Disk In Fluid Research Articles

Mean-field theory of turbulence from the variational principle and its application to the rotation of a thin fluid disk

Mean-field theory of turbulence from the variational principle and its application to the rotation of a thin fluid disk

Formation of a regular sequence of vortex loops around a rotating disk in stratified fluid

A regular system of vortex loops in annular flow behind the edge of a disk of 6–11 cm in diameter rotating in stratified fluid is first visualized using the shadowgraph techniques (classical method with the Foucault knife and the “filament-in-focus” method). Clearly outlined vortex loops are observed in strongly and weakly stratified fluids over a wide range of the angular disk rotation velocities. The dimensions of the vortex flow region depend on the stratification (buoyancy period), the angular velocity of rotation, and the disk diameter. Extended lengthy filaments which form spiral and irregular patterns inside the loops are clearly expressed in the fine flow structure. The filaments connect the neighboring loops. The trajectories of motion of the upper edges of the loops, i.e., the sources of short internal waves, are traced.

Theoretical and experimental study of magneto-rheological fluid disc brake

A Magneto-Rheological (MR) fluid disc brake is a device that transmits a torque by the shear force of MR fluid.The fluid is inserted between the rotating and fixed discs and a magnetic field is imposed on the fluid. In this paper, a complete test rig for an MR fluid disc brake is introduced. Experiments are conducted to measure the braking torque and speed of shaft during braking process and the results are presented at different voltage input to the brake. Also theoretical analysis for both MR brake and the mechanical system is developed and is solved numerically using finite difference method and Matlab software. Effect of current input to the MR brake, viscosity of fluid and design parameters is taken into consideration. A validation of the theoretical results with CFD model is introduced. The experimental results are performed and both angular velocity and the braking torque are obtained as responses during the braking process. A comparison between braking torques obtained from theoretical and experimental work shows agreement when voltage is 2V at speed of 150rpm and also agreement when voltage is 2 and 3V at speed of 250rpm.

A WEAKLY NONLINEAR MODEL FOR THE DAMPING OF RESONANTLY FORCED DENSITY WAVES IN DENSE PLANETARY RINGS

ABSTRACT In this paper, we address the stability of resonantly forced density waves in dense planetary rings. Goldreich & Tremaine have already argued that density waves might be unstable, depending on the relationship between the ring’s viscosity and the surface mass density. In the recent paper Schmidt et al., we have pointed out that when—within a fluid description of the ring dynamics—the criterion for viscous overstability is satisfied, forced spiral density waves become unstable as well. In this case, linear theory fails to describe the damping, but nonlinearity of the underlying equations guarantees a finite amplitude and eventually a damping of the wave. We apply the multiple scale formalism to derive a weakly nonlinear damping relation from a hydrodynamical model. This relation describes the resonant excitation and nonlinear viscous damping of spiral density waves in a vertically integrated fluid disk with density dependent transport coefficients. The model consistently predicts density waves to be (linearly) unstable in a ring region where the conditions for viscous overstability are met. Sufficiently far away from the Lindblad resonance, the surface mass density perturbation is predicted to saturate to a constant value due to nonlinear viscous damping. The wave’s damping lengths of the model depend on certain input parameters, such as the distance to the threshold for viscous overstability in parameter space and the ground state surface mass density.

Asymptotic Analysis to Two Nonlinear Equations in Fluid Mechanics by Homotopy Renormalisation Method

Abstract By the homotopy renormalisation method, the global approximate solutions to Falkner-Skan equation and Von Kármá’s problem of a rotating disk in an infinite viscous fluid are obtained. The homotopy renormalisation method is simple and powerful for finding global approximate solutions to nonlinear perturbed differential equations arising in mathematical physics.

Tribological Classification of Contact Lenses: From Coefficient of Friction to Sliding Work

The coefficient of friction (CoF) has been reported to correlate with clinical comfort of soft contact lenses (SCL). However, a classification in terms of a CoF is not always applicable to soft materials, such as hydrogels, due to the frequently observed nonlinearity between the lateral and the normal forces. An alternative methodology is presented to quantify the tribological characteristics of soft materials under boundary lubrication in terms of average work. Average work was derived from knowledge of the area of contact, the interfacial shear stress, and sliding distance. To illustrate the work concept, three commercially available SCL (n = 10) and rabbit corneas (n = 10) were characterized with regard to lateral force against a biomimetic mucin-coated glass disk in a tear-like fluid, by means of microtribometry. The contact area between the glass disk and the SCL was measured in situ and fitted to an elastic-foundation model of the material. On the cornea, the contact area was observed via the expulsion of a fluorescent marker from the contact region. All SCL materials had significantly (p < 0.05) different values for average work. Furthermore, the interfacial shear stress on the cornea was found to be at least an order of magnitude lower than on any of the SCL. Average work represents a single figure of merit for the lubricious properties of soft materials, such as SCL, that do not show a linear relationship between lateral and normal forces.

Application of the viscosity-expansion method to a rotating thin fluid disk bound by central gravity

Application of the viscosity-expansion method to a rotating thin fluid disk bound by central gravity

Long-Time Behavior for the Two-Dimensional Motion of a Disk in a Viscous Fluid

In this article, we study the long-time behavior of solutions of the two-dimensional fluid-rigid disk problem. The motion of the fluid is modeled by the two-dimensional Navier-Stokes equations, and the disk moves under the influence of the forces exerted by the viscous fluid. We first derive $L^p$-$L^q$ decay estimates for the linearized equations and compute the first term in the asymptotic expansion of the solutions of the linearized equations. We then apply these computations to derive time-decay estimates for the solutions to the full Navier-Stokes fluid-rigid disk system.

Exact relativistic models of perfect fluid disks in a magnetic field

Using the well-known ``displace, cut and reflect'' method we construct thin disks made of a perfect fluid in presence of a magnetic field. The models are based in a magnetic Reissner-Nordstrom metric of Einstein-Maxwell equations for a conformastatic spacetime. The influence of the magnetic field on the matter properties of the disk are analyzed. We also study the motion of charged test particles around the disks. We construct models of perfect fluid disks satisfying all the energy conditions.

Study of Flows Arising from the Torsional Oscillations of Horizontal Disk in an Inhomogeneous Fluid

The high-resolution schlieren visualization of the flow pattern in the stratified fluid near torsional oscillated horizontal disk is performed. Large-scale internal waves, adjacent eddy currents and fine-structure components of the vortex is studied.

Numerical Solution for Accelerated Rotating Disk in a Viscous Fluid

The problem of a disk rotating in a viscous fluid has been investigated. The disk is accelerated with angular velocity proportional to time. Employing suitable similarity transformations the governing partial differential equations are transformed in to ordinary differential form. The resulting equations are solved numerically using SOR method and Simpson’s (1/3) rule. The results have been improved by using Richardson’s extrapolation. The effect of the non-dimensional parameter s which measures unsteadiness is observed on velocity components, skin friction coefficient and torque of the disk.

Broadside mobility of a disk in a viscous fluid near a plane wall with no-slip boundary condition

The broadside motion of a disk in a viscous fluid towards a planar wall with no-slip boundary condition is studied on the basis of the steady-state Stokes equations. It is shown that flow velocity and pressure of the fluid can be found conveniently from a superposition of elementary complex stream functions. The two amplitude functions characterizing the superposition are found from the numerical solution of a pair of integral equations for the axial and radial velocity components at the disk. The numerical procedure converges fast, provided the distance to the plane is not much smaller than the radius of the disk. For small distance the flow is well approximated by lubrication theory.

Response of a circumbinary accretion disc to black hole mass loss

We investigate the evolution of the surface density of a circumbinary accretion disc after the mass loss induced by the merger of two supermassive black holes. We first introduce an analytical model, under the assumption of a disc composed of test particles, to derive the surface density evolution of the disc following the mass loss. The model predicts the formation of sharp density peaks in the disc; the model also allows us to compute the typical timescale for the formation of these peaks. To test and validate the model, we run numerical simulations of the process using the Smoothed Particle Hydrodynamics (SPH) code PHANTOM, taking fluid effects into account. We find good agreement in the shape and position of the peaks between the model and the simulations. In a fluid disc, however, the epicyclic oscillations induced by the mass loss can dissipate, and only some of the predicted peaks form in the simulation. To quantify how fast this dissipation proceeds, we introduce an appropriate parameter, and we show that it is effective in explaining the differences between the analytical, collisionless model and a real fluid disc.

The Effect of Disk Location, Shaft Length and Imbalance on Fluid Induced Rotor Vibrations

Rotors mounted on fluid film bearings can experience fluid induced instabilities that manifest as severe vibration. The setup of the fluid film bearing greatly contributes to the dynamic characteristics of these systems. In this paper, the influence on fluid induced rotor vibrations of shaft length, disk axial separation distance from the fluid film bearing and disk imbalance are studied through experiment and simulation. The basic components of the rotor are the shaft, a disk with variable imbalance, a sleeve and a fluid film bearing. Experimental shaft vibration data was collected and analyzed by computer. The finite element method was used to model the shaft, disk and bearings. The stiffness and damping maps and Campbell diagrams for the system were produced and used to analyze stability. Results indicate that increasing the distance of the disk from the fluid film bearing increases the instability threshold, whereas increasing the shaft length decreases the instability threshold. Changes in shaft length and disk location do not result in noticeable effects on the instability amplitude. Although the mass imbalance produces vibrations in same order as the frequency of shaft rotation, it also affects the oil instability at a frequency of about half that of the shaft rotation.

An Analytical Modeling of the Central Core Flow in a Rotor-Stator System With Several Preswirl Conditions

In the most part of an enclosed rotor-stator system with separated boundary layers, the flow structure is characterized by a central core rotating as a solid body with a constant core-swirl ratio. This behavior is not always observed in an isolated rotor-stator cavity, i.e., without any centripetal or centrifugal throughflow, opened to the atmosphere at the periphery: Recent works have brought to evidence an increasing level of the core-swirl ratio from the periphery to the axis, as in the case of a rotor-stator with superposed centripetal flow. The present work is based on an asymptotical approach in order to provide a better understanding of this process. Assuming that the boundary layers behave as on a single rotating disk in a stationary fluid on the rotor side, and on a stationary disk in a rotating fluid on the stator side, new analytical relations are obtained for the core-swirl ratio, the static pressure on the stator, and also the total pressure at midheight of the cavity. An experimental study is performed: Detailed measurements provide data for several values of the significant dimensionless parameters: 1.14≤10−6×Re≤1.96, 0.05≤G≤0.10, and 0.07≤104×Ek≤2.65. The analysis of the results shows a good agreement between the theoretical solution and the experimental results. The analytical model can be used to provide a better understanding of the flow features. In addition, radial distributions of both core-swirl ratio, dimensionless static pressure on the stator, as well as dimensionless total pressure at midheight of the cavity, which are of interest to the designers, can be computed with an acceptable accuracy knowing the levels of the preswirl coefficient Kp and the solid body rotation swirl coefficient KB.

Tangential oscillations of a circular disk in a viscous stratified fluid

A complete solution is obtained for the wave field generated by the time-harmonic edgewise oscillations of a horizontal circular disk in an incompressible stratified viscous fluid. The linearized equations of viscous internal waves and the no-slip condition on the rigid disk are used to derive sets of dual integral equations for the fluid velocity and vorticity. The dual integral equations are solved by analytic reduction to sets of linear algebraic equations. Asymptotic results confirm that this edgewise motion no longer excites waves in the small-viscosity limit. Broadside oscillations and the effect of density diffusion are also considered.

Tangential oscillations of a circular disk in a viscous stratified fluid

A complete solution is obtained for the wave field generated by the time-harmonic edgewise oscillations of a horizontal circular disk in an incompressible stratified viscous fluid. The linearized equations of viscous internal waves and the no-slip condition on the rigid disk are used to derive sets of dual integral equations for the fluid velocity and vorticity. The dual integral equations are solved by analytic reduction to sets of linear algebraic equations. Asymptotic results confirm that this edgewise motion no longer excites waves in the small-viscosity limit. Broadside oscillations and the effect of density diffusion are also considered.

Slow pressure modes in thin accretion discs

@ABSTRACT Thin accretion discs around massive compact objects can support slow pressure modesof oscillations in the linear regime that have azimuthal wavenumber m = 1. We con-sider finite, flat discs composed of barotropic fluid for various surface density profilesand demonstrate–through WKB analysis and numerical solution of the eigenvalueproblem–that these modes are stable and have spatial scales comparable to the size ofthe disc. We show that the eigenvalue equation can be mapped to a Schr¨odinger-likeequation. Analysisofthis equationshows that all eigenmodes havediscretespectra. Wefind that all the models we have considered support negative frequency eigenmodes;however, the positive eigenfrequency modes are only present in power law discs, albeitfor physically uninteresting values of the power law index β and barotropic index γ.Key words: accretion discs; hydrodynamics; waves; methods: analytical 1 INTRODUCTIONLow-mass discs orbiting massive compact bodies are a fea-ture of many astronomical systems. When the dynamics ofa disc is dominated by the Newtonian gravitational force ofthe central body, the disc may be considered nearly Kep-lerian. In a purely Keplerian potential eccentric orbits donot precess because the orbital frequency is equal to theepicyclic frequency. In a nearly Keplerian disc there is asmall difference between the orbital and epicyclic frequen-cies. This could be due to the self-gravity of the disc, ther-mal pressure in a gas disc, and random motions in a col-lisionless disc. This difference in frequencies manifests as aprecession of eccentric orbits at rates that are small com-pared to the orbital and epicyclic frequencies. Then the discmay be able to support large-scale, slow, lopsided modes(Kato 1983; Sridhar, Syer & Touma 1999; Lee & Goodman1999; Sridhar & Touma 1999). In the linear regime, thesemodes have azimuthal frequency, m = 1, whose first sys-tematic investigation is due to Tremaine (2001). He stud-ied slow modes in various types of discs (fluid, collision-less and softened gravity), with the focus largely on the ef-fect of the self-gravity of the disc. In particular, a WKBanalysis was used to show that the fluid disc can supportlarge-scale slow modes when the Mach number, M, is muchlarger than the Toomre Q parameter (both parameters aredefined in § 2). The assumption behind this analysis is thatthe self-gravity of the disc dominates fluid pressure. How-ever, such is not the case for thin accretion discs aroundwhite dwarfs and neutron stars. Indeed, for a disc around awhite dwarf (Frank, King & Raine 2002), we can estimateM ∼ 50 and Q ∼ 10

Rotationally symmetric flow over a rotating disk

The rotationally symmetric flow over a rotating disk in an incompressible viscous fluid is analyzed by a new method when the fluid at infinity is in a state of rigid rotation (in the same or in the opposite sense) about the same axis as that of the disk. Asymptotic expansions for the velocity field over the entire flow field are obtained for the general class of one-parameter rotationally symmetric flows. This method is further extended to the case when a uniform suction or injection is assumed at the rotating disk. Fluid motion induced by oscillatory suction of small amplitude at the rotating disk is also discussed. An initial-value analysis reveals that resonance is possible only when the angular velocity of the rotating fluid is greater than that of the rotating disk.

Hydroxyapatite/titania nanocomposites derived by combining high-energy ball milling with spark plasma sintering processes

Hydroxyapatite-reinforced nanocomposites with titania nanocrystals addition are prepared by a homogeneous mixing of hydroxyapatite nanoparticles and titania nanocrystals based on high-energy ball milling and spark plasma sintering processes. The microstructural and mechanical properties of the HA/titania composites are studied by X-ray diffractometry analysis, Raman spectrometry, and scanning electron microscopy. The hardness and Young's modulus of the composites are characterized by a nanoindenter and they show that the incorporation of the titania nanocrystals improves the mechanical properties of the composites obviously and the improvement should be ascribed to the main solitary effect of the ceramic as additives as well as a denser composites due to combining high-energy ball milling with spark plasma sintering techniques. The bioactivity of the HA/titania composites is evaluated by immersing the spark plasma sintering (SPS) compact disk in the simulated body fluid (SBF) and the results indicate that the bioactivity of the composites is related to the addition of titania by inducing apatite nucleation on the sample's surface after being immersed in SBF.