Coaxial Disks Research Articles

Added mass of a pair of disks at small separation

Inviscid irrotational flow around a pair of coaxial disks is considered in the limit in which the distance 2h between the disks is small compared to their radius a. The disks have zero thickness and accelerate away from one another along their common axis. The added mass M of each accelerating disk is increased by the presence of the other disk. Analytic predictions are obtained when h/a ≪ 1, with M ~ πa/(8h)-ln(h/a)/2+0.77875+. . . The term O(a/h) can be obtained by means of an inviscid analysis of approximately unidirectional flow within the gap between the disks, but the correction terms have not been reported previously. The irrotational flow problem satisfies Neumann boundary conditions on the surface of the disks, but is otherwise analogous to the Dirichlet problem of the capacitance of a pair of charged disks, which has been the subject of much study and controversy.

Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field

Numerical analyses have been carried out for magnetohydrodynamic flow between a rotating and a stationary disk, whose radii are sufficiently large in comparison with the gap between the two parallel coaxial disks. The gap is filled with an electric conducting fluid and a uniform axial magnetic field is imposed. The magnetic Prandtl number is assumed to be so small that the influence of the induced magnetic field is neglected. The flow depends on both the rotational Reynolds number and the Hartmann number as well as the wall conductance ratios of upper and lower disks. As the Reynolds number increases, the core region of rigid body rotation having slight axial component of velocity is observed between the two boundary layers, whose thickness becomes thinner in proportional to the square root of the Reynolds number. On the other hand, as the Hartmann number increases, the Lorentz force tends to suppress the secondary flow significantly and boundary layer thickness of the azimuthal component of velocity is proportional to the inverse of the Hartmann number. The derived boundary condition for the normal component of electric current density at the interface allows us to obtain similarity solutions for various combinations of each wall conductance ratio and its influence on the flow is quite significant.

Mutual inductance and magnetic force calculations for coaxial bitter disk coils (Pancakes)

Recently Y. Ren and J.T. Conway calculated the mutual inductance and the magnetic force between an ordinary coil and a bitter coil or between two bitter coils. The bitter coil is the coil with inverse radial current density. In this study, the authors calculate the mutual inductance and the magnetic force between two disk coils (pancakes) with inverse radial current density. This coil configuration with proposed current density seems to be similar to bitter coils. Both calculations give the semi-analytical expressions either for mutual inductance or for the magnetic force. Also they derived the self-inductance for the disk coil with radial current density which is obtained in closed form. The results of this method are compared by those obtained by the modified filament method for the presented configuration.

Boiling Heat Transfer Performance in a Spiraling Radial Inflow Microchannel Cold Plate

ABSTRACTThis study presents an experimental exploration of flow boiling heat transfer in a spiraling radial inflow microchannel heat sink. The effect of surface wettability, fluid subcooling, and mass fluxes are considered. The design of the heat sink provides an inward radial swirl flow between parallel, coaxial disks that form a microchannel of 300 microns. The channel is heated on one side, while the opposite side is essentially adiabatic to simulate a heat sink scenario for electronics cooling. To explore the effects of varying surface wetting, experiments were conducted with two different heated surfaces. One was a clean, machined copper surface and the other was a surface coated with zinc oxide nanostructures that are superhydrophilic. During boiling, increased wettability resulted in quicker rewetting and smaller bubble departure diameter, as indicated by reduced temperature oscillations during boiling, and achieving higher maximum heat flux without dryout. The highest heat transfer coefficients were seen in fully developed boiling with low subcooling levels as a result of heat transfer being dominated by nucleate boiling. The highest heat fluxes achieved were during partial subcooled flow boiling at 300 W/cm2 with an average surface temperature of 134° Celsius. Recommendations for electronics cooling applications are also discussed.

Analysis of inductance calculation of coaxial circular filamentary coils, thin‐wall solenoids, and disk coils using inverse hyperbolic functions

This study presents a method using inverse hyperbolic functions to calculate the mutual and self-inductance inductance of coaxial circular filamentary coils, thin-wall solenoids, disk coils, and coils with rectangular cross-section in air. A generalised expression is given for any combination of these coils, which only requires numerical integration with one variable. Singularities are fully analysed and methods to resolve the singularities are presented in this study. When calculating the self-inductance of a disk coil or a thin-wall solenoid, the proposed method involves an improper integral, and the accuracy of the proposed method is limited unless a quadrature rule applicable to improper integrals is applied. For other situations, results by the proposed method were compared with those in the literature, and they were in excellent agreement.

Verification of the standard model of shear stress transport and its modified version that takes into account the streamline curvature and estimation of the applicability of the Menter combined boundary conditions in calculating the ultralow profile drag for an optimally configured cylinder–coaxial disk arrangement

A modification of the popular model of shear stress transport aimed at calculating the separation flow of an incompressible viscous liquid is justified. The modification eliminates the nonphysical pumping of the vortex viscosity in the cores of large-scale vortices. It has been verified with regard to the influence of the streamline curvature on the vortex viscosity by introducing a reciprocal linear function of the turbulent Richardson number with the Isaev–Kharchenko–Usachov constant equal to 0.02.Verification is based on solving the test problem an axisymmetric steady flow about a disk–cylinder tandem with an optimally configured nose, which has an ultralow profile drag for a Reynolds number of 5 × 105. It has been shown that the Menter combined boundary conditions are valid if y+y of the wall does not exceed two.

Stability of a liquid bridge under vibration.

We examine the stability of a vertical liquid bridge between two vertically vibrating, coaxial disks. Assuming that the vibration amplitude and period are much smaller than the mean distance between the disks and the global timescale, respectively, we employ the method of multiple scales to derive a set of asymptotic equations. The set is then used to examine the stability of a bridge of an almost cylindrical shape. It is shown that, if acting alone, gravity is a destabilizing influence, whereas vibration can weaken it or even eliminate altogether. Thus, counter-intuitively, vibration can stabilize an otherwise unstable capillary structure.

Vector potentials for the gravitational interaction of extended bodies and laminas with analytical solutions for two disks

The mutual gravitational potential and the force and torque components between two solid bodies are formulated in terms of vector potentials. If one of the bodies is a flat lamina of arbitrary shape and uniform surface density, then the potential and the force and torque components in Cartesian coordinates are expressed in terms of line integrals. For a uniform circular disk, the various potentials are given explicitly in terms of elliptic integrals, and these are applied to a second homogeneous disk of arbitrary orientation and position relative to the first. Simplified formulas are given for disks with parallel axes. Analytical solutions in terms of elliptic integrals are given for the mutual force and potential for coaxial disks and coplanar disks. Numerical results are given for two disks for a selection of relative orientations and positions.

The effects of ambient impurities on the surface tension

A liquid bridge is a liquid column held captive between two coaxial and parallel solid disks. It is an excellent test bench where measuring the surface tension. In this paper, we used this fluid configuration to examine experimentally the effects of ambient impurities on the surface tension over time. For this purpose, the liquid bridge equilibrium shape was analyzed when the liquid bridge was surrounded by three environments: the uncontrolled ambient, and both air and argon encapsulated in a small glass cover. Ambient contamination produced a sharp decrease of the surface tension of ultra-pure water. The presence of an anionic surfactant in the free surface of an aqueous solution did not inhibit the action of impurities coming from the ambient. Impurities can influence the dynamical behavior of the free surface in flows dominated by the surface tension. Therefore, a careful control of that influence can be crucial in many applications of fluid mechanics.

A Combined Experimental and Numerical Modeling Study of the Deformation and Rupture of Axisymmetric Liquid Bridges under Coaxial Stretching.

The deformation and rupture of axisymmetric liquid bridges being stretched between two fully wetted coaxial disks are studied experimentally and theoretically. We numerically solve the time-dependent Navier-Stokes equations while tracking the deformation of the liquid-air interface using the arbitrary Lagrangian-Eulerian (ALE) moving mesh method to fully account for the effects of inertia and viscous forces on bridge dynamics. The effects of the stretching velocity, liquid properties, and liquid volume on the dynamics of liquid bridges are systematically investigated to provide direct experimental validation of our numerical model for stretching velocities as high as 3 m/s. The Ohnesorge number (Oh) of liquid bridges is a primary factor governing the dynamics of liquid bridge rupture, especially the dependence of the rupture distance on the stretching velocity. The rupture distance generally increases with the stretching velocity, far in excess of the static stability limit. For bridges with low Ohnesorge numbers, however, the rupture distance stay nearly constant or decreases with the stretching velocity within certain velocity windows due to the relative rupture position switching and the thread shape change. Our work provides an experimentally validated modeling approach and experimental data to help establish foundation for systematic further studies and applications of liquid bridges.

Transition to chaotic thermocapillary convection in a half zone liquid bridge

A series of fluid physics microgravity experiments with an enough long run time were performed in the “KIBO,” the Japanese Experiment Module aboard the International Space Station, to examine the transition to chaos of the thermocapillary convection in a half zone liquid bridge of silicone oil with a Prandtl number of 112. The temperature difference between the coaxial disks induced the thermocapillary-driven flow, and we experimentally demonstrated that the flow fields underwent a transition from steady flow to oscillatory flow, and finally to chaotic flow with increasing temperature difference. We obtained the surface temperature time series at the middle of the liquid bridge to quantitatively evaluate the transition process of the flow fields. By Fourier analysis, we further confirmed that the flow fields changed from a periodic, to a quasi-periodic, and finally to a chaotic state. The increasing nonlinearity with the development of the flow fields was confirmed by time-series chaos analysis. The determined Lyapunov exponent and the translation error indicated that the flow fields made transition to the chaotic field with the increasing temperature difference.

Enhanced localized plasmonic characteristics based on coaxial defective-disk and ring structures

We have illustrated the localized plasmonic properties on the terahertz region based on a coaxial disk and ring structure (DRS) and a defective-disk and ring structure (DDRS). Two resonances can be observed in these coaxial structures, which arise from the anti-bonding and bonding resonance hybridization. Compared with DRS, an enhanced localized plasmonic effect at a higher-frequency resonance is observed on the DDRS, which is owing to a new strong dipole mode evoked at the edge of the wedge-shaped slice. Moreover, this higher-frequency localized plasmonic resonance of the DDRS could be further enhanced by increasing the depth of the wedge-shaped slice due to the coupling between the dipole resonance of the deeper wedge-shaped slice and the outer ring. Therefore, by using higher-order enhanced localized plasmonic characteristics, the designed DDRS is promising in sensing, spectroscopy and slow light effect.

Heat and mass transfer analysis in unsteady titanium dioxide nanofluid between two orthogonally moving porous coaxial disks: a numerical study

Study of heat and mass transfer in an unsteady hydromagnetic viscous electrically conducting incompressible water-based nanofluid (containing titanium dioxide nanoparticles) between two orthogonally moving porous coaxial disks with suction and viscous dissipation effects. A combination of iterative and a direct method is employed for solving the sparse systems of linear algebraic equations arising from the finite difference discretization of the quasi-linearized self-similar ordinary differential equations. It has been noticed that the rate of mass transfer at the disks decreases with the permeability Reynolds number; either the disks are approaching or receding. Moreover, the external magnetic field remarkably reduces the fluid velocity and therefore may be used as a controlling agent for the flow.

Instability and associated roll structure of Marangoni convection in high Prandtl number liquid bridge with large aspect ratio

This paper reports the experimental results on the instability and associated roll structures (RSs) of Marangoni convection in liquid bridges formed under the microgravity environment on the International Space Station. The geometry of interest is high aspect ratio (AR = height/diameter ≥ 1.0) liquid bridges of high Prandtl number fluids (Pr = 67 and 207) suspended between coaxial disks heated differentially. The unsteady flow field and associated RSs were revealed with the three-dimensional particle tracking velocimetry. It is found that the flow field after the onset of instability exhibits oscillations with azimuthal mode number m = 1 and associated RSs traveling in the axial direction. The RSs travel in the same direction as the surface flow (co-flow direction) for 1.00 ≤ AR ≤ 1.25 while they travel in the opposite direction (counter-flow direction) for AR ≥ 1.50, thus showing the change of traveling directions with AR. This traveling direction for AR ≥ 1.50 is reversed to the co-flow direction when the temperature difference between the disks is increased to the condition far beyond the critical one. This change of traveling directions is accompanied by the increase of the oscillation frequency. The characteristics of the RSs for AR ≥ 1.50, such as the azimuthal mode of oscillation, the dimensionless oscillation frequency, and the traveling direction, are in reasonable agreement with those of the previous sounding rocket experiment for AR = 2.50 and those of the linear stability analysis of an infinite liquid bridge.

Unsteady Flow between Two Orthogonally Moving Porous Disks

ABSTRACTThe unsteady laminar incompressible flow of a fluid between two orthogonally moving porous coaxial disks is considered numerically. A transformation is used to reduce the governing partial differential equations (PDEs) to a set of nonlinear coupled ordinary differential equations. The effects of physical parameters of interest such as the wall expansion ratio and the permeability Reynolds number on the velocity are discussed in detail.

A Semi-numerical Solution to Problem of Flow Between Two Discs

The steady flow of a viscous incompressible fluid between two coaxial discs, one rotating and other stationary with suction is analysed. We propose a semi-numerical method in which recurrence relations derived allow to generate large number of universal coefficients in small Reynolds number perturbation series of the solution function. The convergence of the series is restricted by a simple pole using some special techniques the region of validity of the series is increased. The results provided are in excellent agreement with pure numerical studies.

11003 二円板間における回転円柱周りの流れ

The present paper presents the rotating instability around a rotating cylinder between two coaxial stationary circular disks experimentally and numerically. In the experiment, the hot-wire measurements were conducted. In the simulations, the calculations were performed which are corresponding to the experimental setup. Then, FFT analysis results indicate the rotating instability of one cell rotating hi the circumferential direction near the rotating cylinder. In the stability Analysis, some vorticities around a rotating cylinder between the stationary disks provided with a complex velocity potential. This system becomes neutrally stable for the stiffness is positive. While this system becomes unstable for damping ratio is negative.

Analytical Solution to the Flow between Two Coaxial Rotating Disks Using HAM

The steady flow of a viscous, incompressible fluid between two infinite rotating coaxial disks is considered in this paper. An exact analytical solution to the problem is given by an effective analytical method called Homotopy Analysis Method (HAM).Three different cases such as one disk rotating with constant angular velocity while the other one is at rest, disks rotating in same as well as opposite sense with different angular velocities are considered and discussed for small Reynolds number.

Forced dynamics of a short viscous liquid bridge

AbstractThe dynamics of an axisymmetric liquid bridge of a fluid of density ${\it\rho}$, viscosity ${\it\mu}$ and surface tension ${\it\sigma}$ held between two co-axial disks of equal radius $e$ is studied when one disk is slowly moved with a velocity $U(t)$. The analysis is performed using a one-dimensional model for thin bridges. We consider attached boundary conditions (the contact line is fixed to the boundary of the disk), neglect gravity and limit our analysis to short bridges such that there exists a stable equilibrium shape. This equilibrium is a Delaunay curve characterized by the two geometric parameters $\ell _{0}=V_{0}/({\rm\pi}e^{3})$ and $S=\ell ({\rm\pi}e^{2})/V_{0}$, where $V_{0}$ is the volume of fluid and $\ell$ the length of the bridge. Our objective is to analyse the departure of the dynamical solution from the static Delaunay shape as a function of $\ell _{0}$, $S$, the Ohnesorge number $Oh={\it\mu}/\sqrt{{\it\rho}{\it\sigma}e}$, and the instantaneous velocity $U(t)$ and acceleration $\partial _{t}U$ (non-dimensionalized using $e$ and ${\it\sigma}/{\it\rho}$) of the disk. Using a perturbation theory for small velocity and acceleration, we show that (i) a non-homogeneous velocity field proportional to $U(t)$ and independent of $Oh$ is present within the bridge; (ii) the area correction to the equilibrium shape can be written as $U^{2}A_{i}+U\,Oh\,A_{v}+\partial _{t}UA_{a}$ where $A_{i}$, $A_{v}$ and $A_{a}$ are functions of $\ell _{0}$ and $S$ only. The characteristics of the velocity field and the shape corrections are analysed in detail. For the case of a cylinder ($S=1$), explicit expressions are derived and used to provide some insight into the break-up that the deformation would induce. The asymptotic results are validated and tested by direct numerical simulations when the velocity is constant and when it oscillates. For the constant velocity case, we demonstrate that the theory provides a very good estimate of the dynamics for a large range of parameters. However, a systematic departure is observed for very small $Oh$ due to the persistence of free eigenmodes excited during the transient. These same eigenmodes also limit the applicability of the theory to oscillating bridges with large oscillating periods. Finally, the perturbation theory is applied to the cylindrical solution of Frankel & Weihs (J. Fluid Mech., vol. 155 (1985), pp. 289–307) obtained for constant velocity $U$ when the contact lines are allowed to move. We show that it can be used to compute the correction associated with acceleration. Finally, the effect of gravity is discussed and shown to modify the equilibrium shape but not the main results obtained from the perturbation theory.

Numerical simulation of unsteady water-based nanofluid flow and heat transfer between two orthogonally moving porous coaxial disks

We present the numerical study of unsteady hydromagnetic (MHD) flow and heat transfer characteristics of a viscous incompressible electrically conducting water-based nanofluid (containing Al2O3 nanoparticles) between two orthogonally moving porous coaxial disks with suction. Different from the classical shooting methodology, we employ a combination of a direct and an iterative method (SOR with optimal relaxation parameter) for solving the sparse systems of linear algebraic equations arising from the FD discretization of the linearized self similar nonlinear ODEs. Effects of the governing parameters on the flow and heat transfer are discussed and presented through tables and graphs. The findings of the present investigation may be beneficial for electronic industry in maintaining the electronic components under effective and safe operational conditions.