Onset of shear thickening in a simple fluid

We report on nonequilibrium molecular-dynamics simulations of the shear-thickening transition in a simple fluid under shear. We relate the shear-thickening transition to the onset of instabilities in the flow profile and to that of dramatic variations in normal stress differences. The dependence of the critical shear rate, which indicates the onset of shear thickening, on density and temperature is rationalized by introducing a ratio between two characteristic times, quantifying the short-time mobility of a particle and the deformation imposed by the applied shear rate, respectively. The shear-thickening transition is shown to occur at a constant value for this ratio for all state points studied. From a structural point of view, this transition is accompanied by the formation of clusters as recently observed in experiments on complex fluids.Received: 26 July 2004, Published online: 21 September 2004PACS: 83.60.Rs Shear rate-dependent structure (shear thinning and shear thickening) - 47.50. + d Non-Newtonian fluid flows - 83.10.Mj Molecular dynamics, Brownian dynamics     

Solution of the Ornstein-Zernike equation for a soft-core Yukawa fluid

A model for simple fluids is proposed in which the radial distribution function has a parametric form appropriate to a soft-core fluid for interparticle separationr R, whereR is some range parameter. Forr > R, the direct correlation function is assumed to be of Yukawa form. The Ornstein-Zernike equation is solved for this system, yielding the radial distribution and the total correlation function for the entire range of interparticle separation. Methods of relating the model fluid to a real fluid by assigning values to the parameters are discussed.Supported by ARGC grant No. B7715646R.     

A numerical examination of shear banding and simple shear non-coaxial flow rules

Strain localisation and shear band formation is frequently observed during the handling and flow of dense phase particulate materials. However, a complete understanding of how shear bands form and what happens inside shear bands is still lacking. In order to address this problem, discrete particle simulations have been carried out to examine the detailed processes that occur at the grain scale associated with the initiation and development of shear bands. To reliably identify the continuum model applicable within a shear band is difficult due to the small number of particles/contacts involved. However, it is normally accepted that the mode of deformation within a shear band is one of simple shear. Consequently, simple shear simulations have been performed in order to determine the evolution of the stress tensor, dilation rate, and the principal directions of stress and strain-rate. It is demonstrated that the corresponding non-coaxial flow rule is equivalent to that suggested by Tatsuoka et al. (Géotechnique 38 148 (1988)). Furthermore, at fully developed flow when there is no further change in volume, the stress and strain-rate directions are coaxial and the flow rule is that proposed by Hill (The Mathematical Theory of Plasticity (Oxford University Press, 1950) p. 294).     

Scalings of elliptic flow for a fluid at finite shear viscosity

Within a parton cascade approach we investigate the scaling of the differential elliptic flow v₂(p_T) with eccentricity _x and system size and its sensitivity to finite shear viscosity. We present calculations for shear viscosity to entropy density ratio η/s in the range from 1/4π up to 1/π, finding that the v₂ saturation value varies by about a factor 2. Scaling of v₂(p_T)/_x is seen also for finite η/s which indicates that it does not prove a perfect hydrodynamical behavior, but is compatible with a plasma at finite η/s. Introducing a suitable freeze-out condition, we see a significant reduction of v₂(p_T) especially at intermediate p_T and for more peripheral collisions. This causes a breaking of the scaling for both v₂(p_T) and the p_T-averaged v₂, while keeping the scaling of v₂(p_T)/v₂. This is in better agreement with the experimental observations and shows as a first indication that the η/s should be significantly lower than the pQCD estimates. We finally point out the necessity to include the hadronization via coalescence for a definite evaluation of η/s from intermediate p_T data.     

Light scattering from a nonequilibrium fluid undergoing a planar Poiseuille-Couette flow

We calculate the Rayleigh-Brillouin spectrum of a Newtonian fluid undergoing a planar Poiseuille-Couette flow using the fluctuating hydrodynamic approach of Landau and Lifshitz. Our results reduce to the corresponding results for pure Couette flow when the pressure gradient is made to vanish. The Brillouin spectrum is obtainable from that appropriate to Couette flow by a simple rescaling. There are very minute corrections which impart asymmetry to the Brillouin lines as well as the Rayleigh line, and these can be selectively picked up by suitable choice of the scattering geometry.     

Pole expansion determination of the pair correlation function of a simple fluid

An explicit expression for the pair correlation function is given in terms of an expansion in the poles of the structure factor S(q) in the complex q plane. This is derived when the direct correlation function or structure factor is of known analytical form. An illustrative example is given for a hard-sphere-Yukawa fluid in the mean spherical approximation.     

Exact moment solution of the Boltzmann equation for uniform shear flow

The Ikenberry-Truesdell exact solution to the Boltzmann equation for Maxwell molecules is revisited. This solution refers to a state characterized by a linear profile of the velocity flow and spatially uniform density and temperature. The solution is extended to include explicit expressions for the fourth-degree moments. It is shown that if the shear rate is larger than a certain critical value, the fourth-degree moments do not reach stationary values, even when the temperature is kept constant. The explicit shear-rate dependence of the moments below this critical value are obtained.     

Observation of droplet size oscillations in a two-phase fluid under shear flow

Experimental observations of droplet size sustained oscillations are reported in a two-phase flow between a lamellar and a sponge phase. Under shear flow, this system presents two different steady states made of monodisperse multilamellar droplets, separated by a shear-thinning transition. At low and high shear rates, the droplet size results from a balance between surface tension and viscous stress, whereas for intermediate shear rates it becomes a periodic function of time. A possible mechanism for such kinds of oscillations is discussed.     

Swinging and tumbling of fluid vesicles in shear flow

The dynamics of fluid vesicles in simple shear flow is studied using mesoscale simulations of dynamically triangulated surfaces, as well as a theoretical approach based on two variables: a shape parameter and the inclination angle, which has no adjustable parameters. We show that, between the well-known tank-treading and tumbling states, a new "swinging" state can appear. We predict the dynamic phase diagram as a function of the shear rate, the viscosities of the membrane and the internal fluid, and the reduced vesicle volume. Our results agree well with recent experiments.     

Calculation of the intrinsic viscosity for dilute polymer solutions undergoing shear flow

Starting from a set of coupled Langevin equations describing the dynamics of flexible (Gaussian) polymer chains in the presence of hydrodynamic interactions and undergoing a weak shear flow, the stress tensor is determined using Kramers formula. With the aid of renormalization group techniques the steady state intrinsic viscosity is extracted toO() (4–d,d being the spatial dimensionality) and to lowest nontrivial order in the flow strength. Within the preaveraging approximation we find a numerically small effect of shear thinning.     

Solution of Ornstein-Zernike equation for wall-particle distribution function

The Ornstein-Zernike (OZ) equation is considered for the wall-particle distribution functiong ₀(x) in the case of a flat, impenetrable wall atx = 0 and a fluid of hard-core particles whose centers are constrained by the wall to occupy the semiinfinite spacex >/2, where is the particle diameter. A solution is given in terms of the wall-particle direct correlation function c₀(x) forx >/2, the bulk-fluid direct correlation function c_B (t), and p_B, the average bulk density. Explicit formulas for the contact surface density, total excess surface density, and the Laplace transform of the fluid density near the wall are given. For mean spherical type approximations, c₀ (x) forx >/2 and c_B (t) are both prescribed functions; for this case, a closed-form solution is obtained. An example is discussed and additional equations that enable one to go beyond the approximations considered above are introduced.Report #270, February 1976.The observation of this paper that the wall-particle problem can be treated using standard Wiener-Hopf techniques was independently made by Percus in his work, which came to our attention too late to be compared to, or incorporated into, our own results here.     

Solution of the Ornstein-Zernike equation for a softcore Yukawa fluid. II. Numerical results

Numerical calculations are reported for the simplest case of the soft-core Yukawa fluid introduced in an earlier paper. Attention is given to the thermodynamic behavior, the correlation functions, and the interparticle potentials found by inverting the structural information using Percus-Yevick and hypernetted chain integration equation approximations.Supported by ARGC grant No. B7715646R.     

Configurational temperature thermostat for fluids undergoing shear flow: application to liquid chlorine

Non-equilibrium molecular dynamics (NEMD) simulations play a major role in characterizing the rheological properties of fluids undergoing shear flow. However, all previous studies of flows in molecular fluids either use an ‘atomic’ thermostat which makes incorrect assumptions concerning the streaming velocity of atoms within their constituent molecules, or they employ a centre of mass kinetic (COM) thermostat which only controls the temperature of relatively few degrees of freedom (3) in complex high molecular weight compounds. In the present paper we show how recently developed configurational expressions for the thermodynamic temperature can be used to develop thermostatting mechanisms which avoid both of these problems. In this work, we propose a thermostat based on a configurational expression for the temperature and apply it to NEMD simulations of chlorine undergoing Couette flow. The results so obtained are compared with those obtained using a COM kinetic thermostat. At equilibrium the properties of systems thermostatted in the two different ways are of course equivalent. We show that the two responses only differ far from equilibrium. In particular, we show that the formation of a string phase for extremely high shear rates is an artefact of the COM thermostat. At the largest shear rates studied with the configurational thermostat, no string phase is observed.     

A new integral equation for the radial distribution function of a hard sphere fluid

Based on a proposal by Shinomoto, a new integral equation is derived for the radial distribution function of a hard-sphere fluid using mainly geometric arguments. This integral equation is solved by a perturbation expansion in the density of the fluid, and the results obtained are compared with those from molecular dynamics simulations and from the Born-Green-Yvon (BGY) and Percus-Yevick (PY) theories. The present theory provides results for the radial distribution function which are intermediate in accuracy between those obtained from the BGY and from the PY theories.     

Distribution function for large velocities of a two-dimensional gas under shear flow

The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform shear flow is studied. First we analyze the shear-rate dependence of the eigenvalues governing the time evolution of the velocity moments derived from the Boltzmann equation. As in the three-dimensional case discussed by us previously, all the moments of degreek⩾4 diverge for shear rates larger than a critical valuea _c ^(k) , which behaves for largek asa _c ^(k) ∼k ⁻¹. This divergence is consistent with an algebraic tail of the formf(V) ∼V ^−4-σ(a), where σ is a decreasing function of the shear rate. This expectation is confirmed by a Monte Carlo simulation of the Boltzmann equation far from equilibrium.     

The concept of effective density and the closed integrofunctional equation for a radial function of a simple liquid

The thermodynamically consistent concept of effective density, similar to the idea of an acting field in statistical thermodynamics, is introduced. Assuming a linear functional relation between the effective and true densities (the packet approximation), a generating functional is constructed and from it, in the usual way, the closed integrofunctional equation for a radial function is obtained. Some properties of the obtained equation are investigated, including its relation to known integral equations, and a possible scheme for its solution, by the method of iteration, is presented.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 96–100, September, 1975.     

On the orientational characteristics and transport coefficients of an oblate spheroidal hematite particle in a simple shear flow

We have developed the basic equation of the orientational distribution function of oblate spheroidal hematite particles with rotational Brownian motion in a simple shear flow under an applied magnetic field. An oblate spheroidal hematite particle has an important characteristic in that it is magnetized in a direction normal to the particle axis. Since a dilute dispersion is addressed in the present study, we have taken into account only the friction force (torque) whilst neglecting the hydrodynamic interactions among the particles. This basic equation has been solved numerically in order that we may investigate the dependence of the orientational distribution on the magnetic field strength, shear rate and rotational Brownian motion and the relationship between the orientational distribution and the transport coefficients such as viscosity and diffusion coefficient. We found that if the effect of the magnetic field is more dominant, the particle inclines in such a way that the oblate surface aligns in the magnetic field direction. If the Peclet number increases and the effect of the shear flow becomes more dominant, the particle inclines such that the oblate surface tilts in the shear flow direction. The viscosity due to the magnetic torque is shown to increase as the magnetic field increases, since the magnetic torque due to the applied magnetic field becomes the more dominant effect. Moreover, the viscosity increase is shown to be more significant for a larger aspect ratio or for a more oblate hematite particle. We have applied the analysis to the problem of particle sedimentation under gravity in the presence of a magnetic field applied in the sedimentation direction. The particles are found to sediment with the oblate surface aligning more significantly in the sedimentation direction as the applied magnetic field strength increases.     

Statistical physics of shear flow: a non-equilibrium problem

Complex fluids are easily and reproducibly driven into non-equilibrium steady states by the action of shear flow. The statistics of the microstructure of non-equilibrium fluids is important to the material properties of every complex fluid that flows, e.g. axle grease on a rotating bearing; blood circulating in capillaries; molten plastic flowing into a mould; the non-equilibrium onion phase of amphiphiles used for drug delivery; the list is endless. Such states are as diverse and interesting as equilibrium states, but are not governed by the same statistics as equilibrium materials. I review some recently discovered principles governing the probabilities of various types of molecular re-arrangements taking place within a sheared fluid. As well as providing new foundations for the study of non-equilibrium matter, the principles are applied to some simple models of particles interacting under flow, showing that the theory exhibits physically convincing behaviour.