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A stochastic Lagrangian model for simulating the dynamics and rheology of a Brownian multi-particle system interacting with a simple liquid medium is presented. The discrete particle model is formulated within the GENERIC framework for Non-Equilibrium Thermodynamics and therefore it satisfies discretely the First/Second Laws of Thermodynamics and the Fluctuation Dissipation Theorem (FDT). Long-range fluctuating hydrodynamics interactions between suspended particles are described by an explicit solvent model. To this purpose, the Smoothed Dissipative Particle Dynamics method is adopted, which is a GENERIC-compliant Lagrangian meshless discretization of the fluctuating Navier–Stokes equations. In dense multi-particle systems, the average inter-particle distance is typically small compared to the particle size and short-range hydrodynamics interactions play a major role. In order to bypass an explicit—computationally costly—solution for these forces, a lubrication correction is introduced based on semi-analytical expressions for spheres under Stokes flow conditions. We generalize here the lubrication formalism to Brownian conditions, where an additional thermal-lubrication contribution needs to be taken into account in a way that discretely satisfies FDT. The coupled lubrication dynamics is integrated in time using a generalized semi-implicit splitting scheme for stochastic differential equations. The model is finally validated for a single particle diffusion as well as for a Brownian multi-particle system under homogeneous shear flow. Results for the diffusional properties as well as the rheological behavior of the whole suspension are presented and discussed.

This study presents an analysis of the dynamics of single and multiple chains of spherical super-paramagnetic beads suspended in a Newtonian fluid under the combined effect of an external rotating magnetic field and a shear flow. Viscosity results depend on two main non-dimensional numbers: the ratio between the shear rate and the magnetic rotation frequency and the ratio between the hydrodynamic and magnetostatic interactions (the Mason number). When the shear rate is smaller than the magnetic field frequency, the chain rotation accelerates the surrounding fluid, reducing the value of the measured suspension viscosity even below that of the solvent. In this regime, shear-thickening is observed. For values of the shear rates comparable to the rotation magnetic frequency, the viscosity reaches a maximum and non-linear coupling effects come up. If the shear rate is increased to values above the rotation frequency, the viscosity decreases and a mild shear-thinning is observed. In terms of the Mason number, the suspension viscosity reduces in line with the literature results reported for fixed magnetic fields, whereas the shear-rate/magnetic-frequency ratio parameters induce a shift of the viscosity curve towards larger values. Results at larger concentrations and multiple chains amplify the observed effects.

The analytical theories derived here for the acoustic load impedance measured by a quartz crystal microbalance (QCM), due to the presence of layers of different types (rigid, elastic and viscous) immersed in a fluid, display generic properties, such as "vanishing mass" and positive frequency shifts, which have been observed in QCM experiments with soft-matter systems. These phenomena seem to contradict the well-known Sauerbrey relation at the heart of many QCM measurements, but here we show that they arise as a natural consequence of hydrodynamics. We compare our one-dimensional immersed plate theory with three-dimensional simulations of rigid and flexible sub-micron-sized suspended spheres, and with experimental results for adsorbed micron-sized colloids which yield a "negative acoustic mass". The parallel behaviour unveiled indicates that the QCM response is highly sensitive to hydrodynamics, even for adsorbed colloids. Our conclusions call for a revision of existing theories based on adhesion forces and elastic stiffness at contact, which should in most cases include hydrodynamics., Comment: 22 pages, 7 figures

We study the rheology of a non-colloidal suspension of rigid spherical particles interacting with a viscoelastic matrix. Three-dimensional numerical simulations under shear flow are performed using the smoothed particle hydrodynamics method and compared with experimental data available in the literature using different constant-viscosity elastic Boger fluids. The rheological properties of the Boger matrices are matched in simulation under viscometric flow conditions. Suspension rheology under dilute to semi-concentrated conditions (i.e. up to solid volume fraction $\unicode[STIX]{x1D719}=0.3$) is explored. It is found that at small Deborah numbers $De$ (based on the macroscopic imposed shear rate), relative suspension viscosities $\unicode[STIX]{x1D702}_{r}$ exhibit a plateau at every concentration investigated. By increasing $De$, shear thickening is observed, which is related to the extensional thickening of the underlying viscoelastic matrix. Under dilute conditions ($\unicode[STIX]{x1D719}=0.05$), numerical results for $\unicode[STIX]{x1D702}_{r}$ agree quantitatively with experimental data in both the $De$ and $\unicode[STIX]{x1D719}$ dependences. Even under dilute conditions, simulations of full many-particle systems with no a priori specification of their spatial distribution need to be considered to recover precisely experimental values. By increasing the solid volume fraction towards $\unicode[STIX]{x1D719}=0.3$, despite the fact that the trend is well captured, the agreement remains qualitative with discrepancies arising in the absolute values of $\unicode[STIX]{x1D702}_{r}$ obtained from simulations and experiments but also with large deviations existing among different experiments. With regard to the specific mechanism of elastic thickening, the microstructural analysis shows that elastic thickening correlates well with the average viscoelastic dissipation function $\unicode[STIX]{x1D703}^{elast}$, requiring a scaling as $\langle \unicode[STIX]{x1D703}^{elast}\rangle \sim De^{\unicode[STIX]{x1D6FC}}$ with $\unicode[STIX]{x1D6FC}\geqslant 2$ to take place. Locally, despite the fact that regions of large polymer stretching (and viscoelastic dissipation) can occur everywhere in the domain, flow regions uniquely responsible for the elastic thickening are well correlated to areas with significant extensional component.

Analytical results for the apparent slip between two spheres in a simple biviscous model of a shear thinning fluid are presented. Velocity profiles and apparent slip lengths along the surfaces are analyzed in order to characterize the physical mechanism. It is shown that in this non-Newtonian model, the effect of shear-thinning limited to high-shear rates in the interstitial regions between close spheres can be alternatively interpreted as the onset of an apparent shear-rate dependent slippage effect. The results of the theory compare well with experiments from the literature showing the presence of surface slip on a particle approaching a planar wall. In terms of implications on suspensions rheology, the present results bridge the ’hidden’ solvent shear-thinning theory [A. Va ́zquez-Quesada et al. , Phys. Rev. Lett., 117, 108001-5 (2016)] with slip-based models presented recently in [M. Kroupa et al., Phys. Chem. Chem. Phys. 19, 5979-5984 (2017)] as a possible explanation on the mechanism behind the shear-thinning in hard-sphere non-Brownian suspensions.

In this paper, a novel semi-implicit lubrication dynamics method that can efficiently simulate dense non-colloidal suspensions is proposed. To reduce the computational cost in the presented methodology, inter-particle lubrication-based forces and torques alone are considered together with a short-range repulsion to enforce finite inter-particle separation due to surface roughness, Brownian forces or other excluded volume effects. Given that the lubrication forces are singular, i.e. scaling inversely with the inter-particle gap, the strategy to expedite the calculations is severely compromised if explicit integration schemes are used, especially at high concentrations. To overcome this issue, an efficient semi-implicit splitting integration scheme to solve for the particles translational and rotational velocities is presented. To validate the proposed methodology, a suspension under simple shear test is simulated in three dimensions and its rheology is compared against benchmark results. To demonstrate the stability/speed-up in the calculations, performance of the proposed semi-implicit scheme is compared against a classical explicit Velocity-Verlet scheme. The predicted viscometric functions for a non-colloidal suspension with a Newtonian matrix are in excellent agreement with the reference data from the literature. Moreover, the presented semi-implicit algorithm is found to be significantly faster than the classical lubrication dynamics methods with Velocity-Verlet integration schemes.

In this work a Smoothed Particle Hydrodynamics model is presented to study rheology and microstructure of a non-colloidal suspension of spherical particles in a Newtonian solvent. The scheme presented in (Bian and Ellero, 2014) is extended to three-dimensions incorporating both normal and tangential short-range interparticle lubrication forces which are solved implicitly with a refined splitting strategy. The scheme allows to bypass prohibitively small time steps generally required for handling divergent lubrication forces and allows to simulate large particle systems. Rheology of a three-dimensional hard-spheres suspension confined in a plane Couette rheometer is investigated for concentrations up to ϕ = 0.5 . Results for the relative suspension viscosity ηr are analyzed against sample size and numerical convergence of the splitting lubrication scheme and compared with available experimental and simulations data. Very close agreement with the experiments is obtained for ηr up to ϕ = 0.35 . At larger concentrations, our results are still unable to explain the significant viscosity increase observed in experiments. Modest shear-thickening is also observed which is related to anisotropy of the particle radial distribution function (RDF) and presence of reversible hydrodynamic aggregates increasing in size with increasingly applied shear rates.

This paper presents a numerical approach to predict the rheology of dense non-colloidal suspensions with a biviscous matrix. A biviscous matrix is characterized as a fluid with two shear rate dependent viscosities i.e. one above and below a critical shear rate γ ˙ c . The methodology is based on the lubrication dynamics which dominantly influence the suspension properties at high values of particle concentration. To efficiently handle the singular lubrication forces in the dense suspensions, a semi-implicit splitting integration scheme is employed. Using the presented approach, three dimensional simulations were performed and the predicted rheology of the suspension with a biviscous matrix is discussed under two regimes: (a) γ ˙ c larger than the macroscopic applied shear rate where fluid slippage effect can be modeled in terms of the non-Newtonian properties of the matrix, and (2) γ ˙ c smaller than the macroscopic applied shear rate where a biviscous model can be seen as a regularization of an apparent yield stress matrix. The results obtained at high γ ˙ c show that the shear thinning of the biviscous matrix in the inter-particle gaps, which can be interpreted as an apparent fluid slipping on the particle surface, provides an alternative mechanism to explain the experimentally observed shear-thinning of non-colloidal suspension with Newtonian matrices. At low γ ˙ c values, the predicted suspension properties and its microstructure corroborates the available experimental results on suspensions with yield stress fluids.

In this work, an analytical model for the behavior of superparamagnetic chains under the effect of\ud a rotating magnetic field is presented. It is postulated that the relevant mechanisms for describing\ud the shape and breakup of the chains into smaller fragments are the induced dipole-dipole magnetic\ud force on the external beads, their translational and rotational drag forces, and the tangential\ud lubrication between particles. Under this assumption, the characteristic S-shape of the chain can\ud be qualitatively understood. Furthermore, based on a straight chain approximation, a novel analytical\ud expression for the critical frequency for the chain breakup is obtained. In order to validate\ud the model, the analytical expressions are compared with full three-dimensional smoothed particle\ud hydrodynamics simulations of magnetic beads showing excellent agreement. Comparison with\ud previous theoretical results and experimental data is also reported.

Experiments and computations were carried out to explore the origins of shear-thinning in non-colloidal suspensions. Two grades of polydimethylsiloxane (silicone oil) and a glycerine/water mixture were used as matrices for the suspensions. The particles were 40 μm diameter polystyrene (PS) and polymethyl methacrylate (PMMA) spheres. We concentrated on 40% volume fraction suspensions where shear-thinning was clear. The silicone oil matrices were nearly Newtonian: at 24 °C the viscosity of the 1.15 Pa s sample showed a 2% drop in viscosity a shear rate of about 3000 s−1, the 13.2 Pa s sample showed a drop of 2% at a shear rate of approximately 100 s−1, and the glycerine/water sample appeared to be Newtonian at least up to 104 s−1. Mild shear-thinning was seen with all suspensions, beginning at shear rates of order 0.1–1 s−1, followed by a rapid reduction of torque in the parallel-plate system at shear rates of 14, 150 and 1000 s−1 respectively with the three matrices. These rapid reductions are ascribed to edge effects. Matching smoothed particle hydrodynamics (SPH) simulations were made. The silicone matrix viscosities were modelled by a Carreau–Yasuda (CY) fit up to shear rates of order 107 s−1. The agreement between computations and experiments is generally good for 40% volume fraction suspensions up to the shear rate where edge effects intervene in the experiments- there are no edge effects in the simulations. This confirms the suggestion by Vazquez-Quesada et al. [1] that ‘hidden’ high shear rates between particles, where the non-Newtonian matrix viscosity comes into play, can result in shear-thinning at the macroscopic level. For the glycerine/water matrix at low shear rates this mechanism does not apply and a separate mechanism based on variable interparticle friction is suggested; the two mechanisms can co-exist.

In this work, the inverse bi-viscous model [Vazquez-Quesada et al., “Planar channel flow of a discontinuous shear-thickening model fluid: Theory and simulation,” Phys. Fluids 29, 103104 (2017)] is used to describe a shear-thickening fluid. An analytical velocity profile in a planar Poiseuille flow is utilized to calculate an approximate solution to the generalized lubrication force between two close spheres interacting hydrodynamically in such a medium. This approximate analytical expression is compared to the exact numerical solution. The flow topology of the shear-thickening transition within the interparticle gap is also shown and discussed in relation to the behaviour of the lubrication force. The present result can allow in the future to perform numerical simulations of dense particle suspensions immersed in a shear-thickening matrix based on an effective lubrication force acting between pairwise interacting particles. This model may find additional values in representing experimental systems consisting of suspensions in shear thickening media, polymer coated suspensions, and industrial systems such as concrete.

A three-dimensional model for a suspension of rigid spherical particles in a Newtonian fluid is presented. The solvent is modeled with smoothed particle hydrodynamics method, which takes into account exactly the long-range multi-body hydrodynamic interactions between suspended spheres. Short-range lubrication forces which are necessary to simulate concentrated suspensions, are introduced pair-wisely based on the analytical solution of Stokes equations for approaching/departing objects. Given that lubrication is singular at vanishing solid particle separations, an implicit splitting integration scheme is used to obtain accurate results and at the same time to avoid prohibitively small simulation time steps. Hydrodynamic interactions between solid particles, at both long-range and short-range limits, are verified against theory in the case of two approaching spheres in a quiescent medium and under bulk shear flow, where good agreements are obtained. Finally, numerical results for the suspension viscosity of a many-particle system are shown and compared with analytical solutions available in the dilute and semi-dilute case as well as with previous numerical results obtained in the concentrated limit.

An analytical solution for the calculation of the normal lubrication force acting between two moving spheres embedded in a shear-thinning fluid represented by a bi-viscous model is provided. The resulting force between the suspended spheres exhibits a consistent transition between the Newtonian constant-viscosity limits and it reduces to the well-known standard Newtonian lubrication theory for viscosity-ratio approaching one. Effects of several physical parameters of the theory are analyzed under relevant physical conditions, i.e., for a prototypical case of two non-colloidal spheres immersed in a non-Newtonian fluid with rheology parameterized by a bi-viscosity model. Topological results for high/low-viscosity regions in the gap between spheres are also analyzed in detail showing a rich phenomenology. The presented model enables the extension of lubrication dynamics for suspensions interacting with non-Newtonian matrices and provides a clean theoretical framework for new numerical computations of flow of dense complex particulate systems.

Shear thinning-a reduction in suspension viscosity with increasing shear rates-is understood to arise in colloidal systems from a decrease in the relative contribution of entropic forces. The shear-thinning phenomenon has also been often reported in experiments with noncolloidal systems at high volume fractions. However its origin is an open theoretical question and the behavior is difficult to reproduce in numerical simulations where shear thickening is typically observed instead. In this letter we propose a non-Newtonian model of interparticle lubrication forces to explain shear thinning in noncolloidal suspensions. We show that hidden shear-thinning effects of the suspending medium, which occur at shear rates orders of magnitude larger than the range investigated experimentally, lead to significant shear thinning of the overall suspension at much smaller shear rates. At high particle volume fractions the local shear rates experienced by the fluid situated in the narrow gaps between particles are much larger than the averaged shear rate of the whole suspension. This allows the suspending medium to probe its high-shear non-Newtonian regime and it means that the matrix fluid rheology must be considered over a wide range of shear rates.

In this work, an analytical solution for the pressure-driven flow of a discontinuous shear-thickening (DST) fluid in a planar channel is presented. In order to model the fluid rheology, a regularized inverse-biviscous model is adopted. This involves a region of finite thickness to model the sharp jump in viscosity, and it is consistent with momentum conservation. In the limit of vanishing thickness, the truly DST behavior is obtained. Analytical results are validated by numerical simulations under steady and start-up flow using the smoothed particle hydrodynamics method. Flow results are investigated and discussed for different values of the model parameters.

This work presents an intermediate resolution model of the hydrodynamics of colloidal particles based on a mixed Eulerian-Lagrangian formulation. The particle is constructed with a small set of overlapping Peskin's Immersed Boundary kernels (blobs) which are held together by springs to build up a particle impenetrable core. Here, we used 12 blobs placed in the vertexes of an icosahedron with an extra one in its center. Although the particle surface is not explicitly resolved, we show that the short-distance hydrodynamic responses (flow profiles, translational and rotational mobilities, lubrication, etc) agree with spherical colloids and provide consistent effective radii. A remarkable property of the present multiblob model is that it naturally presents a "divergent" lubrication force at finite inter-particle distance. This permits to resolve the large viscosity increase at dense colloidal volume fractions. The intermediate resolution model is able to recover highly non-trivial (many-body) hydrodynamics using small particles whose radii are similar to the grid size $h$ (in the range $[1.6-3.2]\,h$). Considering that the cost of the embedding fluid phase scales like the cube of the particle radius, this result brings about a significant computational speed-up. Our code Fluam works in Graphics Processor Units (GPU's) and uses Fast Fourier Transform for the Poisson solver, which further improves its efficiency., 30 pages, 13 figures

A generalization of the Faxén’s theorem to the nonsteady motion of a sphere through a compressible linear viscoelastic fluid in arbitrary flow is presented. From this result, expressions for the velocity autocorrelation function (VAF) and the time-dependent diffusion coefficient of the particle can be obtained. We analyze the behavior of the VAF and the time-dependent diffusion coefficient for different physical regimes of the suspending fluid. The relevance of the theorem to microrheology is discussed, Support of the Spanish Government under Project No. FIS2007-65869-C03-03 is appreciated. A.V.-Q. thanks the projects MODELICO-CM-52009/ESP-1691 from Comunidad de Madrid and “Programa propio de la Investigación de la Universidad Nacional de Educación a Distancia, 2010” for financial support. M.E. financial support provided by the Deutsche Forschungsgemeinschaft (DFG) via the Grant No. EL503/1-2 is also gratefully acknowledged. Support from BIFI is also appreciated

Using Lagrangian simulations of a viscoelastic fluid modeled with an Oldroyd-B constitutive equation, we demonstrate that the flow through a closely spaced linear array of cylinders confined in a channel undergoes a transition to a purely elastic turbulent regime above a critical Weissenberg number ($We$). The high-$We$ regime is characterized by an unsteady motion and a sudden increase in the flow resistance in qualitative agreement with experimental observations. Furthermore, a power-law scaling behavior of the integral quantities as well as enhanced mixing of mass is observed. A stability analysis based on the dynamic mode decomposition method allows us to identify the most energetic modes responsible for the unsteady behavior, which correspond to filamental structures of polymer over- or underextension advected by the main flow preserving their shape. These time-dependent flow features strictly resemble the elastic waves reported in recent numerical simulations.

In this paper a numerical study on the performance of the Smoothed Particle Hydrodynamics method (SPH) in the case of a flow of a viscoelastic liquid around a linear array of cylinders confined in a channel is presented. Numerical convergence in the case of a low Reynolds number Newtonian flow was demonstrated by Ellero et al. in [Int. J. Numer. Methods Eng. 86 (2011) 1027]. Here viscoelastic effects are incorporated in the SPH scheme according to the Oldroyd-B model presented in [Phys. Rev. E 79 (2009) 056707]. Good agreement of the dimensionless drag force acting on the cylinder with literature data is observed for a wide range of Weissenberg numbers We. The case of closely spaced cylinders is also investigated and the impact of We on the solution discussed. It turns out that in this case the Newtonian solution exhibits a stable secondary flow represented by two counter-rotating vortices. When elasticity is considered, above a critical Weissenberg number We c these vortices become unstable, breaking the plane symmetry and producing a quasi-periodic flow of mass in and out of the gap region between cylinders. Correspondingly, the flow becomes increasingly unsteady and a dramatic increase in the cylinder’s drag is observed. The results are in qualitative agreement with experimental observations made on Boger liquids under similar flow conditions.

We present a fluid-particle model for a polymer solution in nonisothermal situations. The state of the fluid particles is characterized by the thermodynamic variables and a configuration tensor that describes the underlying molecular orientation of the polymer molecules. The specification of very simple physical mechanisms inspired by the dynamics of single polymer molecules allows one, with the help of the general equation for nonequilibrium reversible-irreversible coupling (GENERIC) formalism, to derive the equations of motion for a set of fluid particles carrying polymer molecules in suspension. In the simplest case of Hookean dumbbells we recover a fluid-particle version of the Oldroyd-B model in which thermal fluctuations are included consistently. Generalization to more complex viscoelastic models, such as finitely extensible nonlinear elastic Peterlin (FENE-P) model, with the proper introduction of thermal fluctuations is straightforward.

Dissipative particle dynamics (DPD) as a model of fluid particles suffers from the problem that it has no physical scale associated with the particles. Therefore, a DPD simulation requires an ambiguous fine-tuning of the model parameters with the physical parameters. A corrected version of DPD that does not suffer from this problem is smoothed dissipative particle dynamics (SDPD) [P. Espanol and M. Revenga, Phys. Rev. E 67, 026705 (2003)]. SDPD is, in fact, a version of the well-known smoothed particle hydrodynamics method, albeit with the proper inclusion of thermal fluctuations. Here, we show that SDPD produces the proper scaling of the fluctuations as the resolution of the simulation is varied. This is investigated in two problems: the Brownian motion of a spherical colloidal particle and a polymer molecule in suspension.