Nonlocal shear stress for homogeneous fluids

Fluid flow at interfaces driven by thermal gradients

Thermal forces drive several nonequilibrium phenomena able to set a fluid in motion without pressure gradients. Although the most celebrated effect is thermophoresis, also known as Ludwig-Soret effect, probably the simplest example where thermal forces are at play is thermo-osmosis: The motion of a confined fluid exclusively due to the presence of a temperature gradient. We present a concise but complete derivation of the microscopic theory of thermo-osmosis based on linear response theory. This approach is applied to a simple fluid confined in a slab geometry, mimicking the flow through a pore in a membrane separating two fluid reservoirs at different temperatures. We consider both the case of an open channel, where the fluid can flow freely, and that of a closed channel, where mass transport is inhibited and a pressure drop sets in at the boundaries. Quantitative results require the evaluation of generalized transport coefficients, but a preliminary check on a specific prediction of the theory has been successfully performed via nonequilibrium molecular dynamics simulations.

Nonlocal hydrodynamic model for gravity-driven transport in nanochannels

It has been established that Newton’s law of viscosity fails for fluids under strong confinement as the strain-rate varies significantly over molecular length-scales. We thereby investigate if a nonlocal shear stress accounting for the strain-rate of an adjoining region by a convolution relation with a nonlocal viscosity kernel can be employed to predict the gravity-driven isothermal flow of a Weeks–Chandler–Andersen fluid in a nanochannel. We estimate, using the local average density model, the fluid’s viscosity kernel from isotropic bulk systems of corresponding state points by the sinusoidal transverse force method. A continuum model is proposed to solve the nonlocal hydrodynamics whose solutions capture the key features and agree qualitatively with the results of non-equilibrium molecular dynamics simulations, with deviations observed mostly near the fluid–channel interface.

Electropumping of nanofluidic water by linear and angular momentum coupling: theoretical foundations and molecular dynamics simulations

In this article we review the relatively new phenomenon of electropumping in nanofluidic systems, in which nonzero net flow results when polar molecules are rotated by external electric fields. The flow is a consequence of coupling of the spin angular momentum of molecules with their linear streaming momentum. By devising confining surfaces that are asymmetric - specifically one surface is more hydrophobic compared to the other - unidirectional flow results and so pumping can be achieved without the use of pressure gradients. We first cover the historical background to this phenomenon and follow that with a detailed theoretical description of the governing hydrodynamics. Following that we summarise work that has applied this phenomenon to pump water confined to planar nanochannels, semi-functionalised single carbon nanotubes and concentric carbon nanotubes. We also report on the energy efficiency of this pumping technique by comparisons with traditional flows of planar Couette and Poiseuille flow, with the surprising conclusion that electropumping at the nanoscale is some 4 orders of magnitude more efficient than pumping by Poiseuille flow.

Non-local viscosity from the Green-Kubo formula

We study through MD simulations the correlation matrix of the discrete transverse momentum density field in real space for an unconfined Lennard-Jones fluid at equilibrium. Mori theory predicts this correlation under the Markovian approximation from the knowledge of the non-local shear viscosity matrix, which is given in terms of a Green-Kubo formula. However, the running Green-Kubo integral for the non-local shear viscosity does not have a plateau. By using a recently proposed correction for the Green-Kubo formula that eliminates the plateau problem [Español et al., Phys. Rev. E 99, 022126 (2019)], we unambiguously obtain the actual non-local shear viscosity. The resulting Markovian equation, being local in time, is not valid for very short times. We observe that the Markovian equation with non-local viscosity gives excellent predictions for the correlation matrix from a time at which the correlation is around 80% of its initial value. A local in space approximation for the viscosity gives accurate results only after the correlation has decayed to 40% of its initial value.

Modeling of the early stage of atherosclerosis with emphasis on the regulation of the endothelial permeability

In this paper, we develop a mathematical model for the early stage of atherosclerosis, as a chronic inflammatory disease. It includes also processes that are relevant for the "thickening" of the vessel walls, and prepares a more complete model including also the later stages of atherosclerosis. The model consists of partial differential equations: Navier-Stokes equations modeling blood flow, Biot equations modeling the fluid flow inside the poroelastic vessel wall, and convection/chemotaxis-reaction-diffusion equations modeling transport, signaling and interaction processes initiating inflammation and atherosclerosis. The main innovations of this model are: a) quantifying the endothelial permeability to low-density-lipoproteins (LDL) and to the monocytes as a function of WSS, cytokines and LDL on the endothelial surface; b) transport of monocytes on the endothelial surface, mimicking the monocytes adhesion and rolling; c) the monocytes influx in the lumen, as a function of factor increasing monocytopoiesis; d) coupling between Navier-Stokes system, Biot system and convection/chemotaxis-reaction-diffusion equations. Numerical simulations of a simplified model were performed in an idealized two-dimensional geometry in order to investigate the dynamics of endothelial permeability, and the growth and spread of immune cells populations and their dependence in particular on low-density-lipoprotein and wall-shear stress.

Microscopic Slip Boundary Conditions in Unsteady Fluid Flows

An algebraic tail in the Green-Kubo integral for the solid-fluid friction coefficient hampers its use in the determination of the slip length. A simple theory for discrete nonlocal hydrodynamics near parallel solid walls with extended friction forces is given. We explain the origin of the algebraic tail and give a solution of the plateau problem in the Green-Kubo expressions. We derive the slip boundary condition with a microscopic expression for the slip length and the hydrodynamic wall position, and assess it through simulations of an unsteady plug flow.

Nonlocal Effects in Inhomogeneous Flows of Soft Athermal Disks

We numerically investigate nonlocal effects on inhomogeneous flows of soft athermal disks close to but below their jamming transition. We employ molecular dynamics to simulate Kolmogorov flows, in which a sinusoidal flow profile with fixed wave number is externally imposed, resulting in a spatially inhomogeneous shear rate. We find that the resulting rheology is strongly wave-number-dependent, and that particle migration, while present, is not sufficient to describe the resulting stress profiles within a conventional local model. We show that, instead, stress profiles can be captured with nonlocal constitutive relations that account for gradients to fourth order. Unlike nonlocal flow in yield stress fluids, we find no evidence of a diverging length scale.

Discrete hydrodynamics near solid planar walls

We derive, with the projection operator technique, the equations of motion for the time-dependent average of the discrete mass and momentum densities of a fluid confined by planar walls under the assumption that the flow field is translationally invariant along the directions tangent to the walls. Shear flow and sound propagation perpendicular to the walls can be described with the discrete hydrodynamic equations. The interaction with the walls is not given through boundary conditions but rather in terms of impenetrability and friction forces appearing in the discrete hydrodynamic equations. Microscopic expressions for the transport coefficients entering the discrete equations are provided. We further show that the obtained discrete equations can be interpreted as a Petrov-Galerkin finite-element discretization of the continuum equations presented by Camargo et al. [J. Chem. Phys. 148, 064107 (2018)JCPSA60021-960610.1063/1.5010401] when restricted to planar geometries and flows.

Challenges in Nanofluidics—Beyond Navier–Stokes at the Molecular Scale

The fluid dynamics of macroscopic and microscopic systems is well developed and has been extensively validated. Its extraordinary success makes it tempting to apply Navier–Stokes fluid dynamics without modification to systems of ever decreasing dimensions as studies of nanofluidics become more prevalent. However, this can result in serious error. In this paper, we discuss several ways in which nanoconfined fluid flow differs from macroscopic flow. We give particular attention to several topics that have recently received attention in the literature: slip, spin angular momentum coupling, nonlocal stress response and density inhomogeneity. In principle, all of these effects can now be accurately modelled using validated theories. Although the basic principles are now fairly well understood, much work remains to be done in their application.

Nonlocal Elasticity near Jamming in Frictionless Soft Spheres

We use simulations of frictionless soft sphere packings to identify novel constitutive relations for linear elasticity near the jamming transition. By forcing packings at varying wavelengths, we directly access their transverse and longitudinal compliances. These are found to be wavelength dependent, in violation of conventional (local) linear elasticity. Crossovers in the compliances select characteristic length scales, which signify the appearance of nonlocal effects. Two of these length scales diverge as the pressure vanishes, indicating that critical effects near jamming control the breakdown of local elasticity. We expect these nonlocal constitutive relations to be applicable to a wide range of weakly jammed solids, including emulsions, foams, and granulates.

Continuum Nanofluidics

This paper introduces the fundamental continuum theory governing momentum transport in isotropic nanofluidic systems. The theory is an extension of the classical Navier-Stokes equation, and includes coupling between translational and rotational degrees of freedom as well as nonlocal response functions that incorporate spatial correlations. The continuum theory is compared with molecular dynamics simulation data for both relaxation processes and fluid flows, showing excellent agreement on the nanometer length scale. We also present practical tools to estimate when the extended theory should be used. It is shown that in the wall-fluid region the fluid molecules align with the wall, and in this region the isotropic model may fail and a full anisotropic description is necessary.

Nonlocal response functions for predicting shear flow of strongly inhomogeneous fluids. II. Sinusoidally driven shear and multisinusoidal inhomogeneity

We use molecular-dynamics computer simulations to investigate the density, strain-rate, and shear-pressure responses of a simple model atomic fluid to transverse and longitudinal external forces. We have previously introduced a response function formalism for describing the density, strain-rate, and shear-pressure profiles in an atomic fluid when it is perturbed by a combination of longitudinal and transverse external forces that are independent of time and have a simple sinusoidal spatial variation. In this paper, we extend the application of the previously introduced formalism to consider the case of a longitudinal force composed of multiple sinusoidal components in combination with a single-component sinusoidal transverse force. We find that additional harmonics are excited in the density, strain-rate, and shear-pressure profiles due to couplings between the force components. By analyzing the density, strain-rate, and shear-pressure profiles in Fourier space, we are able to evaluate the Fourier coefficients of the response functions, which now have additional components describing the coupling relationships. Having evaluated the Fourier coefficients of the response functions, we are then able to accurately predict the density, velocity, and shear-pressure profiles for fluids that are under the influence of a longitudinal force composed of two or three sinusoidal components combined with a single-component sinusoidal transverse force. We also find that in the case of a multisinusoidal longitudinal force, it is sufficient to include only pairwise couplings between different longitudinal force components. This means that it is unnecessary to include couplings between three or more force components in the case of a longitudinal force composed of many Fourier components, and this paves the way for a highly accurate but tractable treatment of nonlocal transport phenomena in fluids with density and strain-rate inhomogeneities on the molecular length scale.

Nonlocal response functions for predicting shear flow of strongly inhomogeneous fluids. I. Sinusoidally driven shear and sinusoidally driven inhomogeneity

We present theoretical expressions for the density, strain rate, and shear pressure profiles in strongly inhomogeneous fluids undergoing steady shear flow with periodic boundary conditions. The expressions that we obtain take the form of truncated functional expansions. In these functional expansions, the independent variables are the spatially sinusoidal longitudinal and transverse forces that we apply in nonequilibrium molecular-dynamics simulations. The longitudinal force produces strong density inhomogeneity, and the transverse force produces sinusoidal shear. The functional expansions define new material properties, the response functions, which characterize the system's nonlocal response to the longitudinal force and the transverse force. We find that the sinusoidal longitudinal force, which is mainly responsible for the generation of density inhomogeneity, also modulates the strain rate and shear pressure profiles. Likewise, we find that the sinusoidal transverse force, which is mainly responsible for the generation of sinusoidal shear flow, can also modify the density. These cross couplings between density inhomogeneity and shear flow are also characterized by nonlocal response functions. We conduct nonequilibrium molecular-dynamics simulations to calculate all of the response functions needed to describe the response of the system for weak shear flow in the presence of strong density inhomogeneity up to the third order in the functional expansion. The response functions are then substituted directly into the truncated functional expansions and used to predict the density, velocity, and shear pressure profiles. The results are compared to the directly evaluated profiles from molecular-dynamics simulations, and we find that the predicted profiles from the truncated functional expansions are in excellent agreement with the directly computed density, velocity, and shear pressure profiles.

Dependence of the atomic level Green-Kubo stress correlation function on wavevector and frequency: molecular dynamics results from a model liquid

We report on a further investigation of a new method that can be used to address vibrational dynamics and propagation of stress waves in liquids. The method is based on the decomposition of the macroscopic Green-Kubo stress correlation function into the atomic level stress correlation functions. This decomposition, as was demonstrated previously for a model liquid studied in molecular dynamics simulations, reveals the presence of stress waves propagating over large distances and a structure that resembles the pair density function. In this paper, by performing the Fourier transforms of the atomic level stress correlation functions, we elucidate how the lifetimes of the stress waves and the ranges of their propagation depend on their frequency, wavevector, and temperature. These results relate frequency and wavevector dependence of the generalized viscosity to the character of propagation of the shear stress waves. In particular, the results suggest that an increase in the value of the frequency dependent viscosity at low frequencies with decrease of temperature is related to the increase in the ranges of propagation of the stress waves of the corresponding low frequencies. We found that the ranges of propagation of the shear stress waves of frequencies less than half of the Einstein frequency extend well beyond the nearest neighbor shell even above the melting temperature. The results also show that the crossover from quasilocalized to propagating behavior occurs at frequencies usually associated with the Boson peak.

Local shear viscosity of strongly inhomogeneous dense fluids: from the hard-sphere to the Lennard-Jones fluids

This work aims at providing a tractable approach to model the local shear viscosity of strongly inhomogeneous dense fluids composed of spherical molecules, in which the density variations occur on molecular distance. The proposed scheme, which relies on the local density average model, has been applied to the quasi-hard-sphere, the Week-Chandler-Andersen and the Lennard-Jones fluids. A weight function has been developed to deal with the hard-sphere fluid given the specificities of momentum exchange. To extend the approach to the smoothly repulsive potential, we have taken into account that the non-local contributions to the viscosity due to the interactions of particles separated by a given distance are temperature dependent. Then, using a simple perturbation scheme, the approach is extended to the Lennard-Jones fluids. It is shown that the viscosity profiles of inhomogeneous dense fluids deduced from this approach are consistent with those directly computed by non-equilibrium molecular dynamics simulations.

Effects of nanoscale density inhomogeneities on shearing fluids

It is well known that density inhomogeneities at the solid-liquid interface can have a strong effect on the velocity profile of a nanoconfined fluid in planar Poiseuille flow. However, it is difficult to control the density inhomogeneities induced by solid walls, making this type of system unsuitable for a comprehensive study of the effect on density inhomogeneity on nanofluidic flow. In this paper, we employ an external force compatible with periodic boundary conditions to induce the density variation, which greatly simplifies the problem when compared to flow in nonperiodic nanoconfined systems. Using the sinusoidal transverse force method to produce shearing velocity profiles and the sinusoidal longitudinal force method to produce inhomogeneous density profiles, we are able to observe the interactions between the two property inhomogeneities at the level of individual Fourier components. This gives us a method for direct measurement of the coupling between the density and velocity fields and allows us to introduce various feedback control mechanisms which customize fluid behavior in individual Fourier components. We briefly discuss the role of temperature inhomogeneity and consider whether local thermal expansion due to nonuniform viscous heating is sufficient to account for shear-induced density inhomogeneities. We also consider the local Newtonian constitutive relation relating the shear stress to the velocity gradient and show that the local model breaks down for sufficiently large density inhomogeneities over atomic length scales.

How fast does water flow in carbon nanotubes?

The purpose of this paper is threefold. First, we review the existing literature on flow rates of water in carbon nanotubes. Data for the slip length which characterizes the flow rate are scattered over 5 orders of magnitude for nanotubes of diameter 0.81-10 nm. Second, we precisely compute the slip length using equilibrium molecular dynamics (EMD) simulations, from which the interfacial friction between water and carbon nanotubes can be found, and also via external field driven non-equilibrium molecular dynamics simulations (NEMD). We discuss some of the issues in simulation studies which may be reasons for the large disagreements reported. By using the EMD method friction coefficient to determine the slip length, we overcome the limitations of NEMD simulations. In NEMD simulations, for each tube we apply a range of external fields to check the linear response of the fluid to the field and reliably extrapolate the results for the slip length to values of the field corresponding to experimentally accessible pressure gradients. Finally, we comment on several issues concerning water flow rates in carbon nanotubes which may lead to some future research directions in this area.

Local viscosity of a fluid confined in a narrow pore

In this paper, molecular dynamics simulations of a simple Lennard-Jones fluid confined in narrow slit pores and undergoing shear have been performed. The aim is to investigate the effects of density inhomogeneities at the fluid-solid interfaces on the shear viscosity profiles. It has been found that the local viscosity was varying strongly with the distance from the solid walls for both dilute and dense fluid states with oscillations correlated to the density ones. To describe the computed viscosity profiles, we propose a scheme that uses the local average density model, combined with an adequate weight function, for the configurational viscosity and a semiempirical model for the translational viscosity. It is shown that the proposed approach is able to provide viscosity profiles in good agreement with those coming from simulations for different pore widths and for different fluid states (dilute to dense).

Nonlocal viscosity kernel of mixtures

In this Brief Report we investigate the multiscale hydrodynamical response of a liquid as a function of mixture composition. This is done via a series of molecular dynamics simulations in which the wave-vector-dependent viscosity kernel is computed for three mixtures, each with 7-15 different compositions. We observe that the viscosity kernel is dependent on composition for simple atomic mixtures for all the wave vectors studied here; however, for a molecular mixture the kernel is independent of composition for large wave vectors. The deviation from ideal mixing is also studied. Here it is shown that the Lorentz-Berthelot interaction rule follows ideal mixing surprisingly well for a large range of wave vectors, whereas for both the Kob-Andersen and molecular mixtures large deviations are found. Furthermore, for the molecular system the deviation is wave-vector dependent such that there exists a characteristic correlation length scale at which the ideal mixing goes from underestimating to overestimating the viscosity.

Nanoflow hydrodynamics

We show by nonequilibrium molecular dynamics simulations that the Navier-Stokes equation does not correctly describe water flow in a nanoscale geometry. It is argued that this failure reflects the fact that the coupling between the intrinsic rotational and translational degrees of freedom becomes important for nanoflows. The coupling is correctly accounted for by the extended Navier-Stokes equations that include the intrinsic angular momentum as an independent hydrodynamic degree of freedom.

Prediction of fluid velocity slip at solid surfaces

The observed flow enhancement in highly confining geometries is believed to be caused by fluid velocity slip at the solid wall surface. Here we present a simple and highly accurate method to predict this slip using equilibrium molecular dynamics. Unlike previous equilibrium molecular dynamics methods, it allows us to directly compute the intrinsic wall-fluid friction coefficient rather than an empirical friction coefficient that includes all sources of friction for planar shear flow. The slip length predicted by our method is in excellent agreement with the slip length obtained from direct nonequilibrium molecular dynamics simulations.

Molecular transport and flow past hard and soft surfaces: computer simulation of model systems

The equilibrium and flow of polymer films and drops past a surface are characterized by the interface and surface tensions, viscosity, slip length and hydrodynamic boundary position. These parameters of the continuum description are extracted from molecular simulations of coarse-grained models. Hard, corrugated substrates are modelled by a Lennard-Jones solid while polymer brushes are studied as prototypes of soft, deformable surfaces. Four observations are discussed. (i) If the surface becomes strongly attractive or is coated with a brush, the Navier boundary condition fails to describe the effect of the surface independently of the strength and type of the flow. This failure stems from the formation of a boundary layer with an effective, higher viscosity. (ii) In the case of brush-coated surfaces, flow induces a cyclic, tumbling motion of the tethered chain molecules. Their collective motion gives rise to an inversion of the flow in the vicinity of the grafting surfaces and leads to strong, non-Gaussian fluctuations of the molecular orientations. The flow past a polymer brush cannot be described by Brinkman's equation. (iii) The hydrodynamic boundary condition is an important parameter for predicting the motion of polymer droplets on a surface under the influence of an external force. Their steady-state velocity is dictated by a balance between the power that is provided by the external force and the dissipation. If there is slippage at the liquid-solid interface, the friction at the solid-liquid interface and the viscous dissipation of the flow inside the drop will be the dominant dissipation mechanisms; dissipation at the three-phase contact line appears to be less important on a hard surface. (iv) On a soft, deformable substrate like a polymer brush, we observe a lifting-up of the three-phase contact line. Controlling the grafting density and the incompatibility between the brush and the polymer liquid we can independently tune the softness of the surface and the contact angle and thereby identify the parameters for maximizing the deformation at the three-phase contact.

Viscosity kernel of molecular fluids: butane and polymer melts

The wave-vector dependent shear viscosities for butane and freely jointed chains have been determined. The transverse momentum density and stress autocorrelation functions have been determined by equilibrium molecular dynamics in both atomic and molecular hydrodynamic formalisms. The density, temperature, and chain length dependencies of the reciprocal and real-space viscosity kernels are presented. We find that the density has a major effect on the shape of the kernel. The temperature range and chain lengths considered here have by contrast less impact on the overall normalized shape. Functional forms that fit the wave-vector-dependent kernel data over a large density and wave-vector range have also been tested. Finally, a structural normalization of the kernels in physical space is considered. Overall, the real-space viscosity kernel has a width of roughly 3-6 atomic diameters, which means that generalized hydrodynamics must be applied in predicting the flow properties of molecular fluids on length scales where the strain rate varies sufficiently in the order of these dimensions (e.g., nanofluidic flows).

Corrected second-order slip boundary condition for fluid flows in nanochannels

A corrected second-order slip boundary condition is proposed to solve the Navier-Stokes equations for fluid flows confined in parallel-plate nanochannels. Compared with the classical second-order slip boundary condition proposed by Beskok and Karniadakis, the corrected slip boundary condition is not only dependent on the Knudsen number and the tangential momentum accommodation coefficient, but also dependent on the relative position of the slip surface in the Knudsen layer. For the fluid flows in slip-flow regime with the Knudsen number less than 0.3, Couette cell is investigated using molecular-dynamics simulations to verify Newtonian flow behaviors by examining the constitutive relationship between shear stress and strain rate. By comparing the velocity profiles of Poiseuille flows predicted from the Navier-Stokes equations with the corrected slip boundary condition with that from molecular-dynamics simulations, it is found that the flow behaviors in our models can be effectively captured.

An extended analysis of the viscosity kernel for monatomic and diatomic fluids

We present an extended analysis of the wavevector dependent shear viscosity of monatomic and diatomic (liquid chlorine) fluids over a wide range of wavevectors and for a variety of state points. The analysis is based on equilibrium molecular dynamics simulations, which involve the evaluation of transverse momentum density and shear stress autocorrelation functions. For liquid chlorine we present the results in both atomic and molecular formalisms. We find that the viscosity kernel of chlorine in the atomic representation is statistically indistinguishable from that in the molecular representation. The results further suggest that the real space viscosity kernels of monatomic and diatomic fluids depend sensitively on the density, the potential energy function and the choice of fitting function in reciprocal space. It is also shown that the reciprocal space shear viscosity data can be fitted to two different simple functional forms over the entire density, temperature and wavevector range: a function composed of n-Gaussian terms and a Lorentzian-type function. Overall, the real space viscosity kernel has a width of 3-6 atomic diameters, which means that the generalized hydrodynamic constitutive relation is required for fluids with strain rates that vary nonlinearly over distances of the order of atomic dimensions.

Lennard-Jones fluid-fluid interfaces under shear

Using nonequilibrium molecular dynamics simulations on simple Lennard-Jones binary mixtures, we have studied the behavior of planar fluid-fluid interfaces undergoing shear flow. When the miscibility is low enough, a slip together with a partial depletion have been noticed at the interface between the two fluid phases. The slip length can reach a value equal to some molecular diameters and the corresponding interfacial viscosity can be two times smaller than the value in the bulk. It is shown how the omission of this slip may lead to flow-rate misevaluation when dealing with a multiphase flow in a nanoporous medium even for non polymer fluids. In addition, using the simulation results, a simple relation between interfacial tension and interfacial viscosity is proposed for the monoatomic systems studied in this work. Finally, it is shown that the interfacial viscosity cannot be fully accounted for by estimating the local viscosity deduced from the local thermodynamic properties of the interface.

Nonlocal viscous transport and the effect on fluid stress

We demonstrate that, in general, only for fluid flows in which the gradient of the strain rate is constant or zero can the classical Navier-Stokes equations with constant transport coefficients be considered exact. This is typical of two of the most common types of flow: Couette and Poiseuille. For more complicated flow fields in which the streaming velocity involves higher order nonlinear terms, the use of nonlocal constitutive equations gives an exact description of the flow. These constitutive equations involve nonlocal transport kernels. For momentum transport we demonstrate that nonlocality will be significant for any particular flow field if the even moments of the nonlocal viscosity kernel are non-negligible. This corresponds to the condition that the strain rate varies appreciably over the width of the kernel in real space. Such conditions are likely to be dominant for nanofluidic flows.