Engineering droplet navigation through tertiary-junction microchannels

We present an experimental and in silico investigation of path selection by a single droplet inside a tertiary-junction microchannel using oil-in-water as a model system. The droplet was generated at a T-junction inside a microfluidic chip, and its flow behavior as a function of droplet size, streamline position, viscosity, and Reynolds number (Re) of the continuous phase was studied downstream at a tertiary junction having perpendicular channels of uniform square cross section and internal fluidic resistance proportional to their lengths. Numerical studies were performed using the multicomponent lattice Boltzmann method. Both the experimental and numerical results showed good agreement and suggested that at higher Re equal to 3, the flow was dominated by inertial forces resulting in the droplets choosing a path based on their center position in the flow streamline. At lower Re of 0.3, the streamline-assisted path selection became viscous force-assisted above a critical droplet size. As the Re was further reduced to 0.03, or when the viscosity of the dispersed phase was increased, the critical droplet size for transition also decreased. This multivariate approach can in future be used to engineer sorting of cells, e.g., circulating tumor cells (CTCs) allowing early-stage detection of life-threatening diseases.     

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.     

Inertial particle focusing dynamics in a trapezoidal straight microchannel: application to particle filtration

Inertial microfluidics has emerged recently as a promising tool for high-throughput manipulation of particles and cells for a wide range of flow cytometric tasks including cell separation/filtration, cell counting, and mechanical phenotyping. Inertial focusing is profoundly reliant on the cross-sectional shape of channel and its impacts on not only the shear field but also the wall-effect lift force near the wall region. In this study, particle focusing dynamics inside trapezoidal straight microchannels was first studied systematically for a broad range of channel Re number (20 < Re < 800). The altered axial velocity profile and consequently new shear force arrangement led to a cross-lateral movement of equilibration toward the longer side wall when the rectangular straight channel was changed to a trapezoid; however, the lateral focusing started to move backward toward the middle and the shorter side wall, depending on particle clogging ratio, channel aspect ratio, and slope of slanted wall, as the channel Reynolds number further increased (Re > 50). Remarkably, an almost complete transition of major focusing from the longer side wall to the shorter side wall was found for large-sized particles of clogging ratio K ~ 0.9 (K = a/H_min) when Re increased noticeably to ~ 650. Finally, based on our findings, a trapezoidal straight channel along with a bifurcation was designed and applied for continuous filtration of a broad range of particle size (0.3 < K < 1) exiting through the longer wall outlet with ~ 99% efficiency (Re < 100).     

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.     

On (<Emphasis Type="Italic">a</Emphasis>, <Emphasis Type="Italic">d</Emphasis> )-Distance Anti-Magic and 1-Vertex Bimagic Vertex Labelings of Certain Types of Graphs

The results for the corona P _n ?°?P₁ are generalized, which make it possible to state that P _n ?°?P₁ is not an ( a, d)-distance antimagic graph for arbitrary values of a and d. A condition for the existence of an ( a, d)-distance antimagic labeling of a hypercube Q _n is obtained. Functional dependencies are found that generate this type of labeling for Q _n . It is proved by the method of mathematical induction that Q _n is a (2 ⁿ ?+?n???1,?n???2) -distance antimagic graph. Three types of graphs are defined that do not allow a 1-vertex bimagic vertex labeling. A relation between a distance magic labeling of a regular graph G and a 1-vertex bimagic vertex labeling of G?∪?G is established.     

Motion of spin-half particles in the axially symmetric field of naked singularities of the static <Emphasis Type="Italic">q</Emphasis>-metric

Quantum-mechanical motion of a spin-half particle is examined in the axially symmetric fields of static naked singularities formed by a mass distribution with a quadrupole moment (q-metric). The analysis is performed by means of the method of effective potentials of the Dirac equation generalized to the case where radial and angular variables are not separated. If ?1 < q < qlim, |qlim| ? 1, where q is the quadrupolemoment in proper units, the naked singularities do not exclude the existence of stationary bound states of Dirac particles for a prolate mass distribution in the q-metric along the axial axis. For an oblate mass distribution, the naked singularities of the q-metric are separated from a Dirac particle by infinitely large repulsive barriers followed by a potential well which deepens while moving apart from the equator (from θ = θ _min or θ = π ? θ _min) toward the poles. The poles make an exception, and at 0 < q < q*, there are some points θ _i for particle states with j ≥ 3/2.     

Separation and concentration of <Emphasis Type="Italic">Phytophthora ramorum</Emphasis> sporangia by inertial focusing in curving microfluidic flows

We present an integrated microfluidic system for performing isolation and concentration of Phytophthora ramorum pathogens using a chip whose working principle is based on inertial lateral migration in curving flows. The chip was fabricated from multiple layers of thermoplastic polymers and features an embedded spiral separation channel along with peristaltic microvalves for fluidic operation and process control. A pumping system paired with a fully programmable pressure manifold is used to boost concentration levels by recirculating the sample liquid multiple times through the separation chip, making it possible to reduce sample volumes from 10 to 1 mL or less. The system was calibrated using fluorescent polymer particles with a nominal diameter of 30 µm which is comparable to that of P. ramorum sporangia. The separation process has been shown to be highly effective and more than 99% of the beads can be recovered in the concentrated batch. Experiments conducted with P. ramorum sporangia have shown that a 5.3-fold increase in pathogen content with 95% recovery can be achieved using three subsequent concentration cycles. The utility of the method has been validated by processing a sample derived from infested Rhododendron leaves where a 6.1-fold increase in the concentration of P. ramorum has been obtained after four concentration cycles. Although specifically designed and demonstrated for sporangia of P. ramorum, the method and related design rules can easily be extended to other microbial organisms, effectively supporting bioanalytical applications where efficient, high-throughput separation of target species is of primary concern.     

High-throughput,single-stream microparticle focusing using a microchannel with asymmetric sharp corners

A new microfluidic device for fast and high-throughput particle focusing is reported. The particle focusing is based on the combination of inertial lift force effect and centrifugal force effect generated in a microchannel with a series of repeated asymmetric sharp corners on one side of the channel wall. The inertial lift force induces two focused particle streams in the microchannel, and the centrifugal force generated at the sharp corner structures tends to drive the particles laterally away from the corner. With the use of a series of repeated asymmetric sharp corner structures, a single and highly focused particle stream was achieved near the straight channel wall at a wide range of flow rate. In comparison with other hydrodynamic particle focusing methods, this method is less sensitive to the flow rate and can work at a higher flow rate (up to 700 μL/min) and Reynolds number (Re = 129.5). With its simple structure and operation, and high throughput, this method can be potentially used in particle focusing processes in a variety of lab-on-a-chip applications.     

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.     

Protecting query privacy with differentially private <Emphasis Type="Italic">k</Emphasis>-anonymity in location-based services

Nowadays, location-based services (LBS) are facilitating people in daily life through answering LBS queries. However, privacy issues including location privacy and query privacy arise at the same time. Existing works for protecting query privacy either work on trusted servers or fail to provide sufficient privacy guarantee. This paper combines the concepts of differential privacy and k-anonymity to propose the notion of differentially private k-anonymity (DPkA) for query privacy in LBS. We recognize the sufficient and necessary condition for the availability of 0-DPkA and present how to achieve it. For cases where 0-DPkA is not achievable, we propose an algorithm to achieve ??-DPkA with minimized ??. Extensive simulations are conducted to validate the proposed mechanisms based on real-life datasets and synthetic data distributions.     

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.     

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.     

Asymptotic probability of encountering an accidental similarity in the presence of counter examples

Previously, we found the generating function of an accidental resemblance to the b parent examples at m counter examples [1]. In this paper, we restrict ourself to the case where b = 2 with equal success probabilities p in Bernoulli trials for all attributes of each counter example and a success probability р ² for each attribute in an accidental similarity. If the number n of attributes tends to infinity, the success probability is defined as \(p = \sqrt {a/n} \), and m = b√n counter examples are considered, then the probability of the occurrence of an accidental similarity avoiding these m counter examples tends to 1 ? e ^?a ? ae ^?a [1 ? e ^?b√a ]..     

Two- and three-dimensional lattice Boltzmann simulations of particle migration in microchannels

The use of two-dimensional (2D) numerical simulations with a reduced particle-based Reynolds number (Re) for studying particle migration in a microchannel with equally spaced multiple constrictions was investigated. 2D and 3D colloidal lattice Boltzmann (LB) models were used to simulate particle-fluid hydrodynamics. Experiments were conducted with inert microparticles in a creeping flow in a microflow channel with symmetric wall obstacles. Lowering Re in 2D simulations by a factor of R (the dimensionless particle radius in LB simulations) resulted in a close match between numerically computed and experimentally obtained particle velocities, indicating that Re-based dimensional scaling was needed to capture the 3D particle flow dynamics in 2D simulations of experimental data. We captured particle displacement motion in a microchannel with symmetric inline obstacles in 2D simulations, where symmetry in the flow field was broken by local disturbances in the flow field due to particle motion, indicating that asymmetry in channel geometry is not the sole cause for particle displacement motion. Particle acceleration/deceleration around each constriction followed the same pattern, but each constriction acted like a particle accelerator in 2D and 3D simulations, in which particles exhibited progressively higher velocities in each subsequent constriction. Particles migrated across multiple streamlines in converging and diverging flow zones in a creeping flow, which calls into question the use of steady streamlines for calculating transient particle flow. Monotonicity in particle acceleration toward the constriction and deceleration beyond the constriction was broken by interparticle hydrodynamic interactions leading to more pronounced particle migration across multiple streamlines.     

Stability of properties of Kolmogorov complexity under relativization

Assume that a tuple of binary strings \(\bar a\) = 〈a ₁ ..., a _n 〉 has negligible mutual information with another string b. Does this mean that properties of the Kolmogorov complexity of \(\bar a\) do not change significantly if we relativize them to b? This question becomes very nontrivial when we try to formalize it. In this paper we investigate this problem for a special class of properties (for properties that can be expressed by an ?-formula). In particular, we show that a random (conditional on \(\bar a\)) oracle b does not help to extract common information from the strings a _i .     

Role of interface shape on the laminar flow through an array of superhydrophobic pillars

In this study, measurements of the pressure drop and the velocity vector fields through a regular array of superhydrophobic pillars were systematically taken to investigate the role of air–water interface shape on laminar drag reduction. A polydimethylsiloxane microfluidic channel was created with a regular array of apple-core-shaped and circular pillars bridging across the entire channel. Due to the shape and hydrophobicity of the apple-core-shaped pillars, air was trapped on the side of the pillars after filling the microchannel with water. The measurements were taken at a capillary number of Ca = 6.6 × 10^?5. The shape of the air–water interface trapped within the superhydrophobic apple-core-shaped pillars was systematically modified from concave to convex by changing the static pressure within the microchannel. The pressure drop through the microchannel containing the superhydrophobic apple-core-shaped pillars was found to be sensitive to the shape of the air–water interface. For static pressures which resulted in the apple-core-shaped superhydrophobic pillars having a circular cross section, D/D ₀ = 1, a drag reduction of 7% was measured as a result of slip along the air–water interface. At large static pressures, the interface was driven into the apple-core-shaped pillars, resulting in decrease in the effective size of the pillars and an increase in the effective spacing between pillars. When combined with a slip velocity measured to be 10% of the average velocity between the pillars, the result was a pressure drop reduction of 18% compared to the circular pillars at a non-dimensional interface diameter of D/D ₀ = 0.8. At low static pressures, the pressure drop increased significantly as the expanded air–water interface constricted flow through the array of pillars even as large interfacial slip velocity was maintained. At D/D ₀ = 1.1, for example, the pressure drop increased by 17% compared to the circular pillar. This drag increase was the result of an increased form drag due to a decrease in porosity and permeability of the pillar array and a decrease in the skin friction drag due to the presence of the air–water interface. For D/D ₀ = 1.1, the slip velocity was measured to be 45% of the average streamwise velocity between the pillars. When compared to no-slip pillars of similar shape, the drag reduction was found to increase from 6 to 9% with increasing convex curvature of the air–water interface.     

Star reduction among minimal length addition chains

An addition chain for a natural number n is a sequence \({1=a_0 < a_1 < \cdots < a_r=n}\) of numbers such that for each 0 < i ≤ r, a _i  = a _j  + a _k for some 0 ≤ k ≤ j < i. The minimal length of an addition chain for n is denoted by ?(n). If j = i ? 1, then step i is called a star step. We show that there is a minimal length addition chain for n such that the last four steps are stars. Then we conjecture that there is a minimal length addition chain for n such that the last \({\lfloor\frac{\ell(n)}{2}\rfloor}\)-steps are stars. We verify that the conjecture is true for all numbers up to 2¹⁸. An application of the result and the conjecture to generate a minimal length addition chain reduce the average CPU time by 23–29% and 38–58% respectively, and memory storage by 16–18% and 26–45% respectively for m-bit numbers with 14 ≤ m ≤ 22.     

Sorting algal cells by morphology in spiral microchannels using inertial microfluidics

Sub-millimetre phytoplankton (here referred to as algae) exist in a wide variety of shapes and sizes. Measuring algae morphology can be a useful tool for understanding the species dynamics in a body of water, and size-sorting in general is a valuable first step in automated species identification. Here, we demonstrate the sorting of algae by shape and size in a spiral microchannel, in which lift forces and Dean flow drag forces combine to position the cells in a shape-dependent location in the channel cross section. Three species were used for experiments: the high-aspect-ratio cylindrical Monoraphidium griffithii, the prolate spheroidal Cyanothece aeruginosa, and the small spherical Chlorella vulgaris. These results are compared with the sorting of similarly sized polystyrene latex microspheres in the same device over the same range of flow rates. Tests were done at conditions which yielded average Dean numbers over the channel length of 3 < De < 30. At 1.6 mL/min, the 10- and 20-µm microspheres could be separated with an efficiency of 96 %. The best sorting results for the algae were obtained at a flow rate of 3.2 mL/min, which yielded an average Dean number of De = 25 over the channel length. These conditions led to the separation of the Monoraphidium from the differently shaped Cyanothece; these two species could be sorted with a 77 % separation efficiency despite the relatively high polydispersity in cell sizes within each species. The elegance and simplicity of inertial microfluidics make it appropriate for the high-throughput pre-sorting of algae cells upstream of other integrated sensing modalities in a field-deployable device.     

On the <Emphasis Type="Italic">k</Emphasis>-accessibility of cores of <Emphasis Type="Italic">TU</Emphasis>-cooperative games

This paper proposes a strengthening of the author’s core-accessibility theorem for balanced TU-cooperative games. The obtained strengthening relaxes the influence of the nontransitivity of classical domination αv on the quality of the sequential improvement of dominated imputations in a game v. More specifically, we establish the k-accessibility of the core C(α _v ) of any balanced TU-cooperative game v for all natural numbers k: for each dominated imputation x, there exists a converging sequence of imputations x₀, x₁,..., such that x₀ = x, lim x _r ∈ C(α _v ) and x_r?m is dominated by any successive imputation x _r with m ∈ [1, k] and r ≥ m. For showing that the TU-property is essential to provide the k-accessibility of the core, we give an example of an NTU-cooperative game G with a ”black hole” representing a nonempty closed subset B ? G(N) of dominated imputations that contains all the α _G -monotonic sequential improvement trajectories originating at any point x ∈ B.     

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.