petrov-Galerkin based finite element method. Nanosized copper (Cu) particles suspended in water are used with Prandtl number (Pr)=6.9. The computations are carried out at a representative Reynolds number (Re) of 100. The dimensionless cylinder rotation rate, a, is varied between 0 and 2. The range of nanoparticle volume fractions (phi) considered is 0 <= phi <= 5%. Effect of aiding buoyancy is brought about by considering two fixed values of the Richardson number (Ri) as 0.5 and 1.0. A new model for predicting the effective viscosity and thermal conductivity of dilute suspensions of nanoscale colloidal particles is presented. The model addresses the details of the agglomeration-deagglomeration in tune with the pertinent variations in the effective particulate dimensions, volume fractions, as well as the aggregate structure of the particulate system. The total entropy generation is found to decrease sharply with cylinder rotation rates and nanoparticle volume fractions. Increase in nanoparticle agglomeration shows decrease in heat transfer irreversibility. The Bejan number falls sharply with increase in alpha and phi.

Three centered hydrogen bonds of the type C=O...H(N)...X-C in diphenyloxamide derivatives involving halogens and a rotating CF3 group: NMR, QTAIM, NCI and NBO studies

The existence of three centered C=O...H(N)...X-C hydrogen bonds (H-bonds) involving organic fluorine and other halogens in diphenyloxamide derivatives has been explored by NMR spectroscopy and quantum theoretical studies. The three centered H-bond with the participation of a rotating CF3 group and the F...H-N intramolecular hydrogen bonds, a rare observation of its kind in organofluorine compounds, has been detected. It is also unambiguously established by a number of one and two dimensional NMR experiments, such as temperature perturbation, solvent titration, N-15-H-1 HSQC, and F-19-H-1 HOESY, and is also confirmed by theoretical calculations, such as quantum theory of atoms in molecules (QTAIM), natural bond orbital (NBO) and non-covalent interaction (NCI).

Mixed Mode Analysis of Obliquely Oriented Edge Cracked Semicircular Disk Under Compression

A new mixed-mode compression fracture specimen, obliquely oriented edge cracked semicircular disk (OECSD) is analyzed by extending pure opening mode configuration of edge cracked semicircular disk (ECSD) under Hertzian compression. Photoelastic experiments are conducted on two different specimens of OECSD of same size and different crack lengths and inclinations. Finite element method (FEM) is used to solve a number of cases of the problem varying crack length and crack inclination. FE results show a good match with experiments. Inclination of edge crack in OECSD can be so made as to obtain any mode-mixity ratio between zero and one and beyond for any crack length. The new specimen can be used for fracture testing under compression more conveniently than the existing ones in several ways.

Meshless Local Petrov-Galerkin Method for Rotating Euler-Bernoulli Beam

Free vibration problem of a rotating Euler-Bernoulli beam is solved with a truly meshless local Petrov-Galerkin method. Radial basis function and summation of two radial basis functions are used for interpolation. Radial basis function satisfies the Kronecker delta property and makes it simpler to apply the essential boundary conditions. Interpolation with summation of two radial basis functions increases the node carrying capacity within the sub-domain of the trial function and higher natural frequencies can be computed by selecting the complete domain as a sub-domain of the trial function. The mass and stiffness matrices are derived and numerical results for frequencies are obtained for a fixed-free beam and hinged-free beam simulating hingeless and articulated helicopter blades. Stiffness and mass distribution suitable for wind turbine blades are also considered. Results show an accurate match with existing literature.

GRMHD formulation of highly super-Chandrasekhar rotating magnetized white dwarfs: stable configurations of non-spherical white dwarfs

Here we extend the exploration of significantly super-Chandrasekhar magnetized white dwarfs by numerically computing axisymmetric stationary equilibria of differentially rotating magnetized polytropic compact stars in general relativity (GR), within the ideal magnetohydrodynamic regime. We use a general relativistic magnetohydrodynamic (GRMHD) framework that describes rotating and magnetized axisymmetric white dwarfs, choosing appropriate rotation laws and magnetic field profiles (toroidal and poloidal). The numerical procedure for finding solutions in this framework uses the 3 + 1 formalism of numerical relativity, implemented in the open source XNS code. We construct equilibrium sequences by varying different physical quantities in turn, and highlight the plausible existence of super-Chandrasekhar white dwarfs, with masses in the range of 2-3 solar mass, with central (deep interior) magnetic fields of the order of 10(14) G and differential rotation with surface time periods of about 1-10 s. We note that such white dwarfs are candidates for the progenitors of peculiar, overluminous Type Ia supernovae, to which observational evidence ascribes mass in the range 2.1-2.8 solar mass. We also present some interesting results related to the structure of such white dwarfs, especially the existence of polar hollows in special cases.

On the control of rapidly rotating convection by an axially varying magnetic field

The magnetic field in rapidly rotating dynamos is spatially inhomogeneous. The axial variation of the magnetic field is of particular importance because tall columnar vortices aligned with the rotation axis form at the onset of convection. The classical picture of magnetoconvection with constant or axially varying magnetic fields is that the Rayleigh number and wavenumber at onset decrease appreciably from their non-magnetic values. Nonlinear dynamo simulations show that the axial lengthscale of the self-generated azimuthal magnetic field becomes progressively smaller as we move towards a rapidly rotating regime. With a small-scale field, however, the magnetic control of convection is different from that in previous studies with a uniform or large-scale field. This study looks at the competing viscous and magnetic mode instabilities when the Ekman number E (ratio of viscous to Coriolis forces) is small. As the applied magnetic field strength (measured by the Elsasser number Lambda) increases, the critical Rayleigh number for onset of convection initially increases in a viscous branch, reaches an apex where both viscous and magnetic instabilities co-exist, and then falls in the magnetic branch. The magnetic mode of onset is notable for its dramatic suppression of convection in the bulk of the fluid layer where the field is weak. The viscous-magnetic mode transition occurs at Lambda similar to 1, which implies that small-scale convection can exist at field strengths higher than previously thought. In spherical shell dynamos with basal heating, convection near the tangent cylinder is likely to be in the magnetic mode. The wavenumber of convection is only slightly reduced by the self-generated magnetic field at Lambda similar to 1, in agreement with previous planetary dynamo models. The back reaction of the magnetic field on the flow is, however, visible in the difference in kinetic helicity between cyclonic and anticyclonic vortices.

Nonlinear Dynamics of a Rotating Flexible Link

This paper deals with the study of the nonlinear dynamics of a rotating flexible link modeled as a one dimensional beam, undergoing large deformation and with geometric nonlinearities. The partial differential equation of motion is discretized using a finite element approach to yield four nonlinear, nonautonomous and coupled ordinary differential equations (ODEs). The equations are nondimensionalized using two characteristic velocities-the speed of sound in the material and a velocity associated with the transverse bending vibration of the beam. The method of multiple scales is used to perform a detailed study of the system. A set of four autonomous equations of the first-order are derived considering primary resonances of the external excitation and one-to-one internal resonances between the natural frequencies of the equations. Numerical simulations show that for certain ranges of values of these characteristic velocities, the slow flow equations can exhibit chaotic motions. The numerical simulations and the results are related to a rotating wind turbine blade and the approach can be used for the study of the nonlinear dynamics of a single link flexible manipulator.

Nonrotating Beams Isospectral to Tapered Rotating Beams

In this paper, we seek to find nonrotating beams that are isospectral to a given tapered rotating beam. Isospectral structures have identical natural frequencies. We assume the mass and stiffness distributions of the tapered rotating beam to be polynomial functions of span. Such polynomial variations of mass and stiffness are typical of helicopter and wind turbine blades. We use the Barcilon-Gottlieb transformation to convert the fourth-order governing equations of the rotating and the nonrotating beams, from the (x, Y) frame of reference to a hypothetical (z, U) frame of reference. If the coefficients of both the equations in the (z, U) frame match with each other, then the nonrotating beam is isospectral to the given rotating beam. The conditions on matching the coefficients lead to a pair of coupled differential equations. Wesolve these coupled differential equations numerically using the fourth-order Runge-Kutta scheme. We also verify that the frequencies (given in the literature) of standard tapered rotating beams are the frequencies (obtained using the finite-element analysis) of the isospectral nonrotating beams. Finally, we present an example of beams having a rectangular cross-section to show the application of our analysis. Since experimental determination of rotating beam frequencies is a difficult task, experiments can be easily conducted on these isospectral nonrotating beams to calculate the frequencies of the rotating beam.