Power-law Friction Research Articles

Grounding-line flux conditions for marine ice-sheet systems under effective-pressure- dependent and hybrid friction laws

Flux conditions are semi-analytical expressions that can be used to determine the flux at the grounding line of marine ice sheets. In the glaciology literature, such flux conditions have initially been established for the Weertman and Coulomb friction laws. However, the extension to more general and complex friction laws, such as the Budd friction law, for which basal friction depends on both the basal velocity and the effective pressure, is a topic of recent research. Several studies have also shown that hybrid friction laws, which consider a transition between a power-law friction far from the grounding line and a plastic behaviour close to it, were good candidates for improved modelling of marine ice sheets. In this article, we show that the flux conditions previously derived for the Weertman and Coulomb friction laws can be generalised to flux conditions for the Budd friction law with two different effective-pressure models. In doing so, we build a bridge between the results obtained for these two friction laws. We provide a justification for the existence and uniqueness of a solution to the boundary-layer problem based on asymptotic developments. We also generalise our results to hybrid friction laws, based on a parametrisation of the flux condition. Finally, we discuss the validity of the assumptions made during the derivation, and we provide additional explicit expressions for the flux that stay valid when the bedrock slopes are important or when the friction coefficients are relatively small.

Modeling of tow tension fluctuations and parameter optimization during the stable transfer phase for automated fiber placement

The fluctuation in tow tension is critical for Automated Fiber Placement (AFP). However, it has not been widely studied. In this paper, a tension variation equation was developed to represent the tension fluctuation during AFP, and experiments were used to demonstrate the validity of the equation. According to the equation, the degree of bending rigidity, power-law friction, and extensibility influence on the tension change was obtained. Additionally, the effects of bending rigidity and extensibility can be disregarded during AFP. Moreover, automatic wire feeding and laying trials were carried out separately, and proper tensions (T = 3 N) for wire feeding and layup under experimental conditions were obtained. Therefore, the equation provides theoretical support for the structural and parameter optimization of the AFP.

Weak solvability via bipotentials for contact problems with power-law friction

We study a bilateral contact model with power-law friction and a regularized version of it as well, a regularization of the friction law being considered. The behavior of the material is described by means of a subdifferential inclusion. We deliver a two-field variational formulation via bipotentials for the original problem and also for the regularized version. Subsequently, we study the solvability of the variational formulations we propose, based on abstract auxiliary results. Some properties of the solutions, paying attention to the data dependence, are also investigated. Finally, we deliver a convergence result.

The predictive power of ice sheet models and the regional sensitivity of ice loss to basal sliding parameterisations: a case study of Pine Island and Thwaites glaciers, West Antarctica

Abstract. Ice sheet models use a wide range of sliding laws to define a relationship between ice velocity and basal drag, generally comprising some combination of a Weertman-style power law and Coulomb friction. The exact nature of basal sliding is not known from observational data, making assessment of the suitability of different sliding laws difficult. The question of how much this choice could affect predictions of future ice sheet evolution is an important one. Here we conduct a model study of a large sector of the West Antarctic Ice Sheet (WAIS), a particularly critical component of the cryosphere, using a range of sliding parameterisations, and we provide an assessment of the sensitivity of ice loss to the choice of sliding law. We show that, after initialisation, various sliding laws result in broadly similar ranges of sea level contribution over 100 years, with the range primarily dependent on exact parameter values used in each sliding law. Comparing mass loss from Thwaites and Pine Island glaciers and the neighbouring regions reveals significant qualitative geographical differences in the relationship between sliding parameters and the modelled response to changes in forcing. We show that the responses do not necessarily follow universal systematic patterns, and, in particular, higher values of the sliding exponent m do not necessarily imply larger rates of mass loss. Despite differences in the magnitudes of ice loss and rates of change in the system, all of our experiments display broad similarities in behaviour which serve to reinforce the decade-to-century-scale predictive power of ice sheet models, regardless of the choice of basal sliding.

Comparison of ice dynamics using full-Stokes and Blatter–Pattyn approximation: application to the Northeast Greenland Ice Stream

Abstract. Full-Stokes (FS) ice sheet models provide the most sophisticated formulation of ice sheet flow. However, their applicability is often limited due to the high computational demand and numerical challenges. To balance computational demand and accuracy, the so-called Blatter–Pattyn (BP) stress regime is frequently used. Here, we explore the dynamic consequences of using simplified approaches by solving FS and the BP stress regime applied to the Northeast Greenland Ice Stream. To ensure a consistent comparison, we use one single ice sheet model to run the simulations under identical numerical conditions. A sensitivity study to the horizontal grid resolution (from 12.8 to a resolution of 0.1 km) reveals that velocity differences between the FS and BP solution emerge below ∼ 1 km horizontal resolution and continuously increase with resolution. Over the majority of the modelling domain both models reveal similar surface velocity patterns. At the grounding line of the 79∘ North Glacier the simulations show considerable differences whereby the BP model overestimates ice discharge of up to 50 % compared to FS. A sensitivity study to the friction type reveals that differences are stronger for a power-law friction than a linear friction law. Model differences are attributed to topographic variability and the basal drag, in which neglected stress terms in BP become important.

A new model and alternative solutions for the description of double layered flexible elements wrapped around a cylinder

High-strength fiber ropes have a number of advantages over steel ropes, therefore they currently enjoy great research interest in conjunction with rope winches. However, what is lacking in the art are suitable drives, since concepts for steel ropes cannot be easily transferred to high-strength fiber ropes. A promising innovative principle is the belt wrap capstan, although adequate and validated models for predicting optimal rope force with the lowest possible belt force are still missing. This paper aspires to contribute towards closing this gap by presenting a validated model of a double layered belt wrap capstan First, a general law of friction is considered. Then, special focus is given to power law friction. This law of friction implies the zero force phenomenon which is also discussed.

Earthquake dynamic induced by the magma up flow with fractional power law and fractional-order friction.

In this work, we examine the dynamical behaviour of the “single mass-springs” model for earthquake subjected to the strength due to the up flow of magma for the period of volcanism, considering the fractional viscous damping force, the fractional weakening friction and fractional power law of elastic force. The numerical simulation method used in this paper is that of Grünwald-Letnikov based on the generalization of the classical derivative, and the approximately analytical solution obtained by the harmonic balance method. The results have shown that the fractional-order derivative can affect the dynamical properties of fault rock, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. Moreover, the amplitude-frequency equation for the steady-state solution was established. It appears that the resonant amplitude and resonant frequency are strongly dependent on the fractional-order damping r, fractional-order friction q, the fractional deflection 𝛼, the nonlinear stiffness coefficient and the fractional viscous coefficient. We have also shown that, the recurrence time of an event, the duration time of an event and the slip size of an earthquake can be controlled by the fractional-order derivative, the fractional-order deflection and the magnitude of the magma strength. The model allowed us to better interpret the earthquake as a stick-slip motion.

On determining the power-law fluid friction factor in a partially porous channel using the lattice Boltzmann method

In the present work, the power-law fluid flow in a channel partially filled with a porous medium is numerically investigated using the lattice Boltzmann method (LBM). The porous domain, placed in the lower half of the channel, is represented according to a heterogeneous approach by a matrix of solid square disconnected blocks. The apparent viscosity of the power-law fluid is computed by locally varying the LBM relaxation factor. The results show the influence of geometry (porosity, number of obstacles, and hydraulic diameter), inertia (Reynolds number), and fluid properties (power-law index) over the partially porous-to-impermeable channel friction factor ratio. In general, the higher the porosity and the lower the number of obstacles, Reynolds number, and power-law index, the lower the friction factor. Finally, a correlation for the friction factor ratio as a function of the free region hydraulic diameter, permeability, and power-law index is presented for a specific channel configuration.

Theoretical and experimental investigations on the effects of friction, bending rigidity, extensibility, and Poisson's ratio on fabric tensile properties

A three-point tensile model that consists of a noncontact model and a modified capstan model of contact sections, including coupling effects of factors, is established in this study. The tension ratio calculated using the Runge–Kutta method increases along with the extensibility, surface friction coefficient, and radius ratio and is inhibited by the power-law friction ([Formula: see text]). Moreover, the theoretical model in a case with all the factors and frictional modification shows high accuracy with the actual test of the quick-intelligent handling evaluation system, and the Poisson's effect can be negligible, especially with the power-law friction. It has been confirmed that greater surface roughness and thickness (lower radius ratio) with worse extensibility result in tighter fabric tensile properties. Therefore, this work can provide theoretical guidance for the measurement of fabric tensile properties and the evaluation of practical application of fabrics.

Boundary optimal control for antiplane contact problems with power-law friction

• A variational inequality models the antiplane shear deformation of a body in frictional contact with a foundation. • The distance between the solution of the variational inequality and a given target can be minimized by acting with a boundary control force. • The boundary control can be approximated by using linearization and fixed point method. We consider a contact model with power-law friction in the antiplane context. Our study focuses on the boundary optimal control, paying special attention to optimality conditions and computational methods. Depending on the exponent of the power-law friction, we are able to deduce an optimality condition for the original problem or for a regularized version of it. Furthermore, we introduce and analyze a computational technique based on linearization, saddle point theory and a fixed point method. For a slightly modified optimal control problem, some numerical experiments are presented.

Regularized Coulomb Friction Laws for Ice Sheet Sliding: Application to Pine Island Glacier, Antarctica.

The choice of the best basal friction law to use in ice‐sheet models remains a source of uncertainty in projections of sea level. The parameters in commonly used friction laws can produce a broad range of behavior and are poorly constrained. Here we use a time series of elevation and speed data to examine the simulated transient response of Pine Island Glacier, Antarctica, to a loss of basal traction as its grounding line retreats. We evaluate a variety of friction laws, which produces a diversity of responses, to determine which best reproduces the observed speedup when forced with the observed thinning. Forms of the commonly used power law friction provide much larger model‐data disagreement than less commonly used regularized Coulomb friction in which cavitation effects yield an upper bound on basal friction. Thus, adoption of such friction laws could substantially improve the fidelity of large‐scale simulations to determine future sea level.

Stochastic resonance in a fractional oscillator subjected to multiplicative trichotomous noise

In this paper, stochastic resonance in a fractional oscillator with a power-law friction kernel subject to random damping is investigated both theoretically and numerically. The influence of a fluctuating damping is modeled as multiplicative trichotomous noise. The exact expression of the first moment of the system $$'$$ s steady response has been calculated. It is shown that the interplay between multiplicative trichotomous noise and memory effect leads to stochastic resonance in the proposed system. The output amplitude gain (OAG) shows non-monotonic dependence on the driving frequency of the input signal and the characteristics of the noise. Furthermore, a multiresonance-like behavior of the OAG as function of the driving frequency and the inverse-stochastic resonance behavior of the OAG as function of the noise switching rate are observed, which is previously reported and believed to be absent in the case of the non-memory oscillator. Finally, some numerical simulations are performed to support the theoretical analyses.

On some issues in the computational modelling of spacer-filled channels for membrane distillation

This study addresses issues which arise in the computational and experimental modelling of flow and heat/mass transfer in membrane distillation and other processes adopting spacer-filled channels (either planar or spiral wound), but have not been sufficiently clarified in the literature so far. Most of the argumentations presented are based on original computational results obtained by the authors by finite volume simulations; some literature results are also considered. The questions addressed regard the choice of scales for the reduction of data and the definition of dimensionless numbers (Re, f, Nu, Sh); the definition of average heat or mass transfer coefficients; the combined effects of the parameters that characterize the process (spacer pitch to channel height ratio l/H, flow attack angle γ and Reynolds number Re) and the applicability of simple correlations; the influence of the spacer's thermal conductivity. In regard to the influence of the parameters, Re, l/H and γ were found to interact heavily, making a separate-effect analysis impossible and power-law friction or heat/mass transfer correlations of little use. Thermal conduction in the spacer, even for low-conductivity polymeric spacers (λ≈0.15Wm−1K−1), was found to be responsible for up to 10% of the total heat transfer.

An improved Capstan equation including power-law friction and bending rigidity for high performance yarn

The tension of high performance yarn in the weaving region has influence on uniformity and pattern effect of fabric, which is difficult to be predicted in weaving process. In this paper, a modified Capstan equation including bending rigidity and power-law friction effects was developed, and solved numerically. The effects of the bending rigidity and power-law friction on tension ratio were analyzed. The experimental results were compared with the classical Capstan equation and modified one, showing that there was an approximate agreement between the modified tension ratio and experimental values. And therefore, the modified tension ratio could be used to characterize the tension variation and provide theoretical support for precisely controlling of yarn tension.

Power law and composite power law friction factor correlations for laminar and turbulent non-Newtonian open channel flow

Extensive experimental results conducted in a 10-m flume for various types of non-Newtonian fluids spanning a range of cross-sectional open channel shapes are presented and analysed in depth in this work. Open channel flow of non-Newtonian slurries is relevant in mining and chemical engineering applications. This database coupled with the literature data is used to develop the generalised friction factor–Reynolds number correlations in a unified fashion. Much confusion still exists in the literature regarding the definition of non-Newtonian Reynolds numbers. This difficulty is circumvented by considering two widely accepted definitions of the Reynolds number, namely due to Haldenwang et al. (Hydrotransport 15: 15th international conference on the hydraulic transport of solids in pipes, Banff, pp 755–768, 2002) for open channel flow and the modified Metzner–Reed pipe flow Reynolds number adapted for open channel flow. Three different types of purely viscous non-Newtonian fluids in rectangular, trapezoidal, triangular and semi-circular channel shapes were tested. The modelling procedure of Garcia et al. (Int J Multiph Flow 29:1605–1624, 2003) used for pipe flow predictions was extended to the present work. The logistic dose curves based on the Reynolds number proposed by Haldenwang et al. [14] performed better than those based on the adapted Metzner–Reed Reynolds number. Correlations developed can be used for the design of open channels of various shapes to transport non-Newtonian fluids.

Nonlinear Problems with p(·)-Growth Conditions and Applications to Antiplane Contact Models

Abstract We consider a general boundary value problem involving operators of the form div(a(·, ∇u(·)) in which a is a Carathéodory function satisfying a p(·)-growth condition. We are interested on the weak solvability of the problem and, to this end, we start by introducing the Lebesgue and Sobolev spaces with variable exponent, together with their main properties. Then, we state and prove our main existence and uniqueness result, Theorem 3.1. The proof is based on a Weierstrass-type argument. We also consider two antiplane contact problems for nonhomogenous elastic materials of Hencky-type. The contact is frictional and it is modelled with a regularized version of Tresca’s friction law and a power-law friction, respectively. We prove that the problems cast in the abstract setting, then we use Theorem 3.1 to deduce their unique weak solvability.

Diffusion in a Medium with Nonlinear Friction

The Langevin equation describing the Brownian motion of particles is studied in the case when the friction depends nonlinearly on the particle velocity. The intensity \(D(\upsilon )\) of the thermal random force driving the particles then, in equilibrium systems, also depends on the velocity, and the equation of motion should contain an additional ‘spurious’ force proportional to the derivative of \(D(\upsilon )\). This force is chosen in the kinetic representation, for which the stationary probability density for velocities is the Maxwell distribution and simultaneously the generalized Einstein relation is obeyed. A general formula for the diffusion coefficient of the particle is obtained and then specified for various models of dissipative forces studied in the literature, in particular, for the power-law friction. In this case the scaling of the diffusion coefficient with the model parameters is obtained also by a simple dimensional analysis, for systems both at equilibrium and far from it, when \(D\) is usually assumed to be independent of \(\upsilon \).

Sludge pipe flow pressure drop prediction using composite power-law friction factor–Reynolds number correlations based on different non-Newtonian Reynolds numbers

When predicting pressure gradients for the flow of sludges in pipes, the rheology of the fluid plays an important role, especially with increasing concentration of the suspended matter in the sludge. The f-Re relationship is often applied when designing pipelines, but it depends on the rheological parameters of the fluid and what definition of non-Newtonian Reynolds number is used. In this work, a database of 586 ΔP – Q points from tests with 10 different sludges of concentration 3.4 to 7.2% by mass, in 3 test pipe diameters, was established and used to rheologically characterise the sludges as Bingham plastic fluids. Five published definitions of the non-Newtonian Reynolds number were used to create composite power law correlations for the f-Re relationship covering all flow regimes. Pressure gradient predictions based on each cor relation were compared and ranked, based on 2 different statistical estimates of error. The correlations using the MetznerReed Reynolds number (Re MR ) and a Reynolds number proposed by Slatter and Lazarus in 1993 (Re 2 ) yielded the lowest errors in comparison with the experimental values. It is shown that these correlations can be used to predict pressure drop to within ±20% for a given sludge concentration and operating condition.

Two-point approximation to the Kramers problem with coloured noise

We present a method, founded on previous renewal approaches as the classical Wilemski-Fixman approximation, to describe the escape dynamics from a potential well of a particle subject to non-Markovian fluctuations. In particular, we show how to provide an approximated expression for the distribution of escape times if the system is governed by a generalized Langevin equation (GLE). While we show that the method could apply to any friction kernel in the GLE, we focus here on the case of power-law kernels, for which extensive literature has appeared in the last years. The method presented (termed as two-point approximation) is able to fit the distribution of escape times adequately for low potential barriers, even if conditions are far from Markovian. In addition, it confirms that non-exponential decays arise when a power-law friction kernel is considered (in agreement with related works published recently), which questions the existence of a characteristic reaction rate in such situations.

Friction Coefficient (F)-Reynolds Number (Re) Relationship in Non-cohesive Suspended Sediment Laden Flow thorough Pervious Rockfill Dams

Recently pervious rockfill detention dam is used as a flood mitigating structure. Analysis of hydraulics of turbulent flow through this kind of dam is mostly done using Darcy-Weisbach equation. So far, many attempts have been made to study the friction coefficient as a function of Reynolds Number in turbulent flow of clean water through pervious rockfill detention dams, while this subject has remained intact for sediment laden flow of water. In this paper relationship between friction coefficient and Reynolds Number for sediment laden flow through highly pervious rockfill dams was investigated. Required data for a regression analysis obtained by conducting a series of laboratory tests to calibrate and validate a proposed power law friction coefficient-Reynolds Number relationship. A changeable bed slop Plexiglas flume, an adjustable rate sediment feeder and a recirculating flow electro pump system were used in present study. The tests were carried out on four different rectangular laboratory rockfill dams and three different non-cohesive suspended sediments. A power law relationship was obtained with a correlation coefficient of 0.74 using two thirds of laboratory measured friction coefficient and Reynolds Number. The obtained relationship was validated employing the remaining unused data with a Mean Square Error of 0.29 which is an acceptable agreement. A new power law relationship was found between friction coefficient and Reynolds Number in sediment laden flow through pervious rockfill dams. This new relationship is the only one thus has been proposed for the sediment laden flow of water through pervious rockfill dams.