Fiber Bragg Grating Based Stress Measurement for Films Deposited onto Cylindrical Surfaces

Shape of a substrate directly influences the residual stress in thin film coatings. In this study, a method involving Fiber Bragg Grating (FBG) was used to measure residual stress in a film deposited on a cylindrical surface. An FBG has a cylindrical surface and its Bragg wavelength shifts continuously when a film is being deposited on the sensor’s surface. Herein, we calculated the residual strain in the film from the wavelength shift of the Bragg grating by studying the transfer of the residual strain of the cylindrical film to the core of the optical fiber substrate during deposition. By employing the energy method, we derived expressions that related the strain in the core of fiber to the residual strain in single layer films, bilayer films, and multilayer cylindrical films. As an example, we demonstrated a detailed process for testing the stress and the strain distribution across a nickel (Ni) film electrodeposited on the surface of a nickel-phosphorus (Ni-P) alloy-coated optical fiber. The results indicated that the measured strain repeatability was less than 500 μ? and the strain sensitivity was more than ?2 × 10^?3 pm/μ?, when the thickness of the film was less than 5 μm. The negative sign on the strain sensitivity indicated that the tensile strain in the film produced compressive strain in the core of the optical fiber. The FBG sensor system has high test speed, and integrates measurement and signal transmission. This method provides an effective and convenient approach to measure stress in a film deposited on a cylindrical surface.     

Residual Stress Analysis on Thick Film Systems by the Incremental Hole-Drilling Method – Simulation and Experimental Results

The residual stress state in thick film systems, as for example thermal spray coatings, is crucial for many of the component’s properties and for the evaluation of the integrity of the coating under thermal and/or mechanical loading. Therefore it is necessary to be able to determine the local residual stress distribution in the coating, at the interface and in the substrate. The incremental hole-drilling method is a widely used method for measuring residual stress depth profiles, which was already applied for thermally sprayed coatings. But so far no reliable hole-drilling evaluation method exists for layered materials having a stress gradient in depth. The objective was to investigate, how far existing evaluation methods of the incremental hole-drilling method that are only valid for residual stress analysis of homogenous material states can be applied to thick film systems with coating thicknesses between 50 μm and 1000 μm and to point out the application limits for these already existing methods. A systematic Finite Element (FE) study was carried out for coating systems with an axisymmetric residual stress state σ₁?=?σ₂. It is shown that conventional evaluation methods developed for homogeneous, non-layered material states can be successfully applied for a stress evaluation in the substrate and the coating for small and for sufficiently large coating thicknesses, respectively, regardless of the type of evaluation algorithm used i.e. the differential or the integral method. The same accounts for material combinations that have a Young’s modulus ratio close to one, between 0.8 and 1.2. The studies indicated that outside the given ranges case specific calibration must be applied to calculate reliable results. Further, calibration data were determined case specifically for a selected model coating system. The accuracy of a residual stress determination using these case specific calibration data was examined and the sensitivity of the evaluation with respect to an accurate knowledge of the boundary conditions of the coating system i.e. the coating thickness and the Young modulus was studied systematically. Finally, the calibration data were applied on a thermally sprayed aluminium coating on a steel substrate analysis and the results for using the incremental hole drilling method were compared to results from X-ray stress analysis.     

Mathematical analysis of SOFC based on co-ionic conducting electrolyte

In co-ionic conducting solid oxide fuel cell (SOFC), both oxygen ion (O2) and proton (H+) can transport through the electrolyte, generating steam in both the an-ode and cathode. Thus the mass transport phenomenon in the electrodes is quite different from that in conventional SOFC with oxygen ion conducting electrolyte (O-SOFC) or with proton conducting electrolyte (H-SOFC). The generation of steam in both electrodes also affects the concentration over-potential loss and further the SOFC performance. However, no detailed modeling study on SOFCs with co-ionic electrolyte has been reported yet. In this paper, a new mathematical model for SOFC based on co-ionic electrolyte was developed to predict its actual performance considering three major kinds of overpotentials. Ohm’s law and the Butler-Volmer formula were used to model the ion conduction and electrochemical reactions, respectively. The dusty gas model (DGM) was employed to simulate the mass transport processes in the porous electrodes. Parametric simulations were performed to investigate the effects of proton transfer number (tH) and current density (jtotal) on the cell performance. It is interesting to find that the co-ionic conducting SOFC could perform better than O-SOFC and H-SOFC by choosing an appropriate proton transfer number. In addition, the co-ionic SOFC shows smaller difference between the anode and cathode concentration overpotentials than O-SOFC and H-SOFC at certain t H values. The results could help material selection for enhancing SOFC performance.     

Real-Time Stress Measurement in SiO<Subscript>2</Subscript> Thin Films during Electrochemical Lithiation/Delithiation Cycling

Oxide coatings have been shown to improve the cyclic performance of high-energy density electrode materials such as Si. However, no study exists on the mechanical characterization of these oxide coatings. Here, thin film SiO₂ electrodes are cycled under galvanostatic conditions (at C/9 rate) in a half-cell configuration with lithium metal foil as counter/reference electrode, with 1 M LiPF₆ in ethylene carbonate, diethyl carbonate, dimethyl carbonate solution (1:1:1, wt%) as electrolyte. Stress evolution in the SiO₂ thin film electrodes during electrochemical lithiation/delithiation is measured in situ by monitoring the substrate curvature using a multi-beam optical sensing method. Upon lithiation SiO₂ undergoes extensive inelastic deformation, with a peak compressive stress of 3.1 GPa, and upon delithiation the stress became tensile with a peak stress of 0.7 GPa. A simple plane strain finite element model of Si nanotube coated with SiO₂ shell was developed to understand the mechanical response of the core-shell type microstructures under electrochemical cycling; measured stress response was used in the model to represent SiO₂ constitutive behavior while Si was treated as an elastic-plastic material with concentration dependent mechanical properties obtained from the literature. The results reported here provide insights and quantitative understanding as to why the highly brittle SiO₂ coatings are able to sustain significant volume expansion (300%) of Si core without fracture and enhance cyclic performance of Si reported in the literature. Also, the basic mechanical properties presented here are necessary first step for future design and development of durable Si/SiO₂ core shell structures or SiO₂-based electrodes.     

Measurement of the Residual Stress Tensor in a Compact Tension Weld Specimen

Neutron diffraction measurements have been performed to determine the full residual stress tensor along the expected crack path in an austenitic stainless steel (Esshete 1250) compact tension weld specimen. A destructive slitting method was then implemented on the same specimen to measure the stress intensity factor profile associated with the residual stress field as a function of crack length. Finally deformations of the cut surfaces were measured to determine a contour map of the residual stresses in the specimen prior to the cut. The distributions of transverse residual stress measured by the three techniques are in close agreement. A peak tensile stress in excess of 600 MPa was found to be associated with an electron beam weld used to attach an extension piece to the test sample, which had been extracted from a pipe manual metal arc butt weld. The neutron diffraction measurements show that exceptionally high residual stress triaxiality is present at crack depths likely to be used for creep crack growth testing and where a peak stress intensity factor of 35 MPa√m was measured (crack depth of 21 mm). The neutron diffraction measurements identified maximum values of shear stress in the order of 50 MPa and showed that the principal stress directions were aligned to within ~20° of the specimen orthogonal axes. Furthermore it was confirmed that measurement of strains by neutron diffraction in just the three specimen orthogonal directions would have been sufficient to provide a reasonably accurate characterisation of the stress state in welded CT specimens.     

Optimal design of functionally graded plates with thermo-elastic plastic behaviourConfiguration optimale des plaques fonctionnellement graduées à plasticité thermo-élastique

The thermo-elastic plastic behaviour of functionally graded plates under extremal thermal loading at different boundary conditions is considered. The plates consist of two phases – ZrO₂ ceramics and Ti₆Al₄V alloy. The layers are distributed exponentially through the thickness. The mechanical and thermal properties of both materials strongly depend on temperature. The stress–strain behaviour is investigated by the FEM. To predict the stable state of the structures of interest, several failure criteria are applied. Two cost functions are introduced to optimize the design of the plate. The main results are discussed and graphically illustrated. To cite this article: L. Parashkevova et al., C. R. Mecanique 332 (2004).     

Synchronous Full-Field Strain and Temperature Measurement in Tensile Tests at Low,Intermediate and High Strain Rates

Tensile tests with simultaneous full-field strain and temperature measurements at the nominal strain rates of 0.01, 0.1, 1, 200 and 3000 s^?1 are presented. Three different testing methods with specimens of the same thin and flat gage-section geometry are utilized. The full-field deformation is measured on one side of the specimen, using the DIC technique with low and high speed visible cameras, and the full-field temperature is measured on the opposite side using an IR camera. Austenitic stainless steel is used as the test material. The results show that a similar deformation pattern evolves at all strain rates with an initial uniform deformation up to the strain of 0.25–0.35, followed by necking with localized deformation with a maximum strain of 0.7–0.95. The strain rate in the necking regions can exceed three times the nominal strain rate. The duration of the tests vary from 57 s at the lowest strain rate to 197 μs at the highest strain rate. The results show temperature rise at all strain rates. The temperature rise increases with strain rate as the test duration shortens and there is less time for the heat to dissipate. At a strain rate of 0.01 s^?1 the temperature rise is small (up to 48 °C) but noticeable. At a strain rate of 0.1 the temperature rises up to 140 °C and at a strain rate of 1 s^?1 up to 260 °C. The temperature increase in the tests at strain rates of 200 s^?1 and 3000 s^?1 is nearly the same with the maximum temperature reaching 375 °C.     

Experimental and Finite Element Simulation Study of Thermal Relaxation of Residual Stresses in Laser Shock Peened IN718 SPF Superalloy

An integrated experimental and modeling/simulation approach was developed to investigate and secure a quantified knowledge of the impact of high temperature exposures on the stability of residual stresses in a laser shock peened (LSP) high temperature aero-engine alloy, IN718 SPF (super-plastically formed). Single dimple LSP and overlap LSP treatments were carried out utilizing a Nd:Glass laser (λ?=?1.052 μm), and subsequent heat treatments on the LSP-treated coupons were conducted at different temperatures between 550 and 700 °C. A 3-D nonlinear finite element (FE) computational model and the rate-dependent Johnson-Cook material model were calibrated using the experimental results of residual stress from the single dimple LSP and thermal relaxation treatments, and were further extended to the overlap LSP treatment case. Both experimental and FE simulations show that: a) a high level of compressive residual stress (~700 MPa at surface) and residual stress depth (~0.4–0.6 mm) were achieved following LSP, and b) the overlap LSP treatment gave higher residual stress and greater depth. The magnitudes of the initial residual stress (and plastic strain), heating temperature and exposure time were identified as the key parameters controlling the thermal relaxation behavior. The stress relaxation mainly occurs initially before 20 min exposure and the extent of relaxation increases with an increase in temperature and a higher magnitude of the initial as-peened residual stress. In addition, in regions deeper than ~300 μm or after initial thermal exposure where the residual stress was lower than ~300 MPa, stress relaxation was found to be negligible. Kinetic analysis of the experimental thermal relaxation data based on Zener-Wert-Avrami model gave an activation enthalpy of 2.87 to 3.77 eV, which is near that reported in the literatures for volume and/or substitutional solute diffusion in Nickel. These results suggest that thermal relaxation of the LSP-induced residual stress occurs by a creep-like mechanism involving recovery, rearrangement and annihilation of dislocations by climb.     

Dynamic rupture of polymer–metal bilayer plates

Recent theoretical assessments of metal/polymer bilayers indicate a potentially significant delay in the onset of ductile failure modes, especially under dynamic loading, due to strain hardening of the polymer. The response of copper/polyurethane bilayers under dynamic and quasi-static loadings is investigated via static tensile, static bulge forming and dynamic bulge forming tests. Two polyurethanes PU1 and PU2 were chosen with a significant contrast in stiffness and ductility: PU1 has a glass transition temperature T_g close to ?56 °C and at room temperature it has a low modulus, low strength and a high tensile failure strain. In contrast, PU2 has a T_g of 49 °C and at room temperature it has a high modulus and strength but a much smaller tensile failure strain. In most of the tests, the polymer coatings were approximately twice the thickness of the metal layer. Under static loadings (tensile and bulge forming) the PU2 bilayer outperformed the uncoated metal plate of equal mass while the PU1 bilayer had a performance inferior to the equivalent uncoated plate. We attribute this to the fact that the PU2 retards the necking of the copper layer and thus increases its energy absorption capacity while the PU1 coating provides no such synergistic effect. The dynamic bulge forming tests indicate that on an equal mass basis, the dynamic performance of the PU2 bilayers with a weakly bonded polymer coating were comparable to the uncoated plates but intriguingly, when the PU2 was strongly adhered to the copper plates the performance of these bilayers was inferior to that of the uncoated plates. Thus, the coatings do not provide dynamic performance benefits on an equal mass basis. However, it is shown that increasing the mass of a plate by adding a polyurethane layer can improve the performance for a given total blast impulse. Given the ease of applying polyurethane coatings they may provide a practical solution to enhancing the blast resistance of existing metallic structures.     

Measurement of Residual Elastic Strains in a Titanium Alloy Using High Energy Synchrotron X-Ray Diffraction

Residual elastic strains in a bent bar of titanium alloy Ti-6Al-4V were measured using high energy diffraction on station 16.3 at SRS Daresbury. Using a single bounce Laue crystal monochromator, diffraction peaks were collected for reflections (00.2), (10.1), (10.2) and (11.0) from the hcp alpha phase of the titanium alloy. Reference values of the lattice spacing for each of the reflections were found from the diffraction pattern collected from a stress-free sampling volume. The residual elastic strain values calculated on the basis of each reflection were then computed and plotted as a function of position across the bent bar. The average macroscopic residual elastic strain was computed using an averaging procedure taking into account the multiplicity of each reflection. Energy dispersive white beam diffraction from the same bent bar was used to collect diffraction patterns over the range of lattice spacings between 0.8 and 2.2 Å. Detector calibration was carried out using the procedure described in Liu et al. (2005) and detailed interpretation of the energy dispersive profiles was carried out allowing the identification of average residual elastic strains in the two principal phases present in the titanium alloy considered, the α-Ti hcp and the β-Ti bcc phases. Peak-specific residual strain profiles computed on the basis of monochromatic measurements show significant differences reflecting the variation in the elastic and plastic properties with grain orientation, i.e., crystal anisotropy. Using the contrast between the elastic and plastic properties of different directions within the α-Ti hcp lattice, the difference between residual elastic strains measured for (00.2) and (11.0) reflections was plotted, as well as the ‘difference strain’ between (00.2) and (10.1) reflections. These profiles show a good qualitative correlation with the plastic strain profile introduced by inelastic bending that was computed from the analysis of Pawley refinement of the energy-dispersive diffraction measurements.     

<Emphasis Type="Italic">In-situ</Emphasis> Measurement of Crystalline Lattice Strains in Polytetrafluoroethylene

Strain measurements by neutron diffraction are employed as an in situ technique to obtain insight into the deformation modes of crystalline domains in a deformed semi-crystalline polymer. The SMARTS (Spectrometer for MAterials Research at Temperature and Stress) diffractometer has been used to measure the crystalline lattice displacements in polytetrafluoroethylene (PTFE) for crystalline phase IV (at room temperature) in tension and compression and for crystalline phase I (at 60°C) in compression. The chemical structure of PTFE, -(C₂F₄)-_n, makes it ideally suited for investigation by neutron methods as it is free of hydrogen that results in limited penetration depths and poor diffraction acquisition in most polymers. Deformation parallel to the prismatic plane normals is shown to occur by inter-polymer chain compression with a modulus ∼10× bulk, while deformation parallel to the basal plane normal occurs by intra-polymer chain compression with a modulus ∼1000× bulk, corresponding with theoretical values for a PTFE chain modulus. Deformation parallel to the pyramidal plane normals is accommodated by inter-polymer chain shear.     

A study of the generation and creep relaxation of triaxial residual stresses in stainless steel

This paper presents results from a numerical and experimental research programme motivated by the need to predict creep damage generated by multi-axial states of stress in austenitic stainless steels. It has been hypothesized that highly triaxial residual stress fields may be sufficient to promote creep damage in thermally aged components, even in the absence of in-service loads. Two prerequisites to test this hypothesis are the provision of test specimens containing a highly triaxial residual stress field and an accurate knowledge of how this residual stress field relaxes due to creep. Creep damage predictions may then be made for these specimens and compared to damage observed in experiments. This paper provides solutions to both of these prerequisites. Cylindrical and spherical test specimens made from type 316H stainless steel are heated to 850 °C and then quenched in water. Finite element predictions of the residual stress state, validated by extensive neutron diffraction measurements, are presented which confirm the high level of triaxiality present in the specimens. The specimens are then thermally aged at 550 °C and numerical predictions of the residual stress relaxation are given, again validated by extensive neutron diffraction measurements. The results confirm the validity of the creep relaxation models employed. In addition, the results show the influence of specimen size and permit comparisons to be made between three different types of neutron diffractometers.     

The Effect of Grain Size on Deformation and Failure of Ti<Subscript>2</Subscript>AlC MAX Phase under Thermo-Mechanical Loading

An experimental investigation was performed to analyze the effects of grain size on the quasi-static and dynamic behavior of Ti₂AlC. High-density Ti₂AlC samples of three different grain sizes were densified using Spark Plasma Sintering and Pressureless sintering. A servo-hydraulic testing machine equipped with a vertical split furnace, and SiC pushrods, was used for the quasi-static experiments. Also, a Split Hopkinson Pressure Bar (SHPB) apparatus and an induction coil heating system were used for the dynamic experiments. A series of experiments were conducted at temperatures ranging from 25 °C to 1100 °C for strain rates of 10^?4 s^?1 and 400 s^?1. The results show that under quasi-static loading the specimens experience a brittle failure for temperatures below Brittle to Plastic Transition Temperature (BPTT) of 900–1000 °C and large deformation at temperatures above the BPTT. During dynamic experiments, the specimens exhibited brittle failure, with the failure transitioning from catastrophic failure at lower temperatures to graceful failure (softening while bearing load) at higher temperatures, and with the propensity for graceful failure increasing with increasing grain size. The compressive strengths of different grain sizes at a given temperature can be related to the grain length by a Hall-Petch type relation.     

Stable Speckle Patterns for Nano-scale Strain Mapping up to 700 °C

The digital image correlation (DIC) of speckle patterns obtained by vapour-assisted gold remodelling at 200 – 350 °C has already been used to map plastic strains with submicron resolution. However, it has not so far proved possible to use such patterns for testing at high temperatures. Here we demonstrate how a gold speckle pattern can be made that is stable at 700 °C, to study deformation in a commercial TiAl alloy (Ti-45Al-2Nb-2Mn(at%)-0.8 vol% TiB₂). The pattern is made up of a uniformly sized random array of Au islands as small as 15 nm in diameter, depending on reconstruction parameters, with a sufficiently small spacing to be suitable for nano-scale, nDIC, strain mapping at a subset size of 60 × 60 nm². It can be used at temperatures up to 700 °C for many hours, for high cycle fatigue testing for instance. There is good particle attachment to the substrate. It can withstand ultra-sound cleaning, is thermally stable and has a high atomic number contrast for topography-free backscatter electron imaging.     

A summation strain-gage alternative to oblique incidence in photoelastic coatings

A special strain gage (PhotoStress^® Separator Gage, Measurements Group, Inc., Raleigh, NC) designed to measure the sum (? ₁ +? ₂) of the principal strains, is used in conjunction with photoelastic-coating measurements (? ₁ -? ₂) to establish the value of each principal strain (? ₁ and? ₂). The summation strain signal is effectively independent of angular orientation (measurement direction), and by design, the gage negates soldering risks, self-heating, and localized-reinforcement considerations normally associated with strain-gage measurements on plastic parts.     

Measurement and Prediction of Machining Induced Redistribution of Residual Stress in the Aluminium Alloy 7449

The residual stress distributions in two 7449 aluminium alloy rectilinear blocks have been determined using neutron diffraction. Heat treatment included cold water immersion quenching and a period of precipitation hardening. Quenching induced very high magnitude residual stresses into the two blocks. One block was measured in this condition while the other was incrementally machined by milling to half thickness. Neutron diffraction measurements were made on the milled half thickness block at equivalent locations to the unmachined block. This permitted through thickness measurements from both blocks to be compared, revealing the redistribution of residual stresses induced by machining. A square cross section post in the centre of the machined face was left to act as a stress free reference sample. The distortions arising on the face opposite to that being milled were measured using a co-ordinate measuring machine. The residual stresses and distortion arising in the blocks have been compared to finite element analysis prediction and found to generally agree. Material removal only caused distortion and the residual stresses to redistribute; there was no stress relaxation evident.     

Structural and Mechanical Properties of Gastropod Connective and Smooth Muscle Tissue

The pedal integument of terrestrial gastropod Arion rufus is composed mainly of smooth muscle cells (SMCs, 45 %), haemocoelic cavities (36 %), and collagen connective tissue. Using stereological methods, SMC two-dimensional length density (0.12 μm^?1), numerical density (426,000 mm^?3), and mean distance (31 μm) in the cluster were assessed. The average SMC could be approximated by an ellipsoid 72 μm in length with semi-axes of 3 μm. Three-dimensional reconstructions of SMCs and haemocoelic cavities of gastropod tissue were created using serial thick and semi-thin sections. These reconstructions showed the spatial arrangement of individual SMCs within the tissue: longitudinally, perpendicularly, and obliquely oriented to the main axis of the gastropod body. Using uniaxial mechanical loading with linearly increasing load or elongation at various loading rates (2, 10, and 20 mN/min; 2 and 3 mm/min) in transverse and longitudinal orientations to the main gastropod body axis, the Young’s modulus of elasticity for small (23–27 kPa) and large deformations (49–132 kPa) as well as ultimate stress (105–250 kPa) and strain (300–400 %) were determined. There was a trend toward stiffer integument tissue in the longitudinal direction compared to the transversal direction and toward increasing stiffness with loading velocity.     

Perforation of 6082-T651 Aluminum Plates with 7.62 mm APM2 Bullets at Normal and Oblique Impacts

We conducted an experimental study to understand the mechanisms and dominant parameters for 7.62 mm APM2 bullets that perforate 6082-T651 aluminum armor plates at oblique impacts. The 7.62-mm-diameter, 10.7 g, APM2 bullet consists of a brass jacket, lead filler, and a 5.25 g, ogive-nose, hard steel core. The brass and lead were stripped from the APM2 bullets by the targets, so we conducted ballistic experiments with both the APM2 bullets and only the hard steel cores. These projectiles were fired from a rifle to striking velocities between 400 and 1,000 m/s into 20-mm-thick plates at normal impact (β?=?0^o) and at oblique angles of β?=?15^o, 30^o, and 45^o. Measured residual and ballistic-limit velocities for the full bullet and the hard core were within a few percent for normal impact and all oblique angles. Thus, we showed that the perforation process was dominated by the hard steel core of the bullet. In addition, we conducted large strain, compression tests on the 6082-T651 plate material for input to perforation equations derived from a cavity-expansion model for the steel core projectiles. Model predictions were shown to be in good agreement with measured ballistic-limit and residual velocity measurements for β?=?0^o, 15^o, and 30^o. We also presented a scaling law for the bullet that showed the ballistic-limit velocities were proportional to the square root of the product of plate thickness and a material strength term.     

Using Micro-Raman Spectroscopy to Assess MEMS Si/SiO2 Membranes Exhibiting Negative Spring Constant Behavior

We introduce a novel micro-mechanical structure that exhibits two regions of stable linear positive and negative stiffness. Springs, cantilevers, beams and any other geometry that display an increasing return force that is proportional to the displacement can be considered to have a “Hookean” positive spring constant, or stiffness. Less well known is the opposite characteristic of a reducing return force for a given deflection, or negative stiffness. Unfortunately many simple negative stiffness structures exhibit unstable buckling and require additional moving components during deflection to avoid deforming out of its useful shape. In Micro-Electro-Mechanical Systems (MEMS) devices, buckling caused by stress at the interface of silicon and thermally grown SiO₂ causes tensile and compressive forces that will warp structures if the silicon layer is thin enough. The 1 mm² membrane structures presented here utilizes this effect but overcome this limitation and empirically demonstrates linearity in both regions. The Si/SiO₂ membranes presented deflect ~17 μm from their pre-released position. The load deflection curves produced exhibit positive linear stiffness with an inflection point holding nearly constant with a slight negative stiffness. Depositing a 0.05 μm titanium and 0.3 μm layer of gold on top of the Si/SiO₂ membrane reduces the initial deflection to ~13.5 μm. However, the load deflection curve produced illustrates both a linear positive and negative spring constant with a fairly sharp inflection point. These results are potentially useful to selectively tune the spring constant of mechanical structures used in MEMS. The structures presented are manufactured using typical micromachining techniques and can be fabricated in-situ with other MEMS devices.     

Residual mass and flow regimes for the immiscible liquid–liquid displacement in a plane channel

The motion of two immiscible liquids in a plane channel is analyzed for the case in which the flow conditions and the interactions between the liquids and the solid surface maintain the displaced fluid attached to the wall. The Galerkin Finite Element Method is used to compute the velocity field and the configuration of the interface between the two fluids. We compare the residual mass fraction left on the wall with its two counterparts in capillary tubes, namely residual mass fraction and dimensionless layer thickness of the displaced fluid. The main result of this comparison was that although there is a qualitative similarity concerning the layer thickness between the two cases, the residual fraction of mass presented an important difference, showing that when the aspect ratio of the capillary passage is large there is an increase in the displacement efficiency. The thickness of the displaced liquid film attached to the channel walls is a function of the capillary number (Ca) and the viscosity ratio (N_μ). A map of streamlines in the Cartesian space (Ca, N_μ) with the different flow regimes of the problem is presented. We also showed that we can adapt the available analytical results obtained for gas-displacement in capillary tubes to the plane channel case, for low values of Ca.