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This paper is a follow-up to Gérard-Varet and Hillairet (2020) on the derivation of accurate effective models for viscous dilute suspensions. The goal is to identify an effective Stokes equation providing an o(λ2) approximation of the exact fluid–particle system, with λ the solid volume fraction of the particles. This means that we look for an improvement of Einstein’s formula for the effective viscosity in the form μeff​(x)=μ+25​μρ(x)λ+μ2​(x)λ2. Under a separation assumption on the particles, we proved in the article above that if an o(λ2) Stokes effective approximation exists, the correction μ2​ is necessarily given by a mean field limit, which can then be studied and computed under further assumptions on the particle configurations. Roughly, we go here from the conditional result of the article above to an unconditional result: we show that such an o(λ2) Stokes approximation indeed exists, as soon as the mean field limit exists. This includes the case of periodic and random stationary particle configurations.

In this article, we combine a Fast Fourier Transform based computational approach and a supervised machine learning strategy to discover models for the anisotropic effective viscosity of shear-thinning fiber suspensions. Using the Fast Fourier Transform based computational approach, we study the effects of the fiber orientation state and the imposed macroscopic shear rate tensor on the effective viscosity for a broad range of shear rates of engineering process interest. We visualize the effective viscosity in three dimensions and find that the anisotropy of the effective viscosity and its shear rate dependence vary strongly with the fiber orientation state. Combining the results of this work with insights from literature, we formulate four requirements a model of the effective viscosity should satisfy for shear-thinning fiber suspensions with a Cross-type matrix fluid. Furthermore, we introduce four model candidates with differing numbers of parameters and different theoretical motivations, and use supervised machine learning techniques for non-convex optimization to identify parameter sets for the model candidates. By doing so, we leverage the flexibility of automatic differentiation and the robustness of gradient based, supervised machine learning. Finally, we identify the most suitable model by comparing the prediction accuracy of the model candidates on the fiber orientation triangle, and find that multiple models predict the anisotropic shear-thinning behavior to engineering accuracy over a broad range of shear rates.

Aqueous solutions with polymer additives often used to improve the macroscopic sweep efficiency in oil recovery typically exhibit non-Newtonian rheology. In order to predict the Darcy-scale effective viscosity μ eff required for practical applications often, semi-empirical correlations such as the Cannella or Blake–Kozeny correlation are employed. These correlations employ an empirical constant (“ C -factor”) that varies over three orders of magnitude with explicit dependency on porosity, permeability, fluid rheology and other parameters. The exact reasons for this dependency are not very well understood. The semi-empirical correlations are derived under the assumption that the porous media can be approximated by a capillary bundle for which exact analytical solutions exist. The effective viscosity μ eff ( v Darcy ) as a function of flow velocity is then approximated by a cross-sectional average of the local flow field resulting in a linear relationship between shear rate γ and flow velocity. Only with such a linear relationship, the effective viscosity can be expressed as a function of an average flow rate instead of an average shear rate. The local flow field, however, does in general not exhibit such a linear relationship. Particularly for capillary tubes, the velocity is maximum at the center, while the shear rate is maximum at the tube wall indicating that shear rate and flow velocity are rather anti-correlated. The local flow field for a sphere pack is somewhat more compatible with a linear relationship. However, as hydrodynamic flow simulations (using Newtonian fluids for simplicity) performed directly on pore-scale resolved digital images suggest, flow fields for sandstone rock fall between the two limiting cases of capillary tubes and sphere packs and do in general not exhibit a linear relationship between shear rate and flow velocity. This indicates that some of the shortcomings of the semi-empirical correlations originate from the approximation of the shear rate by a linear relationship with the flow velocity which is not very well compatible with flow fields from direct hydrodynamic calculations. The study also indicates that flow fields in 3D rock are not very well represented by capillary tubes.

It is demonstrated that exposing petroleum disperse systems to ultrasound significantly decreases their viscoelastic properties. The operating parameters of exposure of high-viscosity oil to ultrasound were experimentally determined in order to reduce its effective viscosity and pour point. The results of pilot tests of the developed ultrasound module indicate that exposing high-viscosity oil to ultrasound in flow mode can significantly upgrade existing technology for transporting high-viscosity oil.

Owing to its kinetic nature and distinctive computational features, the lattice Boltzmann method for simulating rarefied gas flows has attracted significant research interest in recent years. In this article, a lattice Boltzmann (LB) model is presented to study microchannel flows in the transition flow regime, which have gained much attention because of fundamental scientific issues and technological applications in various micro-electro-mechanical system (MEMS) devices. In the model, a Bosanquet-type effective viscosity is used to account for the rarefaction effect on gas viscosity. To match the introduced effective viscosity and to gain an accurate simulation, a modified second-order slip boundary condition with a new set of slip coefficients is proposed. Numerical investigations demonstrate that the results, including the velocity profile, the non-linear pressure distribution along the channel, and the mass flow rate, are in good agreement with the solution of the linearized Boltzmann equation, the direct simulation Monte Carlo (DSMC) results, and the experimental results over a broad range of Knudsen numbers. It is shown that taking the rarefaction effect on gas viscosity into consideration and employing an appropriate slip boundary condition can lead to a significant improvement in the modeling of rarefied gas flows with moderate Knudsen numbers in the transition flow regime.

We present analytic solutions for transverse flows of a Newtonian fluid within a narrow gap channel over/inside unidirectional fibrous porous media, considering microfluidic applications such as the interfacial slip over a lubrication-infused surface (LIS) and rheometry with corrugated surfaces. Based on the lubrication theory, the effective slip length and Navier-slip boundary condition were derived to allow the flow rate over a fluid/porous interface to be reproduced, and these were validated for a narrow gap of up to 1.5 times of the fiber radius. To model the dual-scale flow in both a fluid channel and porous media, the effective viscosity and stress jump coefficient in the Stokes–Brinkman model with continuous and jump stress conditions, respectively, were also derived analytically by matching the slip velocity and velocity gradient (or stress) at the interface. We show that the effective slip length, effective viscosity, and stress jump coefficient can be written as closed-form solutions as a function of the dimensionless void length and channel height for various porous architectures such as quadrilateral (Quad), compressed hexagonal (Hex1), and equilateral hexagonal (Hex2) fibrous porous architectures. The usefulness of the analytical results was validated with comparative numerical simulations.

With brain white matter can be considered as a periodic fibrous porous medium mainly consisting of axons and interstitial fluid (ISF), the corresponding hydraulic permeability reflects the resistance of ISF flow in the extracellular space (ECS), thus playing a key role in molecular transport and drug delivery. As the ECS exhibits a typical width of 10–80 nm, the ISF flow poses a microscale flow problem with the wettability effect, which may induce a flow with a slip boundary and hence remains elusive. In this study, we idealized brain white matter as a periodic fibrous porous medium to quantify the effect of wettability on its hydraulic permeability, with fluid viscosity and slip boundary duly accounted for. We found that wettability led to enlarged hydraulic permeability by diminishing the effective viscosity and enlarging the slip length. We also found that, for the square arrangement of axons, wettability had a greater effect on perpendicular permeability relative to the parallel one, while for hexagonally arranged axons, the opposite held. Results presented in this study may provide theoretical guidance for cerebral edema treatment regimens and drug delivery.

The present study examines the impact of effective viscosity and suction/injection on the two-dimensional boundary layer flow and heat transfer across a wedge immersed in a porous medium. In this study, we analyze the mechanisms associated with porous media and the fluid, focusing specifically on the viscosity ratio(effective viscosity to the dynamic viscosity) effects. The movement or progression of the fluid outside the boundary layer is acquired in the form of a concept of fluid distance. The governing nonlinear ordinary differential equations are derived from the boundary layer equations with suitable similarity transformations. Two approaches are utilized in this study: comprehensive numerical simulations that solve the nonlinear fully coupled fluid-wedge interaction issue and asymptotic approaches that solve the linearized equation-acquired at a significant distance away from the wedge and a small Prandtl number. A high level of concordance exists between the two methodologies in their predictive capabilities. The velocity and temperature distributions for different favorable pressure gradient and suction parameters are to reduce both momentum and thermal boundary layer thickness, while an opposite scenario is noticed for injection parameters. These results are shown to be a continuation of classical Falkner-Skan flows. The viscosity ratio plays a role in reducing the thickness of the boundary layer, leading to the fluid exhibiting adhesion to the surface of the wedge. Moreover, the effect of permeability-the presence of a porous medium, reduces the thickness of the boundary layer. A comprehensive examination of the outcomes and their associated hydrodynamics concerning the physical parameters is conducted and made in some detail.

The two key parameters of the Brinkman’s model for fluid flow in porous media—permeability and effective viscosity—are determined theoretically. The analytical solution of a 1D problem for the Brinkman equation for the fluid flow in a porous medium between two solid walls is compared with results of the numerical Stokes fully resolved flow model in a two-dimensional periodic cells containing solid inclusions. The boundary value problem for the Stokes flow is solved using the boundary element method. The dependencies of the permeability and effective viscosity on porosity for various types of configurations and shapes of solid inclusions are numerically studied. The product of permeability and effective viscosity is introduced as a new parameter. It is shown that this parameter is almost constant for a wide range of values of porosity excluding the small interval close to the unity value. The analytical estimation confirms the numerically obtained values of the parameter. The obtained dependencies are verified by the solution of the 2D Brinkman flow problem. It is shown that the permeability and effective viscosity determined give good agreement between the pressure fields and flow streamlines of the exact solution of the two-dimensional Brinkman equation and the numerical solution of the Stokes equations. The approximations for the dependencies of permeability and effective viscosity on the porosity are constructed on the basis of the numerical data obtained. The applicability of the Brinkman and Darcy models is discussed.

Lubrication performance dominates the rating life of grease-lubricated pitch bearings. Conventionally, the life modification factor is determined using base oil viscosity, whose validity is rarely verified. This work presents an effective viscosity-based method for life evaluation of wind turbine pitch bearings. The effective viscosity of grease is measured under actual operating conditions, and a comparative study is conducted against the conventional base oil viscosity method. The rationality of the proposed approach is validated by bearing life tests. Results indicate that the life modification factor calculated from effective viscosity agrees significantly better with test data. Adopting effective viscosity can substantially improve the accuracy of bearing life prediction. The proposed method provides a reliable and practical way to assess the lubrication performance and fatigue life of pitch bearings.