Plane Poiseuille Flow Research Articles

Viscoplastic flows in thin superhydrophobic channels

Via numerical computations, we study plane Poiseuille flow of a viscoplastic fluid in channels with the presence of a superhydrophobic (SH) groovy lower wall, where air is trapped inside the SH wall cavities. We assume this liquid/air interface to be flat while it is pinned at the groove edges. Our main focus is on the thin channel limit, for which the channel height is typically smaller than the groove period. To quantify the viscoplastic rheological behaviour, we rely on the Bingham model, via the Papanastasiou regularization method. We consider a no-slip boundary condition at the liquid/solid contacts while we model the liquid/air interface boundary using the Navier slip law. Focusing on the creeping flow limit (i.e. the Reynolds number is fixed to R=0.01), our main flow parameters are the Bingham number (B), the slip number (b), the groove periodicity length (ℓ), and the slip area fraction (φ). Our computational results quantify the effects of these parameters on the flow velocity fields, the unyielded plug zones and their areas, the effective slip lengths, and the friction factors. The shape of the unyielded plug zones are captured, allowing us to classify our flows into four main regimes, based on the characteristics of the unyielded zones, in particular the presence/absence of a broken center plug and/or an SH wall plug. We also find that the effective slip length generally increases with an increase in b, converging to a profile once an SH wall plug appears. Finally, the friction factor decreases with an increase in b, especially at larger B.

Instability of a plane Poiseuille flow bounded between inhomogeneous anisotropic porous layers

Instability of a plane Poiseuille flow bounded between inhomogeneous anisotropic porous layers

POD-based study of turbulent plane Poiseuille flow: comparing structure and dynamics between quasi-linear simulations and DNS

Turbulence in the restricted nonlinear (RNL) dynamics is analysed and compared with direct numerical simulations (DNS) of Poiseuille turbulence at Reynolds number $R=1650$ . The structures are obtained by proper orthogonal decomposition (POD) analysis of the two components of the flow partition used in RNL dynamics: the streamwise mean flow and fluctuations. POD analysis of the streamwise mean flow indicates that the dominant POD modes, in both DNS and RNL dynamics, are roll-streaks harmonic in the spanwise direction. However, we conclude that these POD modes do not occur in isolation but rather are Fourier components of a coherent roll-streak structure. POD analysis of the fluctuations in DNS and RNL dynamics reveals similar complex structures consisting in part of oblique waves collocated with the streak. The origin of these structures is identified by their correspondence to POD modes predicted using a stochastic turbulence model (STM). These predicted POD modes are dominated by the optimally growing structures on the streak, which the STM predicts correctly to be of sinuous oblique wave structure. This close correspondence between the roll-streak structure and the associated fluctuations in DNS, RNL dynamics and the STM implies that the self-sustaining mechanism operating in DNS is essentially the same as that in RNL dynamics, which has been associated previously with optimal perturbation growth on the streak.

Transverse electric-thermal-fluid instabilities in an electromagnetic heat exchanger

Electromagnetic (EM) heat exchangers (HX) are critical components in power beaming applications where EM waves are radiated towards an EM HX, which then converts incident energy into heat or mechanical work. An EM HX consists of a lossy ceramic and a fluid flow that maintains thermal contact with and transfers heat from the ceramic. These materials have loss factors which increase with temperature, so that beyond a critical temperature thermal runaway can take place. Stable characteristic temperatures, which depend on the rate of energy removal from the system, during high-power EM heating of ceramic materials suggest that coolants would be in the gaseous phase. As a first step, we consider a model EM HX system consisting of a horizontal channel containing a viscous, dielectric fluid with a constant coefficient of thermal expansion, bounded from below by a grounded ceramic receiver of finite thickness. The system is subject to plane EM waves, propagating normally to the channel from above, and polarized in the same direction as a plane Poiseuille flow of the coolant. With the Boussinesq approximation, we calculate the base state solution of the system and then investigate the linear stability of this base-state. We find three modes of instability: thermal runaway, Rayleigh-Bénard convection, and a novel instability which we call the fringe-field instability, that takes place in the plane normal to base-state flow direction.

Third-order accurate 13-moment equations for non-continuum transport phenomenon

The derivation of analytical equations of non-continuum macroscopic transport phenomena is underpinned by approximate descriptions of the particle distribution function and is required due to the inability of the Navier–Stokes equations to describe flows at high Knudsen number (Kn ∼ 1). In this paper, we present a compact representation of the second-order correction to the Maxwellian distribution function and 13-moment transport equations that contain fewer terms compared to available moment-based representations. The intrinsic inviscid and isentropic assumptions of the second-order accurate distribution function are then relaxed to present a third-order accurate representation of the distribution function, using which corresponding third-order accurate moment transport equations are derived. Validation studies performed for Grad’s second problem and the force-driven plane Poiseuille flow problem at non-zero Knudsen numbers for Maxwell molecules highlight an improvement over results obtained by using the Navier–Stokes equations and Grad’s 13-moment (G13) equations. To establish the ability of the proposed equations to accurately capture the bulk behavior of the fluid, the results of Grad’s second problem have been validated against the analytical solution of the Boltzmann equation. For the planar Poiseuille flow problem, validations against the direct simulation Monte Carlo method data reveal that, in contrast to G13 equations, the proposed equations are capable of accurately capturing the Knudsen boundary layer.

Stability of a layered reactive channel flow

We analyse the linear stability of a reactive plane Poiseuille flow, where a reactant fluid A overlies another reactant B in a layered fashion within a two-dimensional channel. Both reactants are miscible and have the same viscosity, while upon reaction, they produce either a less or more-viscous product fluid C . The reaction kinetics is of simple A + B → C type, and the production of C occurs across the initial contact line of reactants A and B in a mixed zone of small and finite width. All three fluids have the same density. We demonstrate the effects of various controlling parameters such as the log-mobility ratio, Damköhler number, Schmidt number, Reynolds number, position and thicknesses of the reactive zone on the stability characteristics. We show that a tiny viscosity stratification by the reaction destabilizes the flow at a moderate (10–1000) and even at low Reynolds numbers (0.01–1). The maximum growth occurs for shorter waves than for the Tollmien–Schlichting eigenmode, and the ranges of unstable wavenumbers are wider than that known for non-reactive channel flow systems. In most cases, the instability occurs due to the overlap of the critical layer with the viscosity-stratified layer. Surprisingly for some parameters, it is observed that the reaction can make σ M decrease with increasing Reynolds number.

Numerical study on the hydrodynamic properties of bentonite slurries with Herschel-Bulkley-Papanastasiou rheology model

During the construction of slurry shield tunnels, bentonite slurries play a significant part in maintaining excavation face stability where suitable slurries infiltrate into the stratum to form an impermeable filter cake. To enhance filter cake quality, accurate prediction of the hydrodynamic properties of bentonite slurries is always required during slurry parameters design. The main purpose of this study is to investigate the flow characteristics of bentonite slurries with different polymer contents using computational fluid dynamics (CFD) simulation. Mixtures of sodium bentonite powder, carboxymethyl cellulose (CMC), and water in different ratios were prepared, and a series of indoor rheological tests were conducted. Then an optimized numerical method for slurry flow simulation was developed by incorporating the Herschel- Bulkley-Papanastasiou (HBP) rheology model with the finite volume method (FVM). The proposed model was implanted in OpenFOAM, and then applied to theoretical cases of non-Newtonian plane-Poiseuille flow for model verification. Results show the theoretical and simulation results were in good accordance, with a successful description of the shear thinning characteristics of fluids and no divergency during the numerical solution procedure. Based on it, CFD simulations were implemented to investigate pipe flow characteristics of bentonite slurries with different CMC contents. Results indicate when the inlet velocity is constant, the decrease of pipe diameter can cause a more inhomogeneous flow and the growth of pressure drop; rises in rheological parameters can all increase the energy consumption and reduce the infiltration distance. When the yield stress is 25 Pa, the pressure drop reaches a maximum value of 5.47 kPa, compared with 2.69 kPa for slurries with a yield stress of 5 Pa. Furthermore, the infiltration law of bentonite slurries in pipes essentially conforms to a non-Darcy infiltration with an initial hydraulic gradient i0, which has a positive relationship with yield stress of fluids. i0 increases from 0.082 to 0.414 after adding 0.3% CMC into bentonite slurries. This research can provide significant guidance for the establishment of slurry infiltration theories and relevant numerical simulations.

A thick reaction zone model for premixed flames in two-dimensional channels

Direct interactions between the flow field and the chemical reaction in premixed flames occur when the reaction zone thickness is comparable to, or greater than flow length scales. To study such interactions, a laminar model is considered that has direct bearings to steadily propagating deflagrations in a Hele-Shaw channel with a background plane Poiseuille flow. The study employs asymptotic analyses, pertaining to large activation energy and lubrication theories and considers a distinguished limit where the channel width is comparable to the reaction zone thickness, with account being taken of thermal-expansion and heat-loss effects. The reaction zone structure and burning rates depend on three parameters, namely, the Peclet number, , the Lewis number, and the ratio of channel half-width to reaction zone thickness, . In particular, when the parameter is small wherein the reaction zone is thick, transport processes are found to be controlled by Taylor's dispersion mechanism and an explicit formula for the effective burning speed is obtained. The formula indicates that for , which interestingly coincides with a recent experimental prediction of the turbulent flame speed in a highly turbulent jet flame. The results suggest that the role played by differential diffusion effects is significant both in the laminar and turbulent cases. The reason for the peculiar dependence can be attributed, at least in our laminar model, to Taylor dispersion. Presumably, this dependence may be attributed to a similar but more general mechanism in the turbulent distributed reaction zone regime, rather than to diffusive-thermal curvature effects. The latter effects play however an important role in determining the effective propagation speed for thinner reaction zones, in particular, when is large in our model. It is found that the magnitude of heat losses at extinction, which directly affects the mixture flammability limits, is multiplied by a factor in comparison with those corresponding to the no-flow case in narrow channels.

Characteristics of advection–diffusion–Langmuir adsorption processes in two-dimensional plane Poiseuille flows

The advection–diffusion–Langmuir adsorption processes of a liquid solution, colloid, or suspension occur in many biomedical and chemical engineering fields. The dynamics of the system can be described by the so-called advection–diffusion–reaction (ADR) equations and are greatly influenced by five nondimensional numbers. Up to now, cases over a wider range of parameters have not been thoroughly studied, and the quantitative dependence of the system dynamics on the parameters remains unclear. In this study, we systematically solve the ADR equations in two-dimensional plane Poiseuille flows for cases with selected values of parameters by the finite difference method. We identify two different regimes in terms of the distribution of the maximum adsorption flux and discuss the dominant mechanism of mass transfer and the influences of the nondimensional parameters in each regime. We then propose analytical models to describe the influences of specific parameters on the adsorption equilibrium time. The results of this research may provide a convenient method to identify the dominant processes in the advection–diffusion–Langmuir adsorption system in future studies.

On the Stability and Control of Two-Dimensional Channel Flow

Abstract A direct numerical simulation is used to solve the two-dimensional plane Poiseuille flow for three different stability cases; specifically, they are a supercritical case at a Reynolds number of 10,000, a critical-stable case at a Reynolds number of 5772.22, and subcritical case at a Reynolds number of 1000. The perturbations are developed using the eigenfunctions of Orr–Sommerfeld equation's solution. In many applications, the flow was required to be fully laminar. As a result, a model-based controller is developed for stabilizing the flow field using unsteady strong suction and blowing technique, through distributed slots on the two walls of the channel. This technique is introduced to cancel the propagating wave in the boundary-layers near the two walls. Promising results are achieved for the unstable and critical-stable cases. The stable case is tested. Slight improvements in its growth rate are recorded. In turn, a faster response for the flow field is achieved.

Rheology and structure of elastic capsule suspensions within rectangular channels.

Three-dimensional simulations of the pressure-driven flow dynamics of elastic capsule suspensions within both slit and rectangular cross-section channels are presented. The simulations utilize the Immersed Boundary Method and the Lattice-Boltzmann Method models. The capsule volume fraction is fixed at 0.1 (i.e., a semi-dilute suspension), while the channel Reynolds number (Re), the capillary number (Ca), and the cross-sectional channel dimensions are systematically varied. Comparing results for slit and rectangular channels, it is found that multi-directional confinement hinders inertial focusing due to the capsule-free layers that develop in the two transverse directions. Furthermore, the thicknesses of the capsule-free layers in the two transverse directions differ when the height and width of the channel are not equal. Both the size and aspect ratio of the channel impact the apparent viscosity. It is found that square channels exhibit maximal viscosity and that holding one dimension fixed while increasing or decreasing the other results in a decrease in viscosity. The results therefore represent an expansion of the Fahraeus-Lindqvist effect from 1D cylindrical channels to 2D rectangular channels.

Symmetry-reduced dynamic mode decomposition of near-wall turbulence

Data-driven dimensionality reduction methods such as proper orthogonal decomposition and dynamic mode decomposition have proven to be useful for exploring complex phenomena within fluid dynamics and beyond. A well-known challenge for these techniques is posed by the continuous symmetries, e.g. translations and rotations, of the system under consideration, as drifts in the data dominate the modal expansions without providing an insight into the dynamics of the problem. In the present study, we address this issue for fluid flows in rectangular channels by formulating a continuous symmetry reduction method that eliminates the translations in the streamwise and spanwise directions simultaneously. We demonstrate our method by computing the symmetry-reduced dynamic mode decomposition (SRDMD) of sliding windows of data obtained from the transitional plane-Couette and turbulent plane-Poiseuille flow simulations. In the former setting, SRDMD captures the dynamics in the vicinity of the invariant solutions with translation symmetries, i.e. travelling waves and relative periodic orbits, whereas in the latter, our calculations reveal episodes of turbulent time evolution that can be approximated by a low-dimensional linear expansion.

A note on the slip effects of an Oldroyd 6-constant fluid: Optimal homotopy asymptotic method

We study here the effectiveness of the optimal homotopy asymptotic method (OHAM) in solving non-linear differential equations of non-Newtonian fluids. To this consequence, we consider the Oldroyd 6-constant fluid when it demonstrates slippage between the plate and fluid generating non-linear boundary value problems. The problems of plane Couette flow, generalized Couette flow, and plane Poiseuille flow are considered. Graphs of the results are plotted to show the performance of the method in terms of velocity profile. It is observed that the method is quite easy to implement, having latent potential to handle such kinds of non-linear problems and yield accurate results at minimum to low computational work.

A pairwise hydrodynamic theory for flow-induced particle transport in shear and pressure-driven flows

An exact pairwise hydrodynamic theory is developed for the flow-induced spatial distribution of particles in dilute polydisperse suspensions undergoing two-dimensional unidirectional flows, including shear and planar Poiseuille flows. Coupled diffusive fluxes and a drift velocity are extracted from a Boltzmann-like master equation. A boundary layer is predicted in regions where the shear rate vanishes with thickness set by the radii of the upstream collision cross-sections for pair interactions. An analysis of this region yields linearly vanishing drift velocities and non-vanishing diffusivities where the shear rate vanishes, thus circumventing the source of the singular particle distribution predicted by the usual models. Outside of the boundary layer, a power-law particle distribution is predicted with exponent equal to minus half the exponent of the local shear rate. Trajectories for particles with symmetry-breaking contact interactions (e.g. rough particles, permeable particles, emulsion drops) are analytically integrated to yield particle displacements given by quadratures of hard-sphere (or spherical drop) mobility functions. Using this analysis, stationary particle distributions are obtained for suspensions in Poiseuille flow. The scale for the particle distribution in monodisperse suspensions is set by the collision cross-section of the particles but its shape is almost universal. Results for polydisperse suspensions show size segregation in the central boundary layer with enrichment of smaller particles. Particle densities at the centreline scale approximately with the inverse square root of particle size. A superposition approximation reliably predicts the exact results over a broad range of parameters. The predictions agree with experiments in suspensions up to approximately 20 % volume fraction without fitting parameters.

Comparison of refilling schemes in the free-surface lattice Boltzmann method

Simulating mobile liquid–gas interfaces with the free-surface lattice Boltzmann method (FSLBM) requires frequent re-initialization of fluid flow information in computational cells that convert from gas to liquid. The corresponding algorithm, here referred to as the refilling scheme, is crucial for the successful application of the FSLBM in terms of accuracy and numerical stability. This study compares five refilling schemes that extract information from the surrounding liquid and interface cells by averaging, extrapolating, or assuming one of the three different equilibrium states. Six numerical experiments were performed, covering a broad spectrum of possible scenarios. These include a standing gravity wave, a rectangular and cylindrical dam break, a Taylor bubble, a drop impact into liquid, and a bubbly plane Poiseuille flow. In some simulations, the averaging, extrapolation, and one equilibrium-based scheme were numerically unstable. Overall, the results have shown that the simplest equilibrium-based scheme should be preferred in terms of numerical stability, computational cost, accuracy, and ease of implementation.

Linear stability of Poiseuille flow of viscoelastic fluid in a porous medium

We study the instability of plane Poiseuille flow of the viscoelastic second-order fluid in a homogeneous porous medium. The viscoelastic fluid between two parallel plates is driven by the pressure gradient. The effects of elasticity number E (depends on fluid properties, geometry; E is defined below) and Darcy number Da (gives the permeability of porous medium; Da is defined below) on flow stability are analyzed through the energy method that provides qualitative behavior of flow stability, and the numerical solution of generalized eigenvalue problem that gives the precise upper bound for stability. The plane Poiseuille flow of second-order fluid becomes unstable for increasing elasticity number while preserving Newtonian eigenspectrum up to a certain range of E. For large elasticity number, instability appears as a part of both wall and center modes for all Darcy numbers. We also noticed that along each neutral stability curve, the eigenfunctions are all antisymmetric with a single extremum near the channel walls. When E = 0.0011, we found an additional new elastic mode, which is unstable and also antisymmetric. For E < 0.0011, the neutral curves split into two lobes with different minima. The critical Reynolds number Rec is found to be decreasing (increasing) for higher (lower) values of fluid elasticity (Darcy number). Physical mechanisms are discussed in detail.

Study of Kerosene–Gold–DNA Nanoparticles in a Magnetized Radiative Poiseuille Flow with Thermo-Diffusion Impact

Due to their unique thermal and magnetic characteristics, gold-DNA nanoparticles have a wide spectrum of uses in pharmacology, drug delivery systems, treatment for cancer, and other disciplines. The current problem that analyzes the planar Poiseuille flow consists of gold nanoparticles with a typical fluid based on kerosene. The impact of mixed convection is considered in conjunction with the effects of radiative heat flow and thermo-diffusion (Soret). The numerical technique is utilized to solve the one-dimensional transformed equation for flow phenomena using the built-in MATLAB function bvp5c, with specific fixed values of relevant parameters adjusted. However, for different parameters that are either joint or unique, they are presented in both the surface and two-dimensional plots. It is observed that the particle concentration, as well as the resistive forces, favors greatly influencing the fluid velocity; nevertheless, raising the Peclet number also retards it owing to thermal conductivity retardation. The fluid concentration increases as the Reynolds number increases, but the shear rate decreases. Furthermore, in the conclusion section, the applications for the present research and future scope are discussed.

Three-dimensional simulation of lateral migration of fiber in a laminar channel flow

The lateral migration of a neutrally buoyant fiber in a three-dimensional (3D) channel flow bounded by four walls is numerically studied using a generalized immersed boundary-lattice Boltzmann method. The slender fiber is modeled as an elastic Kirchhoff rod to well reproduce its bending and twisting behaviors. Our simulations first examined the lateral migration of a flexible fiber released at six typical positions. The results show that the fiber accumulate at an isosurface of a specific shear rate value regardless of its initial lateral position. Thus, it would be more physical to characterize fiber accumulation behavior by the accumulation shear rate (γα*) rather than the accumulation position. The fiber basically migrates laterally along the shear rate gradient when there is a discrepancy between the local and the accumulation (target) shear rates. For a fiber of medium flexibility, varying its initial position does not change its γα*, while it could impact its orbits and tumbling plane. Compared to planar Poiseuille flows, more complex fiber orbits with out-of-plane deformations are observable in channel flows, and the fiber could adjust its tumbling plane to adapt to the surrounding shear rate gradient during its lateral migration. Our simulation results further confirm that the tumbling rate of the fiber increases with the increasing local shear rate. The impact of the effective flexibility of the fiber (Λeff) on its lateral migration is also explored by considering a broad range of effective flexibility (Λeff). The results show that the Λeff relates to the γα* non-monotonically, and five monotone intervals are captured. The results would enrich our understanding of fiber accumulation behavior in 3D wall-bounded flows and may also shed light on how we can sort fibers/filaments in microfluidic devices.

A model for estimating the hydraulic conductivity of bentonite under various density conditions

Abstract Bentonite is an important material for low-permeability engineering systems used in dams and hazardous waste facilities. While models to characterize the hydraulic conductivity of bentonite have been developed in previous studies, these models were not applicable to various density conditions for Na- and Ca-bentonite. In this study, we present a new model for estimating the hydraulic conductivity of bentonite applicable to a wide range of density conditions for Na- and Ca-bentonite. In order to consider flow paths in compacted bentonite, a lamination structure of montmorillonite stacks was assumed. Our hydraulic model discriminated interlayer pores and other pores by applying a method for estimating the probability of connected pores and hydraulic coefficients governed by the plane Poiseuille flow equation. The model was consistent with the experimental data investigated in previous studies on the hydraulic conductivity of Na-bentonite and was in good agreement with the data of Ca-bentonite in the range of an effective montmorillonite density (ρem) at 500 kg/m3 ≤ ρem ≤ 1,500 kg/m3 and 600 kg/m3 ≤ ρem, respectively. However, some experimental values on Ca-bentonite under low-density conditions were far higher than the model results in the case where Ca-bentonite forms flow paths of relatively large pores.

A novel high-order solver for simulation of incompressible flows using the flux reconstruction method and lattice Boltzmann flux solver

In this paper, a novel high-order solver using the flux reconstruction (FR) method and the lattice Boltzmann flux solver (LBFS) is proposed for accurately and efficiently simulating nearly incompressible flows. The existing LBFSs are all based on the finite volume (FV) scheme. The present work extends the application of the LBFS for the first time by combining the high-order FR scheme. Compared with the traditional high-order incompressible Navier-stokes (N-S) solvers, no particular techniques are needed for the current FR-LBFS to overcome the difficulty of the pressure-velocity coupling since the present method is a weakly compressible model. Moreover, unlike the traditional N-S solvers where the inviscid and viscous terms are treated separately, the inviscid and viscous fluxes in the current scheme are coupled and computed uniformly in FR-LBFS. In the present method, the common inviscid and viscous fluxes at the cell interface of the FR scheme are evaluated simultaneously by the local reconstruction of lattice Boltzmann equation (LBE) solution from macroscopic flow variables at solution points. Thus the discretization of the second-order partial derivative term is avoided. Compared with the high-order off-lattice Boltzmann methods (OLBM), the FR-LBFS is more stable, efficient and low-storage. No special techniques are needed to remove the stiffness of the governing equations which existing in the discrete velocity Boltzmann equation (DVBE). Compared with the high-order FV-LBFS, the FR-LBFS is compact for parallel computing by avoiding wide stencils on meshes. In addition, the present scheme has lower dissipation and can be more flexible to increase accuracy order. Numerical validations of the proposed method are implemented by simulating (a) Taylor-Green vortex problem, (b) steady plane Poiseuille flow, (c) lid-driven cavity flow, (d) laminar boundary layer and (e) flow past a square cylinder.