Estimation of Chaotic and Regular (Stick–Slip and Slip–Slip) Oscillations Exhibited by Coupled Oscillators with Dry Friction

In this paper, we present a novel approach to quantify regular or chaotic dynamics of either smooth or non-smooth dynamical systems. The introduced method is applied to trace regular and chaotic stick–slip and slip–slip dynamics. Stick–slip and slip–slip periodic and chaotic trajectories are analyzed (for the investigated parameters, a stick–slip dynamics dominates). Advantages of the proposed numerical technique are given.     

The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

A Mathematical Model of Rainfall-Controlled Geothermal Fields

A mathematical model is developed, using a hydrothermal spring concept, for those geothermal fields which are controlled by infiltrated rainfall. Two limiting regimes are identified: when rainfall infiltration is so great that isothermal conditions exist in most of the downflow regions; and when rainfall infiltration is so small that a constant geophysical temperature gradient results. In either case, the ratio of downflow to upflow areas, and the energy output per unit area, are fixed, but the typical size of a geothermal field remains indeterminate. The additional assumption that the horizontal length scale of the downflow region is determined by the depth of the main heat sources, fixes the characteristic mass flows, energy outputs, and the number of geothermal fields in a geothermal region. These model predictions are compared with field data, and are shown to be in approximate agreement with geothermal field characteristics in the Taupo Volcanic Zone of New Zealand. The temperature dependence of water viscosity uses a correlation of Wooding (J. Fluid Mech. 2:273–285, 1957), while the temperature dependence of water density, enthalpy and boiling point use correlations from Elder (Geothermal Systems 508, 1981).     

Critical oscillations of mass–spring systems due to nonsmooth friction

Critical values of the parameters governing the dynamics of simple systems appear when Coulomb friction is not regularized. We explore such systems using a method based on the fact that under constant or analytical data the trajectory exists, is unique and is also sufficiently regular. In fact these properties justify elementary analytical computations on successive time intervals where the condition used to connect the solution from one interval to the other is due to the regularity. Although the systems are simple the dynamics turn out to be quite complex and thus furnish an interesting benchmark for contact dynamics numerical codes. Among other possible applications we choose to present here how to use a mass–spring chain with Coulomb friction to slow down in a progressive and regular manner an oncoming mass with a given initial velocity.     

Variation of the hysteresis loop with the Bouc–Wen model parameters

The Bouc–Wen model for smooth hysteresis has received an increasing interest in the last few years due to the ease of its numerical implementation and its ability to represent a wide range of hysteresis loop shapes. This model consists of a first-order nonlinear differential equation that contains some parameters that can be chosen, using identification procedures, to approximate the behavior of given physical hysteretic system. Despite a large body of literature dedicated to the Bouc–Wen model, the relationship between the parameters that appear in the differential equation and the shape of the obtained hysteresis loop is little understood. The objective of this paper is to fill this gap by analytically exploring this relationship using a new form of the model called the normalized one. The mathematical framework introduced in this study formalizes the vague notion of “loop shape" into precise quantities whose variation with the Bouc–Wen model parameters is analyzed. In light of this analysis, the parameters of Bouc–Wen model are re-interpreted.     

Using spring repulsions to model entanglement interactions in Brownian dynamics simulations of bead–spring chains

We develop a bead–spring Brownian dynamics model for simulating the topological interactions between polymers and thin obstacles and apply this method to electrophoretically translating DNA strands interacting with an immovable post. The use of a bead–spring method allows for the simulation of entanglement interactions of polymer chains too long to be simulated using bead–rod or pearl necklace models. Using stiff “FENE-Fraenkel” springs, we are able to model short chains as well. Our new method determines the shortest distance between a spring and the post, calculates a repulsive force inversely related to this distance using an exponential potential, and corrects for the rare situation when a spring passes beyond the post despite the repulsive interaction. As an example problem, we consider single-chain collisions with a single post in weak electric fields. We explore a wide range of chain lengths (25–1,515 Kuhn steps), and we find that the average delay produced by the collision is a function of both the chain length and the Peclet number. Chains of all lengths reach the same upper limit at high Peclet number, but they follow separate curves with similar slopes at lower Peclet number. Our results are consistent with published results for a 25-Kuhn-step chain at Peclet number Pe = 10. Our new method is a general one that allows us to compute the effects of entanglements in systems with rare entanglements and long chains that cannot be simulated by other more microscopic methods.     

Nonlinear Normal Modes and Their Bifurcations for an Inertially Coupled Nonlinear Conservative System

This work concerns the nonlinear normal modes (NNMs) of a 2 degree-of-freedom autonomous conservative spring–mass–pendulum system, a system that exhibits inertial coupling between the two generalized coordinates and quadratic (even) nonlinearities. Several general methods introduced in the literature to calculate the NNMs of conservative systems are reviewed, and then applied to the spring–mass–pendulum system. These include the invariant manifold method, the multiple scales method, the asymptotic perturbation method and the method of harmonic balance. Then, an efficient numerical methodology is developed to calculate the exact NNMs, and this method is further used to analyze and follow the bifurcations of the NNMs as a function of linear frequency ratio p and total energy h. The bifurcations in NNMs, when near 1:2 and 1:1 resonances arise in the two linear modes, is investigated by perturbation techniques and the results are compared with those predicted by the exact numerical solutions. By using the method of multiple time scales (MTS), not only the bifurcation diagrams but also the low energy global dynamics of the system is obtained. The numerical method gives reliable results for the high-energy case. These bifurcation analyses provide a significant glimpse into the complex dynamics of the system. It is shown that when the total energy is sufficiently high, varying p, the ratio of the spring and the pendulum linear frequencies, results in the system undergoing an order–chaos–order sequence. This phenomenon is also presented and discussed.     

Discrete structures equivalent to nonlinear Dirichlet and wave equations

The Amann–Conley–Zehnder (ACZ) reduction is a global Lyapunov–Schmidt reduction for PDEs based on spectral decomposition. ACZ has been applied in conjunction to diverse topological methods, to derive existence and multiplicity results for Hamiltonian systems, for elliptic boundary value problems, and for nonlinear wave equations. Recently, the ACZ reduction has been translated numerically for semilinear Dirichlet problems and for modeling molecular dynamics, showing competitive performances with standard techniques. In this paper, we apply ACZ to a class of nonlinear wave equations in , attaining to the definition of a finite lattice of harmonic oscillators weakly nonlinearly coupled exactly equivalent to the continuum model. This result can be thought as a thermodynamic limit arrested at a small but finite scale without residuals. Reduced dimensional models reveal the macroscopic scaled features of the continuum, which could be interpreted as collective variables.      

Variational principles and conservation laws to the Burridge–Knopoff equation

We generate conservation laws for the Burridge–Knopoff equation which model nonlinear dynamics of earthquake faults by a new conservation theorem proposed recently by Ibragimov. One can employ this new general theorem for every differential equation (or systems) and derive new local and nonlocal conservation laws. Nonlocal conservation laws comprise nonlocal variables defined by the adjoint equations to the Burridge–Knopoff equation.     

Synaptic Coupling Between Two Electronic Neurons

An electrical circuit is proposed to realize an unidirectional coupling between two cells, mimicking chemical synaptic coupling. Each cell represents the FitzHugh–Nagumo (FHN) model of neuron with a modified exitability (MFHN). We present experimental results on frequency doublings and on the chaotic dynamics depending on the coupling strength in a master–slave configuration. In all experiments, we stress the influence of the coupling strength on the control of the slave neuron.     

An approximate solution for the Couette–Poiseuille flow of the Giesekus model between parallel plates

An approximate analytical solution is derived for the Couette–Poiseuille flow of a nonlinear viscoelastic fluid obeying the Giesekus constitutive equation between parallel plates for the case where the upper plate moves at constant velocity, and the lower one is at rest. Validity of this approximation is examined by comparison to the exact solution during a parametric study. The influence of Deborah number (De) and Giesekus model parameter (α) on the velocity profile, normal stress, and friction factor are investigated. Results show strong effects of viscoelastic parameters on velocity profile and normal stress. In addition, five velocity profile types were obtained for different values of α, De, and the dimensionless pressure gradient (G).     

Transient Heat and Moisture Flow Around Heat Source Buried in an Unsaturated Half Space

This article presents solutions for the transient heat and moisture transport due to both disk heat source and cylindrical heat source buried in an unsaturated half space. The solutions are presented in Hankel–Laplace transform domain and in dimensionless style. Coupled effect of thermally driven moisture transport is especially investigated because of its importance to alter the flow field in low-permeability medium. Parametric study has been performed to assess the effects of five independent dimensionless parameters on flow field. The stability and accuracy of the present solutions are demonstrated from the comparison between the results obtained from these solutions and those by using a well-established finite element code CODE_BRIGHT. Despite the simplified assumptions required in order to obtain analytical solutions in Hankel–Laplace transform domain, the results incorporate the main mechanisms involved in the coupled thermo-hydraulic (T-H) problem, and they may be eventually used for validation purposes.     

Predictive Capabilities of an Improved Cubic k–ε Model for Inert Steady Flows

Through an improved ε transport equation, a major quality enhancement of the cubic k–ε model, earlier developed in[13], is obtained. The ε-equation of [13],yielding good results for wall-bounded and rotating flows, is combined with the one derived by Shih et al. [20], which produces good results for free shear flows (e.g. the plane jet–round jet anomaly is resolved).Results are presented for the following flows: fully developed stationary and rotating channel and pipe, backward-facing step, sudden pipe expansion, smooth channel expansion and contraction, plane and round jet. Heat transfer predictions in turbulent impinging jets are also discussed. Accurate results are obtained for the mean flow quantities for all test cases, without case dependent model tuning. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the Range of Applicability of the Reissner–Mindlin and Kirchhoff–Love Plate Bending Models

We show that the Reissner–Mindlin plate bending model has a wider range of applicability than the Kirchhoff–Love model for the approximation of clamped linearly elastic plates. Under the assumption that the body force density is constant in the transverse direction, the Reissner–Mindlin model solution converges to the three-dimensional linear elasticity solution in the relative energy norm for the full range of surface loads. However, for loads with a significant transverse shear effect, the Kirchhoff–Love model fails. This revised version was published online in June 2006 with corrections to the Cover Date.     

Rheological characterization and constitutive modeling of bread dough

Bread dough (a flour–water system) has been rheologically characterized using a parallel-plate, an extensional, and a capillary rheometer at room temperature. Based on the linear and nonlinear viscoelastic and viscoplastic data, two constitutive equations have been applied, namely a viscoplastic Herschel–Bulkley model and a viscoelastoplastic K–BKZ model with a yield stress. For cases where time effects are unimportant, the viscoplastic Herschel–Bulkley model can be used. For cases where transient effects are important, it is more appropriate to use the K-BKZ model with the addition of a yield stress. Finally, the wall slip behavior of dough was studied in capillary flow, and an appropriate slip law was formulated. These models characterize the rheological behavior of bread dough and constitute the basic ingredients for flow simulation of dough processing, such as extrusion, calendering, and rolling.     

Prediction of momentum and scalar transport in turbulent swirling flows with an objective Reynolds-stress transport closure

The accurate prediction of turbulent swirling flows requires the use of a differential Reynolds-stress transport model to close the time-averaged Navier–Stokes equations. The performance of such model is largely determined by the way in which the fluctuating pressure–strain correlations are approximated. A number of alternative approximations are available, all of which depend explicitly on the mean vorticity tensor. Such dependence renders a constitutive relation inconsistent with the principle of Material Frame Indifference (MFI). In this paper, an objective model (i.e. one which is consistent with MFI) for the pressure–strain correlations is presented. This model, which was developed using Tensor Representation Theory, has fewer terms than the conventional alternatives and is therefore easier to implement in computational codes. Moreover, the model was calibrated to correctly reproduce the relative stress levels in both free and wall-bounded flows without the need to employ wall-damping corrections. The performance of this model is assessed using experimental data from both weakly- and strongly-swirled jets. Comparisons are also made with results obtained using three widely-used alternative models for the pressure–strain correlations. It is found that the objective model, although simpler in formulation than the others, yields results that are generally in closer correspondence with the data. The paper also reports on the prediction of mass transfer in a swirling jet. The case considered was that of a co-axial, strongly-swirled flow with an outer annular air stream and an inner helium jet. Swirl was imparted to the outer stream only. The concentration of helium was predicted using a differential scalar-flux transport closure. Close agreement was obtained with the measured concentrations. Analysis of the predicted mass fluxes revealed that the turbulent diffusivity is strongly anisotropic in this flow.     

'T-RANS' Simulation of Deterministic Eddy Structure in Flows Driven by Thermal Buoyancy and Lorentz Force

The paper reports on the application of the Time-dependent Reynolds-Averaged Navier–Stokes (T-RANS) approach to analysing the effects of magnetic force and bottom-wall configuration on the reorganisation of a large coherent structure and its role in the transport processes in Rayleigh–Bénard convection. The large-scale deterministic motion is fully resolved in time and space, whereas the unresolved stochastic motion is modelled by a `subscale' model for which the conventional algebraic stress/flux expressions were used, closed with the low-Re number (k)-(ε)-(θ²) three-equation model. The applied method reproduces long-term averaged mean flow properties, turbulence second moments, and all major features of the coherent roll/cell structure in classic Rayleigh–Bénard convection in excellent agreement with the available DNS and experimental results. Application of the T-RANS approach to Rayleigh–Bénard convection with wavy bottom walls and a superimposed magnetic field yielded the expected effects on there organisation of the eddy structure and consequent modifications of the mean and turbulence parameters and wall heat transfer. This revised version was published online in August 2006 with corrections to the Cover Date.     

Droplet dynamics in sub-critical complex flows

A newly designed eccentric cylinder device has been used to study the deformation and orientation of single Newtonian droplets immersed in an immiscible Newtonian liquid in a controlled complex flow field. Optical microscopy coupled with image acquisition analysis allows monitoring the dynamics of droplets flowing in the gap between the eccentric cylinders. Throughout the experiments, the flow intensity was kept below the critical conditions for droplet break-up. The experimental results are compared with predictions which are obtained using the transient form of the phenomenological model of Maffettone and Minale (J Non-Newtonian Fluid Mech 78:227–241, 1998; J Non-Newtonian Fluid Mech 84:105–106, 1999), incorporating a flow type parameter that accounts for the relative amount of elongational effects in the flow field and adapting the capillary number to mixed flows. For all the sub-critical flows studied here, good agreement was found between model predictions and experimental data, providing, for the first time, a quantitative assessment of drop shape predictions in complex flows.     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

A pragmatic approach for deriving constitutive equations endowed with pom–pom attributes

The most important rheological and mathematical features of the pom–pom model are presently used to compare and improve other constitutive models such as the Giesekus and Phan-Thien–Tanner models. A pragmatic methodology is selected that allows derivation of simple constitutive equations, which are suited to possible software implementation. Alterations to the double convected pom–pom, Phan-Thien–Tanner and Giesekus models are proposed and assessed in rheometric flows by comparing model predictions to experimental data.