Rough Disk Research Articles

Orthogonal versus covariant Lyapunov vectors for rough hard disc systems

The Oseledec splitting of the tangent space into covariant subspaces for a hyperbolic dynamical system is numerically accessible by computing the full set of covariant Lyapunov vectors. In this paper, the covariant Lyapunov vectors, the orthogonal Gram–Schmidt vectors and the corresponding local (time-dependent) Lyapunov exponents are analyzed for a planar system of rough hard discs (RHDS). These results are compared to respective results for a smooth-hard-disc system (SHDS). We find that the rotation of the discs deeply affects the Oseledec splitting and the structure of the tangent space. For both the smooth and rough hard discs, the stable, unstable and central manifolds are transverse to each other, although the minimal angle between the unstable and stable manifolds of the RHDS typically is very small. Both systems are hyperbolic. However, the central manifold is precisely orthogonal to the rest of the tangent space only for the smooth-particle case and not for the rough discs. We also demonstrate that the rotations destroy the Hamiltonian (symplectic) character for the RHDS.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.

Heat and mass transfer of the flow due to a rotating rough and porous disk

The present paper is an extension of the traditional Von Karman swirling flow problem where the rotating disk surface admits partial slip in the presence of a uniform suction or injection. Using the Karman similarity transformations, the partial differential equations governing the heat and flow motion are transformed into a system of ordinary differential equations which are then treated numerically. The existence of numerical results are verified through a theoretical analysis and also they are supported by asymptotic expressions in the case of large suction limit. Effects of wall roughness and temperature jump on the heat and mass transfer are further discussed.

Water wave radiation by a submerged rough disc

A thin circular body is submerged below the free surface of deep water. The problem is reduced to a hypersingular integral equation over the boundary of the body. Using a perturbation method, the problem is then reformulated by a sequence of simpler hypersingular equations over a flat disc making it well suited for an efficient previously used solution method. The first order approximation is computed and the hydrodynamic force due to heaving radiation motion are presented in terms of the added mass and damping coefficients for a polynomial cap and for a rough disc, modelled by a superposition of sinusoidal surfaces defined by randomly generated parameters. The solution exhibits larger maxima associated with smaller volume of submergence of the body. A slight shift of the damping coefficient maxima to lower frequencies is noticed for the caps. Rough discs with similar statistical properties exhibit different behaviours. Thus, it is the exact specific form of the rough disc that dictates the hydrodynamic force.

Electromagnetically induced transparency with variable fluctuation time of photon number of pseudothermal probe light

We investigate the electromagnetically induced transparency (EIT) in Rb atoms interacting with a pseudothermal probe light exhibiting a Bose–Einstein photon number distribution. The EIT spectrum in the 5S1/2−5P1/2 transition of the Λ-type system of the 87Rb atom was observed using a coupling laser and the pseudothermal probe light generated by passing the probe laser though a rotating rough disk. From the EIT spectrum with pseudothermal light, we experimentally demonstrated that the fluctuation of the number of photons was the same as the superposition of frequency noise of the probe light. The coherence time of the pseudothermal probe light was estimated from the measured EIT spectra. We confirmed that the experimental results as a function of the rotating speed of the rough disk were in good agreement with the numerically calculated results.

Spinning rough disc moving in a rarefied medium

We study the Magnus effect: deflection of the trajectory of a spinning body moving in a gas. It is well known that in rarefied gases, the inverse Magnus effect takes place, which means that the transversal component of the force acting on the body has opposite signs in sparse and relatively dense gases. The existing works derive the inverse effect from non-elastic interaction of gas particles with the body. We propose another (complementary) mechanism of creating the transversal force owing to multiple collisions of particles in cavities of the body surface. We limit ourselves to the two-dimensional case of a rough disc moving through a zero-temperature medium on the plane, where reflections of the particles from the body are elastic and mutual interaction of the particles is neglected. We represent the force acting on the disc and the moment of this force as functionals depending on ‘shape of the roughness’, and determine the set of all admissible forces. The disc trajectory is determined for several simple cases. The study is made by means of billiard theory, Monge–Kantorovich optimal mass transport and by numerical methods.

Force acting on a rough disk spinning in a flow of noninteracting particles

Force acting on a rough disk spinning in a flow of noninteracting particles

Microscale disk-induced gas displacement with and without slip

Displacements of gas flows, both with and without slip, are described for rotation-induced flows in a C-shaped fluid chamber passage formed between a rotating disk and a stationary surface, with a height h of 13.3 µm. Included are accommodation coefficients for the stationary smooth wall, smooth disk surface, medium rough disk surface rough disk surface. Flow rate and pressure rise magnitudes deduced using these accommodation coefficients, and simplified forms of the Navier–Stokes equations, are consistent with experimental data over rotational speeds of 200–1200 rpm, pressure increases of 0–400 Pa, net flow rates of 0–100 µl min−1, Knudsen numbers of 0.0056 and 0.0158, average roughness magnitudes of 0.01– 1.1 µm and working fluids of air and helium. All situations investigated consider a large roughness size compared to the molecular mean free path. For a particular normalized fluid chamber pressure rise P*, the normalized volumetric flow rate Q* with slip is generally less than the value with no slip. Lower P* magnitudes are also generally present at a particular value of Q*, in general, as the working fluid is changed from air to helium and the Knudsen number increases. The slopes of dimensional pressure variations with rotational speed decrease as disk surface roughness levels increase, as values of the accommodation coefficient decrease and, thus, as near-wall slip velocity magnitudes increase.

Experimental and theoretical evaluation of friction at contacting magnetic storage slider–disk interfaces

To understand better the friction force and wear processes at contacting slider–disk interfaces, we have developed an experimental method for measuring and a theoretical method for calculating the friction force. For this study, a slider with a 1500 μm2 contact pad located at the recording head is burnished against a relatively rough disk (∼12 A rms), which ensures smooth sliding. In the experimental method, the friction force is measured as the disk is spun-down to bring the slider–disk interface into an increasing degree of contact. A modified air bearing code is used to determine the experimental normal contact force for each friction measurement. In the theoretical method, the friction force and other relevant interfacial forces are calculated using an improved sub-boundary lubrication (ISBL) rough surface model. The friction force calculation in this model is based on the force needed to induce yielding of the individual disk asperities contacting the flat surface of the contact pad without any assumption of the coefficient of friction. Good agreement is found between the measured and theoretical friction vs. normal contact force curves, indicating that the model is capturing the essential origins of friction at this interface. The model also provides valuable insights into how wear particles may be generated at this contacting slider–disk interface.

Electrostatic microactuator for HDD tracking servo: compatibility with contact recording

Using an electrostatic micro-actuator for moving the slider body relative to suspension is a promising technique for achieving the extremely high track densities expected for future disk drives. To achieve, however, the small magnetic spacing necessary for these high recording densities, the recording head may have run in continuous contact with the disk surface by the time a micro-actuator technology needs to be implemented. So, a major concern is that the contact friction forces will cause an unacceptable amount of off-track motion due the flexible micro-actuator. This paper shows how minimal off-track motion of a contact slider with a microactuator can be achieved by using a suitably rough disk and a damping fluid in the micro-actuator.

Self-assembly behaviour of amphiphilic diblock copolymer in selective solvents studied by synchrotron small-angle x-ray scattering

The aggregation behaviour of styrene-vinyl benzoic acid (PSm-b-PVBAn) amphiphilic diblock copolymers in selective solvents with different m and n was investigated by synchrotron small-angle x-ray scattering (SAXS). We have carried out a detailed analysis of scattering intensity, dimension, shape and microstructure of the diblock copolymers of narrow distribution in water, methanol, ethanol and isopropanol selective solvents, respectively. We have found that the aggregation behaviour of the copolymer depends on the nature of the solvent and the micelle forms flat disc objects with the ratio of radius ω=0.4. The average radius gyration Rg of the copolymer decreases as solvents change from isopropanol to ethanol and to methanol and increases with increasing pH in aqueous solution, but decreases with the addition of CoCl2 in ethanol solvent. The scattering intensity of diblock copolymer micelle follows I(h)∝h-α in different selective solutions, suggesting that the PSm-b-PVBAn coils have self-similar structure behaviour or a fractal structure in the selective solvents. All of these revealed that the aggregation behaviour of the diblock copolymer changes dramatically with experimental condition in the selective solvent. The increase of mass fractal dimension (Dm) from 2.12 to 2.47 indicates that the copolymer chain changes from a swollen coil to a rather compact disc in the course of changing solvents, decreasing surface fractal dimension (Ds) from 2.98 to 2.58 indicates that the copolymer micelle change from a rather rough surface to a smooth form in the course of increasing pH in aqueous solutions and increasing Dm and Ds from 2.29 to 2.35 and 2.70 to 2.90, respectively, indicates the shrinkage of copolymer micelle to a rather compact and rough disc form by adding CoCl2 in ethanol solvents.

Design Consideration of Contact/Near-Contact Sliders Based on a Rough Surface Contact Model

We numerically investigated contact characteristics of a contact pad with a rough disk surface and the possibility of contact/near-contact sliders, using a single-degree-of-freedom (1-DOF) slider and a random wavy surface model with random roughness. Contact characteristics of a contact pad are numerically calculated based on a modified Greenwood-Williamson model, considering the bulk deformation of the surface due to all other asperity contact forces. It was found that contact stiffness and other characteristics are mainly determined by asperity contact, to the extent that the contact pad penetrates into the upper standard deviation of asperity peak height. However, the contact stiffness tends to approach a constant value as the pad penetrates into the average asperity height because the bulk deformation becomes predominant. From the numerical simulations of a 1-DOF air bearing slider model in contact and near-contact regimes over a random wavy surface with random roughness, a typical example of design condition of disk surface waviness in terms of the tracking ability and wear durability are shown, and the possibility and difficulty of a contact/near-contact slider is discussed. Finally, we analyzed meniscus effects on the contact characteristics and found a hysteresis process of the touch down and take off of a slider due to the meniscus force.

The Role of Induced Air Shear on the Development of a Conducting Fluid Film over a Rough Spinning Disk in the Presence of a Transverse Magnetic Field

The effect of air shear on the development of a thin conducting fluid film on a rough rotating disk has been analysed. A uniform magnetic field acts parallel to the axis of the rough rotating disk. The numerical solution of the governing equations of motion using the initial and boundary conditions have been obtained by a finite difference method. The effect of air shear on the temporal evolution of the free surface of the fluid and on the retention of the liquid lubricant has been analysed.

Effects of Surface Micro-Geometry On the Lift-Off Speed of an EHL Contact

With decreasing speed the film thickness in an EHL contact decreases. Below a certain speed asperity contact will take place and gradually the contact enters the mixed lubrication regime. Vice-versa, with increasing speed beyond a certain speed the film thickness has reached a level where asperity contacts have become so rare that the contact will be in the full film regime. It seems only logical to expect that the speed at which the first (or last) significant asperity interactions start to take place is influenced by the micro-geometry of the surfaces. In experiments performed on a two-disk rig under conditions of pure rolling, using one very smooth and one rough disk it was indeed observed that the “lift-off” speed defined as the speed above which full film lubrication prevails, differed significantly for surfaces with a different micro-geometry. The test results can be seen as ranking for the surface micro-geometries in terms of their film generating capability. In this paper the question is addressed if a ranking as observed in the tests can be predicted in advance, using the load conditions and the measured surface micro-geometry as input, without having to resort to full-scale numerical simulations of any sort. Based on the amplitude reduction formula proposed by Venner et al. (2000), for roughness in contacts under pure rolling a model is constructed that, given the load condition and a measured surface micro-geometry, determines the deformed micro-geometry and subsequently a measure of “probability of contact.” For a given contact this measure can be plotted as a function of speed to obtain a theoretical “lift-off” curve. For the different types of surface micro-geometries used in the tests such a curve is compared with the experimental results, showing a promising agreement in ranking. Presented as a Society of Tribologists and Lubrication Engineers Paper at the STLE/ASME Tribology Conference in San Francisco, CA October 21–24, 2001

Flow of thin liquid film over a rough rotating disk in the presence of a transverse magnetic field

The development of a flow of a viscous conducting fluid over a rough spinning disk in the presence of a transverse magnetic field has been analysed for different patterns of surface roughness of the disk and different initial distributions of the height of the liquid lubricant. The numerical solution of the governing equation of motion subject to initial and boundary conditions has been obtained by a finite-difference method. The temporal evolution of the free surface of the fluid and the rate of retention of the liquid lubricant on the spinning disk have been obtained for different values of the two parameters M , the Hartmann number and Nratio, the ratio of the surface tension effect to the centrifugation effect. In the absence of the magnetic field, the results have been observed to agree with those of [6]. It has been observed that the effect of surface roughness is to enhance the relative volume of the fluid retained on the spinning disk and this is further enhanced by the presence of the magnetic field.

An Inverse Approach to the Validation of Pressure Predictions in Rough Elastohydrodynamic Contacts

A simple experimental technique is developed that allows the pressures and stresses predicted by EHL analyses to be checked. A soft, rough steel disc is run, under controlled conditions, against a harder, smooth counterface. The pressures generated cause the soft disc to plastically deform. Once deformation ceases the residual profile may be measured and used as input to an EHL solver. The calculated pressures are then used to determine the stress distributions as the rough surface passes through the conjunction. After allowing for the build up of residual stress the maximum von Mises’ stress should be equal to the yield strength of the disc. This provides an accurate, quantitative check on the theoretical values.

Contact interaction of cylindrical bodies with allowance for the surface roughness factors

The problem of the elastic contact interaction between a rough disk and a rough plane with a round cut-out is solved with allowance for the microgeometry of their surfaces. This makes it possible to clarify the effect of the main parameters of the problem, on the stress state in the examined joint. A comparison of results from an analysis of the stress state in the contact region for various combinations of the elastic characteristics of interacting smooth bodies and results of well-known studies confirms the high effectiveness of the approach proposed.

Effect of Surface Roughness and Composition on Costochondral Chondrocytes is Dependent on Cell Maturation State

During endochondral bone formatin, as occurs in fracture healing, chondrocytes are one of the first cells to see an implant surface. We tested the hypothesis that chemical composition and surface roughness affect chondrocyte differentiation matrix synthesis, and local factor production and that the nature of the response is dependent on the state of maturation of the cells. To do this, we harvested rat growth zone and resting zone chondrocytes and examined their tesponce to smooth and rough disk surfaces manufactured from either commercially pure titanium or titanium alloy. profilometry, scanning electron microscopy, Auger spectroscopy, and Fourier transform infrared spectroscopy were used to characterize the surfaces. Average roughness values were 0.22 μm for smooth titanium surfaces, 0.23 μm for smooth titanium alloy surfaces, 4.24 μm for rough titanium surfaces. and 3.20 μm for rough titanium alloy surface. Cells were grown on the different disk surfaces until the cultures had reached confluence on plastic. The effect of the surfaces was determined by assaying cell number and [3H]thymidine incorporation as measurs of cell proliferatin, cell layer and cell alkaline phosphatase specific activity as markers of differentiation, and collagen production and [35S]sulfate incorporation as indicators of extracellular matrix production. In addition, the synthesis of prostaglandin E2 and transforming growth factor-β were examined to measure changes in local factor synthesis. In growth Zone and resting Zone cultures, cell number and [3H]thymidine incorporation were decreased on rough surfaces: however, this effect was greater on commercially pure titanium surfaces. Cell layer and cell alkaline phosphatase specific activity were decreased in resting zone cells grown on rough surfaces. Cell alkaline phosphatase specific activity in growth zone cells was decreased on rough surfaces, whereas cell layer alkaline phosphatase specific activity was increased only in growth zone cells growth on rough commercially pure titanium surfaces, Resting Zone cell collagen production was decreased only on rough commercially pure titanium, whereas in growth zone cells, collagen production was increased. Increased prostaglandin E2 release into the media was found for growth zone and resting zone cell cultures on the disks with rough surfaces. The observed effect was greater on rough commercially pure titanium. Production of transforming growth factor-β by resting zones was similarly affected. whereas an increase in its production by growth zone cells was measured only on rough commercially pure titanium. These results indicate that surface roughness affects chondrocyte proliferation, differentiation, matrix synthesis. and local factor production and that these parameters are also affected by chemical composition. Furthermore. the nature and extent of the cell response is dependent on cell maturation. The overriding variable in response to an implant material, however, appears to be roughness of the surface.

A comprehensive kinetic meniscus model for prediction of long-term static friction

In some interfaces, the static friction force increases with an increase in the duration of stationary contact. In this study, a comprehensive kinetic meniscus model is proposed to explain this phenomenon, both for a single asperity and multiple asperity contacts at a liquid mediated interface. It is found that the static friction increases up to a certain equilibrium time after which it remains constant. The equilibrium time is dependent on the liquid film thickness, the liquid viscosity and the contact geometry. The developed model is applied to a rough textured disk and a smooth disk in the head-disk interface of a computer hard disk drive.

Effect of surface roughness and composition on costochondral chondrocytes is dependent on cell maturation state.

During endochondral bone formation, as occurs in fracture healing, chondrocytes are one of the first cells to see an implant surface. We tested the hypothesis that chemical composition and surface roughness affect chondrocyte differentiation, matrix synthesis, and local factor production and that the nature of the response is dependent on the state of maturation of the cells. To do this, we harvested rat growth zone and resting zone chondrocytes and examined their response to smooth and rough disk surfaces manufactured from either commercially pure titanium or titanium alloy. Profilometry, scanning electron microscopy, Auger spectroscopy, and Fourier transform infrared spectroscopy were used to characterize the surfaces. Average roughness values were 0.22 microm for smooth titanium surfaces, 0.23 microm for smooth titanium alloy surfaces, 4.24 microm for rough titanium surfaces, and 3.20 microm for rough titanium alloy surfaces. Cells were grown on the different disk surfaces until the cultures had reached confluence on plastic. The effect of the surfaces was determined by assaying cell number and [3H]thymidine incorporation as measures of cell proliferation, cell layer and cell alkaline phosphatase specific activity as markers of differentiation, and collagen production and [35S]sulfate incorporation as indicators of extracellular matrix production. In addition, the synthesis of prostaglandin E2 and transforming growth factor-beta were examined to measure changes in local factor synthesis. In growth zone and resting zone cultures, cell number and [3H]thymidine incorporation were decreased on rough surfaces; however, this effect was greater on commercially pure titanium surfaces. Cell layer and cell alkaline phosphatase specific activity were decreased in resting zone cells grown on rough surfaces. Cell alkaline phosphatase specific activity in growth zone cells was decreased on rough surfaces, whereas cell layer alkaline phosphatase specific activity was increased only in growth zone cells grown on rough commercially pure titanium surfaces. Resting zone cell collagen production was decreased only on rough commercially pure titanium, whereas in growth zone cells, collagen production was increased. Increased prostaglandin E2 release into the media was found for growth zone and resting zone cell cultures on the disks with rough surfaces. The observed effect was greater on rough commercially pure titanium. Production of transforming growth factor-beta by resting zones was similarly affected, whereas an increase in its production by growth zone cells was measured only on rough commercially pure titanium. These results indicate that surface roughness affects chondrocyte proliferation, differentiation, matrix synthesis, and local factor production and that these parameters are also affected by chemical composition. Furthermore, the nature and extent of the cell response is dependent on cell maturation. The overriding variable in response to an implant material, however, appears to be roughness of the surface.

Kinetic Meniscus Model for Prediction of Rest Stiction

The time dependence of static friction is an important aspect of tribological studies whereby the static friction increases with an increase in the time of stationary contact (rest static friction or rest stiction). This effect is commonly observed in computer disk drives especially at high humidities, in which the static friction increases rapidly with an increase in rest time between some head slider and the disk surfaces. For the first time, a comprehensive kinetic meniscus model is proposed to explain this phenomenon, both for a single asperity and multiple asperity contacts at a liquid mediated interface. It is found that the static friction increases up to a certain equilibrium time with a power law relationship after which it remains constant. The equilibrium time is dependent on the liquid film thickness, the liquid viscosity and the contact geometry. The model developed is applied to a rough textured disk, a polished disk and a super smooth disk. To study the effect of the scan size on the meniscus force, the static friction is calculated by using roughness parameters obtained from measurements at different scan sizes.