Pressure-driven Poiseuille Flow Research Articles

On pressure-driven Poiseuille flow with non-monotonic rheology.

Shear thickening fluids are liquids that stiffen as the applied stress increases. If many of these types of fluids follow a monotonic rheological curve, some experimental and numerical studies suggest that certain fluids, like cornstarch, may exhibit a non-monotonic, S-shaped rheology. Such non-monotonic behavior has however proved very difficult to observe experimentally in classical rheometer. To explain such difficulties, the possible presence of vorticity banding in the rheometer has been considered. To prevent such instabilities, we use a capillary rheometer, which is a cylindrical tube, measuring the flow rate versus the applied pressure drop. With this setup, we indeed observe a non-monotonic behavior: the flow rate increases monotonically at low pressure drops up to a maximum, after which it abruptly decreases to an almost constant flow rate regardless of further increases in pressure drop. This maximum-jump-plateau behavior occurs over a wide range of concentrations and is reproducible without hysteresis, which is in agreement with an S-shaped rheology. However, the obtained flow versus pressure difference function does not agree with the classical Wyart-Cates rheological model, which predicts an S-shaped non-monotonic function, but with neither a jump nor a plateau. To understand this jump-plateau behavior, we remark that any rheological model would establish a relationship between the flow rate and the local pressure gradient, but not the total pressure drop. We thus discuss and analyze the implications of having an S-shaped non-monotonic flow rate-pressure gradient in Poiseuille flow. In particular, we discuss the possibility of a non-uniform pressure gradient in the direction of the flow, i.e., a kind of streamwise banding. The key issue is then the selection of the gradient pressure distribution along the tube. One solution could arise from an analogy of this problem with the spinodal decomposition. It, however, leads to an increase in flow rate with up to a plateau between two values of as determined by the Maxwell construction. To account for the bump-jump behavior, we have implemented a simple dynamical stochastic version of the Wyart-Cates model, where the thickening occurs with a characteristic time. As a result, with increasing the total pressure drop, the flow rate increases monotonically up to a maximum value. Beyond this point, the flow rate drops abruptly to a lower value, forming a slowly decreasing plateau. This behavior is likely to account for the maximum-jump-plateau observed in the experiments. We also show that in such a system, the final state is quite sensitive to the initial state of the fluid, especially its homogeneity. Our results then demonstrate that the mere presence of a non-monotonic rheological curve is sufficient to predict the occurrence of stress banding in the streamwise direction and a plateau flow rate, even if the suspension remains homogeneous.

Electrophoresis of a small sphere in a cylinder of finite length.

The pressure due to electrophoretic motion of a charged colloidal sphere in a fluid-filled circular cylinder is determined in the limit in which the sphere radius is small compared with that of the cylinder. If the ends of the cylinder are open, pressure-driven Poiseuille flow occurs, but the magnitude of this flow is shown to be small when the cylinder is long compared to its radius. It is concluded that the flow has little effect upon electrophoretic velocities, unlike when the diameter of the sphere is comparable to that of the cylinder in which case the Poiseuille flow increases electrophoretic velocities and creates long-range interactions betweenspheres.

A discrete unified gas-kinetic scheme for multi-species rarefied flows

A discrete unified gas kinetic scheme (DUGKS) is developed for multi-species flow in all flow regimes based on the Andries-Aoki-Perthame (AAP) kinetic model. Although the species collision operator in the AAP model conserves fully the mass, momentum, and energy for the mixture, it does not conserve the momentum and energy for each species due to the inter-species collisions. In this work, the species collision operator is decomposed into two parts: one part is fully conservative for the species and the other represents the excess part. With this decomposition, the kinetic equation is solved using the Strang-splitting method, in which the excess part of the collision operator is treated as a source, while the kinetic equation with the species conservative part is solved by the standard DUGKS. Particularly, the time integration of the source term is realized by either explicit or implicit Euler scheme. By this means, it is easy to extend the scheme to gas mixtures composed of Maxwell or hard-sphere molecules, while the previous DUGKS [Zhang Y, Zhu L, Wang R et al, Phys Rev E 97(5):053306, 2018] of binary gases was only designed for Maxwell molecules. Several tests are performed to validate the scheme, including the shock structure under different Mach numbers and molar concentrations, the Couette flow under different mass ratios, and the pressure-driven Poiseuille flow in different flow regimes. The results are compared with those from other reliable numerical methods based on different models. And the influence of molecular model on the flow characteristics is studied. The results also show that the present DUGKS with implicit source discretization is more stable and preferable for gas mixture problems involving different flow regimes.

Taming Taylor-Aris dispersion through chaotic advection

Taylor-Aris dispersion represents an undesired phenomenon in pressure-driven liquid chromatography, often responsible for the unchecked increase of the Height Equivalent of the Theoretical Plate (HETP) when high throughput operating conditions are sought. Previous work on the subject showed how it is possible to contain the augmented dispersion in empty microchannels by inducing cross-sectional velocity components yielding an overall helical structure of the flow streamlines. Here, we explore the possibility of further reducing axial dispersion by devising flow conditions that give rise to chaotic streamlines. A three-dimensional steady flow generated by the combination of a pressure-driven Poiseuille flow and an electroosmotically-induced spatially periodic flow is used as a case study. Brenner’s macrotransport approach is used to predict the axial dispersion coefficient of a diffusing solute in flows possessing regular, partially chaotic and globally chaotic kinematic features. Accurate Lagrangian-stochastic simulations of particle ensembles are used to validate the predictions obtained through Brenner’s paradigm. Our findings suggest that the Taylor-Aris phenomenon can be altogether suppressed in the limit of globally chaotic kinematics. A theoretical interpretation of this outcome is developed by combining macrotransport theory with established results of the spectral approach to mixing in advecting-diffusing chaotic flows.

Optical investigation of dense suspensions with non-Newtonian matrix under pulsating pumping

Pressure-driven Poiseuille flow of a bidisperse, non-colloidal, dense granular suspension with non-Newtonian matrix under pulsatile pumping is studied experimentally. The flow properties are recorded using a transmitted light optical system. The setup allows to derive velocity profiles with a high spatial and temporal resolution. Statements can be made about the near-wall region, which are difficult to achieve experimentally in dense suspensions. The results are compared in detail with velocity profiles based on theoretical models from the field of suspension rheology and particle migration under the conditions present in the experiment. • Spatially and temporally resolved velocity profiles of a pulsating pipe flow. • Complete comparison of experimental data with rheological and particle migration models. • Determination of wall slip coefficients and comparison with predictions of theoretical models. • Evaluation of velocity profiles during acceleration and deceleration phases (pulsating pumping). • Investigation of different particle distributions with regard to the evolving velocity profiles and wall slip.

How inertial lift affects the dynamics of a microswimmer in Poiseuille flow

The transport of motile microorganisms is strongly influenced by fluid flows that are ubiquitous in biological environments. Here we demonstrate the impact of fluid inertia. We analyze the dynamics of a microswimmer in pressure-driven Poiseuille flow, where fluid inertia is small but non-negligible. Using perturbation theory and the reciprocal theorem, we show that in addition to the classical inertial lift of passive particles, the active nature generates a ‘swimming lift’, which we evaluate for neutral and pusher/puller-type swimmers. Accounting for fluid inertia engenders a rich spectrum of complex dynamics including bistable states, where tumbling coexists with stable centerline swimming or swinging. The dynamics is sensitive to the swimmer’s hydrodynamic signature and goes well beyond the findings at vanishing fluid inertia. Our work will have non-trivial implications on the transport and dispersion of active suspensions in microchannels.

An exact non-equilibrium extrapolation scheme for pressure and velocity boundary conditions with large gradients in the lattice Boltzmann method

In this work, an exact non-equilibrium extrapolation (eNEQ) scheme for velocity and pressure boundary conditions in the lattice Boltzmann method is proposed. Based on the non-equilibrium extrapolation (NEQ) scheme, well designed parameters are introduced to correct the distribution functions. Numerical results of velocity and pressure driven Poiseuille flows demonstrate that the present eNEQ scheme is of second-order spatial accuracy for both velocity and pressure boundary conditions. In addition, a series of other numerical simulation results show that in some cases with the large pressure or velocity gradient at the boundary, the eNEQ scheme can well ensure the accuracy of the calculation results, while the NEQ scheme performs struggle due to the adoption of the extrapolation scheme. On the basis of retaining the advantages of the original NEQ scheme, the present eNEQ scheme can be used to implement the velocity and pressure boundary conditions exactly.

Contribution of Particle–Wall Distance and Rotational Motion of a Single Confined Elliptical Particle to the Effective Viscosity in Pressure-Driven Plane Poiseuille Flows

Rheological properties of the suspension flow, especially effective viscosity, partly depend on spatial arrangement and motion of suspended particles. It is important to consider effective viscosity from the microscopic point of view. For elliptical particles, the equilibrium position of inertial migration in confined state is unclear, and there are few studies on the relationship between dynamics of suspended particles and induced local effective viscosity distribution. Contribution of a single circular or elliptical particle flowing between parallel plates to the effective viscosity was studied, focusing on the particle–wall distance and particle rotational motion using the two-dimensional regularized lattice Boltzmann method and virtual flux method. As a result, confinement effects of the elliptical particle on the equilibrium position of inertial migration were summarized using three definitions of confinement. In addition, the effects of particle shape (aspect ratio and confinement) on the effective viscosity were assessed focusing on the particle–wall distance. The contribution of particle shape to the effective viscosity was found to be enhanced when the particle flowed near the wall. Focusing on the spatial and temporal variation of relative viscosity evaluated from wall shear stress, it was found that the spatial variation of the local relative viscosity was larger than temporal variation regardless of the aspect ratio and particle–wall distance.

Spatial Temporal Anisotropic Transport of the Microalgae Chlorella Vulgaris in the Microfluidic Channel

Transport of biomaterials in confined geometry exhibit complex dynamics, both in spatial and temporal scales. In this work, we examined the transport behavior of microalgae Chlorella Vulgaris inside the microfluidic channel under pressure-driven Poiseuille flow environment. The microalgae system is treated as the spherical naturally buoyant particles. The algae particles trajectories are visually traced using particle imaging techniques and the sample paths are resolved in stream wise (flow) and perpendicular directions. In order to understand boundary wall effects on the flow, we partitioned the microfluidic channel into three different regions, namely the center region and two near-wall boundary regions based on the velocity flow profile. Time-averaged mean square displacement (MSD), probability density function (PDF), skewness, and kurtosis of finite ensembles of particle trajectories are determined for these regions. In addition to the spatial dependencies, we also examined the transient characteristics of the algae transport at early-time and long-time. We found the existence of the mixed types for transport dynamics irrespective of flow region separation often assumed in many laminar flow simulations. This finding will be useful for optimization of mixing of algae culture in micro-scale photobiore actor as the productivity of cell growth depends critically on the cell dispersion or transport

A novel median dual finite volume lattice Boltzmann method for incompressible flows on unstructured grids

A novel median dual finite volume lattice Boltzmann method (FV-LBM) for the accurate simulation of incompressible flows on unstructured grids is presented in this paper. The finite volume method is adopted to discretize the discrete velocity Boltzmann equation (DVBE) on median dual control volumes (CVs). In the previous studies on median dual FV-LBMs, the fluxes for each partial face have to be computed separately. In the present second-order scheme, we assume the particle distribution functions (PDFs) to be constant for all faces grouped around a particular edge. The fluxes are then evaluated using the low-diffusion Roe scheme at the midpoint of the edge, and the PDFs at the faces of the CV are obtained through piecewise linear reconstruction of the left and right states. The gradients of the PDFs are computed with the Green–Gauss approach. The presented scheme is validated on four benchmark flows: (a) pressure driven Poiseuille flow; (b) the backward-facing step flow with [Formula: see text], 100, 200 and 300; (c) the lid-driven flow with [Formula: see text] and 1000; and (d) the steady viscous flow past a circular cylinder with [Formula: see text], 20 and 40.

Molecular insight into the boundary conditions of water flow in clay nanopores

Understanding the underlying mechanisms of water flow in clay nanopores plays an important role in groundwater hydrology and shale production. In this work, we use molecular dynamic (MD) simulations to study water flow in various clay nanopores under a typical reservoir condition (323 K and 20 MPa) and investigate the boundary conditions to describe the water flow by using a hydrodynamic flow model. We found that only the first water adsorption layer is immobile, while other water molecules can flow with a slight slip velocity on the first water adsorption layer for a pressure-driven Poiseuille flow. In addition, the critical thickness is from 0.34 to 0.4 nm, which is almost half of the values applied in previous theoretical models. Such small difference in the critical thickness could result in large discrepancies on the water flow prediction. Our work should shed lights into water flow mechanisms in clay nanopores and provide reliable boundary conditions for the hydrodynamic flow models to predict water flow in unconventional reservoirs.

Hydrodynamic dispersion due to a variety of flow velocity profiles in a porous-walled microfluidic channel

A reduced-order model for the advective-dispersive mass transfer due to a variety of flow velocity profiles in a porous-walled microfluidic channel is developed in this study. The fluid flow in the microfluidic channel is described by a generalized class of the Poiseuille equation proposed by Amatore et al. (2009), which includes the bluntness parameter for exhibiting the level of bluntness of the velocity profile. This flow formulation represents all the traditional velocity profiles for the purely pressure-driven (Poiseuille) flow, the combined pressure-driven and electro-osmotic flows, and the purely electro-osmotic flow (EOF) in the microfluidic channel. The resulting reduced-order model delivers the mass transfer coefficients, including the hydrodynamic dispersion and the effective advection coefficients, which are functions of the Peclet number and the bluntness parameter. The results reveal that the hydrodynamic dispersion decreases when the flow varies from the pressure-driven to the electro-osmotic. In other words, the smaller the bluntness parameter the smaller the hydrodynamic dispersion. The diffusive, transient, and advective mass transfer regimes can be identified from the ratio of the hydrodynamic dispersion coefficient in the porous-walled microfluidic channel to the one in the nonporous-walled microfluidic channel. It is found that the mass transfer between the microfluidic channel and the porous medium should be included in determination of the hydrodynamic dispersion coefficient due to a variety of flow velocity profiles in a porous-walled microfluidic channel for the transient and the advective regimes. The results also show that the smaller the bluntness parameter the slower the mass transfer. Furthermore, the mass transfer in a porous-walled microfluidic channel is slower than the mass transfer in a nonporous-walled microfluidic channel.

Characterisation of minimal-span plane Couette turbulence with pressure gradients

The turbulence statistics and dynamics in the spanwise-minimal plane Couette flow with pressure gradients, so-called, Couette-Poiseuille (C-P) flow, are investigated using direct numerical simulation. The large-scale motion is limited in the spanwise box dimension as in the minimal-span channel turbulence of Flores & Jiménez (Phys. Fluids, vol. 22, 2010, 071704). The effect of the top wall, where normal pressure-driven Poiseuille flow is realised, is distinguished from the events on the bottom wall, where the pressure gradient results in mild or almost-zero wall-shear stress. A proper scaling of turbulence statistics in minimal-span C-P flows is presented. Also the ‘shear-less’ wall-bounded turbulence, where the Corrsin shear parameter is very weak compared to normal wall-bounded turbulence, represents local separation, which is also observed as spanwise streaks of reversed flow in full-size plane C-P turbulence. The local separation is a multi-scale event, which grows up to the order of the channel height even in the minimal-span geometry.

Crustal deformation induced by mantle dynamics: insights from models of gravitational lithosphere removal

Mantle-based stresses have been proposed to explain the occurrence of deformation in the interior regions of continental plates, far from the effects of plate boundary processes. We examine how the gravitational removal of a dense mantle lithosphere root may induce deformation of the overlying crust. Simplified numerical models and a theoretical analysis are used to investigate the physical mechanisms for deformation and assess the surface expression of removal. Three behaviours are identified: (1) where the entire crust is strong, stresses from the downwelling mantle are efficiently transferred through the crust. There is little crustal deformation and removal is accompanied by surface subsidence and a negative free-air gravity anomaly. Surface uplift and increased free-air gravity occur after the dense root detaches. (2) If the mid-crust is weak, the dense root creates a lateral pressure gradient in the crust that drives Poiseuille flow in the weak layer. This induces crustal thickening, surface uplift and a minor free-air gravity anomaly above the root. (3) If the lower crust is weak, deformation occurs through pressure-driven Poiseuille flow and Couette flow due to basal shear. This can overthicken the crust, producing a topographic high and a negative free-air gravity anomaly above the root. In the latter two cases, surface uplift occurs prior to the removal of the mantle stress. The modeling results predict that syn-removal uplift will occur if the crustal viscosity is less than ∼10^(21) Pa s, corresponding to temperatures greater than ∼400–500 °C for a dry and felsic or wet and mafic composition, and ∼900 °C for a dry and mafic composition. If crustal temperatures are lower than this, lithosphere removal is marked by the formation of a basin. These results can explain the variety of surface expressions observed above areas of downwelling mantle. In addition, observations of the surface deflection may provide a way to constrain the vertical rheological structure of the crust.

An Analytical Solution to Neumann-Type Mixed Boundary Poiseuille Microfluidic Flow in Rectangular Channel Cross-Sections (Slip/No-Slip) including a Numerical Technique to Derive It

In most microfluidic applications, pressure-driven Poiseuille flow in a contained cross-section with no-slip boundary conditions is the underlying fluid-mechanical model. Solutions for this problem exist for many known cross-sections. We have recently demonstrated a simple method to solve the relevant Poisson equation using a finite difference scheme in a spreadsheet analysis tool such as Microsoft Excel. The numerical solutions obtained from such a spreadsheet are close-to-exact to the analytical solutions with errors on the order of only a few percent. However, there are numerous applications in microfluidics for which the no-slip boundary condition is not valid. Examples include drag-reducing air-retaining surfaces as well as open-channel flow. For these scenarios few to no analytical models exist. In this paper, we derive an analytical model for mixed boundary conditions (slip/no-slip) in two dimensions in a rectangular channel cross-section. We also demonstrate that the equivalent numerical solution can be derived conveniently by adaption of the spreadsheet. In general, mixed boundary-type flow scenarios are especially difficult to solve analytically whereas numerical solutions can be derived using Microsoft Excel within seconds.

Numerics made easy: solving the Navier–Stokes equation for arbitrary channel cross-sections using Microsoft Excel

The fluid mechanics of microfluidics is distinctively simpler than the fluid mechanics of macroscopic systems. In macroscopic systems effects such as non-laminar flow, convection, gravity etc. need to be accounted for all of which can usually be neglected in microfluidic systems. Still, there exists only a very limited selection of channel cross-sections for which the Navier-Stokes equation for pressure-driven Poiseuille flow can be solved analytically. From these equations, velocity profiles as well as flow rates can be calculated. However, whenever a cross-section is not highly symmetric (rectangular, elliptical or circular) the Navier-Stokes equation can usually not be solved analytically. In all of these cases, numerical methods are required. However, in many instances it is not necessary to turn to complex numerical solver packages for deriving, e.g., the velocity profile of a more complex microfluidic channel cross-section. In this paper, a simple spreadsheet analysis tool (here: Microsoft Excel) will be used to implement a simple numerical scheme which allows solving the Navier-Stokes equation for arbitrary channel cross-sections.

Microfluidic flow of cholesteric liquid crystals.

We explore the rheology and flow-induced morphological changes of cholesteric liquid crystal patterns subject to Poiseuille flow within a slab geometry, and under different anchoring conditions at the wall. Our focus is particularly on the behaviour of "Cholesteric Fingers of the first kind" and of Blue Phase II. Depending on the applied pressure gradient, we observe a number of dynamic regimes with different rheological properties. Our results provide the first insight into the flow response of cholesteric phases with fully two- or three-dimensional director field patterns and normal and planar degenerate anchoring conditions as commonly realised in experiments. They are also of high relevance for a fundamental understanding of complex liquid crystals in confinement and an important step towards future microfluidic applications that are based on cholesteric liquid crystals.

Cessation of viscoplastic Poiseuille flow in a square duct with wall slip

We solve numerically the cessation of the pressure-driven Poiseuille flow of a Bingham plastic under the assumption that slip occurs along the wall following a generalized Navier-slip law involving a non-zero slip yield stress. In order to avoid the numerical difficulties caused by their inherent discontinuities, both the constitutive and the slip equations are regularized by means of exponential (Papanastasiou-type) regularizations. As with one-dimensional Poiseuille flows, in the case of Navier slip (zero slip yield stress), the fluid slips at all times, the velocity becomes and remains plug before complete cessation, and the theoretical stopping time is infinite. The cessation of the plug flow is calculated analytically. No stagnant regions appear at the corners when Navier slip is applied. In the case of slip with non-zero slip yield stress, the fluid may slip everywhere or partially at the wall only in the initial stages of cessation depending on the initial condition. Slip ceases at a critical time after which the flow decays exponentially and the stopping times are finite in agreement with theory. The combined effects of viscoplasticity and slip are investigated for wide ranges of the Bingham and slip numbers and results showing the evolution of the yielded and unyielded regions are presented. The numerical results also showed that the use of regularized equations may become problematic near complete cessation or when the velocity profile becomes almost plug.

Apparent Viscosity of Active Nematics in Poiseuille Flow

A Leslie-Erickson continuum hydrodynamic for flowing active nematics has been used to characterize active particle systems such as bacterial suspensions. The behavior of such a system under a plane pressure-driven Poiseuille flow is analyzed. When plate anchoring is tangential and normal, we find the apparent viscosity formula indicating a significant difference between tangential anchoring and normal anchoring conditions for both active rodlike and discoid nematics.

Effects of Convection on Isotachophoresis of Electrolytes

In this study, the form of the analytes distribution in isotachophoresis (ITP) in the presence of a convective flow is analyzed in a wide rectangular microchannel. The imposed convection is considered due to a mismatch of electroosmotic (EO) slip velocity of electrolytes of different electrophoretic mobilities. We compute the two-dimensional (2D) Nernst–Planck equations coupled with the Navier–Stokes equations for fluid flow and an equation for electric field. We use a control volume method along with a higher-order upwind scheme to capture the sharp variation of variables in the transition zones. The convection of electrolytes produces a smearing effect on the steepness of the electric field and ion distribution in the interface between two adjacent electrolytes in the ITP process. The dispersion of the interface in plateau-mode and the sample in peak-mode is analyzed through the second- and third-order moments. The dispersion due to nonuniform EO flow (EOF) of electrolytes is found to be different from the case when the dispersion is considered only due to an external pressure driven Poiseuille flow. The nonuniform EOF of electrolytes produces less dispersion and skewness in the sample distribution when the molecular diffusivity of the sample ionic species is close to the harmonic mean of the diffusivity of adjacent electrolytes. We find that the EOF may become advantageous in separating two analytes of close diffusivity. Our results show that the one-dimensional (1D) Taylor–Aris model is suitable to predict the dispersed ITP when the average convection speed of electrolytes is in the order of the ITP speed.