Droplet generation in a microchannel with a controllable deformable wall

We report the droplet generation behavior of a microfluidic droplet generator with a controllable deformable membrane wall using experiments and analytical model. The confinement at the droplet generation junction is controlled by using external pressure, which acts on the membrane, to generate droplets smaller than junction size (with other parameters fixed) and stable and monodispersed droplets even at higher capillary numbers. A non-dimensional parameter, i.e., controlling parameter K _p, is used to represent the membrane deformation characteristics due to the external pressure. We investigate the effect of the controlled membrane deformation (in terms of K _p), viscosity ratio λ and flow rate ratio r on the droplet size and mobility. A correlation is developed to predict droplet size in the controllable deformable microchannel in terms of the controlling parameter K _p, viscosity ratio λ and flow rate ratio r. Due to the deflection of the membrane wall, we demonstrate that the transition from the stable dripping regime to the unstable jetting regime is delayed to a higher capillary number Ca (as compared to rigid droplet generators), thus pushing the high throughput limit. The droplet generator also enables generation of droplets of sizes smaller than the junction size by adjusting the controlling parameter.     

Study on microchannel flows with a sudden contraction–expansion at a wide range of Knudsen number using lattice Boltzmann method

Numerical simulations have been performed on the pressure-driven rarefied flow through channels with a sudden contraction–expansion of 2:1:2 using isothermal two and three-dimensional lattice Boltzmann method (LBM). In the LBM, a Bosanquet-type effective viscosity and a modified second-order slip boundary condition are used to account for the rarefaction effect on gas viscosity to cover the slip and transition flow regimes, that is, a wider range of Knudsen number. Firstly, the in-house LBM code is verified by comparing the computed pressure distribution and flow pattern with experimental ones measured by others. The verified code is then used to study the effects of the outlet Knudsen number Kn _o , driving pressure ratio P _i /P _o , and Reynolds number Re, respectively, varied in the ranges of 0.001–1.0, 1.15–5.0, and 0.02–120, on the pressure distributions and flow patterns as well as to document the differences between continuum and rarefied flows. Results are discussed in terms of the distributions of local pressure, Knudsen number, centerline velocity, and Mach number. The variations of flow patterns and vortex length with Kn _o and Re are also documented. Moreover, a critical Knudsen number is identified to be Kn _oc  = 0.1 below and above which the behaviors of nonlinear pressure profile and velocity distribution and the variations of vortex length with Re upstream and downstream of constriction are different from those of continuum flows.     

Inertial migration of single particle in a square microchannel over wide ranges of <Emphasis Type="Italic">Re</Emphasis> and particle sizes

Inertial migration of particles has been widely used in inertial microfluidic systems to passively manipulate cells/particles. However, the migration behaviors and the underlying mechanisms, especially in a square microchannel, are still not very clear. In this paper, the immersed boundary-lattice Boltzmann method (IB-LBM) was introduced and validated to explore the migration characteristics and the underlying mechanisms of an inertial focusing single particle in a square microchannel. The grid-independence analysis was made first to highlight that the grid number across the thin liquid film (between a particle and its neighboring channel wall) was of significant importance in accurately capturing the migrating particle’s dynamics. Then, the inertial migration of a single particle was numerically investigated over wide ranges of Reynolds number (Re, from 10 to 500) and particle sizes (diameter-to-height ratio a/H, from 0.16 to 0.5). It was interesting to find that as Re increased, the channel face equilibrium (CFE) position moved outward to channel walls at first, and then inflected inwards to the channel center at high Re (Re?>?200). To account for the physical mechanisms behind this behavior, the secondary flow induced by the inertial focusing single particle was further investigated. It was found that as Re increased, two vortices appeared around the particle and grew gradually, which pushed the particle away from the channel wall at high Re. Finally, a correlation was proposed based on the numerical data to predict the critical length L_c (defined to describe the size of fluid domain that was strongly influenced by the particle) according to the particle size a/H and Re.     

Effects of topological changes in microchannel geometries on the asymmetric breakup of a droplet

Passive asymmetric breakups of a droplet could be done in many microchannels of various geometries. In order to study the effects of different geometries on the asymmetric breakup of a droplet, four types of asymmetric microchannels with the topological equivalence of geometry are designed, which are T-90, Y-120, Y-150, and I-180 microchannels. A three-dimensional volume of fluid multiphase model is employed to investigate the asymmetric rheological behaviors of a droplet numerically. Three regimes of rheological behaviors as a function of the capillary numbers Ca and the asymmetries As defined by As = (b1 ? b2)/(b1 + b2) (where b1 and b2 are the widths of two asymmetric sidearms) have been observed. A power law model based on three major factors (Ca, As and the initial volume ratio r ₀) is employed to describe the volume ratio of two daughter droplets. The analysis of pressure fields shows that the pressure gradient inside the droplet is one of the major factors causing the droplet translation during its asymmetric breakup. Besides the above similarities among various microchannels, the asymmetric breakup in them also have some slight differences as various geometries have different enhancement or constraint effects on the translation of the droplet and the cutting action of flows. It is disclosed that I-180 microchannel has the smallest critical capillary number, the shortest splitting time, and is hardest to generate satellite droplets.     

Demarcating wetting states in textured microchannels under flow conditions by Poiseuille number

In this work, flow friction in microchannels decorated with micropillars was investigated experimentally, with an interest to understand the wetting transition through two simple means: Poiseuille number and scaling laws. Different wetting states were demarcated by qualitatively assessing the behaviour of Poiseuille number (Po = f·Re, where f is friction factor and Re is Reynolds number), which are further corroborated by confocal microscopy-based measurements and numerical simulations. The wetting transition ensued smoothly with an increase in Re, independent of the gas fraction (a ratio of area covered by the liquid–gas interface to the total projected area), for moderate gas fractions, whereas an early breakdown of the Cassie–Baxter state occurred irrespective of Re at high gas fractions. Additionally, the scaling laws were found to correlate well with the underlying state of the flow. Our observations revealed that the liquid–gas interface exhibits a partial slip, contrary to the common notion that it is shear free. It is inferred that an increase in effective flow area leads to a reduction in flow friction in textured microchannels. The present work underlines three important outcomes. The first is the identification of wetting states in flow conditions shown by tracking the Poiseuille number. The second is that the liquid–gas interface is deduced to behave like a partial slip boundary. The third is that a textured microchannel can be worse than an enlarged dimension microchannel.     

Deterministic lateral displacement (DLD) in the high Reynolds number regime: high-throughput and dynamic separation characteristics

Recent progress in the development of biosensors has created a demand for high-throughput sample preparation techniques that can be easily integrated into microfluidic or lab-on-a-chip platforms. One mechanism that may satisfy this demand is deterministic lateral displacement (DLD), which uses hydrodynamic forces to separate particles based on size. Numerous medically relevant cellular organisms, such as circulating tumor cells (10–15 µm) and red blood cells (6–8 µm), can be manipulated using microscale DLD devices. In general, these often-viscous samples require some form of dilution or other treatment prior to microfluidic transport, further increasing the need for high-throughput operation to compensate for the increased sample volume. However, high-throughput DLD devices will require a high flow rate, leading to an increase in Reynolds numbers (Re) much higher than those covered by existing studies for microscale (≤?100 µm) DLD devices. This study characterizes the separation performance for microscale DLD devices in the high-Re regime (10?<?Re?<?60) through numerical simulation and experimental validation. As Re increases, streamlines evolve and microvortices emerge in the wake of the pillars, resulting in a particle trajectory shift within the DLD array. This differs from previous DLD works, in that traditional models only account for streamlines that are characteristic of low-Re flow, with no consideration for the transformation of these streamlines with increasing Re. We have established a trend through numerical modeling, which agrees with our experimental findings, to serve as a guideline for microscale DLD performance in the high-Re regime. Finally, this new phenomenon could be exploited to design passive DLD devices with a dynamic separation range, controlled simply by adjusting the device flow rate.     

A linear-time algorithm for finding Hamiltonian (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">t</Emphasis>)-paths in odd-sized rectangular grid graphs with a rectangular hole

A grid graph \(G_{\mathrm{g}}\) is a finite vertex-induced subgraph of the two-dimensional integer grid \(G^\infty \). A rectangular grid graph R(m, n) is a grid graph with horizontal size m and vertical size n. A rectangular grid graph with a rectangular hole is a rectangular grid graph R(m, n) such that a rectangular grid subgraph R(k, l) is removed from it. The Hamiltonian path problem for general grid graphs is NP-complete. In this paper, we give necessary conditions for the existence of a Hamiltonian path between two given vertices in an odd-sized rectangular grid graph with a rectangular hole. In addition, we show that how such paths can be computed in linear time.     

Inertia-driven enhancement of mixing efficiency in microfluidic cross-junctions: a combined Eulerian/Lagrangian approach

Mixing of a diffusing species entrained in a three-dimensional microfluidic flow-focusing cross-junction is numerically investigated at low Reynolds numbers, \(1 \le Re \le 150\), for a value of the Schmidt number representative of a small solute molecule in water, \(Sc = 10^3\). Accurate three-dimensional simulations of the steady-state incompressible Navier–Stokes equations confirm recent results reported in the literature highlighting the occurrence of different qualitative structures of the flow geometry, whose range of existence depends on Re and on the ratio, R, between the volumetric flowrates of the impinging currents. At low values of R and increasing Re, the flux tube enclosing the solute-rich stream undergoes a topological transition, from the classical flow-focused structure to a multi-branched shape. We here show that this transition causes a nonmonotonic behavior of mixing efficiency with Re at constant flow ratio. The increase in efficiency is the consequence of a progressive compression of the cross-sectional diffusional lengthscale, which provides the mechanism sustaining the transversal Fickian flux even when the Peclet number, \({Pe=Re \, Sc}\), characterizing mass transport, becomes higher due to the increase in Re. The quantitative assessment of mixing efficiency at the considerably high values of the Peclet number considered (\(10^3 \le Pe \le 1.5 \times 10^5\)) is here made possible by a novel method of reconstruction of steady-state cross-sectional concentration maps from velocity-weighted ensemble statistics of noisy trajectories, which does away with the severe numerical diffusion shortcomings associated with classical Eulerian approaches to mass transport in complex 3d flows.     

Study of flow behaviors of droplet merging and splitting in microchannels using Micro-PIV measurement

Droplet merging and splitting are important droplet manipulations in droplet-based microfluidics. However, the fundamental flow behaviors of droplets were not systematically studied. Hence, we designed two different microstructures to achieve droplet merging and splitting respectively, and quantitatively compared different flow dynamics in different microstructures for droplet merging and splitting via micro-particle image velocimetry (micro-PIV) experiments. Some flow phenomena of droplets different from previous studies were observed during merging and splitting using a high-speed microscope. It was also found the obtained instantaneous velocity vector fields of droplets have significant influence on the droplets merging and splitting. For droplet merging, the probability of droplets coalescence (η) in a microgroove is higher (50% < η < 92%) than that in a T-junction microchannel (15% < η < 50%), and the highest coalescence efficiency (η = 92%) comes at the two-phase flow ratio e of 0.42 in the microgroove. Moreover, compared with a cylinder obstacle, Y-junction bifurcation can split droplets more effectively and the droplet flow during splitting is steadier. The results can provide better understanding of droplet behaviors and are useful for the design and applications of droplet-based microfluidics.     

Generation of droplets in the T-junction with a constriction microchannel

Droplet microfluidics plays an essential role in science and technology with various applications such as chemical engineering, environment, energy and other fields. T-junction with a constriction microchannel is designed for the controlled production of monodisperse microdroplets, which could produce droplets with the same size under a lower flow resistance. The influence of the microchannel structure, operating conditions, and physical properties on the dispersion rules is systematically investigated by combinations of micro-particle image velocimetry (Micro-PIV), high-speed camera and numerical simulation. Compared to the traditional T-junction channel, the T-junction with a constriction microchannel can generate smaller droplets whose size conforms to the size prediction formula of the traditional T-junction channel. It is found that the velocity vector of the T-junction with a constriction microchannel is faster than that in the T-junction channel at each stage of droplet generation. The droplet size is mainly based on the Ca number, the flow rate ratio and viscosity ratio of the continuous phases in our channel, and the range of the index of Ca with the droplet size is found. The constriction width has a significant influence on the dispersion rule, as there is a decreasing tendency for the droplet size with reducing constriction width.     

On the Advice Complexity of the k-server Problem Under Sparse Metrics

We consider the k-Server problem under the advice model of computation when the underlying metric space is sparse. On one side, we introduce Θ(1)-competitive algorithms for a wide range of sparse graphs. These algorithms require advice of (almost) linear size. We show that for graphs of size N and treewidth α, there is an online algorithm that receives O (n(log α + log log N))^* bits of advice and optimally serves any sequence of length n. We also prove that if a graph admits a system of μ collective tree (q, r)-spanners, then there is a (q + r)-competitive algorithm which requires O (n(log μ + log log N)) bits of advice. Among other results, this gives a 3-competitive algorithm for planar graphs, when provided with O (n log log N) bits of advice. On the other side, we prove that advice of size Ω(n) is required to obtain a 1-competitive algorithm for sequences of length n even for the 2-server problem on a path metric of size N ≥ 3. Through another lower bound argument, we show that at least \(\frac {n}{2}(\log \alpha - 1.22)\) bits of advice is required to obtain an optimal solution for metric spaces of treewidth α, where 4 ≤ α < 2k.     

Hydrodynamic focusing and interdistance control of particle-laden flow for microflow cytometry

Single-file focusing and minimum interdistance of micron-size objects in a sample is a prerequisite for accurate flow cytometry measurements. Here, we report analytical models for predicting the focused width of a sample stream b as a function of channel aspect ratio α, sheath-to-sample flow rate ratio f and viscosity ratio λ in both 2D and 3D focusing. We present another analytical model to predict spacing between an adjacent pair of objects in a focused sample stream as a function of sample concentration C, mobility ? of the objects in the prefocused and postfocused regions and flow rate ratio f in both 2D and 3D flow focusing. Numerical simulations are performed using Ansys Fluent VOF model to predict the width of sample stream in 2D and 3D hydrodynamic focusing for different sample-to-sheath viscosity ratios, aspect ratios and flow rate ratios. Experiments are performed on both planar and three-dimensional devices fabricated in PDMS to demonstrate focusing of sample stream and spacing of polystyrene beads in the unfocused and focused stream at different sample concentrations C. The predictions of the analytical model and simulations are compared with experimental data, and a good match is found (within 12 %). Further, mobility of objects is experimentally studied in 2D and 3D focusing, and the spread of the mobility data is used as tool for the demonstration of particle focusing in flow cytometer applications.     

Continuity of the Effective Delay Operator for Networks Based on the Link Delay Model

This paper is concerned with a dynamic traffic network performance model, known as dynamic network loading (DNL), that is frequently employed in the modeling and computation of analytical dynamic user equilibrium (DUE). As a key component of continuous-time DUE models, DNL aims at describing and predicting the spatial-temporal evolution of traffic flows on a network that is consistent with established route and departure time choices of travelers, by introducing appropriate dynamics to flow propagation, flow conservation, and travel delays. The DNL procedure gives rise to the path delay operator, which associates a vector of path flows (path departure rates) with the corresponding path travel costs. In this paper, we establish strong continuity of the path delay operator for networks whose arc flows are described by the link delay model (Friesz et al., Oper Res 41(1):80–91, 1993; Carey, Networks and Spatial Economics 1(3):349–375, 2001). Unlike the result established in Zhu and Marcotte (Transp Sci 34(4):402–414, 2000), our continuity proof is constructed without assuming a priori uniform boundedness of the path flows. Such a more general continuity result has a few important implications to the existence of simultaneous route-and-departure-time DUE without a priori boundedness of path flows, and to any numerical algorithm that allows convergence to be rigorously analyzed.     

Negation-Limited Complexity of Parity and Inverters

In negation-limited complexity, one considers circuits with a limited number of NOT gates, being motivated by the gap in our understanding of monotone versus general circuit complexity, and hoping to better understand the power of NOT gates. We give improved lower bounds for the size (the number of AND/OR/NOT) of negation-limited circuits computing Parity and for the size of negation-limited inverters. An inverter is a circuit with inputs x ₁,…,x _n and outputs ¬ x ₁,…,¬ x _n . We show that: (a) for n=2 ^r ?1, circuits computing Parity with r?1 NOT gates have size at least 6n?log?₂(n+1)?O(1), and (b) for n=2 ^r ?1, inverters with r NOT gates have size at least 8n?log?₂(n+1)?O(1). We derive our bounds above by considering the minimum size of a circuit with at most r NOT gates that computes Parity for sorted inputs x ₁≥???≥x _n . For an arbitrary r, we completely determine the minimum size. It is 2n?r?2 for odd n and 2n?r?1 for even n for ?log?₂(n+1)??1≤r≤n/2, and it is ?3n/2??1 for r≥n/2. We also determine the minimum size of an inverter for sorted inputs with at most r NOT gates. It is 4n?3r for ?log?₂(n+1)?≤r≤n. In particular, the negation-limited inverter for sorted inputs due to Fischer, which is a core component in all the known constructions of negation-limited inverters, is shown to have the minimum possible size. Our fairly simple lower bound proofs use gate elimination arguments in a somewhat novel way.     

Numerical simulation of heat transfer enhancement by elastic turbulence in a curvy channel

Although many investigations on elastic turbulence have been conducted in recent years, two major research topics still call for in-depth mechanistic investigations. Specifically, one is heat transfer performance affected by elastic turbulence; the other is so-called high Weissenberg number problem (HWNP) in numerical simulation of viscoelastic fluid flow. Taking these two topics into account simultaneously, the coupled problem becomes heat transfer characteristic of viscoelastic fluid in elastic turbulence at high Weissenberg number (Wi) and very low Reynolds number (Re). In this work, we implement numerical simulations by embedding log-conformation reformulation algorithm into the open-source software OpenFOAM. The heat transfer process of viscoelastic fluid flow in a three-dimensional (3D) curvy channel is simulated over a wide range of Wi. For the first time, significant heat transfer enhancement induced by elastic turbulence in a curvy channel at high Wi was identified numerically. When Wi is above the critical value of O(1), the heat transfer performance is found to be dramatically improved by elastic turbulence and then approaches a saturation. From the transient analysis of flow motions in the axial and cross sections, it can be seen that the flow twists and wiggles in the curvy channel and the field synergy effect of viscoelastic fluid flow becomes more intensive than that of Newtonian fluid flow. These effects give rise to the extremely irregular flow motions in the cross section and consequently lead to heat transfer enhancement.     

Influence of adjusting the inlet channel confluence angle on mixing behaviour in inertial microfluidic mixers

Recent drive for high-throughput microfluidic systems has triggered tremendous research effort to develop efficient, high-throughput microfluidic mixers. In particular, inducing a fluid–fluid collision at high flow rate in microfluidic channel has been suggested as an effective strategy to enhance mixing. However, previous studies using T-shaped microfluidic mixers showed that, in addition to fluid–fluid collision, the confluence angle of fluid stream in microfluidic channel also has a dramatic effect on mixing. This study suggests the possibility to enhance mixing by simply changing the inlet confluence angle of the streams. In this work, we assess the mixing behaviour of microfluidic mixers with variable inlet confluence angle with the Reynolds number (Re) range of 2.83–566. It is shown that the increase in inlet confluence angle enables the reduction of Re required for complete mixing. Simulation results demonstrate that increasing the confluence angle facilitates the interaction of vortices in mixers to induce an enhanced mixing. We further demonstrate that the increased interaction of vortices also prompts the turbulent emulsification where a significant reduction in emulsion size is observed for each mixer with increased inlet confluence angle at same Re.     

Embedding cycles and paths on solid grid graphs

Despite many algorithms for embedding graphs on unbounded grids, only a few results on embedding graphs on restricted grids have been published. In this paper, we study the problem of embedding paths and cycles on solid grid graphs. We show that a cycle of length k is unit-length embeddable on a solid grid graph G if k is an even integer between four and the length of the longest cycle of G. In addition, our result shows that a path of length k is unit-length embeddable on G, between its two given vertices s and t, if \(k\le L\) and \(k\equiv L (\mathrm{mod}\ 2)\), in which L is the length of the longest path between s and t. Our presented two algorithms show that such embeddings can be found in linear time for cycles and quadratic time for paths, with respect to the size of graph G. In the case of rectangular grid graphs, the running time of the algorithms can be improved to O(k) and O\((k^2)\), respectively. In addition, we extend our results to \(m\times n\times o\) 3D grids. A application of our result is in the interconnection network mapping in parallel processing.     

Efficient heat transfer enhancement by elastic turbulence with polymer solution in a curved microchannel

Heat and mass transfer in microscale flows are limited due to extremely low Reynolds number (Re). In a curved microchannel, however, complex flow behaviors, such as elastic instability and elastic turbulence, can be induced via viscoelastic fluid at vanishingly low-Re conditions, which is of great potential to enhance the heat transfer performance. The influence of elastic instabilities and turbulence on heat dissipation of exothermic components is experimentally investigated in this study. The heat transfer performance of both viscoelastic (polymer solutions) and Newtonian (sucrose solutions) fluid flows in a curved microchannel with a square cross section is experimentally characterized. Titanium–platinum (Ti–Pt) thin films embedded at the bottom wall of the polydimethylsiloxane (PDMS) microchannel serve as both microheater and temperature sensor. For viscoelastic fluids, the spectrum of outlet temperature fluctuation in broad frequency (f) region fits the power law of f ^?1.1. Heat transfer enhancement due to the elastic turbulence in a curved microchannel is thereby identified by the drastic growth of the Nusselt number (Nu, the ratio of convective to conductive heat transfer normal to the boundary) with the increase in the Weissenberg number (Wi, the ratio of elastic stress to viscous stress). The mechanism of heat transfer enhanced by the convection effect of elastic turbulence is also elucidated.     

Optimal Path Embedding in the Exchanged Crossed Cube

The (s + t + 1)-dimensional exchanged crossed cube, denoted as ECQ(s, t), combines the strong points of the exchanged hypercube and the crossed cube. It has been proven that ECQ(s, t) has more attractive properties than other variations of the fundamental hypercube in terms of fewer edges, lower cost factor and smaller diameter. In this paper, we study the embedding of paths of distinct lengths between any two different vertices in ECQ(s, t). We prove the result in ECQ(s, t): if s ≥ 3, t ≥ 3, for any two different vertices, all paths whose lengths are between \( \max \left\{9,\left\lceil \frac{s+1}{2}\right\rceil +\left\lceil \frac{t+1}{2}\right\rceil +4\right\} \) and 2 ^s+t+1 ? 1 can be embedded between the two vertices with dilation 1. Note that the diameter of ECQ(s, t) is \( \left\lceil \frac{s+1}{2}\right\rceil +\left\lceil \frac{t+1}{2}\right\rceil +2 \). The obtained result is optimal in the sense that the dilations of path embeddings are all 1. The result reveals the fact that ECQ(s, t) preserves the path embedding capability to a large extent, while it only has about one half edges of CQ _n .     

Symmetric Masks for In-fill Pixel Interpolation on Discrete <Emphasis Type="Italic">p</Emphasis>:<Emphasis Type="Italic">q</Emphasis> Lattices

A 2D p:q lattice contains image intensity entries at pixels located at regular, staggered intervals that are spaced p rows and q columns apart. Zero values appear at all other intermediate grid locations. We consider here the construction, for any given p:q, of convolution masks to smoothly and uniformly interpolate values across all of the intermediate grid positions. The conventional pixel-filling approach is to allocate intensities proportional to the fractional area that each grid pixel occupies inside the boundaries formed by the p:q lines. However, these area-based masks have asymmetric boundaries, flat interior values and may be odd or even in size. Where edges, lines or points are in-filled, area-based p:q masks imprint intensity patterns that recall p:q because the shape of those masks is asymmetric and depends on p:q. We aim to remove these “memory” artefacts by building symmetric p:q masks. We show here that smoother, symmetric versions of such convolution masks exist. The coefficients of the masks constructed here have simple integer values whose distribution is derived purely from symmetry considerations. We have application for these symmetric interpolation masks as part of a precise image rotation algorithm which disguises the rotation angle, as well as to smooth back-projected values when performing discrete tomographic image reconstruction.