Non-rectangular Cross-section Channels Research Articles

Asymptotic similarity solutions for particle-driven gravity currents in non-rectangular cross-section channels of power-law form

Abstract We consider a high-Reynolds particle-driven gravity current (GC) propagating in channel above a horizontal boundary. The bottom and the top of the channel are at z = 0 , H and its cross-section is given by − f ( z ) ≤ y ≤ f ( z ) for 0 ≤ z ≤ H , where f ( z ) is a power-law function f ( z ) = z α . We focus attention on the similarity stage of propagation of the current using a shallow-water (SW) model. While it is possible to derive the analytical similarity solutions for the homogeneous GC (for which the density of the current remains constant during the propagation), such similarity-solutions do not exist for the particle-driven GCs. We extend the similarity solutions of homogeneous GC to particle-driven currents by developing an asymptotic expansions and derive approximations of such solutions for channels of typical rectangular, triangular and parabolic forms. Comparison with the numerical solution of the SW equations shows that the leading-order asymptotic results are a good approximation in the domain of expected validity. However, out of bounds of this domain the numerical solutions systematically depart from the first-order asymptotic solutions, suggesting the need to include higher-order terms in the asymptotic series.

Size-Dependent Inertial Focusing Position Shift and Particle Separations in Triangular Microchannels.

A recent study of inertial microfluidics within nonrectangular cross-section channels showed that the inertial focusing positions changes with cross-sectional shapes; therefore, the cross-sectional shape can be a useful control parameter for microfluidic particle manipulations. Here, we conducted detail investigation on unique focusing position shift phenomena, which occurs strongly in channels with the cross-sectional shape of the isosceles right triangle. The top focusing positions shift along the channel walls to the direction away from the apex with increasing Reynolds number and decreasing particle size. A larger particle with its center further away from the side walls experiences shear gradient lift toward the apex, which leads to an opposite result with changes of Reynolds and particle size. The focusing position shift and the subsequent stabilization of corner focusing lead to changes in the number of focusing positions, which enables a novel method for microparticle separations with high efficiency (>95%) and resolution (<2 μm). The separation method based on equilibrium focusing; therefore, the operation is simple and no complex separation optimization is needed. Moreover, the separation threshold can be easily modulated with flow rate adjustment. Rare cell separation from blood cell was successfully demonstrated with spiked MCF-7 cells in blood by achieving the yield of ∼95% and the throughput of ∼106 cells/min.

Polydisperse particle-driven gravity currents in non-rectangular cross section channels

We consider a high-Reynolds-number gravity current generated by polydisperse suspension of n types of particles distributed in a fluid of density ρi. Each class of particles in suspension has a different settling velocity. The current propagates along a channel of non-rectangular cross section into an ambient fluid of constant density ρa. The bottom and top of the channel are at z = 0, H, and the cross section is given by the quite general form −f1(z) ≤ y ≤ f2(z) for 0 ≤ z ≤ H. The flow is modeled by the one-layer shallow-water equations obtained for the time-dependent motion. We solve the problem by a finite-difference numerical code to present typical height h, velocity u, and mass fractions of particle (concentrations) (ϕ( j), j = 1, …, n) profiles. The runout length of suspensions in channels of power-law cross sections is analytically predicted using a simplified depth-averaged “box” model. We demonstrate that any degree of polydispersivity adds to the runout length of the currents, relative to that of equivalent monodisperse currents with an average settling velocity. The theoretical predictions are supported by the available experimental data. The present approach is a significant generalization of the particle-driven gravity current problem: on the one hand, now the monodisperse current in non-rectangular channels is a particular case of n = 1. On the other hand, the classical formulation of polydisperse currents for a rectangular channel is now just a particular case, f(z) = const., in the wide domain of cross sections covered by this new model.

Particle-driven gravity currents in non-rectangular cross section channels

We consider a high-Reynolds-number gravity current generated by suspension of heavier particles in fluid of density ρi, propagating along a channel into an ambient fluid of the density ρa. The bottom and top of the channel are at z = 0, H, and the cross section is given by the quite general −f1(z) ≤ y ≤ f2(z) for 0 ≤ z ≤ H. The flow is modeled by the one-layer shallow-water equations obtained for the time-dependent motion which is produced by release from rest of a fixed volume of mixture from a lock. We solve the problem by the finite-difference numerical code to present typical height h(x, t), velocity u(x, t), and volume fraction of particles (concentration) ϕ(x, t) profiles. The methodology is illustrated for flow in typical geometries: power-law (f(z) = zα and f(z) = (H − z)α, where α is positive constant), trapezoidal, and circle. In general, the speed of propagation of the flows driven by suspensions decreases compared with those driven by a reduced gravity in homogeneous currents. However, the details depend on the geometry of the cross section. The runout length of suspensions in channels of power-law cross sections is analytically predicted using a simplified depth-averaged “box” model. The present approach is a significant generalization of the classical gravity current problem. The classical formulation for a rectangular channel is now just a particular case, f(z) = const., in the wide domain of cross sections covered by this new model.

Gravity currents in non-rectangular cross-section channels: extensions to non-Boussinesq systems

We consider the propagation of a gravity current of density $$\rho _c$$ from a lock length $$x_0$$ and height $$h_0$$ into an ambient fluid of density $$\rho _a$$ in a horizontal channel of height $$H$$ along the horizontal coordinate x. The bottom and top of the channel are at $$z = 0, H$$ , and the cross-section is given by the quite general $$f_{1}(z)\le y \le f_{2}(z)$$ for $$0 \le z \le H$$ . When the Reynolds number is large, the resulting flow is governed by the parameters $$\rho _{c}/\rho _{a}$$ , $$H^{*} = H/h_{0}$$ and $$f(z)=f_{1}(z)+f_{2}(z)$$ . We show that the shallow-water one-layer model, combined with a Benjamin-type front condition, provides a versatile formulation for the thickness $$h$$ and speed $$u$$ of the current. The results cover in a continuous manner the range of light $$\rho _{c}/\rho _{a} \ll 1$$ , Boussinesq $$\rho _{c}/\rho _{a} \approx 1$$ and heavy $$\rho _{c}/\rho _{a} \gg 1$$ currents in a fairly wide range of depth ratio in various cross-section geometries. We obtain analytical solutions for the initial dam-break stage of propagation with constant speed, which appears for any cross-section geometry, and derive explicitly the trend for small and large values of the governing parameters. For large time, $$t$$ , a self-similar propagation is feasible only for heavy currents in channels with $$f(z)=bz^{\alpha }$$ cross-sections or for light currents in channels of $$f(z) = (H-z)^\alpha $$ , with $$t^{(2+2\alpha )/(3+2\alpha )}$$ . The methodology is illustrated for non-Boussinesq flow in typical geometries: power-law ( $$f(z) = z^{\alpha }$$ and $$f(z) = (H-z)^{\alpha }$$ , where $$\alpha $$ is positive constant), trapezoidal and circle (full or sector). The present approach is a significant generalization of the classical non-Boussinesq gravity current problem. The classical formulation for a rectangular (or laterally unbounded) channel is now just a particular case, $$f(z) = const.$$ , in the wide domain of cross-sections covered by this new model.

Publisher's Note: “Gravity currents in non-rectangular cross-section channels: Analytical and numerical solutions of the one-layer shallow-water model for high-Reynolds-number propagation” [Phys. Fluids 25, 026601 (2013)

First Page

Gravity currents in non-rectangular cross-section channels: Analytical and numerical solutions of the one-layer shallow-water model for high-Reynolds-number propagation

We consider the propagation of a high-Reynolds-number gravity current at the bottom of a horizontal channel along the horizontal coordinate x. The bottom and top of the channel are at z = 0, H, and the cross-section is given by the general −f1(z) ⩽ y ⩽ f2(z) for 0 ⩽ z ⩽ H. We use a one-layer, Boussinesq, shallow-water formulation to solve the time-dependent motion produced by release from rest of a fixed volume of fluid from a lock. The dependent variables are the position of the horizontal interface, h(x, t), and the speed (averaged over the area of the current), u(x, t). The non-rectangular cross-section geometry enters the formulation via f(h) and integrals of f(z) and zf(z), where f(z) = f1(z) + f2(z) is the width of the channel. For a given geometry f(z), the only input parameter in the lock-release problem is the height ratio H/h0 of ambient to lock. In general, the solution is obtained by a finite-difference numerical code. Analytical results are derived for the initial dam-break slumping motion, and for the long-time self-similar phase. The model is illustrated for various cross-section shapes: power-law (f(z) = bzα, where b, α are positive constants), trapezoidal, V-shaped valley, and circle-segment. In addition to the quantitative results, the qualitative similarities and differences between the classical rectangular and the general non-rectangular channel have been elucidated: in the first case there is always an initial “slumping” stage of propagation with speed uN = const., followed by a stage of decreasing speed, then a phase of self-similar behavior during which uN decreases like time to some power. In the non-rectangular cross-section channel, the first two stages appear, but the self-similar phase is attained only when the cross-section shape is given by a power-law. When the cross-section of the channel expands upwards, the speed of propagation is larger, and the decay of speed after the slumping stage is weaker, than in the classical rectangular counterpart. The theoretical results are in good agreement with previously published experimental data, but a sharp comparison is not feasible because the experiments were performed in full-depth-lock configurations where the one-layer model is not expected to be accurate.

Fabrication of circular microfluidic channels by combining mechanical micromilling and soft lithography

The fabrication of microfluidic channels with complex three-dimensional (3D) geometries presents a major challenge to the field of microfluidics, because conventional lithography methods are mainly limited to rectangular cross-sections. In this paper, we demonstrate the use of mechanical micromachining to fabricate microfluidic channels with complex cross-sectional geometries. Micro-scale milling tools are first used to fabricate semi-circular patterns on planar metallic surfaces to create a master mold. The micromilled pattern is then transferred to polydimethylsiloxane (PDMS) through a two-step reverse molding process. Using these semi-circular PDMS channels, circular cross-sectioned microchannels are created by aligning and adhering two channels face-to-face. Straight and serpentine-shaped microchannels were fabricated, and the channel geometry and precision of the metallic master and PDMS molds were assessed through scanning electron microscopy and non-contact profilometry. Channel functionality was tested by perfusion of liquid through the channels. This work demonstrates that micromachining enabled soft lithography is capable of fabricating non-rectangular cross-section channels for microfluidic applications. We believe that this approach will be important for many fields from biomimetics and vascular engineering to microfabrication and microreactor technologies.

Lock-Exchange Flows in Non-Rectangular Cross-Section Channels

Lock-exchange flows driven by density differences in non-rectangular cross-section channels are investigated in situations that resemble estuaries, navigation canals and hydraulic engineering structures. A simple analytical model considering stratified flows suggests practical relationships corroborated by results of laboratory experiments carried out in a straight channel of triangular cross-section.