Second-order gaseous slip flow models in long circular and noncircular microchannels and nanochannels

This paper significantly extends previous studies to the transition regime by employing the second-order slip boundary conditions. A simple analytical model with second-order slip boundary conditions for a normalized Poiseuille number is proposed. The model can be applied to either rarefied gas flows or apparent liquid slip flows. The developed simple models can be used to predict the Poiseuille number, mass flow rate, tangential momentum accommodation coefficient, pressure distribution of gaseous flow in noncircular microchannels and nanochannels by the research community for the practical engineering design of microchannels and nanochannels. The developed second-order models are preferable since the difficulty and “investment” is negligible compared with the cost of alternative methods such as molecular simulations or solutions of Boltzmann equation. Navier–Stokes equations with second-order slip models can be used to predict quantities of engineering interest such as the Poiseuille number, tangential momentum accommodation coefficient, mass flow rate, pressure distribution, and pressure drop beyond its typically acknowledged limit of application. The appropriate or effective second-order slip coefficients include the contribution of the Knudsen layers in order to capture the complete solution of the Boltzmann equation for the Poiseuille number, mass flow rate, and pressure distribution. It could be reasonable that various researchers proposed different second-order slip coefficients because the values are naturally different in different Knudsen number regimes. It is analytically shown that the Knudsen’s minimum can be predicted with the second-order model and the Knudsen value of the occurrence of Knudsen’s minimum depends on inlet and outlet pressure ratio. The compressibility and rarefaction effects on mass flow rate and the curvature of the pressure distribution by employing first-order and second-order slip flow models are analyzed and compared. The condition of linear pressure distribution is given.     

Rarefaction and compressibility effects on steady and transient gas flows in microchannels

The main theoretical and experimental results from the literature about steady pressure-driven gas microflows are summarized. Among the different gas flow regimes in microchannels, the slip flow regime is the most frequently encountered. For this reason, the slip flow regime is particularly detailed and the question of appropriate choice of boundary conditions is discussed. It is shown that using second-order boundary conditions allows us to extend the applicability of the slip flow regime to higher Knudsen numbers that are usually relevant to the transition regime.The review of pulsed flows is also presented, as this kind of flow is frequently encountered in micropumps. The influence of slip on the frequency behavior (pressure gain and phase) of microchannels is illustrated. When subjected to sinusoidal pressure fluctuations, microdiffusers reveal a diode effect which depends on the frequency. This diode effect may be reversed when the depth is shrunk from a few hundred to a few m.Thermally driven flows in microchannels are also described. They are particularly interesting for vacuum generation using microsystems without moving parts.     

Analysis of Stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves

Superhydrophobic surfaces have been demonstrated to be capable of reducing fluid resistance in micro- and nanofluidic applications. The objective of this paper is to present analytical solutions for the Stokes flow through microchannels employing superhydrophobic surfaces with alternating micro-grooves and ribs. Results are presented for both cases where the micro-grooves are aligned parallel and perpendicular to the flow direction. The effects of patterning the grooves on one or both channel walls are also analyzed. The reduction in fluid resistance has been quantified in terms of a dimensionless effective slip length, which is found to increase monotonically with the shear-free fraction and the periodic extent of each groove–rib combination normalized by the channel half-height. Asymptotic relationships have been derived for the normalized effective slip length corresponding to large and small limiting values of the shear-free fraction and the normalized groove–rib period. A detailed comparison has been made between transverse and longitudinal grooves, patterned on one or both channel walls, to assess their effectiveness in terms of enhancing the effective slip length. These comparisons have been carried out for small and large limiting values, as well as finite values of the shear-free fraction and normalized groove–rib period. Results for the normalized effective slip length corresponding to transverse and longitudinal grooves are further applied to model the Stokes flow through microchannels employing superhydrophobic surfaces containing a periodic array of micro-grooves inclined at an angle to the direction of the applied pressure gradient. Results are presented for the normalized effective slip lengths parallel to the direction of the applied pressure gradient and the normalized cross flow rate perpendicular to the direction of the applied pressure gradient.     

Optimal design of multi-channel microreactor for uniform residence time distribution

Multi-channel microreactors can be used for various applications that require chemical or electrochemical reactions in either liquid, gaseous or multi phase. For an optimal control of the chemical reactions, one key parameter for the design of such microreactors is the residence time distribution of the fluid, which should be as uniform as possible in the series of microchannels that make up the core of the reactor. Based on simplifying assumptions, an analytical model is proposed for optimizing the design of the collecting and distributing channels which supply the series of rectangular microchannels of the reactor, in the case of liquid flows. The accuracy of this analytical approach is discussed after comparison with CFD simulations and hybrid analytical-CFD calculations that allow an improved refinement of the meshing in the most complex zones of the flow. The analytical model is then extended to the case of microchannels with other cross-sections (trapezoidal or circular segment) and to gaseous flows, in the continuum and slip flow regimes. In the latter case, the model is based on second-order slip flow boundary conditions, and takes into account the compressibility as well as the rarefaction of the gas flow.     

Slip model for the ultra-thin gas-lubricated slider bearings of an electrostatic micromotor in MEMS

A new slip model derived by molecular dynamics has been used to investigate the ultra-thin gas-lubricated slider bearings beneath the three bushings of an electrostatic micromotor in micro-electro-mechanical systems (MEMS). Modified Reynolds equation is proposed based on the modified slip model. Analytical solutions for flow rate, pressure distribution, load carrying capacity and streamwise location using the modified Reynolds equation are obtained and compared with the results gained from those in the literature. It demonstrates that the new second-order slip model is of greater accuracy than that predicted by the first-order, second-order slip models and MMGL model and produces a good approximation to variable hard sphere (VHS) and variable soft sphere (VSS) models, which agree well with the solution obtained from the linearized Boltzmann equation. It is indicated that the slip effect reduces the pressure distribution and load carrying capacity, and shifts the streamwise location of the load carrying capacity, which should not be ignored to study the step-shaped slider bearings in micromotors for MEMS devices.     

Development of a rapid-response flow-control system using MEMS microvalve arrays

A method for providing high-resolution gas flow control using microelectromechanical systems (MEMS) has been developed and tested. The micromachined component consists of an array of 61 synchronized microvalves operating in parallel. A number of tests were conducted on microvalves of various designs to characterize their operation. The best performing of these was used with a prototype flow controller. Additionally, a mathematical model of the flow system and controller was derived to predict the response of the system to various changes in operating conditions. This work describes the design, modeling, and testing of a compact, stand-alone mass flow controller (MFC) to demonstrate high resolution, fast response flow control using MEMS microvalves. The device consists of a microvalve array packaged with a micro flow sensor and a microprocessor-based control system. The high bandwidth of microvalves allows an atypical flow control architecture. The controller regulates a pulsewidth-modulated (PWM) signal sent to the valve array and is capable of both open- and closed-loop control. A mathematical model was also developed to predict the dynamic performance of the system under various operating conditions. Additional advantages of the MEMS flow-control system include low-power consumption, low fabrication costs, and scalable precision.     

Two-Dimensional Analysis of Electrokinetically Driven Out-of-Plane Vortices in a Microchannel Liquid Flow Using Patterned Surface Charge

The technology developed for photolithographically patterning the electric surface charge to be negative, positive, or neutral enables the realization of complex liquid flows even in straight and uniform microchannels with extremely small Reynolds number. A theoretical model to analyze a steady incompressible electrokinetically driven two-dimensional liquid flow in a microchannel with an inhomogeneous surface charge under externally applied electric field is derived. The flow field is obtained analytically by solving the biharmonic equation with the Helmholtz-Smoluchowski slip boundary condition using the Fourier series expansion method. The model has been applied to study three basic out-of-plane vortical flow fields: single vortex and a train of corotating and a series of counterrotating vortex pairs. For model verification, the solution for the single vortex has been tested against numerical computations based on the full Navier-Stokes equations revealing the dominant control parameters. Two interesting phenomena have been observed in out-of-plane multivortex dynamics: merging of corotating vortices and splitting of counterrotating vortices. The criteria for the onset of both phenomena are discussed     

Analytical investigation of the feasibility of sacrificial microchannel sealing for Chip-Scale Atomic Magnetometers

An alkali metal vapor cell is a crucial component of the highly sensitive Chip Scale Atomic Magnetometers (CSAMs) that are increasingly deployed in a variety of electronic devices. Herein, we propose a novel microfabrication technique utilizing an array of microchannels at a bonded interface, to enable gas feedthrough for evacuation of unwanted gases from a vapor cell and subsequent introduction of an inert gas, followed by permanent sealing of the microchannels by reflow of a glass frit. The characteristics of glass frit reflow are analyzed to investigate the feasibility of using microchannels formed either on a silicon substrate, or embedded in a glass frit layer, with four different cross-sectional shapes considered. Prior to modeling the microchannels for simulation, the minimum cross-sectional size of a microchannel that fulfills gas feedthrough requirements was calculated and a value of 10 μm was determined based on a flow conductance model. The sealing of the microchannels was simulated using the finite element method (FEM) and the results revealed that flow resistance is a crucial design factor. Thus, embedded microchannel designs were more suitable for the proposed sealing technique than microchannel designs fabricated in silicon.     

Semi-analytical solutions for electroosmotic flows with interfacial slip in microchannels of complex cross-sectional shapes

In this article, we investigate the implications of electroosmosis with interfacial slip on electrohydrodynamic transport in microchannels having complex (yet symmetric) cross-sectional shapes, by employing a generic semi-analytical approach. We also devise an approximate technique of flow rate prediction under these conditions, using a combined consideration of electroosmotic slip (under thin electrical double layer limits) and Navier slip conditions (originating out of confinement-induced hydrophobic interactions) at the fluid–solid interface. We further assess the effectiveness of the approximate solutions in perspective of the exact solutions, as a parametric function of the relative thickness of the electrical double layer with respect to the channel hydraulic diameter. We illustrate the underlying consequences through examples of elliptic, polygonal, point star-shaped, and annular microchannel cross sections.     

Subcritical crack growth in silicon MEMS

New experimental techniques need to be developed to address fundamental materials issues in MEMS. Experimental protocols developed for macroscale testing are not necessarily applicable, and an understanding of the behavior of macroscale specimens cannot necessarily be relied upon to predict the behavior of microscale MEMS structures. An experimental protocol for studying slow crack growth in MEMS materials has been developed, and this protocol has been used to show that polycrystalline silicon (polysilicon) MEMS are susceptible to stress corrosion cracking. Using a model of the nonlinear dynamics of a specimen allowed an estimation of crack length and crack closure from the frequency response of the specimen. The procedure can resolve 1-nm crack extensions and crack growth rates below 10^-13 m/s. Crack closure, which has a pronounced effect on the dynamics of this nonlinear system, may be associated with the native oxide that grows on the faces of the crack. The data show that subcritical crack growth in polysilicon MEMS is driven by the synergistic effects of water and stress. In contrast to macroscale stress corrosion cracking behavior, a clear relationship between crack growth rate, stress intensity and humidity has not been found. Micrographs suggest that the crack path is transgranular     

Pressure-driven electrokinetic slip flows of viscoelastic fluids in hydrophobic microchannels

This work investigates the steady-state slip flow of viscoelastic fluids in hydrophobic two-dimensional microchannels under the combined influence of electro-osmotic and pressure gradient forcings with symmetric or asymmetric zeta potentials at the walls. The Debye–Hückel approximation for weak potential is assumed, and the simplified Phan-Thien-Tanner model was used for the constitutive equation. Due to the different hydrophobic characteristics of the microchannel walls, we study the influence of the Navier slip boundary condition on the fluid flow, by considering different slip coefficients at both walls and varying the electrical double-layer thickness, the ratio between the applied streamwise gradients of electric potential and pressure, and the ratio of the zeta potentials. For the symmetric case, the effect of the nonlinear Navier slip model on the fluid flow is also investigated.     

A study of PZT valveless micropump with asymmetric obstacles

The results of a study on a new type of PZT valveless micropump with asymmetric obstacles are presented in this paper. The valveless micropump was made through a MEMS fabrication process by using simply only one photo mask. Asymmetric obstacles are used for the flow directing device instead of the diffuser/nozzle elements used in previous studies. In this study, numerical simulations were also carried out to evaluate the design and the performance of the new micropump. The main feature of the present micropump is that it has a uniform cross-section area across the micro-channel, which gives many advantages. The differential pressure head and the pumping flow rate can be adjusted easily by using obstacles of different shapes and changing the PZT operating frequency without changing the dimensions of the micro-channel. In this experiment, the performance of the micropump was evaluated by measuring the pressure head difference and the flow rate as the input voltage ranged from 20 to 40 V, a range much lower than those in previous studies. The pumping pressure can reach a maximum of 1.2 kPa, and the maximum net volume flow rate is 156 μl/min. These test data indicate that this micropump fulfills the demands for most micro-fluidic systems. Moreover, the present device can be easily applied to complex systems with combinations of several pumps and microchannels in the future.     

Infrared thermal velocimetry in MEMS-based fluidic devices

Most MEMS (microelectromechanical system) devices are made of silicon which is transparent at infrared wavelengths. Utilizing this infrared transparency of silicon, infrared thermal velocimetry was developed to measure the velocity in MEMS based fluidic devices. The method uses an infrared laser to generate a short heating pulse in a flowing liquid. An infrared camera records the radiative images from the heated flowing liquid and the steady flow velocity is obtained from consecutive radiative images. A wide range of the velocity (1 cm/s-1 m/s or higher) in silicon (or other materials that are transparent to infrared radiation) microchannels can be measured. Numerical simulations have been carried out and are in good agreement with the experiments. Parametric studies have been carried out for different channel dimensions and laser characteristics.     

Gaseous slip flow in long microchannels

An analytic and experimental investigation into gaseous flow with slight rarefaction through long microchannels is undertaken. A two-dimensional (2-D) analysis of the Navier-Stokes equations with a first-order slip-velocity boundary condition demonstrates that both compressibility and rarefied effects are present in long microchannels. By undertaking a perturbation expansion in ϵ, the height-to-length ratio of the channel, and using the ideal gas equation of state, it is shown that the zeroth-order analytic solution for the streamwise mass flow corresponds well with the experimental results. Also, the effect of slip upon the pressure distribution is derived, and it is obtained that this slip velocity leads directly to a wall-normal migration of mass. The fabrication of wafer-bonded microchannels that possess well-controlled surface structure is described, and a means for accurately measuring the mass how through the channels is presented. Experimental results obtained with this mass-flow measurement technique for streamwise helium mass flow through microchannels 52.25-μm wide, 1.33-μm deep, and 7500-μm long for a pressure range of 1.6-4.2 atmospheres (outlet pressures at atmospheric) are presented and shown to compare favorably with the analysis     

Numerical simulation of capsule deformation in simple shear flow

The transient deformation of two-dimensional non-circular and three-dimensional non-spherical capsules in simple shear flow is studied numerically, using the hybrid immersed boundary and multi-block lattice Boltzmann method recently proposed by the present authors. The capsules are modeled as Newtonian liquid drops enclosed by elastic membranes; the liquids inside and outside the capsule have the same physical properties. The present results show important different behaviors between two-dimensional and three-dimensional capsules in shear flow. For two-dimensional non-circular capsules without considering the membrane bending rigidity, or considering the bending with the minimum bending-energy configurations (shapes at which the bending-energy has a global minimum) being uniform-curvature shapes, the capsules will always achieve the steady tank treading motion (a capsule deforms to a steady shape with a steady inclination and the membrane rotates around the liquid inside). However, for three-dimensional non-spherical capsules without membrane bending rigidity, such a steady mode does not exist; with the shear rate decreasing, the three-dimensional capsules’ motion changes from swinging mode (a capsule undergoes periodic shape deformation and inclination oscillation while its membrane is rotating around the liquid inside) to flipping mode. The deformation of two-dimensional capsules, with their initial non-circular shapes taken as the minimum bending-energy configurations, is also considered. It is quite interesting to find such two-dimensional capsules behave qualitatively similar to three-dimensional capsules: there is a swinging-to-flipping transition induced by lowering the shear rate.     

Transient electroosmotic slip flow of fractional Oldroyd-B fluids

The electroosmotic flow of a fractional Oldroyd-B fluid in a circular microchannel is studied. The linear Navier slip velocity model is used as the chosen slip boundary condition. Exact solutions for the electric potential and transient velocity are established by means of Laplace and finite Hankel transforms. And the velocity was presented as a sum of the steady part and the unsteady one. The corresponding solutions for the fractional Maxwell fluid, fractional second grade fluid, and Newtonian fluid can also be obtained from our results. Finally, numerical results for the fluid flow are obtained and some useful conclusions are drawn. Our results may be useful for the prediction of the flow behavior of viscoelastic fluids in microchannels and can benefit the design of microfluidic devices.     

A two-port model for wave propagation along a long circular microchannel

A compact model for oscillatory flow in a long microchannel with a circular cross-section is derived from the linearised Navier–Stokes equations. The resulting two-port model includes the effects of viscosity due to rarefied gas in the slip flow regime, inertia, compressibility and losses due to heat exchange. Both an acoustic impedance T network and an acoustic admittance Π network are presented for implementation in system level and circuit simulation tools. Also, reduced T and Π networks with constant component values are given to be used in the low frequency region. They are useful in time domain simulations, too. To verify the analytical model, simulations with a harmonic finite element solver for acoustic viscous flow are performed for microchannels exploiting the axisymmetry. The simulation results with both open and closed outlet conditions are compared with the two-port model with excellent agreement. Contribution of the slip conditions and the accuracy of the simple model are demonstrated.     

Liquid film thickness measurement underneath a gas slug with miniaturized sensor matrix in a microchannel

Measurement of liquid film thickness is essential for understanding the dynamics of two-phase flow in microchannels. In this work, a miniaturized sensor matrix with impedance measurement and MEMS technology to measure the thin liquid film underneath a bubble in the air–water flow in a horizontal microchannel has been developed. This miniaturized sensor matrix consists of 5 × 5 sensors where each sensor is comprised of a transmitter and a receiver electrode concentrically. The dimension and performance of the sensor electrodes were optimized with simulation results. The maximum diameter of the sensor ring is 310 µm, allowing a measurable range of liquid film thickness up to 83 µm. These sensors were distributed on the surface of a wafer with photolithography technology, covering a total length of 8 mm and a width of 2 mm. A spatial resolution of 0.5 × 2.0 mm² and a temporal resolution of 5 kHz were achieved for this sensor matrix with a measurement accuracy of 0.5 µm. A series of microchannels with different heights were used in the calibration in order to achieve the signal-to-thickness characteristics of each sensor. This delicate sensor matrix can provide detailed information on the variation of film thickness underneath gas–water slug directly, accurately and dynamically.     

Effect of patchwise slip on fluid flow

In this paper, we show that large connected slip patches (hydrophobic patches) are a necessity to induce macroscopic slip effects of water flow in microchannels. For this purpose, the 2D fluid flow between a planar stationary surface with alternating stick and slip patches and a parallel planar surface moving with a constant relative velocity has been studied by computer simulations based on Navier–Stokes equations. A slip patch is defined as the slipping length in a 2D system or a slip area of the surface in a 3D system. The simulations reveal that the ratio (size of each slip patch)/(distance between the two parallel interfaces) has profound effect on the viscous stress on the moving surface when this ratio is around and above one. However, when the ratio is much below one, the effect of the slip patches are minor, even if the area fraction of slip patches are higher than 50 %. Obviously, the stick patches adjacent to the slip patches act as effective barriers, preventing the fluid velocity to increase near the surface with alternating stick and slip patches. The obtained results are scalable and applicable on all length scales, with an exception for narrow channels in the subnano regime, i.e. <1 nm where specific effects as the atomistic composition and the nanostructure of the wall as well as the interactions between the wall and the water molecules have an effect.