Figure 7 - uploaded by Billy Dean Todd
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P xy and ( ˙ γ ) as functions of y for average fluid densities of ρ = 0.442 and channel width L = 5.1.
Source publication
We present a detailed analysis of a hydrodynamic constitutive model recently applied to study the non-local viscosity of highly confined inhomogeneous fluids (Zhang et al 2004 J. Chem. Phys. 121 10778, Zhang et al 2005 J. Chem. Phys. 122 219901). This model makes the assumption that, for pore widths significantly greater than the width of the visco...
Contexts in source publication
Context 1
... an example of what the strain rate and stress profiles look like, in figure 7 we plot ˙ γ and P yx as functions of position in the channel. The fluid density is 0.442 and the fluid is confined to a narrow channel of length L = 5.1. ...
Context 2
... expression says that the stress is a linear function of position plus some nonlinear terms that are important only in the regions near the walls. This is seen to be the case, as observed in figure 7. If there were no walls, the stress would simply be a linear function of y (i.e., the convolution of an infinite ramp is another infinite ramp whose slope is given by the integral of the kernel). ...
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Citations
... [12][13][14] We, however, aim for simpler techniques to estimate the nanochannel flow solving Eq. (2), which for an inhomogeneous fluid requires the viscosity kernel to be position dependent. Past attempts to extract such a kernel of bounded systems have been inaccurate due to abundant challenges, 3,15,16 while deriving the nonlocal viscosity kernel of homogeneous fluids via the STF method [17][18][19][20] or the Green-Kubo formula 21 has been extensively demonstrated. Applying the local average density model (LADM), [22][23][24] a set of such isotropic kernels over a range of thermodynamic variables can be used as an approximate substitute to the viscosity kernel of an inhomogeneous system of the fluid displaying the same range of local thermodynamic variables in space. ...
... The real-space nonlocal viscosity kernels are then extracted in Fig. 1(b) by an inverse Fourier transform of Eqs. (15) and (16). For the Gaussian fit, this can be done analytically and results in When using Eq. ...
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 (WCA) fluid in a nanochannel. We estimate, using the local average density model (LADM), the fluid's viscosity kernel from isotropic bulk systems of corresponding state points by the Sinusoidal Transverse Force (STF) 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 (NEMD) simulations, with deviations observed mostly near the fluid-channel interface.
... Finally, Cadusch et al. 36 show that the use of a non-local translation invariant kernel is not exempt of problems in high density fluids in strong nanoconfinement. They state "the fundamental theoretical challenge that remains is to include the position dependence into the kernel so that it becomes a genuinely inhomogeneous function of space and also to appropriately model the boundary conditions at the fluid-wall interface, including stick/slip boundary conditions." ...
... In fact, it is possible to find the explicit expression of λ g and λ P by performing the momentum integrals in Eq. (36). One obtains ...
Density Functional Theory (DFT) is a successful and well-established theory for the study of the structure of simple and complex fluids at equilibrium. The theory has been generalized to dynamical situations when the underlying dynamics is diffusive as in, for example, colloidal systems. However, there is no such a clear foundation for Dynamic DFT (DDFT) for the case of simple fluids in contact with solid walls. In this work, we derive DDFT for simple fluids by including not only the mass density field but also the momentum density field of the fluid. The standard projection operator method based on the Kawasaki-Gunton operator is used for deriving the equations for the average value of these fields. The solid is described as featureless under the assumption that all the internal degrees of freedom of the solid relax much faster than those of the fluid (solid elasticity is irrelevant). The fluid moves according to a set of non-local hydrodynamic equations that include explicitly the forces due to the solid. These forces are of two types, reversible forces emerging from the free energy density functional, and accounting for impenetrability of the solid, and irreversible forces that involve the velocity of both the fluid and the solid. These forces are localized in the vicinity of the solid surface. The resulting hydrodynamic equations should allow one to study dynamical regimes of simple fluids in contact with solid objects in isothermal situations.
... This viscous term involving 3.3 Highlights 55 second derivatives is the usual viscosity term of the Navier-Stokes equations, which is here expressed in a non-local form. The use of non-local viscosities has been advocated recently in the field of nano-hydrodynamics [120,121]. The fourth order viscosity tensor η rr is given in terms of the correlation of the fluctuations of the fluid stress tensor. ...
In this work, we derive from first principles the equations of hydrodynamics near a solid wall, valid for the study of the nanoscale. We generalize Dynamic Density Functional Theory (DDFT) by including not only the mass density field as in usual approaches to DDFT, but also the momentum density field of the fluid. The solid is described as featureless under the assumption that all the internal degrees of freedom of the solid relax much faster than those of the fluid. In this new theory, the fluid moves according to a set of non-local hydrodynamic equations that include explicitly the forces due to the solid. These forces are of two types, reversible forces emerging from the free energy density functional, and accounting for impenetrability, and irreversible forces that involve the velocity of both the fluid and the solid. These forces are localized in the vicinity of the solid surface. The non-locality of the equations is due to the non-locality of the transport coefficients, which are given explicitly in terms of Green-Kubo formulae. We particularize this general hydrodynamic DDFT for simple fluids to the case of slit nanopores with planar flow configurations. In this simple geometry, only a reduced number of non-local transport coefficients (wall friction, slip friction, and viscous friction) are needed in this planar configuration. In the planar geometry, the continuum hydrodynamic equations for a fluid in a slit nanopore are discretized into bins. This allows us both, to compute explicitly the Green-Kubo expressions for the non-local transport coefficients, and to solve numerically the continuum hydrodynamic equations. The Green-Kubo formulae are computed from the time correlations of the force density on each bin that the solid exerts on the fluid, and the stress tensor of each bin . The phase functions trajectories are obtained from extensive Equilibrium Molecular Simulations by time span of 54ns. Thes Green-Kubo transport coefficients are subsequently used for the explicit numerical solution of the discrete hydrodynamic equations. Two initial non-equilibrium profiles of the plug and Poiseuille flow form are allowed to decay towards equilibrium. Non-Equilibrium Molecular Dynamics simulations and the predicted flow from the discrete hydrodynamic equation are then compared, with excellent agreement.Finally, we show that the present theory leads, in the limit of macroscopic flows, to a microscopic derivation of the Navier slip boundary conditions for the usual Navier-Stokes hydrodynamics, in which the slip length is given in microscopic terms., Resumen: En este trabajo, hemos derivado de primeros principios las ecuaciones hidrodinámicas para un fluido cercano a una pared sólida, válido para el estudio en la nanoescala. Generalizamos la Teoría Dinámica del Funcional de la Densidad (Dynamic Density Functional Theory (DDFT)), incluyendo no sólo el campo de densidad de masa como es habitual en el método de DDFT, sino también incluimos el campo de densidad de momento del fluido. Todas las moléculas del solido sólido se describen como un solo cuerpo (sin dinamicas en la nanoescala), bajo la suposición de que todos sus grados internos de libertad son más rápidos que los del fluido. El fluido se mueve de acuerdo con un conjunto de ecuaciones hidrodinámicas no locales que incluyen explícitamente las fuerzas debido al sólido. Estas fuerzas son de dos tipos, fuerzas reversibles que emergen de un funcional de densidad de energía libre, que da cuenta de la impenetrabilidad del fluido en el solido y las fuerzas irreversibles que involucran la velocidad tanto del fluido como del sólido. Estas fuerzas se localizan en la proximidad de la superficie sólida. La no localidad de las ecuaciones hidrodinámicas se debe a la no localidad de los coeficientes de transporte, que están dados por fórmulas de Green-Kubo. Hemos particularizado la teoría generalizada DDFT hidrodinámica obtenida previamente para un fluido simple en un nanoporo plano con configuraciones de flujo planar. Para esta geometría simple sólo se requiere un número reducido de coeficientes de transporte no locales (la fricción de la pared, la fricción “slip” y la fricción viscosa). Las ecuaciones hidrodinámicas continuas de un fluido en un nanoporo plano se han discretizado en capas. Esto posibilita dos cosas. Por una parte, nos permite calcular las expresiones explicitas de GreenKubo para los coeficientes de transporte no locales. Por otra, nos permite resolver numéricamente las ecuaciones hidrodinámicas de variables continuas. Los coeficientes tipo Green-Kubo se calculan a partir de las trayectorias de funciones de fase como la densidad de fuerza que ejerce el sólido sobre cada capa del fluido, y el tensor de tensión del fluido en cada capa. Las trayectorias de las funciones de fase se obtienen a partir de simulaciones de dinámica molecular (MD) en equilibrio para un periodo de tiempo equivalente a 54ns. Una vez calculados los coeficientes de transporte de Green-Kubo, estos se usan para obtener una solución numérica explícita de las ecuaciones hidrodinámicas. A partir de simulaciones de MD hemos obtenido los perfiles de la evolución temporal del fluido en condición de no equilibrio para validar el modelo mesoscópico aquí presentado. Las ecuaciones hidrodinámicas resultantes nos permiten estudiar dos regímenes dinámicos de fluidos simples en el flujo de plug y Poiseuille en nanoporos planos en situaciones isotérmicas con excelente concordancia entre las predicciones de la teoría y las simulaciones directas por MD en no equilibrio. Finalmente, mostramos que esta nueva teoría hidrodinámica permite, en el límite en el que los flujos son macroscópicos, deducir las condiciones de contorno de deslizamiento de Navier para las ecuaciones ordinarias de Navier-Stokes, en la que la longitud de deslizamiento está dada a través de una expresión microscópica.
... In the past few decades, many physical phenomena have been formulated into nonlocal mathematical models (see [1,2,8,7,15], [5,6,9], [11], [12], [14], [21] and references therein), and they have been extensively studied by many authors. In particular, Pérez-Llanos and Rossi ( [21]) studied the following non-local diffusion-like equation ...
We present new blow-up results for nonlocal reaction-diffusion equations with nonlocal nonlinearities. The nonlocal source terms we consider are of several types, and are relevant to various models in physics and engineering. They may involve an integral of an unknown function, either in space, in time, or both in space and time, or they may depend on localized values of the solution. We first show the existence and uniqueness of the solution to problem relying on contraction mapping fixed point theorem. Then, the comparison principles for problem are established through a standard method. Finally, for the radially symmetric and non-increasing initial data, we give a complete classification in terms of global and single point blow-up according to the parameters. Moreover, the blow-up rates are also determined in each case.
... At the nano-scale the fluid molecules form layers parallel to an interface, which causes the strain rate to vary rapidly within several molecular diameters [13]. These large variations mean the stress no longer has local linear behaviour [14,15]. ...
... We can now use Eqs. (10) and (14) to calculate the expected value of h 2 0 =2h 2 1 . When comparing this to the optimum found by the enhanced CFD we get excellent agreement, as highlighted in Fig. 9. ...
We demonstrate that a computational fluid dynamics (CFD) model enhanced with molecular-level information can accurately predict unsteady nano-scale flows in non-trivial geometries, while being efficient enough to be used for design optimisation. We first consider a converging–diverging nano-scale channel driven by a time-varying body force. The time-dependent mass flow rate predicted by our enhanced CFD agrees well with a full molecular dynamics (MD) simulation of the same configuration, and is achieved at a fraction of the computational cost. Conventional CFD predictions of the same case are wholly inadequate. We then demonstrate the application of enhanced CFD as a design optimisation tool on a bifurcating two-dimensional channel, with the target of maximising mass flow rate for a fixed total volume and applied pressure. At macro scales the optimised geometry agrees well with Murray’s Law for optimal branching of vascular networks; however, at nanoscales, the optimum result deviates from Murray’s Law, and a corrected equation is presented.
... The application of this relation is not straightforward [89,90] as it is unclear how the convolution should be performed at the wall where the support of the kernel goes beyond the boundary and is unknown [89,90]. Recently, Dalton et al. [91] used a sinusoidal longitudinal force (SLF), also introduced by Hoang and Galliero [87], to control the density variation in a periodic system. ...
... The application of this relation is not straightforward [89,90] as it is unclear how the convolution should be performed at the wall where the support of the kernel goes beyond the boundary and is unknown [89,90]. Recently, Dalton et al. [91] used a sinusoidal longitudinal force (SLF), also introduced by Hoang and Galliero [87], to control the density variation in a periodic system. ...
This paper introduces the fundamental continuum theory governing momentum
transport in isotropic nanofluidic flows. The theory is an extension to the
classical Navier-Stokes equation, which includes coupling between translational
and rotational degrees of freedom, as well as non-local response functions that
incorporates 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.
... For nano-confined fluids below the continuum limit, however, the strain rate can vary rapidly within several molecular diameters due to oscillations in the density; therefore, the stress becomes non-local. For quickly varying strain rates in homogeneous fluids, the non-local stress is calculable , but for confined fluids, this is an unsolved problem [see for example Cadusch et al. (2008); Todd (2005)]. In addition, tests of continuum-fluid equation performance at the nanoscale have been restricted to extremely simple geometries, typically Poiseuille flow [e.g. ...
We present a procedure for using molecular dynamics (MD) simulations to provide essential fluid and interface properties for subsequent use in computational fluid dynamics (CFD) calculations of nanoscale fluid flows. The MD pre-simulations enable us to obtain an equation of state, constitutive relations, and boundary conditions for any given fluid/solid combination, in a form that can be conveniently implemented within an otherwise conventional Navier–Stokes solver. Our results demonstrate that these enhanced CFD simulations are then capable of providing good flow field results in a range of complex geometries at the nanoscale. Comparison for validation is with full-scale MD simulations here, but the computational cost of the enhanced CFD is negligible in comparison with the MD. Importantly, accurate predictions can be obtained in geometries that are more complex than the planar MD pre-simulation geometry that provides the nanoscale fluid properties. The robustness of the enhanced CFD is tested by application to water flow along a (15,15) carbon nanotube, and it is found that useful flow information can be obtained.
... Under these conditions it is appropriate to relate the shear stress to the strain rate via a nonlocal viscosity kernel. Cadusch et al. [21] showed that a translationally invariant nonlocal viscosity kernel can be used to approximate the position-dependent kernel in a confined fluid when the confinement channel is significantly wider then the kernel width. For highly confined fluids, where the density inhomogeneities are significant across the whole channel, the translationally invariant kernel is no longer adequate and the stress may depend nonlocally on both the strain rate and the density. ...
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.
... A number of papers [79,86,327,328] in the last years showed local viscosity calculations from shear stress -strain rate relations as a function of location. For example, Todd et al. [293], Cadusch et al. [329] and Todd and Hansen [294] compared local and non-local constitutive relations in narrow rectangular channels with Weeks-Chandler-Andersen (WCA) atoms [77]. ...
... This means that the local shear viscosity is not just a linear combination of the local shear stress and strain rate, but can be a more complicated relation, for example one that contains an additional field or one that is nonlocal in space and time. The possibility of a spatially nonlocal shear viscosity is considered in several studies [293,294,329]. Finding a suitable kernel or other expression for shear viscosity for confined fluids is still an open problem and is not studied here. ...
... Note that the existence of a local relation is not guaranteed. The possibility of a non-local viscosity kernel, which is suggested in Refs.[293,329], is not studied here. ...
... Non-local theories have been invoked also, not only in the contest of the generalized hydrodynamics, [1,2,12,10], but also in the modeling of the stress in Brownian suspensions of rigid fibres, [38] as well as in the study of phase transitions, [8,22], and traveling waves, [9,15], in transport in plasmas, [31], image processing, [11,32], dynamics of population, [13], etc. ...
In this paper we study the blow-up phenomenon for the non-local pp-laplacian equation with a reaction term, ut(x,t)=∫AJ(x−y)|u(y,t)−u(x,t)|p−2(u(y,t)−u(x,t))dy+uq(x,t),x∈Ω,t∈[0,T], with Dirichlet conditions (A=RN,u≡0 in RN∖ΩRN∖Ω) or Neumann conditions (A=ΩA=Ω). Those problems are the non-local analogous to the equation vt=Δpv+vqvt=Δpv+vq with the corresponding conditions. We determine in both cases which are the global existence exponents, that coincide with the exponents of the corresponding local problem for Neumann boundary conditions. However, we observe differences with respect to the global existence exponents of the local Dirichlet problem. Moreover, we show that the blow-up rate is the same as the one that holds for the ODE ut=uqut=uq, that is, limt↗T(T−t)1q−1‖u(⋅,t)‖∞=(1q−1)1q−1. We also find some differences between the local and the non-local models concerning the blow-up sets. Precisely we show that regional blow-up is not possible for non-local problems. Finally, we include some numerical experiments which illustrate our results.
