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.     

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.     

Numerical simulation of gas flow in rough microchannels: hybrid kinetic–continuum approach versus Navier–Stokes

The flow field in a rough microchannel is numerically analyzed using a hybrid solver, dynamically coupling kinetic and Navier–Stokes solutions computed in rarefied and continuum subareas of the flow field, respectively, and a full Navier–Stokes solver. The rough surface is configured with triangular roughness elements, with a maximum relative roughness of 5 % of the channel height. The effects of Mach number, Knudsen number (or Reynolds number) and roughness height are investigated and discussed in terms of Poiseuille number and mass flow rate. Discrepancies between full Navier–Stokes and hybrid solutions are analyzed, assessing the range of validity of Navier–Stokes equations provided with first-order slip boundary conditions for modeling gas flow along a rough surface. Results indicate that the roughness increases Poiseuille number and decreases mass flux in comparison with those for the smooth microchannel. Increasing rarefaction results in further enhancement of roughness effect. At the same time, the compressibility effect is more noticeable than the roughness one, although the compressibility effect is alleviated by increase in the rarefaction. It was found that, although the Navier–Stokes solution of the flow in a smooth channel is accurate up to Kn = 0.1, when relative roughness height is higher than 1.25 % significant errors already appear at Kn = 0.02.     

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.     

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.     

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.     

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.     

Pressure drop of three-phase liquid–liquid–gas slug flow in round microchannels

In this paper we present a model for the calculation of pressure drop of three-phase liquid–liquid–gas slug flow in microcapillaries of a circular cross section. Introduced models consist of terms attributing for frictional and interfacial pressure drop, incorporating the presence of a stagnant thin film at the wall of the channel. Different formulations of the interfacial pressure drop equation were employed, using expressions developed by Bretherton (J Fluid Mech 10:166–188, 1961), Warnier et al. (Microfluid Nanofluid 8:33–45, 2010) or Ratulowski and Chang (Phys Fluids A 1:1642–1655, 1989). Models were validated experimentally using oleic acid–water–nitrogen and heptane–water–nitrogen three-phase flows in round Teflon or Radel R microchannels of 254- and 508-µm nominal inner diameter, for capillary numbers Ca _b between 10^?4 and 4.9 × 10^?1 and Reynolds numbers Re between 0.095 and 300. Best agreement between measured and calculated values of pressure drop, with relative error between ?22 and 19 % or ?20 and 16 %, is reached for Warnier’s or Ratulowski and Chang’s interfacial pressure drop equation, respectively. The results prove that three-phase slug flow pressure drop can be successfully predicted by extending existing two-phase slug flow correlations. Good agreement of Bretherton’s equation was reached only at lower Ca numbers, indicating that an extension of the interfacial pressure drop equation as performed by Warnier et al. (Microfluid Nanofluid 8:33–45, 2010) or Ratulowski and Chang (Phys Fluids A 1:1642–1655, 1989) for higher capillary numbers is necessary. Additionally it was demonstrated that pressure drop increases substantially if dry slug flow occurs or if microchannels with significant surface roughness are employed. Those influences were not accounted for in the models presented.     

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.     

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.     

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.     

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.     

Effects of geometry factors on microvortices evolution in confined square microcavities

Recently, microcavities have become a central feature of diverse microfluidic devices for many biological applications. Thus, the flow and transport phenomena in microcavities characterized by microvortices have received increasing research attention. It is important to understand thoroughly the geometry factors on the flow behaviors in microcavities. In an effort to provide a design guideline for optimizing the microcavity configuration and better utilizing microvortices for different applications, we investigated quantitatively the liquid flow characteristics in different square microcavities located on one side of a main straight microchannel by using both microparticle image velocimetry (micro-PIV) and numerical simulation. The influences of the inlet Reynolds numbers (with relatively wider values Re?=?1–400) and the hydraulic diameter of the main microchannel (D_H?=?100, 133 μm) on the evolution of microvortices in different square microcavities (100, 200, 400 and 800 μm) were studied. The evolution and characteristic of the microvortices were investigated in detail. Moreover, the critical Reynolds numbers for the emergence of microvortices and the transformation of flow patterns in different microcavities were determined. The results will provide a useful guideline for the design of microcavity-featured microfluidic devices and their applications.     

Microcapillary electrophoresis chip with a bypass channel for autonomous compensation of hydrostatic pressure flow

The application of chip-based microcapillary electrophoresis (µCE) to determine the electrophoretic mobility of molecules and particles has been intensively studied in the last two decades. Balancing the hydrostatic pressure between both ends of the microchannel is essential for free-zone electrophoresis and highly accurate measurement. This balancing operation appears simple on a macroscale (e.g., >?10^?3 m); however, on a microscale (e.g., 10^?6–10^?3 m), it is not straightforward because of the complexity of the interface dynamics at the meniscus. The hydrostatic pressure flow is unstable because of the small size of the microchannel, which is smaller than a single droplet of water. In this study, a µCE chip design was proposed by adding an extra bypass channel to balance the fluid level of the two open reservoirs and inhibit the generation of hydrostatic pressure flow within the microchannel. The fluid behaviors in the microchannel and current and voltage (I–V) characterization were experimentally studied. In addition, a numerical simulation of the electroosmotic flow and hydrostatic flow in the µCE chip was performed. The comparison between the µCE chip with and without the bypass channel showed that the bypass channel did not produce a disturbance in the microchannel for the electrophoretic measurement. The simple microchannel design enabled autonomous compensation of the hydrostatic pressure from the instability of the meniscus, and thus improved the usability of the chip-based µCE chip and the accuracy in the electrophoretic measurement.     

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.     

Ambiguity and Communication

The ambiguity of a nondeterministic finite automaton (NFA) N for input size n is the maximal number of accepting computations of N for inputs of size n. For every natural number k we construct a family \((L_{r}^{k}\;|\;r\in \mathbb{N})\) of languages which can be recognized by NFA’s with size k?poly(r) and ambiguity O(n ^k ), but \(L_{r}^{k}\) has only NFA’s with size exponential in r, if ambiguity o(n ^k ) is required. In particular, a hierarchy for polynomial ambiguity is obtained, solving a long standing open problem (Ravikumar and Ibarra, SIAM J. Comput. 19:1263–1282, 1989, Leung, SIAM J. Comput. 27:1073–1082, 1998).     

Quantum and Classical Query Complexities of Local Search Are Polynomially Related

Let f be an integer valued function on a finite set V. We call an undirected graph G(V,E) a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V,E) find a vertex x∈V such that f(x) is not bigger than any value that f takes on some neighbor of x. The complexity of the algorithm is measured by the number of questions of the form “what is the value of f on x?” We show that the deterministic, randomized and quantum query complexities of the problem are polynomially related. This generalizes earlier results of Aldous (Ann. Probab. 11(2):403–413, [1983]) and Aaronson (SIAM J. Comput. 35(4):804–824, [2006]) and solves the main open problem in Aaronson (SIAM J. Comput. 35(4):804–824, [2006]).     

Independence Numbers of Random Subgraphs of Some Distance Graph

The distance graph G(n, 2, 1) is a graph where vertices are identified with twoelement subsets of {1, 2,..., n}, and two vertices are connected by an edge whenever the corresponding subsets have exactly one common element. A random subgraph G _p (n, 2, 1) in the Erd?os–Rényi model is obtained by selecting each edge of G(n, 2, 1) with probability p independently of other edges. We find a lower bound on the independence number of the random subgraph G_1/2(n, 2, 1).     

Parallelization Strategy for Elementary Morphological Operators on Graphs: Distance-Based Algorithms and Implementation on Multicore Shared-Memory Architecture

This article focuses on the (unweighted) graph-based mathematical morphology operators presented in Cousty et al. (CVIU 117(4):370–385, 2013). These operators depend on a size parameter that specifies the number of iterations of elementary dilations/erosions. Thus, the associated running times increase with the size parameter, the algorithms running in \(O(\lambda .n)\) time, where n is the size of the underlying graph and \(\lambda \) is the size parameter. In this article, we present distance maps that allow us to recover (by thresholding) all considered dilations and erosions. The algorithms based on distance maps allow the operators to be computed with a single linear O(n) time iteration, without any dependence to the size parameter. Then, we investigate a parallelization strategy to compute these distance maps. The idea is to build iteratively the successive level-sets of the distance maps, each level-set being traversed in parallel. Under some reasonable assumptions about the graph and sets to be dilated, our parallel algorithm runs in \(O(n/p + K \log _2 p)\) where n, p, and K are the size of the graph, the number of available processors, and the number of distinct level-sets of the distance map, respectively. Then, implementations of the proposed algorithm on a shared-memory multicore architecture are described and assessed on datasets of 45 images and 6 textured three-dimensional meshes, showing a reduction of the processing time by a factor up to 55 over the previously available implementations on a 8-core architecture.