Spectral bounds and basis results for non-self-adjoint pencils, with an application to Hagen-Poiseuille flow

We obtain eigenvalue enclosures and basisness results for eigen- and associated functions of a non-self-adjoint unbounded linear operator pencil A−λBA−λB in which BB is uniformly positive and the essential spectrum of the pencil is empty. Both Riesz basisness and Bari basisness results are obtained. The results are applied to a system of singular differential equations arising in the study of Hagen–Poiseuille flow with non-axisymmetric disturbances.

Bifurcation contrasts between plane Poiseuille flow and plane Magnetohydrodynamic flow

The stability characteristics of an incompressible viscous pressure-driven flow of an electrically conducting fluid between two parallel boundaries in the presence of a transverse magnetic field are compared and contrasted with those of Plane Poiseuille flow (PPF). Assuming that the outer regions adjacent to the fluid layer are perfectly electrically insulating, the appropriate boundary conditions are applied. The eigenvalue problems are then solved numerically to obtain the critical Reynolds number Rec and the critical wave number ac in the limit of small Hartmann number (M) range to produce the curves of marginal stability. The non-linear two-dimensional travelling waves that bifurcate by way of a Hopf bifurcation from the neutral curves are approximated by a truncated Fourier series in the streamwise direction. Two and three dimensional secondary disturbances are applied to both the constant pressure and constant flux equilibrium solutions using Floquet theory as this is believed to be the generic mechanism of instability in shear flows. The change in shape of the undisturbed velocity profile caused by the magnetic field is found to be the dominant factor. Consequently the critical Reynolds number is found to increase rapidly with increasing M so the transverse magnetic field has a powerful stabilising effect on this type of flow.

Small-angle x-ray scattering study of flow alignment of a thermotropic liquid crystal in the nematic and smectic phases

WThe capillary flow alignment of the thermotropic liquid crystal 4-n-octyl-4′-cyanobiphenyl in the nematic and smectic phases is investigated using time-resolved synchrotron small-angle x-ray scattering. Samples were cooled from the isotropic phase to erase prior orientation. Upon cooling through the nematic phase under Poiseuille flow in a circular capillary, a transition from the alignment of mesogens along the flow direction to the alignment of layers along the flow direction (mesogens perpendicular to flow) appears to occur continuously at the cooling rate applied. The transition is centered on a temperature at which the Leslie viscosity coefficient α3 changes sign. The configuration with layers aligned along the flow direction is also observed in the smectic phase. The transition in the nematic phase on cooling has previously been ascribed to an aligning-nonaligning or tumbling transition. At high flow rates there is evidence for tumbling around an average alignment of layers along the flow direction. At lower flow rates this orientation is more clearly defined. The layer alignment is ascribed to surface-induced ordering propagating into the bulk of the capillary, an observation supported by the parallel alignment of layers observed for a static sample at low temperatures in the nematic phase.

Cell exclusion in couette flow:evaluation through flow visualisation and mechanical forces

Cell exclusion is the phenomenon whereby the hematocrit and viscosity of blood decrease in areas of high stress. While this is well known in naturally occurring Poiseuille flow in the human body, it has never previously been shown in Couette flow, which occurs in implantable devices including blood pumps. The high-shear stresses that occur in the gap between the boundaries in Couette flow are known to cause hemolysis in erythrocytes. We propose to mitigate this damage by initiating cell exclusion through the use of a spiral-groove bearing (SGB) that will provide escape routes by which the cells may separate themselves from the plasma and the high stresses in the gap. The force between two bearings (one being the SGB) in Couette flow was measured. Stained erythrocytes, along with silver spheres of similar diameter to erythrocytes, were visualized across a transparent SGB at various gap heights. A reduction in the force across the bearing for human blood, compared with fluids of comparable viscosity, was found. This indicates a reduction in the viscosity of the fluid across the bearing due to a lowered hematocrit because of cell exclusion. The corresponding images clearly show both cells and spheres being excluded from the gap by entering the grooves. This is the first time the phenomenon of cell exclusion has been shown in Couette flow. It not only furthers our understanding of how blood responds to different flows but could also lead to improvements in the future design of medical devices.

Lattice Boltzmann method for flow and heat transfer in microgeometries

Recent technological developments have made it possible to design various microdevices where fluid flow and heat transfer are involved. For the proper design of such systems, the governing physics needs to be investigated. Due to the difficulty to study complex geometries in micro scales using experimental techniques, computational tools are developed to analyze and simulate flow and heat transfer in microgeometries. However, conventional numerical methods using the Navier-Stokes equations fail to predict some aspects of microflows such as nonlinear pressure distribution, increase mass flow rate, slip flow and temperature jump at the solid boundaries. This necessitates the development of new computational methods which depend on the kinetic theory that are both accurate and computationally efficient. In this study, lattice Boltzmann method (LBM) was used to investigate the flow and heat transfer in micro sized geometries. The LBM depends on the Boltzmann equation which is valid in the whole rarefaction regime that can be observed in micro flows. Results were obtained for isothermal channel flows at Knudsen numbers higher than 0.01 at different pressure ratios. LBM solutions for micro-Couette and micro-Poiseuille flow were found to be in good agreement with the analytical solutions valid in the slip flow regime (0.01 < Kn < 0.1) and direct simulation Monte Carlo solutions that are valid in the transition regime (0.1 < Kn < 10) for pressure distribution and velocity field. The isothermal LBM was further extended to simulate flows including heat transfer. The method was first validated for continuum channel flows with and without constrictions by comparing the thermal LBM results against accurate solutions obtained from analytical equations and finite element method. Finally, the capability of thermal LBM was improved by adding the effect of rarefaction and the method was used to analyze the behavior of gas flow in microchannels. The major finding of this research is that, the newly developed particle-based method described here can be used as an alternative numerical tool in order to study non-continuum effects observed in micro-electro-mechanical-systems (MEMS).

Lattice Boltzmann Method for Flow and Heat Transfer in Microgeometries

Recent technological developments have made it possible to design various microdevices where fluid flow and heat transfer are involved. For the proper design of such systems, the governing physics needs to be investigated. Due to the difficulty to study complex geometries in micro scales using experimental techniques, computational tools are developed to analyze and simulate flow and heat transfer in microgeometries. However, conventional numerical methods using the Navier-Stokes equations fail to predict some aspects of microflows such as nonlinear pressure distribution, increase mass flow rate, slip flow and temperature jump at the solid boundaries. This necessitates the development of new computational methods which depend on the kinetic theory that are both accurate and computationally efficient. In this study, lattice Boltzmann method (LBM) was used to investigate the flow and heat transfer in micro sized geometries. The LBM depends on the Boltzmann equation which is valid in the whole rarefaction regime that can be observed in micro flows. Results were obtained for isothermal channel flows at Knudsen numbers higher than 0.01 at different pressure ratios. LBM solutions for micro-Couette and micro-Poiseuille flow were found to be in good agreement with the analytical solutions valid in the slip flow regime (0.01 < Kn < 0.1) and direct simulation Monte Carlo solutions that are valid in the transition regime (0.1 < Kn < 10) for pressure distribution and velocity field. The isothermal LBM was further extended to simulate flows including heat transfer. The method was first validated for continuum channel flows with and without constrictions by comparing the thermal LBM results against accurate solutions obtained from analytical equations and finite element method. Finally, the capability of thermal LBM was improved by adding the effect of rarefaction and the method was used to analyze the behavior of gas flow in microchannels. The major finding of this research is that, the newly developed particle-based method described here can be used as an alternative numerical tool in order to study non-continuum effects observed in micro-electro-mechanical-systems (MEMS).

Computer simulations of soft particles in flow

Understanding the dynamics of blood cells is a crucial element to discover biological mechanisms, to develop new efficient drugs, design sophisticated microfluidic devices, for diagnostics. In this work, we focus on the dynamics of red blood cells in microvascular flow. Microvascular blood flow resistance has a strong impact on cardiovascular function and tissue perfusion. The flow resistance in microcirculation is governed by flow behavior of blood through a complex network of vessels, where the distribution of red blood cells across vessel cross-sections may be significantly distorted at vessel bifurcations and junctions. We investigate the development of blood flow and its resistance starting from a dispersed configuration of red blood cells in simulations for different hematocrits, flow rates, vessel diameters, and aggregation interactions between red blood cells. Initially dispersed red blood cells migrate toward the vessel center leading to the formation of a cell-free layer near the wall and to a decrease of the flow resistance. The development of cell-free layer appears to be nearly universal when scaled with a characteristic shear rate of the flow, which allows an estimation of the length of a vessel required for full flow development, $l_c \approx 25D$, with vessel diameter $D$. Thus, the potential effect of red blood cell dispersion at vessel bifurcations and junctions on the flow resistance may be significant in vessels which are shorter or comparable to the length $l_c$. The presence of aggregation interactions between red blood cells lead in general to a reduction of blood flow resistance. The development of the cell-free layer thickness looks similar for both cases with and without aggregation interactions. Although, attractive interactions result in a larger cell-free layer plateau values. However, because the aggregation forces are short-ranged at high enough shear rates ($\bar{\dot{\gamma}} \gtrsim 50~\text{s}^{-1}$) aggregation of red blood cells does not bring a significant change to the blood flow properties. Also, we develop a simple theoretical model which is able to describe the converged cell-free-layer thickness with respect to flow rate assuming steady-state flow. The model is based on the balance between a lift force on red blood cells due to cell-wall hydrodynamic interactions and shear-induced effective pressure due to cell-cell interactions in flow. We expect that these results can also be used to better understand the flow behavior of other suspensions of deformable particles such as vesicles, capsules, and cells. Finally, we investigate segregation phenomena in blood as a two-component suspension under Poiseuille flow, consisting of red blood cells and target cells. The spatial distribution of particles in blood flow is very important. For example, in case of nanoparticle drug delivery, the particles need to come closer to microvessel walls, in order to adhere and bring the drug to a target position within the microvasculature. Here we consider that segregation can be described as a competition between shear-induced diffusion and the lift force that pushes every soft particle in a flow away from the wall. In order to investigate the segregation, on one hand, we have 2D DPD simulations of red blood cells and target cell of different sizes, on the other hand the Fokker-Planck equation for steady state. For the equation we measure force profile, particle distribution and diffusion constant across the channel. We compare simulation results with those from the Fokker-Planck equation and find a very good correspondence between the two approaches. Moreover, we investigate the diffusion behavior of target particles for different hematocrit values and shear rates. Our simulation results indicate that diffusion constant increases with increasing hematocrit and depends linearly on shear rate. The third part of the study describes development of a simulation model of complex vascular geometries. The development of the model is important to reproduce vascular systems of small pieces of tissues which might be gotten from MRI or microscope images. The simulation model of the complex vascular systems might be divided into three parts: modeling the geometry, developing in- and outflow boundary conditions, and simulation domain decomposition for an efficient computation. We have found that for the in- and outflow boundary conditions it is better to use the SDPD fluid than DPD one because of the density fluctuations along the channel of the latter. During the flow in a straight channel, it is difficult to control the density of the DPD fluid. However, the SDPD fluid has not that shortcoming even in more complex channels with many branches and in- and outflows because the force acting on particles is calculated also depending on the local density of the fluid.

Sloshing and slamming oscillations in collapsible channel flow

We consider laminar high-Reynolds-number flow through a finite-length planar channel, where a portion of one wall is replaced by a thin massless elastic membrane that is held under longitudinal tension T and subject to an external pressure distribution. The flow is driven by a fixed pressure drop along the full length of the channel. We investigate the global stability of two-dimensional Poiseuille flow using a method of matched local eigenfunction expansions, which is compared to direct numerical simulations. We trace the neutral stability curve of the primary oscillatory instability of the system, illustrating a transition from high-frequency ‘sloshing’ oscillations at high T to vigorous ‘slamming’ motion at low T . Small-amplitude sloshing at high T can be captured using a low-order eigenmode truncation involving four surface-based modes in the compliant segment of the channel coupled to Womersley flow in the rigid segments. At lower tensions, we show that hydrodynamic modes contribute increasingly to the global instability and we demonstrate a change in the mechanism of energy transfer from the mean flow, with viscous effects being destabilising. Simulations of finite-amplitude oscillations at low T reveal a generic slamming motion, in which the the flexible membrane is drawn close to the opposite rigid wall before rapidly recovering. A simple model is used to demonstrate how fluid inertia in the downstream rigid channel segment, coupled to membrane curvature downstream of the moving constriction, together control slamming dynamics.

Verification of a mixed high-order accurate DNS code for laminar turbulent transition by the method of manufactured solutions

This paper presents results on a verification test of a Direct Numerical Simulation code of mixed high-order of accuracy using the method of manufactured solutions (MMS). This test is based on the formulation of an analytical solution for the Navier-Stokes equations modified by the addition of a source term. The present numerical code was aimed at simulating the temporal evolution of instability waves in a plane Poiseuille flow. The governing equations were solved in a vorticity-velocity formulation for a two-dimensional incompressible flow. The code employed two different numerical schemes. One used mixed high-order compact and non-compact finite-differences from fourth-order to sixth-order of accuracy. The other scheme used spectral methods instead of finite-difference methods for the streamwise direction, which was periodic. In the present test, particular attention was paid to the boundary conditions of the physical problem of interest. Indeed, the verification procedure using MMS can be more demanding than the often used comparison with Linear Stability Theory. That is particularly because in the latter test no attention is paid to the nonlinear terms. For the present verification test, it was possible to manufacture an analytical solution that reproduced some aspects of an instability wave in a nonlinear stage. Although the results of the verification by MMS for this mixed-order numerical scheme had to be interpreted with care, the test was very useful as it gave confidence that the code was free of programming errors. Copyright (C) 2009 John Wiley & Sons, Ltd.

Using the GPU to design complex profile extrusion dies

In the present work the benefits of using graphics processing units (GPU) to aid the design of complex geometry profile extrusion dies, are studied. For that purpose, a3Dfinite volume based code that employs unstructured meshes to solve and couple the continuity, momentum and energy conservation equations governing the fluid flow, together with aconstitutive equation, was used. To evaluate the possibility of reducing the calculation time spent on the numerical calculations, the numerical code was parallelized in the GPU, using asimple programing approach without complex memory manipulations. For verificationpurposes, simulations were performed for three benchmark problems: Poiseuille flow, lid-driven cavity flow and flow around acylinder. Subsequently, the code was used on the design of two real life extrusion dies for the production of a medical catheter and a wood plastic composite decking profile. To evaluate the benefits, the results obtained with the GPU parallelized code were compared, in terms of speedup, with a serial implementation of the same code, that traditionally runs on the central processing unit (CPU). The results obtained show that, even with the simple parallelization approach employed, it was possible to obtain a significant reduction of the computation times.

Gravity-driven slug motion in capillary tubes

The velocity of a liquid slug falling in a capillary tube is lower than predicted for Poiseuille flow due to presence of menisci, whose shapes are determined by the complex interplay of capillary, viscous, and gravitational forces. Due to the presence of menisci, a capillary pressure proportional to surface curvature acts on the slug and streamlines are bent close to the interface, resulting in enhanced viscous dissipation at the wedges. To determine the origin of drag-force increase relative to Poiseuille flow, we compute the force resultant acting on the slug by integrating Navier-Stokes equations over the liquid volume. Invoking relationships from differential geometry we demonstrate that the additional drag is due to viscous forces only and that no capillary drag of hydrodynamic origin exists (i.e., due to hydrodynamic deformation of the interface). Requiring that the force resultant is zero, we derive scaling laws for the steady velocity in the limit of small capillary numbers by estimating the leading order viscous dissipation in the different regions of the slug (i.e., the unperturbed Poiseuille-like bulk, the static menisci close to the tube axis and the dynamic regions close to the contact lines). Considering both partial and complete wetting, we find that the relationship between dimensionless velocity and weight is, in general, nonlinear. Whereas the relationship obtained for complete-wetting conditions is found in agreement with the experimental data of Bico and Quere [J. Bico and D. Quere, J. Colloid Interface Sci. 243, 262 (2001)], the scaling law under partial-wetting conditions is validated by numerical simulations performed with the Volume of Fluid method. The simulated steady velocities agree with the behavior predicted by the theoretical scaling laws in presence and in absence of static contact angle hysteresis. The numerical simulations suggest that wedge-flow dissipation alone cannot account for the entire additional drag and that the non-Poiseuille dissipation in the static menisci (not considered in previous studies) has to be considered for large contact angles.

Development of filter media treatments for liquid filtration

Woven monofilament, multifilament, and spun yarn filter media have long been the standard media in liquid filtration equipment. While the energy for a solid-liquid separation process is determined by the engineering work, it is the interface between the slurry and the equipment - the filter media - that greatly affects the performance characteristics of the unit operation. Those skilled in the art are well aware that a poorly designed filter medium may endanger the whole operation, whereas well-performing filter media can make the operation smooth and economical. As the mineral and pulp producers seek to produce ever finer and more refined fractions of their products, it is becoming increasingly important to be able to dewater slurries with average particle sizes around 1 ¿m using conventional, high-capacity filtration equipment. Furthermore, the surface properties of the media must not allow sticky and adhesive particles to adhere to the media. The aim of this thesis was to test how the dirt-repellency, electrical resistance and highpressure filtration performance of selected woven filter media can be improved by modifying the fabric or yarn with coating, chemical treatment and calendering. The results achieved by chemical surface treatments clearly show that the woven media surface properties can be modified to achieve lower electrical resistance and improved dirt-repellency. The main challenge with the chemical treatments is the abrasion resistance and, while the experimental results indicate that the treatment is sufficiently permanent to resist standard weathering conditions, they may still prove to be inadequately strong in terms of actual use.From the pressure filtration studies in this work, it seems obvious that the conventional woven multifilament fabrics still perform surprisingly well against the coated media in terms of filtrate clarity and cake build-up. Especially in cases where the feed slurry concentration was low and the pressures moderate, the conventional media seemed to outperform the coated media. In the cases where thefeed slurry concentration was high, the tightly woven media performed well against the monofilament reference fabrics, but seemed to do worse than some of the coated media. This result is somewhat surprising in that the high initial specific resistance of the coated media would suggest that the media will blind more easily than the plain woven media. The results indicate, however, that it is actually the woven media that gradually clogs during the coarse of filtration. In conclusion, it seems obvious that there is a pressure limit above which the woven media looses its capacity to keep the solid particles from penetrating the structure. This finding suggests that for extreme pressures the only foreseeable solution is the coated fabrics supported by a strong enough woven fabric to hold thestructure together. Having said that, the high pressure filtration process seems to follow somewhat different laws than the more conventional processes. Based on the results, it may well be that the role of the cloth is most of all to support the cake, and the main performance-determining factor is a long life time. Measuring the pore size distribution with a commercially available porometer gives a fairly accurate picture of the pore size distribution of a fabric, but failsto give insight into which of the pore sizes is the most important in determining the flow through the fabric. Historically air, and sometimes water, permeability measures have been the standard in evaluating media filtration performance including particle retention. Permeability, however, is a function of a multitudeof variables and does not directly allow the estimation of the effective pore size. In this study a new method for estimating the effective pore size and open pore area in a densely woven multifilament fabric was developed. The method combines a simplified equation of the electrical resistance of fabric with the Hagen-Poiseuille flow equation to estimate the effective pore size of a fabric and the total open area of pores. The results are validated by comparison to the measured values of the largest pore size (Bubble point) and the average pore size. The results show good correlation with measured values. However, the measured and estimated values tend to diverge in high weft density fabrics. This phenomenon is thought to be a result of a more tortuous flow path of denser fabrics, and could most probably be cured by using another value for the tortuosity factor.

Une nouvelle méthode smoothed particle hydrodynamics : simulation des interfaces immergées et de la dynamique Brownienne des molécules avec des interactions hydrodynamiques

Dans cette thèse, nous présentons une nouvelle méthode smoothed particle hydrodynamics (SPH) pour la résolution des équations de Navier-Stokes incompressibles, même en présence des forces singulières. Les termes de sources singulières sont traités d'une manière similaire à celle que l'on retrouve dans la méthode Immersed Boundary (IB) de Peskin (2002) ou de la méthode régularisée de Stokeslets (Cortez, 2001). Dans notre schéma numérique, nous mettons en oeuvre une méthode de projection sans pression de second ordre inspirée de Kim et Moin (1985). Ce schéma évite complètement les difficultés qui peuvent être rencontrées avec la prescription des conditions aux frontières de Neumann sur la pression. Nous présentons deux variantes de cette approche: l'une, Lagrangienne, qui est communément utilisée et l'autre, Eulerienne, car nous considérons simplement que les particules SPH sont des points de quadrature où les propriétés du fluide sont calculées, donc, ces points peuvent être laissés fixes dans le temps. Notre méthode SPH est d'abord testée à la résolution du problème de Poiseuille bidimensionnel entre deux plaques infinies et nous effectuons une analyse détaillée de l'erreur des calculs. Pour ce problème, les résultats sont similaires autant lorsque les particules SPH sont libres de se déplacer que lorsqu'elles sont fixes. Nous traitons, par ailleurs, du problème de la dynamique d'une membrane immergée dans un fluide visqueux et incompressible avec notre méthode SPH. La membrane est représentée par une spline cubique le long de laquelle la tension présente dans la membrane est calculée et transmise au fluide environnant. Les équations de Navier-Stokes, avec une force singulière issue de la membrane sont ensuite résolues pour déterminer la vitesse du fluide dans lequel est immergée la membrane. La vitesse du fluide, ainsi obtenue, est interpolée sur l'interface, afin de déterminer son déplacement. Nous discutons des avantages à maintenir les particules SPH fixes au lieu de les laisser libres de se déplacer. Nous appliquons ensuite notre méthode SPH à la simulation des écoulements confinés des solutions de polymères non dilués avec une interaction hydrodynamique et des forces d'exclusion de volume. Le point de départ de l'algorithme est le système couplé des équations de Langevin pour les polymères et le solvant (CLEPS) (voir par exemple Oono et Freed (1981) et Öttinger et Rabin (1989)) décrivant, dans le cas présent, les dynamiques microscopiques d'une solution de polymère en écoulement avec une représentation bille-ressort des macromolécules. Des tests numériques de certains écoulements dans des canaux bidimensionnels révèlent que l'utilisation de la méthode de projection d'ordre deux couplée à des points de quadrature SPH fixes conduit à un ordre de convergence de la vitesse qui est de deux et à une convergence d'ordre sensiblement égale à deux pour la pression, pourvu que la solution soit suffisamment lisse. Dans le cas des calculs à grandes échelles pour les altères et pour les chaînes de bille-ressort, un choix approprié du nombre de particules SPH en fonction du nombre des billes N permet, en l'absence des forces d'exclusion de volume, de montrer que le coût de notre algorithme est d'ordre O(N). Enfin, nous amorçons des calculs tridimensionnels avec notre modèle SPH. Dans cette optique, nous résolvons le problème de l'écoulement de Poiseuille tridimensionnel entre deux plaques parallèles infinies et le problème de l'écoulement de Poiseuille dans une conduite rectangulaire infiniment longue. De plus, nous simulons en dimension trois des écoulements confinés entre deux plaques infinies des solutions de polymères non diluées avec une interaction hydrodynamique et des forces d'exclusion de volume.

An improved finite element space for discontinuous pressures

We consider incompressible Stokes flow with an internal interface at which the pressure is discontinuous, as happens for example in problems involving surface tension. We assume that the mesh does not follow the interface, which makes classical interpolation spaces to yield suboptimal convergence rates (typically, the interpolation error in the L(2)(Omega)-norm is of order h(1/2)). We propose a modification of the P(1)-conforming space that accommodates discontinuities at the interface without introducing additional degrees of freedom or modifying the sparsity pattern of the linear system. The unknowns are the pressure values at the vertices of the mesh and the basis functions are computed locally at each element, so that the implementation of the proposed space into existing codes is straightforward. With this modification, numerical tests show that the interpolation order improves to O(h(3/2)). The new pressure space is implemented for the stable P(1)(+)/P(1) mini-element discretization, and for the stabilized equal-order P(1)/P(1) discretization. Assessment is carried out for Poiseuille flow with a forcing surface and for a static bubble. In all cases the proposed pressure space leads to improved convergence orders and to more accurate results than the standard P(1) space. In addition, two Navier-Stokes simulations with moving interfaces (Rayleigh-Taylor instability and merging bubbles) are reported to show that the proposed space is robust enough to carry out realistic simulations. (c) 2009 Elsevier B.V. All rights reserved.