Power-law Friction Research Articles

A lower bound to the survival probability and an approximate first passage time distribution for Markovian and non-Markovian dynamics in phase space

We derive a very general expression of the survival probability and the first passage time distribution for a particle executing Brownian motion in full phase space with an absorbing boundary condition at a point in the position space, which is valid irrespective of the statistical nature of the dynamics. The expression, together with the Jensen's inequality, naturally leads to a lower bound to the actual survival probability and an approximate first passage time distribution. These are expressed in terms of the position-position, velocity-velocity, and position-velocity variances. Knowledge of these variances enables one to compute a lower bound to the survival probability and consequently the first passage distribution function. As examples, we compute these for a Gaussian Markovian process and, in the case of non-Markovian process, with an exponentially decaying friction kernel and also with a power law friction kernel. Our analysis shows that the survival probability decays exponentially at the long time irrespective of the nature of the dynamics with an exponent equal to the transition state rate constant.

Capstan equation including bending rigidity and non-linear frictional behavior

Abstract More rigorous analyses have been conducted in improving the classical capstan equation by including both the rod bending rigidity and a power-law friction (in place of the Amonton’s law) into the formula. During the analyses, the power-law exponent was used as the indicator for the non-linear friction behavior, whereas both the capstan radius ratio and the incoming load incline angle were used as the parameters reflecting the bending rigidity. Our analyses indicate complex relationships those parameters have in influencing the tension ratio or tension transmitting efficiency, a thorough parametric study illustrates clearly such interconnections. Since in most cases, the bending rigidity of the tension member is non-negligible and non-linear friction is more common, the new results should be more useful in practical applications.

Power-law friction in closely packed granular materials

In order to understand the nature of friction in dense granular materials, a discrete element simulation on granular layers subjected to isobaric plain shear is performed. It is found that the friction coefficient increases as the power of the shear rate, the exponent of which does not depend on the material constants. Using a nondimensional parameter that is known as the inertial number, the power law can be cast in a generalized form so that the friction coefficients at different confining pressures collapse on the same curve. We show that the volume fraction also obeys a power law.

Experimental investigation of void fraction and flow patterns in coiled flow inverter

Coiled flow inverter is an innovative device, which works on the principle of complete flow inversion and centrifugal force. It was fabricated by introducing 90° bends in helical coils. It has two-fold advantage of intensifying the convective transfer processes (i.e. increase heat and mass transfer coefficients) and also provide increased transfer area per unit volume of space. It is very compact in size as compared to straight tubes and straight helix. Hence, it is a step towards process intensification. An experimental study was conducted for estimating the gas void fraction and flow patterns for gas–liquid two-phase flowing through this device. The coils were prepared using transparent PVC tubing of 0.005–0.015 m internal diameter with coil diameter varying from 0.08 to 0.2 m. A wide range of gas and liquid flow rates, 8.33 × 10 −5 to 0.001 and 3.33 × 10 −6 to 0.001 m 3/s respectively were investigated. Sixteen coiled flow inverters of different geometric configurations were tested. Various flow patterns observed were stratified, slug, plug, wavy and churn flow. An empirical correlation for void fraction for laminar, transition and turbulent regimes were developed. Data from 7582 gas–liquid flow experiments in coiled flow inverter have been processed for power law and composite power law friction factor correlations to flow regime determination. Separate power law correlations for laminar and turbulent flows were obtained for all flow regimes in the database and also for different flow patterns. The results were compared with Lockhart–Martinelli correlation and Hughmark's correlation.

Kramers model with a power-law friction kernel: Dispersed kinetics and dynamic disorder of biochemical reactions

Kramers' model for the rate of chemical reaction is generalized to explain the phenomena of dispersed kinetics and dynamic disorder in biochemical reactions, by incorporating the newly observed power-law friction kernel into the generalized Langevin equation for a one-dimensional reaction ordinate. This new model accounts for time scale overlap between conformational and chemical dynamics, and quantitatively describes the multi-exponential kinetics and memory effects of fluctuating rate constants, which have been revealed by recent single-molecule experiments.

Power law and composite power law friction factor correlations for laminar and turbulent gas–liquid flow in horizontal pipelines

Data from 2435 gas–liquid flow experiments in horizontal pipelines, taken from different sources, including new data for heavy oil are compiled and processed for power law and composite power law friction factor correlations. To our knowledge this is the largest database so far published in the literature; it includes the widest range of operational conditions and fluid properties for two-phase friction factor correlations. Separate power laws for laminar and turbulent flows are obtained for all flows in the database and also for flows sorted by flow pattern. Composite analytical expressions for the friction factor covering both laminar and turbulent flows are obtained by fitting the transition region between laminar and turbulent flow with logistic dose curves. Logistic dose curves lead to rational fractions of power laws which reduce to the power laws for laminar flow when the Reynolds number is low and to turbulent flow when the Reynolds number is large. The Reynolds number appropriate for gas–liquid flows in horizontal pipes is based on the mixture velocity and the liquid kinematic viscosity. The definition of the Fanning friction factor for gas–liquid flow used in this study is based on the mixture velocity and density. Error estimates for the predicted vs. measured friction factor together with standard deviation for each correlation are presented. The correlations in this study are compared with previous correlations, homogeneous models and mechanistic models most commonly used for gas–liquid flow in pipelines. Since different authors use different definitions for friction factors and Reynolds numbers, comparisons of the predicted pressure drop for each and every data point in the database are presented. Our correlations predict the pressure drop with much greater accuracy than those presented by previous authors.

Densification Forming of Alumina Powder—Effects of Power Law Creep and Friction

High temperature forming processes of alumina powder compacts were analyzed by using constitutive equations which are capable of predicting densification and grain growth under diffusional creep and power law creep. Experimental results for alumina powder compacts were compared with finite element calculations by using the constitutive equations. The effects of friction between alumina powder compact and punches during sinter forging of alumina powder compacts were also investigated. Densification mechanism maps of alumina powder, which can be used for the optimization of various process variables, were constructed under hot pressing and general states of stresses.

Diffusion on two-dimensional percolation clusters: Influence of cluster anisotropy.

This paper investigates the influence of geometric anisotropy of two-dimensional percolation clusters on diffusion. Monte Carlo simulation indicates that the ensemble of all random walks starting from any given origin is typically anisotropic as determined by the local geometry of the cluster. Single-particle (or tracer) diffusion is described by the position-correlation matrix (PCM). A mean-field equation is developed for the PCM of the random walker in terms of the moment of inertia tensor of the cluster as a function of the chemical distance from the origin. In contrast, the ensemble of all random walks starting from all possible origins, weighted by the stationary probabilities, leads to essentially isotropic diffusion due to self-averaging over many regions within the cluster. The diffusion of an ensemble of particles in the latter case may be characterized by the velocity-correlation matrix (VCM). It is proved that the VCM is symmetric when a detailed balance condition holds and stationary initial conditions are used. An evolution equation for the VCM is constructed using a generalized Langevin equation with a power-law friction kernel. The kernel is a phenomenological characterization of the diffusion process in a fractal medium.