Pressure-driven Poiseuille Flow Research Articles

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.

Transport of a dilute active suspension in pressure-driven channel flow

Confined suspensions of active particles show peculiar dynamics characterized by wall accumulation, as well as upstream swimming, centreline depletion and shear trapping when a pressure-driven flow is imposed. We use theory and numerical simulations to investigate the effects of confinement and non-uniform shear on the dynamics of a dilute suspension of Brownian active swimmers by incorporating a detailed treatment of boundary conditions within a simple kinetic model where the configuration of the suspension is described using a conservation equation for the probability distribution function of particle positions and orientations, and where particle–particle and particle–wall hydrodynamic interactions are neglected. Based on this model, we first investigate the effects of confinement in the absence of flow, in which case the dynamics is governed by a swimming Péclet number, or ratio of the persistence length of particle trajectories over the channel width, and a second swimmer-specific parameter whose inverse measures the strength of propulsion. In the limit of weak and strong propulsion, asymptotic expressions for the full distribution function are derived. For finite propulsion, analytical expressions for the concentration and polarization profiles are also obtained using a truncated moment expansion of the distribution function. In agreement with experimental observations, the existence of a concentration/polarization boundary layer in wide channels is reported and characterized, suggesting that wall accumulation in active suspensions is primarily a kinematic effect that does not require hydrodynamic interactions. Next, we show that application of a pressure-driven Poiseuille flow leads to net upstream swimming of the particles relative to the flow, and an analytical expression for the mean upstream velocity is derived in the weak-flow limit. In stronger imposed flows, we also predict the formation of a depletion layer near the channel centreline, due to cross-streamline migration of the swimming particles towards high-shear regions where they become trapped, and an asymptotic analysis in the strong-flow limit is used to obtain a scale for the depletion layer thickness and to rationalize the non-monotonic dependence of the intensity of depletion upon flow rate. Our theoretical predictions are all shown to be in excellent agreement with finite-volume numerical simulations of the kinetic model, and are also supported by recent experiments on bacterial suspensions in microfluidic devices.

Shear-stress-induced conformational changes of von Willebrand factor in a water-glycerol mixture observed with single molecule microscopy.

The von Willebrand factor (VWF) is a human plasma protein that plays a key role in the initiation of the formation of thrombi under high shear stress in both normal and pathological situations. It is believed that VWF undergoes a conformational transition from a compacted, globular to an extended form at high shear stress. In this paper, we develop and employ an approach to visualize the large-scale conformation of VWF in a (pressure-driven) Poiseuille flow of water-glycerol buffers with wide-field single molecule fluorescence microscopy as a function of shear stress. Comparison of the imaging results for VWF with the results of a control with λ-phage double-stranded DNA shows that the detection of individual VWF multimers in flow is feasible. A small fraction of VWF multimers are observed as visibly extended along one axis up to lengths of 2.0 μm at high applied shear stresses. The size of this fraction of molecules seems to exhibit an apparent dependency on shear stress. We further demonstrate that the obtained results are independent of the charge of the fluorophore used to label VWF. The obtained results support the hypothesis of the conformational extension of VWF in shear flow.

Channel flow of a tensorial shear-thinning Maxwell model: Lattice Boltzmann simulations.

We discuss pressure-driven channel flow for a model of shear-thinning glass-forming fluids, employing a modified lattice-Boltzmann (LB) simulation scheme. The model is motivated by a recent microscopic approach to the nonlinear rheology of colloidal suspensions and captures a nonvanishing dynamical yield stress and the appearance of normal-stress differences and a flow-induced pressure contribution. The standard LB algorithm is extended to deal with tensorial, nonlinear constitutive equations of this class. The new LB scheme is tested in 2D pressure-driven channel flow and reproduces the analytical steady-state solution. The transient dynamics after startup and removal of the pressure gradient reproduce a finite stopping time for the cessation flow of yield-stress fluids in agreement with previous analytical estimates.

Revisiting the analytical solution for temperature in Poiseuille flow of an ideal gas

The discrepancy between the numerical solution to the Navier-Stokes equations for a pressure-driven Poiseuille flow in an ideal gas and the classical analytical solution for temperature is examined. A minimum temperature at the center is observed in the numerical solution to the Navier-Stokes equations but the analytical solution predicts a maximum temperature at the center. It is found that the DpDt term should not be neglected in the reduced energy equation in the analytical method when the fluid under consideration is an ideal gas. When the DpDt term is retained, a minimum instead of a maximum temperature arises at the center in the new analytical solution as well. On the other hand, when the fluid is a liquid, the reduced energy equation and the solution assume the classical form, and thus the solution in fact has a maximum at the center.

Nonequilibrium gaseous heat transfer in pressure-driven plane Poiseuille flow

Nonequilibrium heat and mass transfer in a pressure-driven plane Poiseuille flow is investigated using the direct simulation Monte Carlo method from the early slip to the free molecular regime. Our investigations reveal several nonintuitive, nonequilibrium thermal flow patterns, including expansion cooling near the walls, a nonconstant pressure profile, and counter-gradient heat transfer along the channel center-line. A bimodal trend in the tangential heat flux is found in the slip and the early transition regime. In the upper transition and free molecular regime, the net heat flow in the entire channel is largely unidirectional and in the opposite direction of mass flow. However, in the slip and the early transition regime, a two-way heat flow is observed in the channel as the normal heat flux profile plays a key role in determining the net gaseous heat flow direction. Moreover, the heat flow rate profile exhibits a maximum value at an intermediate value of Knudsen number. The effects of incomplete surface accommodation on nonequilibrium heat flow are also investigated in this work. It is shown that for very low values of the accommodation coefficient, the gaseous heat flow direction is reversed and is consistently in the direction of mass flow.

Convergence proof of the DSMC method and the Gas-Kinetic Unified Algorithm for the Boltzmann equation

This paper investigates the convergence proof of the Direct Simulation Monte Carlo (DSMC) method and the Gas-Kinetic Unified Algorithm in simulating the Boltzmann equation. It can be shown that the particle velocity distribution function obtained by the DSMC method converges to a modified form of the Boltzmann equation, which is the equation of the gas-kinetic unified algorithm to directly solve the molecular velocity distribution function. Their convergence is derived through mathematical treatment. The collision frequency is presented using various molecular models and the local equilibrium distribution function is obtained by Enskog expansion using the converged equation of the DSMC method. These two expressions agree with those used in the unified algorithm. Numerical validation of the converging consistency between these two approaches is illustrated by simulating the pressure driven Poiseuille flow in the slip transition flow regime and the two-dimensional and three-dimensional flows around a circular cylinder and spherical-cone reentry body covering the whole flow regimes from low speed micro-channel flow to high speed non-equilibrium aerothermodynamics.

Viscosity calculation of a nanoparticle suspension confined in nanochannels

The kinetic properties of the pressure-driven Poiseuille flow in nanochannels with and without nanoparticles were studied with a nonequilibrium molecular dynamics simulation. To allow the fluid to dissipate heat, the boundary was kept at a constant temperature. Pure fluid simulations were taken as references and also used to study the fluid-wall interfacial interaction effects. The viscosity profiles of the fluid were calculated on the basis of velocity profiles and known applied shear stress. We present the relationship between the viscosity increase and particle loading. The role of channel wall-fluid wetting properties on the flow and viscosity was also investigated.

Instability of the two-dimensional Poiseuille flow between elastic plates

The pressure-driven Poiseuille flow of a viscous incompressible fluid between two parallel plates is considered. Within the framework of the triple-deck theory, the elasticity of the walls is shown to have a stabilizing effect on the flow.

Novel surface acoustic wave (SAW)-driven closed PDMS flow chamber

In this article, we demonstrate a novel microfluidic flow chamber driven by surface acoustic waves. Our device is a closed loop channel with an integrated acoustic micropump without external fluidic connections that allows for the investigation of small fluid samples in a continuous flow. The fabrication of the channels is particularly simple and uses standard milling and PDMS molding. The micropump consists of gold electrodes deposited on a piezoelectric substrate employing photolithography. We show that the pump generates a pressure-driven Poiseuille flow, investigate the acoustic actuation mechanism, characterize the flow profile for different channel geometries, and evaluate the driving pressure, efficiency and response time of the acoustic micropump. The fast response time of our pump permits the generation of non-stationary flows. To demonstrate the versatility of the device, we have pumped a red blood cell suspension at a physiological rate of 60 beats/min.

Entropy generation under the effect of suction/injection

This work investigates entropy generation in a steady flow of viscous incompressible fluids between two infinite parallel porous plates. The fluid temperature variation is due to asymmetric heating of the porous plates as well as viscous dissipation. Two different physical situations are discussed with their entropy generation profiles: (i) Couette flow with suction/injection and (ii) pressure-driven Poiseuille flow with suction/injection. In each case, closed form expressions for entropy generation number and Bejan number are derived in dimensionless form by using the expressions for velocity and temperature which are derived by solving the resulting momentum and energy equations by the method of undetermined coefficient. The effect of the governing parameters on velocity, temperature, entropy generation and Bejan number are extensively discussed with the help of graphs. It is interesting to remark that entropy generation number increases with suction on one porous plate while it decreases on the other porous plate with injection.

Shear banding and interfacial instability in planar Poiseuille flow

Motivated by the need for a theoretical study in a planar geometry that can easily be implemented experimentally, we study the pressure driven Poiseuille flow of a shear banding fluid. After discussing the “basic states” predicted by a one-dimensional calculation that assumes a flat interface between the bands, we proceed to demonstrate such an interface to be unstable with respect to the growth of undulations along it. We give results for the growth rate and wavevector of the most unstable mode that grows initially, as well as for the ultimate flow patterns to which the instability leads. We discuss the relevance of our predictions to the present state of the experimental literature concerning interfacial instabilities of shear banded flows, in both conventional rheometers and microfluidic channels.

Kramers’ problem and the Knudsen minimum: a theoretical analysis using a linearized 26-moment approach

A set of linearized 26 moment equations, along with their wall boundary conditions, are derived and used to study low-speed gas flows dominated by Knudsen layers. Analytical solutions are obtained for Kramers’ defect velocity and the velocity-slip coefficient. These results are compared to the numerical solution of the BGK kinetic equation. From the analysis, a new effective viscosity model for the Navier–Stokes equations is proposed. In addition, an analytical expression for the velocity field in planar pressure-driven Poiseuille flow is derived. The mass flow rate obtained from integrating the velocity profile shows good agreement with the results from the numerical solution of the linearized Boltzmann equation. These results are good for Knudsen numbers up to 3 and for a wide range of accommodation coefficients. The Knudsen minimum phenomenon is also well captured by the present linearized 26-moment equations.

Long wavelength convection, Poiseuille–Couette flow in the low-viscosity asthenosphere and the strength of plate margins

SUMMARY Mixed heated 3-D mantle convection simulations with a low-viscosity asthenosphere reveal relatively short and long wavelength regimes with different scalings in terms of surface velocity and surface heat flux and show that mantle flow in the lithosphere–asthenosphere region is a Poiseuille–Couette flow. The Poiseuille/Couette velocity magnitude ratio, D/U, allows us to characterize solid-state flow in the asthenosphere and to predict the regime transition. The transition from dominantly pressure-driven Poiseuille flow at shorter wavelengths to dominantly shear-driven Couette flow at long wavelengths depends on the relative strength of lithosphere and asthenosphere and is associated with a switch in the dominant resistance to convective motion. In the Poiseuille regime significant resistance is provided by plate-bending, whereas in the Couette regime most resistance is due to vertical shear in the bulk mantle. The Couette case corresponds to classical scaling ideas for mantle convection whereas the Poiseuille case, with asthenospheric velocities exceeding surface velocities, is an example of a sluggish lid mode of mantle convection that has more recently been invoked for thermal history models of the Earth. Our simulations show that both modes can exist for the same level of convective vigour (i.e. Rayleigh number) but at different convective wavelengths. Additional simulations with temperature- and yield-stress dependent viscosity show consistent behaviour and suggest an association of the regime crossover with the relative strength of plate margins. Our simulations establish a connection between the strength of plate margins, solid-state flow in the asthenosphere and the wavelength of mantle convection. This connection suggests that plate tectonics in the sluggish lid mode is wavelength dependent and potentially more robust than previously envisioned.

A high-order moment approach for capturing non-equilibrium phenomena in the transition regime

The method of moments is employed to extend the validity of continuum-hydrodynamic models into the transition-flow regime. An evaluation of the regularized 13 moment equations for two confined flow problems, planar Couette and Poiseuille flows, indicates some important limitations. For planar Couette flow at a Knudsen number of 0.25, they fail to reproduce the Knudsen-layer velocity profile observed using a direct simulation Monte Carlo approach, and the higher-order moments are not captured particularly well. Moreover, for Poiseuille flow, this system of equations creates a large slip velocity leading to significant overprediction of the mass flow rate for Knudsen numbers above 0.4. To overcome some of these difficulties, the theory of regularized moment equations is extended to 26 moment equations. This new set of equations highlights the importance of both gradient and non-gradient transport mechanisms and is shown to overcome many of the limitations observed in the regularized 13 moment equations. In particular, for planar Couette flow, they can successfully capture the observed Knudsen-layer velocity profile well into the transition regime. Moreover, this new set of equations can correctly predict the Knudsen layer, the velocity profile and the mass flow rate of pressure-driven Poiseuille flow for Knudsen numbers up to 1.0 and captures the bimodal temperature profile in force-driven Poiseuille flow. Above this value, the 26 moment equations are not able to accurately capture the velocity profile in the centre of the channel. However, they are able to capture the basic trends and successfully predict a Knudsen minimum at the correct value of the Knudsen number.

Steady-state performance of an infinitely fast reaction in a three-dimensional open Stokes flow

We investigate the steady-state performance of a single irreversible mixing-controlled reaction between segregated reactant streams entering the annular gap between coaxial cylinders that can rotate independently. The three-dimensional Stokes flow results from the superposition of a pressure-driven Poiseuille flow along the mixer axis, and a cross-sectional Couette flow generated by the steady rotation of the inner and outer cylinders. Flow protocols associated with different geometries and different conditions of the rotating cylinders are considered. For each protocol, we analyze the axial behavior of reaction yield as a function of the Peclet number, quantifying the relative importance of convective vs. diffusive transport mechanisms. Following a well-established approach developed in the context of closed bounded flows, we connect the steady-state reactor performance with the spectral (eigenvalue–eigenfunction) structure of the advection–diffusion operator restricted to the system cross-section. For slim channels, this approach indicates that reaction performance is controlled by the real part, say - Λ ( Λ > 0 ), of the dominant eigenvalue of the advection–diffusion operator, and therefore the quantitative analysis of reaction performance consists of determining how Λ depends on the Peclet number for the assigned protocol. The presence of a nonuniform axial flow makes the spectral analysis different from previous studies focusing on transient mixing in closed two-dimensional autonomous flows in that it transforms the standard eigenvalue–eigenfunction formulation of the problem into a generalized form, where the axial velocity plays the role of weight function for the generic eigenvalue–eigenfunction. This difference is not just formal, as it can give rise to a variety of convection-enhanced regimes which cannot be observed in the standard eigenvalue formulation. Identifying the onset conditions and the range of existence of these regimes could greatly improve the rational design of geometry and operating conditions of micro and ordinary lengthscale continuous reactors operating under laminar flow conditions.

On Weak Plane Couette and Poiseuille Flows of Rigid Rod and Platelet Ensembles

Films and molds of nematic polymer materials are notorious for heterogeneity in the orientational distribution of the rigid rod or platelet macromolecules. Predictive tools for structure length scales generated by shear-dominated processing are vitally important: both during processing because of flow feedback phenomena such as shear thinning or thickening, and postprocessing since gradients in the rod or platelet ensemble translate to nonuniform composite properties and to residual stresses in the material. These issues motivate our analysis of two prototypes for planar shear processing: drag-driven Couette and pressure-driven Poiseuille flows. Hydrodynamic theories for high aspect ratio rod and platelet macromolecules in viscous solvents are well developed, which we apply in this paper to model the coupling between short-range excluded volume interactions, anisotropic distortional elasticity (unequal elasticity constants), wall anchoring conditions, and hydrodynamics. The goal of this paper is to generalize scaling properties of steady flow molecular structures in slow Couette flows with equal elasticity constants (M. G. Forest et al., J. Rheol., 48 (2004), pp. 175-192) in several ways: to contrast isotropic and anisotropic elasticity; to compare Couette versus Poiseuille flow; and to consider dynamics and stability of these steady states within the asymptotic model equations.

Electro-osmotic flow of a model electrolyte

Electro-osmotic flow is studied by nonequilibrium molecular dynamics simulations in a model system chosen to elucidate various factors affecting the velocity profile and facilitate comparison with existing continuum theories. The model system consists of spherical ions and solvent, with stationary, uniformly charged walls that make a channel with a height of 20 particle diameters. We find that hydrodynamic theory adequately describes simple pressure-driven (Poiseuille) flow in this model. However, Poisson-Boltzmann theory fails to describe the ion distribution in important situations, and therefore continuum fluid dynamics based on the Poisson-Boltzmann ion distribution disagrees with simulation results in those situations. The failure of Poisson-Boltzmann theory is traced to the exclusion of ions near the channel walls resulting from reduced solvation of the ions in that region. When a corrected ion distribution is used as input for hydrodynamic theory, agreement with numerical simulations is restored. An analytic theory is presented that demonstrates that repulsion of the ions from the channel walls increases the flow rate, and attraction to the walls has the opposite effect. A recent numerical study of electro-osmotic flow is reanalyzed in the light of our findings, and the results conform well to our conclusions for the model system.

DNA separation by EFFF in a microchannel

This paper theoretically explores the application of electric field flow fractionation (EFFF) for the size-based separation of DNA strands in a microchannel. An axial electric field cannot separate DNA strands in solution because the electrical mobility of the strands is independent of the length. However, lateral electric fields coupled with an axial Poiseuille flow can separate the DNA strands of different sizes. By using regular perturbation analysis, we obtain the effective diffusivity and the mean velocity of the DNA molecules that are undergoing a pressure driven Poiseuille flow in a 2D channel in presence of a lateral electric field. The mean velocities and the dispersion coefficients are then utilized to determine the scaling for length of the channel and the time required for separation of DNA molecules in different parameter regimes. The results show that EFFF can separate DNA strands in the range of 10 kbp that differ in size by about 2.5 kbp in about half an hour in a 1 cm long channel. While DNA strands can be separated by EFFF, the performance of devices based on EFFF seems to be at best comparable to other techniques such as entropic trapping.

Micro- and nanoscale non-ideal gas Poiseuille flows in a consistent Boltzmann algorithm model

The direct simulation Monte Carlo method in the consistent Boltzmann algorithm model has been developed and expanded for non-ideal gas predictions. The enhanced collision rate factor is determined by considering the excluded molecular volume and shadowing/screening effects based on the Enskog theory. The parameter for the attraction strength is also determined by comparison with the classical thermodynamics theory. Different pressure-driven gas Poiseuille flows in micro- and nanoscale channels are investigated. The van der Waals effect leads to a higher mass flow rate and different friction and heat transfer characteristics on the wall surface, compared to the results in the perfect gas model. The results also show that the van der Waals effect is dependent not only on the pressure but also on the channel size. A higher driving pressure or a smaller channel size will result in a larger van der Waals gas effect.