First-principles constitutive equation for suspension rheology

Using mode-coupling theory, we derive a constitutive equation for the nonlinear rheology of dense colloidal suspensions under arbitrary time-dependent homogeneous flow. Generalizing previous results for simple shear, this allows the full tensorial structure of the theory to be identified. Macroscopic deformation measures, such as the Cauchy-Green tensors, thereby emerge. So does a direct relation between the stress and the distorted microstructure, illuminating the interplay of slow structural relaxation and arbitrary imposed flow. We present flow curves for steady planar and uniaxial elongation and compare these to simple shear. The resulting nonlinear Trouton ratios point to a tensorially nontrivial dynamic yield condition for colloidal glasses.     

Oscillating viscosity in a lyotropic lamellar phase under shear flow

We report some time-dependent behavior of lyotropic lamellar phase under shear flow. At fixed stress, near a layering instability, the system presents an oscillating shear rate. We build up a new stress versus shear rate diagram that includes temporal behavior. This diagram is made of two distinct branches of stationary states which correspond, respectively, to disordered and ordered multilamellar vesicle phases. When increasing the shear stress, prior to the transition to the ordered structural state, sustained oscillations of the viscosity are recorded. They correspond to periodic structural change of the entire sample between a disordered and a ordered state of multilamellar vesicles.     

Lubricated sliding dynamics: Flow factors and Stribeck curve

We study the fluid flow at the interface between elastic solids with randomly rough surfaces. We derive (approximate) analytical expressions for the fluid flow factors which enter in the equation describing the fluid flow, and for the frictional shear stress factors which enter in the equation for the frictional shear stress. Numerical results for a rubber cylinder with surface roughness sliding on a flat lubricated substrate, under “low” and “high” pressure conditions, are presented and discussed. Finally we discuss the role of the fluid-induced elastic deformations of the surface roughness profile.     

Exact Solutions for the Flow of Fractional Maxwell Fluid in Pipe-Like Domains

This paper presents an analysis of unsteady flow of incompressible fractional Maxwell fluid filled in the annular region between two infinite coaxial circular cylinders. The fluid motion is created by the inner cylinder that applies a longitudinal time-dependent shear stress and the outer cylinder that is moving at a constant velocity. The velocity field and shear stress are determined using the Laplace and finite Hankel transforms. Obtained solutions are presented in terms of the generalized G and R functions. We also obtain the solutions for ordinary Maxwell fluid and Newtonian fluid as special cases of generalized solutions. The influence of different parameters on the velocity field and shear stress is also presented using graphical illustration. Finally, a comparison is drawn between motions of fractional Maxwell fluid, ordinary Maxwell fluid and Newtonian fluid.     

Creep and fluidity of a real granular packing near jamming

We study the internal dynamical processes taking place in a granular packing below yield stress. At all packing fractions and down to vanishingly low applied shear, a logarithmic creep is observed. The experiments are analyzed using a viscoelastic model which introduces an internal, time-dependent, fluidity variable. For all experiments, the creep dynamics can be rescaled onto a unique curve which displays jamming at the random-close-packing limit. At each packing fraction, we measure a stress corresponding to the onset of internal granular reorganization and a slowing down of the creep dynamics before the final yield.     

Why does shear banding behave like first-order phase transitions? Derivation of a potential from a mechanical constitutive model

Numerous numerical and experimental evidence suggest that shear banding behavior looks like first-order phase transitions. In this paper, we demonstrate that this correspondence is actually established in the so-called non-local diffusive Johnson-Segalman model (the DJS model), a typical mechanical constitutive model that has been widely used for describing shear banding phenomena. In the neighborhood of the critical point, we apply the reduction procedure based on the center manifold theory to the governing equations of the DJS model. As a result, we obtain a time evolution equation of the flow field that is equivalent to the time-dependent Ginzburg-Landau (TDGL) equations for modeling thermodynamic first-order phase transitions. This result, for the first time, provides a mathematical proof that there is an analogy between the mechanical instability and thermodynamic phase transition at least in the vicinity of the critical point of the shear banding of DJS model. Within this framework, we can clearly distinguish the metastable branch in the stress-strain rate curve around the shear banding region from the globally stable branch. A simple extension of this analysis to a class of more general constitutive models is also discussed. Numerical simulations for the original DJS model and the reduced TDGL equation is performed to confirm the range of validity of our reduction theory.     

Collective oscillations of vortex lattices in rotating Bose-Einstein condensates

The complete low-energy collective-excitation spectrum of vortex lattices is discussed for rotating Bose-Einstein condensates by solving the Bogoliubov-de Gennes equation, yielding, e.g., the Tkachenko mode recently observed at JILA. The totally symmetric subset of these modes includes the transverse shear, common longitudinal, and differential longitudinal modes. We also solve the time-dependent Gross-Pitaevskii equation to simulate the actual JILA experiment, obtaining the Tkachenko mode and identifying a pair of breathing modes. Combining both approaches allows one to unambiguously identify every observed mode.     

Elastic capsules in shear flow: Analytical solutions for constant and time-dependent shear rates

We investigate the dynamics of microcapsules in linear shear flow within a reduced model with two degrees of freedom. In previous work for steady shear flow, the dynamic phases of this model, i.e. swinging, tumbling and intermittent behaviour, have been identified using numerical methods. In this paper, we integrate the equations of motion in the quasi-spherical limit analytically for time-constant and time-dependent shear flow using matched asymptotic expansions. Using this method, we find analytical expressions for the mean tumbling rate in general time-dependent shear flow. The capsule dynamics is studied in more detail when the inverse shear rate is harmonically modulated around a constant mean value for which a dynamic phase diagram is constructed. By a judicious choice of both modulation frequency and phase, tumbling motion can be induced even if the mean shear rate corresponds to the swinging regime. We derive expressions for the amplitude and width of the resonance peaks as a function of the modulation frequency.     

Mechanical responses and stress fluctuations of a supercooled liquid in a sheared non-equilibrium state

A steady shear flow can drive supercooled liquids into a non-equilibrium state. Using molecular dynamics simulations under steady shear flow superimposed with oscillatory shear strain for a probe, non-equilibrium mechanical responses are studied for a model supercooled liquid composed of binary soft spheres. We found that even in the strongly sheared situation, the supercooled liquid exhibits surprisingly isotropic responses to oscillating shear strains applied in three different components of the strain tensor. Based on this isotropic feature, we successfully constructed a simple two-mode Maxwell model that can capture the key features of the storage and loss moduli, even for highly non-equilibrium state. Furthermore, we examined the correlation functions of the shear stress fluctuations, which also exhibit isotropic relaxation behaviors in the sheared non-equilibrium situation. In contrast to the isotropic features, the supercooled liquid additionally demonstrates anisotropies in both its responses and its correlations to the shear stress fluctuations. Using the constitutive equation (a two-mode Maxwell model), we demonstrated that the anisotropic responses are caused by the coupling between the oscillating strain and the driving shear flow. Due to these anisotropic responses and fluctuations, the violation of the fluctuation-dissipation theorem (FDT) is distinct for different components. We measured the magnitude of this violation in terms of the effective temperature. It was demonstrated that the effective temperature is notably different between different components, which indicates that a simple scalar mapping, such as the concept of an effective temperature, oversimplifies the true nature of supercooled liquids under shear flow. An understanding of the mechanism of isotropies and anisotropies in the responses and fluctuations will lead to a better appreciation of these violations of the FDT, as well as certain consequent modifications to the concept of an effective temperature.     

Dynamics of nuclear fluid: (I). Foundation

Starting with the time-dependent Hartree-Fock (TDHF) formulation of the many-body problem, we cast the equation into a set of conservation laws of classical type. Besides the equation of continuity, TDHF leads to an equation of motion which is analogous to the Euler equation in classical fluid dynamics. The forces do not come from the collective kinetic stress alone, but also from a density-dependent chemical potential, the surface tensional force which depends on density differences and the Coulomb interaction. With an assumed Navier-Stokes generalization of the stress tensor, such a set of differential equations provides a powerful tool for the study of complicated collective motions of nuclear systems such as those involved in heavy-ion reactions and nuclear fission. In the static case, the equation of motion leads to the Thomas-Fermi model of a finite nucleus as formulated by Bethe.     

On the description of dissipative collective motion

We use a recently developed time-dependent projection method to describe the dissipation of collective motion coupled to an intrinsic system. The underlying physical picture is similar to that of the linear response approach. Our approach is, however, different from the conceptual point of view. We do not resort to a quasistatic picture but use instead a time-dependent projector. Furthermore, we project on a model space which includes the intrinsic hamiltonian in addition to the collective subspace. In this way we obtain a Fokker-Planck equation for the collective variables which is coupled to a transport equation describing the evolution of the temperature of the intrinsic system.     

A minimal model for vorticity and gradient banding in complex fluids

A general phenomenological reaction-diffusion model for flow-induced phase transitions in complex fluids is presented. The model consists of an equation of motion for a nonconserved composition variable, coupled to a Newtonian stress relation for the reactant and product species. Multivalued reaction terms allow for different homogeneous phases to coexist with each other, resulting in banded composition and shear rate profiles. The one-dimensional equation of motion is evolved from a random initial state to its final steady state. We find that the system chooses banded states over homogeneous states, depending on the shape of the stress constitutive curve and the magnitude of the diffusion coefficient. Banding in the flow gradient direction under shear rate control is observed for shear-thinning transitions, while banding in the vorticity direction under stress control is observed for shear-thickening transitions. Received 1 April 2001 and Received in final form 16 June 2001     

Two-spin dynamic autocorrelations and energy diffusion in the Heisenberg paramagnet

We establish the kinetic equation for the two-spin time-dependent correlation function in the Weiss limit. Considering this equation in the limit of small wavenumbers, we obtain an explicit expression for the energy diffusion coefficient. We also discuss some difficulties connected with overcounting problems of the Weiss approximation.     

Shear flow in the two-body Boltzmann gas

We present the analytic solution of the two-particle Boltzmann equation for hard disks undergoing isothermal shear, within the relaxation time approximation. This system has a steady state for all values of the strain rate. The fluid is shear thinning and exhibits normal stress differences. This solution is compared with the collision free solution of the Liouville equation. The non-equilibrium entropy is also calculated.     

Nonlocal shear stress for homogeneous fluids

It has been suggested that for fluids in which the rate of strain varies appreciably over length scales of the order of the intermolecular interaction range, the viscosity must be treated as a nonlocal property of the fluid. The shear stress can then be postulated to be a convolution of this nonlocal viscosity kernel with the strain rate over all space. In this Letter, we confirm that this postulate is correct by a combination of analytical and numerical methods for an atomic fluid out of equilibrium. Furthermore, we show that a gradient expansion of the nonlocal constitutive equation gives a reasonable approximation to the shear stress in the small wave vector limit.     

Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators

We construct a time-dependent double well potential as an exact spectral equivalent to the explicitly time-dependent negative quartic oscillator with a time-dependent mass term. Defining the unstable anharmonic oscillator Hamiltonian on a contour in the lower-half complex plane, the resulting time-dependent non-Hermitian Hamiltonian is first mapped by an exact solution of the time-dependent Dyson equation to a time-dependent Hermitian Hamiltonian defined on the real axis. When unitary transformed, scaled and Fourier transformed we obtain a time-dependent double well potential bounded from below. All transformations are carried out non-perturbatively so that all Hamiltonians in this process are spectrally exactly equivalent in the sense that they have identical instantaneous energy eigenvalue spectra.     

Fractional nonlinear diffusion equation and first passage time

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the fractional nonlinear diffusion equation with a space- and time-dependent diffusion coefficient subject to absorbing boundaries and the initial condition. We obtain explicit analytical expression for the probability distribution, the first passage time distribution, the mean first passage time, and the mean squared displacement corresponding to different time-dependent diffusion coefficient. In addition, we compare our results for the first passage time distribution and the mean first passage time with the one obtained by usual linear diffusion equation with time-dependent diffusion coefficient.     

Phase-ordering dynamics of binary mixtures with field-dependent mobility in shear flow

The effect of shear flow on the phase-ordering dynamics of a binary mixture with field-dependent mobility is investigated. The problem is addressed in the context of the time-dependent Ginzburg-Landau equation with an external velocity term, studied in self-consistent approximation. Assuming a scaling ansatz for the structure factor, the asymptotic behavior of the observables in the scaling regime can be analytically calculated. All the observables show log-time periodic oscillations which we interpret as due to a cyclical mechanism of stretching and break-up of domains. These oscillations are damped as consequence of the vanishing of the mobility in the bulk phase. Received 13 April 1999     

The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion

We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution.     

Sheared solid materials

We present a time-dependent Ginzburg-Landau model of nonlinear elasticity in solid materials. We assume that the elastic energy density is a periodic function of the shear and tetragonal strains owing to the underlying lattice structure. With this new ingredient, solving the equations yields formation of dislocation dipoles or slips. In plastic flow high-density dislocations emerge at large strains to accumulate and grow into shear bands where the strains are localized. In addition to the elastic displacement, we also introduce the local free volumem. For very smallm the defect structures are metastable and long-lived where the dislocations are pinned by the Peierls potential barrier. However, if the shear modulus decreases with increasingm, accumulation ofm around dislocation cores eventually breaks the Peierls potential leading to slow relaxations in the stress and the free energy (aging). As another application of our scheme, we also study dislocation formation in two-phase alloys (coherency loss) under shear strains, where dislocations glide preferentially in the softer regions and are trapped at the interfaces.