Disk Of Finite Thickness Research Articles

Modification alpha formalism of Shakura–Sunyaev for the coefficient of turbulent viscosity in an astrophysical disk of finite thickness

In the approximation of one-fluid hydrodynamics, a closed system of Favre-averaged magneto-hydrodynamic equations is formulated, intended for the numerical simulation of compressible turbulent flows of electrically conductive media in the presence of a magnetic field. Special emphasis is paid to the method of obtaining, within the framework of irreversible thermodynamics, the constitutive relations for the turbulent flux heat and the total (kinetic plus magnetic) tensor of turbulent stresses. A new approach to modeling the coefficient of turbulent kinematic viscosity for an astrophysical disk is proposed, which takes into account the influence of an external and generated magnetic field, as well as the processes of convective heat transfer on turbulence in a stratified layer of finite thickness, and thereby modifies the Shakura–Sunyaev alpha formalism developed by for a thin disk and widely used in the astrophysical literature.

Investigation of the Motion of a Heavy Body of Revolution on a Perfectly Rough Plane by the Kovacic Algorithm

Investigation of various problems of mechanics and mathematical physics is reduced to the solution of second-order linear differential equations with variable coefficients. In 1986, the American mathematician J. Kovacic proposed an algorithm for solution of a second-order linear differential equation in the case where the solution can be expressed in terms of so-called Liouville functions. If a linear second-order differential equation has no Liouville solutions, the Kovacic algorithm also allows one to ascertain this fact. In this paper, we discuss the application of the Kovacic algorithm to the problem of the motion of a heavy body of revolution on a perfectly rough horizontal plane. The existence of Liouville solutions of the problem is examined for the cases where the rolling body is an infinitely thin disk, a disk of finite thickness, a dynamically symmetric torus, a paraboloid of revolution, and a spindle-shaped body.

Disjoining pressure between disks of finite thickness with dispersion interaction

Local normal pressure has been calculated for dispersion forces inside a circular finite slit between two identical cylindrical disks. Corrections to the disjoining pressure have been obtained for the thicknesses of the interacting disks as compared with infinitely extended cylinders. The obtained corrections have been shown to be most significant, provided that the disk thickness is smaller than the slit size.

Numerical Simulation of Transient Thermal Transport on a Rotating Disk Under Partially Confined Laminar Liquid Jet Impingement

Abstract This paper considers the transient conjugate heat transfer characterization of a partially confined liquid jet impinging on a rotating and uniformly heated solid disk of finite thickness and radius. A constant heat flux was imposed at the bottom surface of the solid disk at t=0, and heat transfer was monitored for the entire duration of the transient until the steady state condition was reached. Calculations were done for a number of disk materials using water as the coolant, covering a range of Reynolds numbers (225–900), Ekman numbers (7.08×10−5−∞), nozzle-to-target spacing (β=0.25–1.0), confinement ratios (rp/rd=0.2–0.75), disk thicknesses to nozzle diameter ratios (b/dn=0.25–1.67), and solid to fluid thermal conductivity ratios (36.91–697.56). It was found that a higher Reynolds number decreases the time to achieve the steady state condition and increases the local and average Nusselt number. The duration of the transient increases with the increment of the Ekman number and disk thickness, and the reduction in the thermal diffusivity of the disk material.

Transient Conjugate Heat Transfer During Free Liquid Jet Impingement on a Rotating Solid Disk

This article presents the transient conjugate heat transfer characterization of a free liquid jet impinging on a rotating solid disk of finite thickness and radius. Calculations were done for a range of Reynolds number (500–1400), Ekman number (6.62 × 10−5–∞), disk thicknesses to nozzle diameter ratio (0.17–1.67), and solid to fluid thermal conductivity ratio (36.91–697.56) using water as the coolant. It was detected that the duration of the transient increased with disk thickness and decreased with Reynolds number and thermal conductivity ratio. A correlation for average Nusselt number for the transient heat transfer is presented.

Higher-order acoustic diffraction by edges of finite thickness

A cw solution of acoustic diffraction by a three-sided semi-infinite barrier or a double edge, where the width of the midplanar segment is finite and cannot be ignored, involving all orders of diffraction is presented. The solution is an extension of the asymptotic formulas for the double-edge second-order diffraction via amplitude and phase matching given by Pierce [A. D. Pierce, J. Acoust. Soc. Am. 55, 943-955 (1974)]. The model accounts for all orders of diffraction and is valid for all kw, where k is the acoustic wave number and w is the width of the midplanar segment and reduces to the solution of diffraction by a single knife edge as w-->0. The theory is incorporated into the deformed edge solution [Stanton et al., J. Acoust. Soc. Am. 122, 3167 (2007)] to model the diffraction by a disk of finite thickness, and is compared with laboratory experiments of backscattering by elastic disks of various thicknesses and by a hard strip. It is shown that the model describes the edge diffraction reasonably well in predicting the diffraction as a function of scattering angle, edge thickness, and frequency.

A Comprehensive Study of Conjugate Heat Transfer During Free Liquid Jet Impingement on a Rotating Disk

The aim of this computational study is to characterize convective heat transfer for a free liquid jet impinging on a rotating and uniformly heated solid disk of finite thickness and radius. Calculations are done for a number of disk materials and working fluids covering a range of Reynolds number (445–1,800), Ekman number (2.21 × 10−5–2.65 × 10−4), nozzle-to-target spacing (β = 0.55–5.0), disk thicknesses-to-nozzle diameter ratio (0.167–1.67), disk-to-nozzle diameter ratio (2.11–6.33), Prandtl number (1.29–124.44), and solid-to-fluid thermal conductivity ratio (36.91–2222). A generalized correlation for average Nusselt number is developed from numerical results. The simulation results compare reasonably well with available experimental data.

Steady flow generated by the differential rotation of a porous disk of finite thickness

When a body of fluid bounded by a porous disk of finite thickness is disturbed from a state of rigid rotation by an enhanced (or reduced) angular velocity of the disk, a few authors followed Darcy’s model and observed that the centrifugal pumping occurs through the entire porous layer regarded as a convection zone. The shear stress can develop only at the edge of the porous layer. We use a porous disk of high permeability that allows the fluid in the porous disk to deform in response to the changing angular velocity. Based on the Birkman’s model, we solve for the steady non-linear flow and observe that there arises (i) a convection zone of nearly uniform angular velocity at the boundary (within the porous layer) and (ii) a transition zone adjacent to the convection zone which provides a smooth transition to the interior. This makes the model relevant to some astrophysical situations as described by some authors [1, 3]. The two point boundary value problem is solved subject to the boundary conditions, the far field conditions, and the matching conditions at the fluid-porous medium interface. The solution is obtained using a numerical procedure known as the method of Adjoints.

Spiral soliton solution for disk galaxies

In this paper, weakly nonlinear dynamics of spiral galaxies is studied, using reductive perturbation method. One primarily aims at the derivation of possible soliton solution for two dimensional geometry, in the state of marginal stability. In order to use proper coordinate transformation, it was necessary to analyze stability of the linearized system of equations, and to define proper parameter regime. Such parameter regime is in agreement with the observational data, too. The influence of finite-thickness of the galaxy disk on dispersive properties of the system is studied, extending approximate solution of Poisson?s equation. For both cases, infinitesimally thin disk and disk of finite thickness, the same type of NLS equation is derived, but with different coefficients for nonlinear and dispersive terms. This means that corresponding soliton solutions have different properties. By comparing soliton properties with observational data it is possible to control validity of approximation for different geometry of the model.

Electrical Interaction between Two Thin Disks of Finite Thickness

The interaction between two identical, thin disks of finite thickness is investigated theoretically. We show that the thickness of a disk may play a significant role in the estimation of the electrical and the van der Waals interaction energies between two disks. These interaction energies depend largely on the surface conditions of a disk, its thickness, and the relative orientation between two disks. If the basal plane of a disk is negatively charged, and its edge face positively charged, then for a fixed total amount of charge on the edge face, the thicker the disk, the greater the absolute electrical interaction energy. This implies that neglecting the thickness of a disk will underestimate the absolute electrical interaction energy. If the masses of disks and the separation distance between them are fixed, the thinner the disks, the greater the absolute van der Waals interaction energy. The results for the total interaction energy between two disks reveal that neglecting the thickness of a disk may r...

The Structure of Cooling Fronts in Accretion Disks

Recent work has shown that the speed of the cooling front in soft X-ray transients may be an important clue in understanding the nature of accretion disk viscosity. In a previous paper (Vishniac & Wheeler), we derived the scaling law for the cooling front speed. Here we derive a similarity solution for the hot inner part of disks undergoing cooling. This solution is exact in the limit of a thin disk, power-law opacities, and a minimum hot state column density, which is an infinitesimal fraction of the maximum cold state density. For a disk of finite thickness, the largest error is in the ratio of the mass flow across the cooling front to the mass flow at small radii. Comparison to the numerical simulations of Cannizzo et al. indicates that the errors in other parameters do not exceed (csF/rFΩF)q, that is, the ratio of the sound speed at the disk midplane to its orbital velocity, evaluated at the cooling front, to the qth power. Here q ≈ ½. Its precise value is determined by the relevant hot state opacity law and the functional form of the dimensionless viscosity.

Generation and maintenance of bisymmetric spiral magnetic fields and interaction with spiral density waves

Abstract The paper consists of two parts. The first introduces the dynamo equation into a rotating gaseous disk of finite thickness and then searches for its solution for the generation and maintenance of large-scale bisymmetric spiral (BSS) magnetic fields. We determine numerically the dynamo strength and vertical thickness of the gaseous disk which are necessary for the BSS magnetic fields to rotate as a wave over large area of the disk. Next we present linearized equations of motion for the self-gravitating disk gas under the Lorentz force due to the BSS magnetic fields. Since the angular velocity of the BSS field is very close to that of the spiral density wave, a nearly-resonant interaction is caused between these two waves to produce large-amplitude condensation of gas in a double-spiral way. The BSS magnetic field is considered as a promising agency to trigger and maintain the spiral density wave.

Corotation torques in the solar nebula - The cutoff function

The behavior of high-order corotation resonances in a disk of finite thickness is examined. The torque exerted at an mth-order resonance is determined by employing a vertically averaged disturbing function, and the ratio of this torque to that exerted on a cold, two-dimensional disk is identified as the so-called torque cutoff function. This function is then used to calculate contributions from the corotation torques to eccentricity variations of a perturber's orbit assumed orbiting in the disk.

Accretion disk oscillations - A local analysis in a disk of finite thickness

Two types of oscillations are observed to occur in dwarf novae: 'coherent' and 'quasi-periodic' oscillations. These may be associated with the pulsation of the white dwarf or the accretion disk components of the dwarf nova. Here a local (short-wavelength) analysis is utilized to study the oscillation of a self-consistent, two-dimensional model of an accretion disk. The linearized equations describing adiabatic, inviscid, nonaxisymmetric oscillations are used to derive a fifth-order algebraic equation for the (complex) pulsation frequency of the disk. The solutions of this equation for various values of the wavevector k reveal that the disk is capable of supporting (1) a pair of high-frequency acoustic modes (p-modes); (2) a pair of intermediate-frequency modes which may share the characteristics of internal gravity waves (g-modes) and inertial waves; and (3) a mode associated with a dynamical instability (purely imaginary frequency). The role played by the shear in determining the stability or instability of these modes is also considered. Finally, the global oscillation frequencies of the disk are discussed.

On the structure of normal spiral galaxies with finite disk thickness part II: Three-dimensional asymptotic density-potential relation

Abstract A three-dimensional asymptotic density-potential relation is established for a spiral galaxy with a disk of finite thickness. This relation is accurate to the order /kr/−3, where k is the radial wave number and r is the radial distance from the center of the galaxy. In the case of small disk thickness, some simpler and more explicit relations are obtained with the accuracy of order /kr/−2, which is usually sufficient for the discussion of spiral mode. A density-potential relation, accurate to the order /kr/−3, is also obtained for the zero-thickness disk. To the order /kr/−2, this relation reduces to that given earlier by Bertin and Mark (1979), but the present derivation is simpler. The dynamical effect of the density-potential relation for a disk of finite thickness is investigated in Yue (1981).

A Constraint on the Velocity Dispersion of the Missing Mass in the Solar Neighborhood

The stability of an N-component stellar disk with finite thickness is examined with the gas dynamical approximation. The dispersion relation for marginal stability is obtained. This dispersion relation for N=2 is applied to the missing mass problem in the solar neighborhood, where two components represent the observed mass component and the miss­ ing mass component in the solar neighborhood. From the requirement that the Galactic disk should be locally stable, it is found that the velocity dispersion of the missing mass component should be greater than about 25km/sec. The stability of an infinitesimally thin disk is also investigated and compared with the disk of finite thickness. § l. Introduction It is well known that the observed mass density in the solar neighborhood is about a half of the dynamical mass density. 1l A solution to this problem has once been given by \Veistrop 2l and by Murray and Sanduleak. 3l They have claimed from their observations that the missing mass in the solar neighborhood consists of late .lYf-type dwarfs. However, recent controversies on this problem 4l, 5l have re­ vealed that their results are doubtful, and we must consider that the missing mass has not yet been identified. It may consist of fainter M-type dwarfs, white dwarfs and/or more exotic objects such as black holes. On the other hand, the density wave theory 6l prevails nowadays among the theories of spiral arms. According to this theory, unless the velocity dispersion is greater than Toomre's minimum dispersion,n the Galactic disk breaks into pieces and well-ordered spiral structures cannot be seen. 8l Usually the Galactic disk is considered to satisfy Toomre's criterion on the assumption that the velocity dispersion of the missing mass is the same as that of the observed stars. 6l However, as the velocity dispersion of the missing mass is unknown, the stellar disk of at least two components should be examined to check the stability of the Galactic disk. Here the two components represent the observed mass component and the missing mass component. Alternatively, the requirement that the Galactic disk should be locally stable will give a constraint on the velocity dispersion of the missing mass. To clarify this problem the local stability of an N-component stellar disk is studied in this paper. The N-component stellar disk is approximated as an N­ component gaseous system each component of which is isothermal and has a differ­ ent velocity dispersion from the others. Each component is assumed to interact

Circuit Analysis of Eddy Currents

Maxwell's equations govern the eddy-current phenomenon, and are the starting point of this analysis. It is shown that Maxwell's equations, as applied to steady-state ac conditions, can be transformed to a Fredholm-type integral equation in eddy-current density. In turn, it is demonstrated that the method of subareas can be used to solve the Fredholm equation. This approach leads to the familiar circuit-analysis concepts of resistance and inductance in finite coupled circuits. The coupledcircuit method can be utilized in cases of complex, mixed boundary conditions without difficulty, as is illustrated by an example of the eddy-current losses in a conducting disk of finite thickness and finite radius, in the presence of a current-carrying loop. Experimental data is presented which confirms the theory, for a range of disk thicknesses. References are included to previous work.

Improved Slotted Disk Type Velocity Selector for Molecular Beams

A slotted disk type mechanical velocity selector, 10 cm long and 16 cm in diameter, consisting of six disks has been constructed and tested. The velocity resolution corresponding to the half-intensity width is 0.047; no velocity sidebands are transmitted. The effective fractional open time to the incident beam is 0.35. At the highest rotor speed used (17 000 rpm) the transmitted velocity is 1.05×105 cm/sec. A general theoretical analysis of the transmission and resolution of selectors with disks of finite thickness is presented. The velocity distribution is reported for a molecular beam effusing from an oven equipped with Zacharias corrugated foil exit canals.

Thermal Stresses in Infinite Elastic Disks

Abstract A direct method of solving problems of thermal stress in a disk of finite thickness and infinite radius is discussed in this paper.