Pressure-driven Channel Flow Research Articles

Experimental Study of Relationship between Wave Interfacial Instabilities and Mechanical Strength of Three-Layer Polymer Structure

In this research, an experimental apparatus has been developed for observing interfacial stability and deformation of multilayer pressure driven channel flows. The interface instability of the co-extrusion flow of polyethylene and polypropylene is studied experimentally in a slit geometry. This is done by introducing disturbances of controlled wave length and amplitude on three-layer symmetric (A-B-A) polymer melts as well as performing a series of extrudate mechanical testing. In this study variations of the mechanical properties as well as wave interlocking have been related to the conformation of the interfacial waves (IW). By investigating the growing (IW) and tensile stress of extrudate samples a relationship between interfacial instability (II) and mechanical properties of polypropylene (PP) and high density polyethylene (HDPE) has been established. It has been shown that instabilities are associated with IW, and it turns out that IW amplitude is known as a mechanism for controlling the strength of three layer polymer products. It is shown that the mechanism of interfacial strength is related to interfacial instabilities as well as the interfacial strength. By considering that the instability is controlled by its dominant mode. It is shown that there is ability to forecast the quality of final products in co-extrusion process.

Effects of wall-heating on the linear instability characteristics of pressure-driven two-layer channel flow

The linear stability analysis of a pressure-driven two-layer channel flow of two immiscible, Newtonian and incompressible fluids is considered. The walls of the channel are maintained at different constant temperatures and Nahme's law is applied to model the temperature dependence of the fluid viscosity. A modified Orr–Sommerfeld equation for the disturbance streamfunction coupled to a linearized energy equation is derived and solved using a spectral collocation method. Our results indicate that increasing the dimensionless top wall temperature has a non-monotonic effect on the linear stability characteristics. We also found that increasing the thermal conductivity and density ratios stabilise the flow for the set of parameter values considered; the viscosity ratio has a non-monotonic effect on the maximal growth rate. An energy ‘budget’ analysis shows that the most dangerous mode is of ‘interfacial’ type.

Stability and energy budget of pressure-driven collapsible channel flows

AbstractAlthough self-excited oscillations in collapsible channel flows have been extensively studied, our understanding of their origins and mechanisms is still far from complete. In the present paper, we focus on the stability and energy budget of collapsible channel flows using a fluid–beam model with the pressure-driven (inlet pressure specified) condition, and highlight its differences to the flow-driven (i.e. inlet flow specified) system. The numerical finite element scheme used is a spine-based arbitrary Lagrangian–Eulerian method, which is shown to satisfy the geometric conservation law exactly. We find that the stability structure for the pressure-driven system is not a cascade as in the flow-driven case, and the mode-2 instability is no longer the primary onset of the self-excited oscillations. Instead, mode-1 instability becomes the dominating unstable mode. The mode-2 neutral curve is found to be completely enclosed by the mode-1 neutral curve in the pressure drop and wall stiffness space; hence no purely mode-2 unstable solutions exist in the parameter space investigated. By analysing the energy budgets at the neutrally stable points, we can confirm that in the high-wall-tension region (on the upper branch of the mode-1 neutral curve), the stability mechanism is the same as proposed by Jensen & Heil. Namely, self-excited oscillations can grow by extracting kinetic energy from the mean flow, with exactly two-thirds of the net kinetic energy flux dissipated by the oscillations and the remainder balanced by increased dissipation in the mean flow. However, this mechanism cannot explain the energy budget for solutions along the lower branch of the mode-1 neutral curve where greater wall deformation occurs. Nor can it explain the energy budget for the mode-2 neutral oscillations, where the unsteady pressure drop is strongly influenced by the severely collapsed wall, with stronger Bernoulli effects and flow separations. It is clear that more work is required to understand the physical mechanisms operating in different regions of the parameter space, and for different boundary conditions.

Self-consistent particle simulation of model-stabilized colloidal suspensions

Dynamics of model-stabilized colloidal suspensions were investigated by the self-consistent particle simulation method (SC), a new simulation algorithm that takes into account the interaction between the particles and suspending fluid. In this method, the fluid-particle interaction is introduced self-consistently by combining the finite element method (FEM) for fluid motion with Brownian dynamics (BD) for particle dynamics. To validate the reliability of the proposed algorithm, the shear dynamics of the stable particle suspensions were investigated. Relative viscosity and microstructure as a function of dimensionless shear rate at different volume fractions were in good agreement with previous observations. The robustness of the method was also verified through numerical convergence test. The effect of the fluid-particle interaction was well represented in simulations of two model problems, pressure-driven channel flow and rotating Couette flow. Plug-shaped velocity profile was observed in pressure-driven channel flow, which arised from shear thinning behavior of the stable suspension. In rotating Couette flow, shear banded nonlinear flow profile was observed. Although full hydrodynamic interaction (HI) was not rigorously taken into account, it successfully captured the macroscopic structure-induced flow field. It also takes advantage of the geometrical adaptability of FEM and computational efficiency of BD. We expect this newly developed simulation platform to be useful and efficient for probing the complex flow dynamics of particle systems as well as for practical applications in the complex flow of complex fluids.

Three-dimensional convective and absolute instabilities in pressure-driven two-layer channel flow

Three-dimensional convective and absolute instabilities in pressure-driven two-layer channel flow

Two-fluid pressure-driven channel flow with wall deposition and ageing effects

Two-fluid, stratified pressure-driven channel flow is studied in the limit of small viscosity ratios. Cases are considered in which the core fluid undergoes phase separation that results in the ‘precipitation’ of a distinct phase and the formation of a wall layer; these situations are common in the oil industry where ‘fouling’ deposits are formed during the flow. The thickness of this layer increases as a result of continual deposition through Stefan-like fluxes, which are related to the phase behaviour of the core fluid through a chemical equilibria model that treats the fluid as a bi-component mixture. The deposit also undergoes an ‘ageing’ process whereby its viscosity increases due to the build-up of internal structure; the latter is modelled here via a Coussot-type relation. Lubrication theory is used in the wall layer and an integral balance in the core fluid wherein inertial effects are important. By choosing appropriate semi-parabolic velocity and temperatures closures for the laminar flow in the channel core, and a closure relation for the wall layer rheology, evolution equations are derived that describe the flow dynamics. In the presence of ageing but absence of deposition, it is demonstrated how the time-varying deposit rheology alters the wave dynamics; for certain parameter ranges, these effects can give rise to the formation of steep waves and what appears to be finite-time ‘blow-up’. With both ageing and deposition, the spatio-temporal evolution of the deposit is shown together with the increase in the deposition rate with increasing temperature difference between the wall and the inlet.

Interfacial instability of pressure-driven channel flow for a two-species model of entangled wormlike micellar solutions

Abstract We examine the linear stability of the one dimensional inhomogeneous (shear banded) pressure-driven flow through a rectilinear microchannel predicted by the VCM model (Vasquez et al., A network scission model for wormlike micellar solutions I: model formulation and homogeneous flow predictions, J. Non-Newton. Fluid Mech., 144:122–139, 2007). The VCM model is a microstructural network model that incorporates the breakage and reforming of two elastically-active species (a long species ‘A’ and a shorter species ‘B’). The model consists of a set of coupled nonlinear partial differential equations describing the two micellar species, which relax due to reptative and Rousian stress-relaxation mechanisms as well as breakage events. The model includes nonlocal effects arising from stress-microstructure diffusion and we investigate the effect of these nonlocal terms on the linear stability of the pressure-driven flow. Calculation of the full eigenspectrum shows that the mode of instability is a sinuous (odd) interfacial mode, in agreement with previous calculations for the shear-banded Johnson–Segalman model (Fielding and Wilson, Shear banding and interfacial instability in planar Poiseuille flow, J. Non-Newton. Fluid Mech., 165:196–202, 2010). Increased diffusion, or smaller characteristic channel dimensions, smoothes the kink in the velocity profile that develops at the shear band and progressively reduces the spectrum of unstable modes. For sufficiently large diffusion this smoothing effect eliminates the instability entirely and restabilizes the base (shear-banded) velocity profile.

Bingham fluid simulation with the incompressible lattice Boltzmann model

The Bingham fluid flow is numerically studied using the lattice Boltzmann method by incorporating the Papanastasiou exponential modification approach. The He–Luo incompressible lattice Boltzmann model is employed to avoid numerical instability usually encountered in non-Newtonian fluid simulations due to a strong non-linear relationship between the shear rate tensor and the rate-of-strain tensor. First, the value of the regularization parameter in Bingham fluid mimicking is analyzed and a method to determine the value is proposed. Then, the model is validated by pressure-driven planar channel flow and planar sudden expansion flow. The velocity profiles for the pressure-driven planar channel flow are in good agreement with analytical solutions. The calculated reattachment lengths for a 2:1 planar sudden expansion flow also agree well with the available data. Finally, the Bingham flow over a cavity is studied, and the streamlines and yielded/unyielded regions are discussed.

Three-dimensional linear instability in pressure-driven two-layer channel flow of a Newtonian and a Herschel–Bulkley fluid

The three-dimensional linear stability characteristics of pressure-driven two-layer channel flow are considered, wherein a Newtonian fluid layer overlies a layer of a Herschel–Bulkley fluid. We focus on the parameter ranges for which Squire’s theorem for the two-layer Newtonian problem does not exist. The modified Orr–Sommerfeld and Squire equations in each layer are derived and solved using an efficient spectral collocation method. Our results demonstrate the presence of three-dimensional instabilities for situations where the square root of the viscosity ratio is larger than the thickness ratio of the two layers; these “interfacial” mode instabilities are also present when density stratification is destabilizing. These results may be of particular interest to researchers studying the transient growth and nonlinear stability of two-fluid non-Newtonian flows. We also show that the “shear” modes, which are present at sufficiently large Reynolds numbers, are most unstable to two-dimensional disturbances.

Linear and nonlinear spatio-temporal instability in laminar two-layer flows

The linear and nonlinear spatio-temporal stability of an interface separating two Newtonian fluids in pressure-driven channel flow at moderate Reynolds numbers is analysed both theoretically and numerically. A linear, Orr–Sommerfeld-type analysis shows that most of such systems are unstable. The transition to an absolutely unstable regime is investigated, and is shown to occur in an intermediate range of Reynolds numbers and ratios of the thicknesses of the two layers, for near-density matched fluids with a viscosity contrast. A critical Reynolds number is found for transition from convective to absolute instability of relatively thin films. Results obtained from direct numerical simulations (DNSs) of the Navier–Stokes equations for long channels using a diffuse-interface method elucidate that waves generated by random noise at the inlet show that, near the inlet, waves are formed and amplified strongly, leading to ligament formation. Successive waves coalesce with each other further downstream, resulting in longer larger-amplitude waves further downstream. In the linearly absolute regime, the characteristics of the spatially growing wave near the inlet agree with that of the saddle point as predicted by the linear theory. The transition point from a convective to an absolute regime predicted by linear theory is also in agreement with a sharp change in the value of a healing length obtained from the DNSs.

Numerical simulation of non-isothermal pressure-driven miscible channel flow with viscous heating

We study the pressure-driven, non-isothermal miscible displacement of one fluid by another in a horizontal channel with viscous heating. We solve the continuity, Navier–Stokes, and energy conservation equations coupled to a convective-diffusion equation for the concentration of the more viscous fluid. The viscosity is assumed to depend on the concentration as well as the temperature, while density contrasts are neglected. Our transient numerical simulations demonstrate the development of ‘roll-up’ of the ‘interface’ separating the fluids and vortical structures whose intensity increases with the temperature of the invading fluid. This brings about fluid mixing and accelerates the displacement of the fluid originally occupying the channel. Increasing the level of viscous heating gives rise to high-temperature, low-viscosity near-wall regions. The increase in viscous heating retards the propagation of the invading fluid but accelerates the ultimate displacement of the resident fluid.

Microchannel flows with superhydrophobic surfaces: Effects of Reynolds number and pattern width to channel height ratio

Superhydrophobic surfaces are widely adopted for reducing the flow resistance in microfluidic channels. The structures on the superhydrophobic surfaces may consist of longitudinal grooves, transverse grooves, posts, holes, etc. In this paper their effective slip performances are systematically studied and compared in detail through numerical simulations. The numerical results show that channel wall confinement effects have a positive influence on the effective slip length for square posts and longitudinal grooves, and a negative influence for square holes and transverse grooves. Square posts, holes, and transverse grooves all exhibit deteriorating effective slip performances at higher Reynolds numbers, while the effective slip performance of longitudinal grooves remains independent of the Reynolds number. For small pattern width to channel height ratios and at low Reynolds numbers, for low shear-free fractions, the effective slip length of square posts is equivalent of that of transverse grooves, and both geometries yield effective slip lengths which are in turn lower than those of square holes and longitudinal grooves. With increasing shear-free fractions, the effective slip length of square posts surpasses that of square holes and longitudinal grooves, but it becomes lower than that of longitudinal grooves at high Reynolds numbers or large pattern width to channel height ratios. Scaling laws for the effective slip length of superhydrophobic surfaces with square posts, square holes, and transverse grooves have previously been reported for shear-driven flows. This study extends the validity of these scaling laws to pressure-driven channel flows, even at high Reynolds numbers. The findings in this study serve as a useful guide for applications involving the reduction in flow resistance in microchannels containing superhydrophobic surfaces.

Particle migration and suspension structure in steady and oscillatory plane Poiseuille flow

A structure-tensor-based model is used to compute the microstructure and velocity field of concentrated suspensions of hard spheres in a fully developed, pressure-driven channel flow. The model is comprised of equations governing conservation of mass and momentum in the bulk suspension, conservation of particles, and conservation of momentum in the particle phase. The equations governing the relation between structure and stress in hard-sphere suspensions were developed previously and were shown to reproduce quantitatively results obtained by Stokesian dynamics simulations of linear shear flows. In nonhomogeneous, pressure-driven flows, the divergence of the particle contribution to the stress is nonzero and acts as a body force that causes particles to migrate across streamlines. Under steady conditions, the model predicts that the resulting migration causes particles to move to the center of the channel, where the concentration approaches the maximum packing for hard-sphere suspensions. In oscillatory flow, the behavior depends strongly on the amplitude of the strain. For oscillations with large strains, the particles migrate to the channel center. However, when the strain is small, the maximum concentration is located either at a position between the channel center and walls or, in the limit of very small strains, at the wall. The migration to the wall induced by small-strain oscillation occurs in conjunction with the suspension microstructure becoming ordered. This behavior agrees qualitatively with experimental observations reported in the literature. However, the predicted rate of migration toward the wall in the simulations is significantly slower than what is observed experimentally.

Pressure-driven miscible two-fluid channel flow with density gradients

We study the effect of buoyancy on pressure-driven flow of two miscible fluids in inclined channels via direct numerical simulations. The flow dynamics are governed by the continuity and Navier–Stokes equations, without the Boussinesq approximation, coupled to a convective-diffusion equation for the concentration of the more viscous fluid through a concentration-dependent viscosity and density. The effect of varying the density ratio, Froude number, and channel inclination on the flow dynamics is examined, for moderate Reynolds numbers. We present results showing the spatiotemporal evolution of the flow together with an integral measure of mixing.

Dispersion of a buoyant plume in a turbulent pressure-driven channel flow

Direct numerical simulations of the turbulent dispersion of a buoyant line of hot fluid released at the inlet of a plane channel flow are reported ( Re τ = 180, Gr = 10 7 and Pr = 0.7). Results of turbulent dispersion of a neutrally buoyant scalar and mixed convection flow are also included. The buoyancy force induces a vertical movement that, although small in mean, exhibits a significant fluctuation in the vertical velocity component and deflects the plume with the consequent loss of symmetry found in the neutrally buoyant results. The modification of the budgets for the time averaged momentum and heat transport equations reflects the rearranging of the different contributions induced by the buoyancy force.

Erratum: “Linear instability of pressure-driven channel flow of a Newtonian and a Herschel–Bulkley fluid” [Phys. Fluids 19, 122101 (2007)

2 1 � U 1 =0 . This change produced minor corrections to Figs. 2, 5a and 5b ,6 a and 6b ,7 a and 7b ,8 a and 8b ,1 0b ,1 3b, and 15b, as well as Tables I and II in the original paper. The captions of all figures and tables and conclusions of the

Assessment of algorithms for the no‐slip boundary condition in the lattice Boltzmann equation of BGK model

AbstractThree kinds of algorithms for the lattice Boltzmann equation of the BGK model in the implementation of the no‐slip boundary condition on the wall are assessed by using the analytical formula for the slip velocity of the fully developed pressure‐driven channel flow. It is shown that the bounce‐back algorithm results in the spatial accuracy of 1st order, except for the case when the wall is located at half way between the two grid lines. The interpolation scheme proposed by Yu et al. (Prog. Aerospace Sci. 2003; 39:329–367) and the similar one by Bouzidi et al. (Phys. Fluids 2001; 13(11):3452–3459) are of 2nd order, but the error increases quadratically with the relaxation time. The extrapolation scheme of Guo et al. (Phys. Fluids 2002; 14(6):2007–2010) is also shown to be of 2nd order, and the error level increases linearly with the relaxation time, but it turns out that this scheme is unstable for a certain range of parameter values. Numerical experiments with various parameter sets have been performed to obtain the stability diagram. Three algorithms are then applied to a circular‐Couette flow and their performance is also studied in terms of the numerical accuracy and stability. Copyright © 2008 John Wiley & Sons, Ltd.

Linear instability of pressure-driven channel flow of a Newtonian and a Herschel-Bulkley fluid

The linear stability characteristics of pressure-driven two-layer channel flow are considered, wherein a Newtonian fluid layer overlies a layer of a Herschel-Bulkley fluid. A pair of coupled Orr-Sommerfeld eigenvalue equations are derived and solved using an efficient spectral collocation method for cases in which unyielded regions are absent. An asymptotic analysis is also carried out in the long-wave limit, the results of which are in excellent agreement with the numerical predictions. Our analytical and numerical results indicate that increasing the dimensionless yield stress, prior to the formation of unyielded plugs below the interface, is destabilizing. Increasing the shear-thinning tendency of the lower fluid is stabilizing.

Numerical simulation of the onset of slug initiation in laminar horizontal channel flow

Results are presented for the initiation of slug-type structures from stratified 2D, two-layer pressure-driven channel flow. Good agreement is obtained with an Orr–Sommerfeld-type stability analysis for the growth rate and wave speed of very small disturbances. The numerical results elucidate the non-linear evolution of the interface shape once small disturbances have grown substantially. It is shown that relatively short waves (which are the most unstable according to linear theory) saturate when the length of the periodic domain is equally short. In longer domains, coalescence of short waves of small-amplitude is shown to lead to large-amplitude long waves, which subsequently exhibit a tendency towards slug formation. The non-uniform distribution of the interfacial shear stress is shown to be a significant mechanism for wave growth in the non-linear regime.

Interfacial dynamics in pressure-driven two-layer laminar channel flow with high viscosity ratios

The large-scale dynamics of an interface separating two immiscible fluids in a channel is studied in the case of large viscosity contrasts. A long-wave analysis in conjunction with the Kármán-Polhausen method to approximate the velocity profile in the less viscous fluid is used to derive a single equation for the interface. This equation accounts for the presence of interfacial stress, capillarity, and viscous retardation as well as inertia in the less viscous fluid layer where the flow is considered to be quasistatic; the equation is shown to reduce to a Benney-type equation and the Kuramoto-Sivashinskiy equation in the relevant limits. The solutions of this equation are parametrized by an initial thickness ratio h0 and a dimensionless parameter S , which measures the relative significance of inertial to capillary forces. A parametric continuation technique is employed, which reveals that nonuniqueness of periodic solutions is possible in certain regions of (h0,S) space. Transient numerical simulations are also reported, whose results demonstrate good agreement with the bifurcation structure obtained from the parametric continuation results.