Log in

Relevant bibliographies by topics / Knudsen Layer / Journal articles

To see the other types of publications on this topic, follow the link: Knudsen Layer.

Journal articles on the topic 'Knudsen Layer'

Author: Grafiati

Published: 4 June 2021

Last updated: 18 June 2025

Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles

Select a source type:

Consult the top 50 journal articles for your research on the topic 'Knudsen Layer.'

Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.

You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.

Browse journal articles on a wide variety of disciplines and organise your bibliography correctly.

1

GALVIN, J. E., C. M. HRENYA, and R. D. WILDMAN. "On the role of the Knudsen layer in rapid granular flows." Journal of Fluid Mechanics 585 (August 7, 2007): 73–92. http://dx.doi.org/10.1017/s0022112007006489.

Full text

Abstract:

A combination of molecular dynamics simulations, theoretical predictions and previous experiments are used in a two-part study to determine the role of the Knudsen layer in rapid granular flows. First, a robust criterion for the identification of the thickness of the Knudsen layer is established: a rapid deterioration in Navier–Stokes order prediction of the heat flux is found to occur in the Knudsen layer. For (experimental) systems in which heat flux measurements are not easily obtained, a rule-of-thumb for estimating the Knudsen layer thickness follows, namely that such effects are evident within 2.5 (local) mean free paths of a given boundary. Secondly, comparisons of simulation and experimental data with Navier–Stokes order theory are used to provide a measure as to when Knudsen-layer effects become non-negligible. Specifically, predictions that do not account for the presence of a Knudsen layer appear reliable for Knudsen layers collectively composing up to 20% of the domain, whereas deterioration of such predictions becomes apparent when the domain is fully comprised of the Knudsen layer.

APA, Harvard, Vancouver, ISO, and other styles

2

Ikeda, Akihiko, Masuaki Matsumoto, Shohei Ogura, Tatsuo Okano, and Katsuyuki Fukutani. "Knudsen layer formation in laser induced thermal desorption." Journal of Chemical Physics 138, no. 12 (2013): 124705. http://dx.doi.org/10.1063/1.4795827.

Full text

APA, Harvard, Vancouver, ISO, and other styles

3

GU, XIAO-JUN, and DAVID R. EMERSON. "A high-order moment approach for capturing non-equilibrium phenomena in the transition regime." Journal of Fluid Mechanics 636 (September 25, 2009): 177–216. http://dx.doi.org/10.1017/s002211200900768x.

Full text

Abstract:

The method of moments is employed to extend the validity of continuum-hydrodynamic models into the transition-flow regime. An evaluation of the regularized 13 moment equations for two confined flow problems, planar Couette and Poiseuille flows, indicates some important limitations. For planar Couette flow at a Knudsen number of 0.25, they fail to reproduce the Knudsen-layer velocity profile observed using a direct simulation Monte Carlo approach, and the higher-order moments are not captured particularly well. Moreover, for Poiseuille flow, this system of equations creates a large slip velocity leading to significant overprediction of the mass flow rate for Knudsen numbers above 0.4. To overcome some of these difficulties, the theory of regularized moment equations is extended to 26 moment equations. This new set of equations highlights the importance of both gradient and non-gradient transport mechanisms and is shown to overcome many of the limitations observed in the regularized 13 moment equations. In particular, for planar Couette flow, they can successfully capture the observed Knudsen-layer velocity profile well into the transition regime. Moreover, this new set of equations can correctly predict the Knudsen layer, the velocity profile and the mass flow rate of pressure-driven Poiseuille flow for Knudsen numbers up to 1.0 and captures the bimodal temperature profile in force-driven Poiseuille flow. Above this value, the 26 moment equations are not able to accurately capture the velocity profile in the centre of the channel. However, they are able to capture the basic trends and successfully predict a Knudsen minimum at the correct value of the Knudsen number.

APA, Harvard, Vancouver, ISO, and other styles

4

Satayev, Marat, Abdugani Azimov, Arnold Brener, Nina Alekseeva, and Zulfia Shakiryanova. "Model of Selectivity of Membrane Processes and Dissolution of Impurities in a Membrane Pore in a Medium with Surface-Active Micelles." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 120, no. 1 (2024): 151–75. http://dx.doi.org/10.37934/arfmts.120.1.151175.

Full text

Abstract:

It is shown that concentration polarization affects the size of the Knudsen layer and the mechanism of mass transfer in the Knudsen layer is established. Based on the analysis of the selectivity patterns of ultrafiltration membranes, an equation was obtained for calculating the total probability of particle drift through the boundary diffusion layer and penetration into the Knudsen layer region and an equation for calculating the diffusion coefficient determined by the value of the derivative of the chemical potential in concentration. An equation is derived for determining the flow of particles to the membrane surface in the Knudsen layer by its thickness using the free path length of particles, the average velocity of thermal motion of molecules, the average residence time of particles in the Knudsen layer. The probability of particles passing through the pore is estimated, taking into account the influence of the activation energy for the passage of the solvent and the ratio of the interparticle distance in the pore to the free path length, and the selectivity of the membrane is estimated. An equation is obtained for calculating the length of the dissolution front in a micellar solution inside a membrane pore. An equation is obtained for calculating the kinetic factor in a micellar solution inside a membrane pore, taking into account the mass transfer coefficient along the interface, the diffusion coefficient, as well as the concentration of micelles in the medium and the solubilization rate constant. An effective design of a membrane apparatus for water purification is proposed.

APA, Harvard, Vancouver, ISO, and other styles

5

Davidovits, Seth, and Nathaniel J. Fisch. "Fusion utility in the Knudsen layer." Physics of Plasmas 21, no. 9 (2014): 092114. http://dx.doi.org/10.1063/1.4895477.

Full text

APA, Harvard, Vancouver, ISO, and other styles

6

Bhagat, Apurva, Harshal Gijare, and Nishanth Dongari. "Modeling of Knudsen Layer Effects in the Micro-Scale Backward-Facing Step in the Slip Flow Regime." Micromachines 10, no. 2 (2019): 118. http://dx.doi.org/10.3390/mi10020118.

Full text

Abstract:

The effect of the Knudsen layer in the thermal micro-scale gas flows has been investigated. The effective mean free path model has been implemented in the open source computational fluid dynamics (CFD) code, to extend its applicability up to slip and early transition flow regime. The conventional Navier-Stokes constitutive relations and the first-order non-equilibrium boundary conditions are modified based on the effective mean free path, which depends on the distance from the solid surface. The predictive capability of the standard `Maxwell velocity slip—Smoluchwoski temperature jump’ and hybrid boundary conditions `Langmuir Maxwell velocity slip—Langmuir Smoluchwoski temperature jump’ in conjunction with the Knudsen layer formulation has been evaluated in the present work. Simulations are carried out over a nano-/micro-scale backward facing step geometry in which flow experiences adverse pressure gradient, separation and re-attachment. Results are validated against the direct simulation Monte Carlo (DSMC) data, and have shown significant improvement over the existing CFD solvers. Non-equilibrium effects on the velocity and temperature of gas on the surface of the backward facing step channel are studied by varying the flow Knudsen number, inlet flow temperature, and wall temperature. Results show that the modified solver with hybrid Langmuir based boundary conditions gives the best predictions when the Knudsen layer is incorporated, and the standard Maxwell-Smoluchowski can accurately capture momentum and the thermal Knudsen layer when the temperature of the wall is higher than the fluid flow.

APA, Harvard, Vancouver, ISO, and other styles

7

Su, Wei, Qi Li, Yonghao Zhang, and Lei Wu. "Temperature jump and Knudsen layer in rarefied molecular gas." Physics of Fluids 34, no. 3 (2022): 032010. http://dx.doi.org/10.1063/5.0086076.

Full text

Abstract:

The temperature jump problem in rarefied molecular (diatomic and polyatomic) gases is investigated based on a one-dimensional heat conduction problem. The gas dynamics is described by a kinetic model, which is capable of recovering the general temperature and thermal relaxation processes predicted by the Wang–Chang Uhlenbeck equation. Analytical formulations for the temperature jump coefficient subject to the classical Maxwell gas–surface interaction are derived via the Chapman–Enskog expansion. Numerically, the temperature jump coefficient and the Knudsen layer function are calculated by matching the kinetic solution to the Navier–Stokes prediction in the Knudsen layer. Results show that the temperature jump highly depends on the thermal relaxation processes: the values of the temperature jump coefficient and the Knudsen layer function are determined by the relative quantity of the translational thermal conductivity to the internal thermal conductivity; and a minimum temperature jump coefficient emerges when the translational Eucken factor is 4/3 times of the internal one. Due to the exclusion of the Knudsen layer effect, the analytical estimation of the temperature jump coefficient may possess large errors. A new formulation, which is a function of the internal degree of freedom, the Eucken factors, and the accommodation coefficient, is proposed based on the numerical results.

APA, Harvard, Vancouver, ISO, and other styles

8

Albright, B. J., Kim Molvig, C. K. Huang, et al. "Revised Knudsen-layer reduction of fusion reactivity." Physics of Plasmas 20, no. 12 (2013): 122705. http://dx.doi.org/10.1063/1.4833639.

Full text

APA, Harvard, Vancouver, ISO, and other styles

9

Cornubert, R., D. d'Humières, and D. Levermore. "A Knudsen layer theory for lattice gases." Physica D: Nonlinear Phenomena 47, no. 1-2 (1991): 241–59. http://dx.doi.org/10.1016/0167-2789(91)90295-k.

Full text

APA, Harvard, Vancouver, ISO, and other styles

10

Liu, Wenbin, Jinbai Zhang, and Chun-Hian Lee. "DSMC Study of Strong Shear Nonequilibrium Phenomenon in Hypersonic Knudsen-Layer Flows." Journal of Physics: Conference Series 2285, no. 1 (2022): 012036. http://dx.doi.org/10.1088/1742-6596/2285/1/012036.

Full text

Abstract:

Abstract A series of plane Couette flows of rarefied argon gas under different flowfield settings are simulated by the direct simulation Monte Carlo method. The relaxation process of strong shear nonequilibrium is systematically studied. The results show a qualitatively platform-shaped distribution of molecular streamwise velocity under the conditions of small wall spacing (roughly in the Knudsen layer), and large shear rate and wall speed. The analysis of this phenomenon will help understand the relaxation process of hypersonic Knudsen-layer flow and provide some references for future modelling research.

APA, Harvard, Vancouver, ISO, and other styles

11

Xu, Mingtian. "Slip boundary condition of heat flux in Knudsen layers." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2161 (2014): 20130578. http://dx.doi.org/10.1098/rspa.2013.0578.

Full text

Abstract:

In a Knudsen layer with thickness comparable to the mean free path, collisions between heat carriers and solid walls play an important role in nanoscale heat transports. An interesting question is that whether these collisions also induce the slip of heat flow similar to the velocity slip condition of the rarefied gases on solid walls. In this work based on the discrete Boltzmann transport equation, the slip boundary condition of heat flux on solid walls in the Knudsen layer is established. This result is exemplified by the slip boundary condition of heat flux in nanowires, which has been proposed in a phenomenological way.

APA, Harvard, Vancouver, ISO, and other styles

12

Fan, Yuwei, Jun Li, Ruo Li, and Zhonghua Qiao. "Resolving Knudsen layer by high-order moment expansion." Continuum Mechanics and Thermodynamics 31, no. 5 (2019): 1313–37. http://dx.doi.org/10.1007/s00161-019-00749-3.

Full text

APA, Harvard, Vancouver, ISO, and other styles

13

Zhan, Shiyuan, Yuliang Su, Mingjing Lu, et al. "Effect of Surface Type on the Flow Characteristics in Shale Nanopores." Geofluids 2021 (March 8, 2021): 1–12. http://dx.doi.org/10.1155/2021/6641922.

Full text

Abstract:

The underlying mechanism of shale gas migration behavior is of great importance to understanding the flow behavior and the prediction of shale gas flux. The slippage of the methane molecules on the surface is generally emphasized in nanopores in most predicted methods currently. In this work, we use molecular dynamic (MD) simulations to study the methane flow behavior in organic (graphene) and inorganic (quartz) nanopores with various pore size. It is observed that the slippage is obvious only on the graphene nanopores and disappeared on the quartz surface. Compared with the traditional Navier-Stokes equation combined with the no-slip boundary, the enhancement of the gas flux is nonnegligible in the graphene nanopores and could be neglected in the quartz nanopores. In addition, the flux contribution ratios of the adsorption layer, Knudsen layer, and the bulk gas are analyzed. In quartz nanopores, the contributions of the adsorption layer and the Knudsen layer are slight when the pore size is larger than 10 nm. It is also noted that even if the Knudsen number is the same, the flow mode may be various with the effect of the pore surface type. Our work should give molecular insights into gas migration mechanisms in organic and inorganic nanopores and provide important reference to the prediction of the gas flow in various types of shale nanopores.

APA, Harvard, Vancouver, ISO, and other styles

14

Allan, Adam M., and Gary Mavko. "The effect of adsorption and Knudsen diffusion on the steady-state permeability of microporous rocks." GEOPHYSICS 78, no. 2 (2013): D75—D83. http://dx.doi.org/10.1190/geo2012-0334.1.

Full text

Abstract:

Microporous rocks are being increasingly researched as novel exploration and development technologies facilitate production of the reserves confined in the low-permeability reservoir. The ability to numerically estimate effective permeability is pivotal to characterizing the production capability of microporous reservoirs. In this study, a novel methodology is presented for estimating the steady-state effective permeability from FIB-SEM volumes. We quantify the effect of a static adsorbed monolayer and Knudsen diffusion on effective permeability as a function of pore pressure to better model production of microporous rock volumes. The adsorbed layer is incorporated by generating an effective pore geometry with a pore pressure-dependent layer of immobile voxels at the fluid-solid interface. Due to the steady-state nature of this study, surface diffusion within the adsorbed layer and topological variations of the layer within pores are neglected, potentially resulting in underestimation of effective permeability over extended production time periods. Knudsen diffusion and gas slippage is incorporated through computation of an apparent permeability that accounts for the rarefaction of the pore fluid. We determine that at syn-production pore pressures, permeability varies significantly as a function of the phase of the pore fluid. Simulation of methane transport in micropores indicates that, in the supercritical regime, the effect of Knudsen diffusion relative to adsorption is significantly reduced resulting in effective permeability values up to 10 nanodarcies ([Formula: see text]) less or 40% lower than the continuum prediction. Contrastingly, at subcritical pore pressures, the effective permeability is significantly greater than the continuum prediction due to rarefaction of the gas and the onset of Knudsen diffusion. For example, at 1 MPa, the effective permeability of the kerogen body is five times the continuum prediction. This study demonstrates the importance of, and provides a novel methodology for, incorporating noncontinuum effects in the estimation of the transport properties of microporous rocks.

APA, Harvard, Vancouver, ISO, and other styles

15

Prozorova, Evelina. "Model of Boundary Conditions on Metal Surfaces for Rarefied Gas." MOLECULAR SCIENCES AND APPLICATIONS 4 (July 4, 2024): 34–41. http://dx.doi.org/10.37394/232023.2024.4.4.

Full text

Abstract:

Currently, the classical theory discusses issues related to the sliding of liquid and gas along a wall at low external flow velocities. These questions become especially relevant when the surface size is reduced to the nanoscale. The article discusses the formation of sliding conditions and an adsorption layer for an ideal crystalline surface. For gas, the Knudsen layer is proposed to be divided into two parts: an adjacent layer with a thickness of several molecular interaction radii, in which molecules do not collide with each other, and a layer in which the Chapman-Enskog method is defined. The solution for this layer can be found by the small parameter method. For water, there is no Knudsen layer, but adhesion and the formation of a thin stationary layer are possible. Various possible causes of slipping are discussed. The formation of a dislocation from a point defect near the surface, which is a vacancy, is considered. An analysis of the causes of pore clogging during water movement near the surface was carried out. The emphasis is on the change in stress in the metal, taking into account the influence of the moment that occurs when the position of the molecules changes.

APA, Harvard, Vancouver, ISO, and other styles

16

Erofeev, A. I., M. N. Kogan, and O. G. Fridlender. "Quasiequilibrium Knudsen boundary layer on a nonisothermal porous body." Fluid Dynamics 45, no. 1 (2010): 134–46. http://dx.doi.org/10.1134/s0015462810010151.

Full text

APA, Harvard, Vancouver, ISO, and other styles

17

Szalmás, L. "Knudsen layer theory for high-order lattice Boltzmann models." Europhysics Letters (EPL) 80, no. 2 (2007): 24003. http://dx.doi.org/10.1209/0295-5075/80/24003.

Full text

APA, Harvard, Vancouver, ISO, and other styles

18

Skorov, Yu V., and H. Rickman. "Simulation of gas flow in a cometary Knudsen layer." Planetary and Space Science 46, no. 8 (1998): 975–96. http://dx.doi.org/10.1016/s0032-0633(97)00226-2.

Full text

APA, Harvard, Vancouver, ISO, and other styles

19

Sibold, D., and H. M. Urbassek. "Formation of a knudsen layer in electronically induced desorption." Applied Physics B Photophysics and Laser Chemistry 55, no. 4 (1992): 391–96. http://dx.doi.org/10.1007/bf00333088.

Full text

APA, Harvard, Vancouver, ISO, and other styles

20

Goniak, Remy, and Georges Duffa. "Corrective term in wall slip equations for Knudsen layer." Journal of Thermophysics and Heat Transfer 9, no. 2 (1995): 383–84. http://dx.doi.org/10.2514/3.677.

Full text

APA, Harvard, Vancouver, ISO, and other styles

21

Berger, Anne, Michael Striednig, Christoph Simon та Hubert A. Gasteiger. "Determination of the τ/ε-Ratio for Gas Diffusion Substrates and Microporous Layers in a Proton Exchange Membrane Fuel Cell". Journal of The Electrochemical Society 172, № 1 (2025): 014508. https://doi.org/10.1149/1945-7111/ada63e.

Full text

Abstract:

An understanding of the GDL properties is crucial for high-current-density operation of proton exchange membrane fuel cells (PEMFCs). The parameters porosity ( ε ) and tortuosity ( τ ) directly link the theoretical diffusivity in free space and the effective diffusivity in the structure. The τ / ε -ratio is therefore an important descriptor for the gas diffusion in a porous network. This study characterizes the τ / ε -ratio for gas diffusion layer substrate (GDL-S) materials from two suppliers (Toray, Freudenberg) and for one microporous layer (MPL) using limiting current measurements in an operating fuel cell. However, only the application of a new analysis method that considers the contribution from Knudsen diffusion allows for the τ / ε -determination of GDL-S materials which have carbon black added to the structure (e.g., Freudenberg GDL-S); this method can then also be used for the determination of the τ / ε -ratio of porous networks containing small pores like MPLs. Here, we assess the contribution of pressure-independent Knudsen diffusion to the overall diffusion and introduce a mixed diffusion coefficient, which includes both Knudsen and molecular diffusion. This analysis serves as a guideline to estimate the contribution of the respective diffusive processes for gas diffusion materials utilized in PEMFCs, where pore sizes resulting in different diffusion regimes are present.

APA, Harvard, Vancouver, ISO, and other styles

22

Ying, Ruoxian, and Michael H. Peters. "Hydrodynamic interaction of two unequal-sized spheres in a slightly rarefied gas: resistance and mobility functions." Journal of Fluid Mechanics 207 (October 1989): 353–78. http://dx.doi.org/10.1017/s0022112089002612.

Full text

Abstract:

The problem of the hydrodynamic interaction of two unequal-sized spheres in a slightly rarefied gas is treated following the singular perturbation scheme of Sone & Onishi (1978), valid at small, but finite, particle Knudsen numbers. In this method the solution to the linearized BGKW transport equation governing the gas molecular motion consists of two parts: one describing a Knudsen layer where the actual microscopic boundary conditions are applied and the other describing a Hilbert region where the Stokes equations of continuum hydrodynamics hold. The Knudsen-layer solution establishes the ‘slip’ boundary conditions for the Stokes equations. Here we clearly distinguish between particle ‘slip’ due to the type of boundary conditions and particle ‘slip’ due to lengthscale effects as measured by the Knudsen number. The present analysis has been carried out to first order in particle Knudsen number for the case of diffuse reflective molecular boundary conditions. General relationships between the first- and zero-order velocity fields, both of which are written in the form of Lamb's (1932) solution to the Stokes equation, are established. It is illustrated how these general relationships can be used to determine the force and torque acting on a single sphere translating and rotating in a slightly rarefied gas. Finally, we have treated the two-sphere problem in a slightly rarefied gas using the twin multipole expansion method of Jeffrey & Onishi (1984). Here again, general relationships are established between the solutions of the first-order fluid velocity field and the zero-order velocity field, the latter being shown to recover Jeffrey & Onishi's results for stick boundary conditions. These general relationships are subsequently used to determine the complete resistance and mobility matrices of the two-sphere system. The symmetric properties of the resistance and mobility matrices are demonstrated for slip boundary conditions, in agreement with the general proof of Landau & Lifshitz (1980) and Bedeaux, Albano & Mazur (1977).

APA, Harvard, Vancouver, ISO, and other styles

23

Kelly, Roger, and R. W. Dreyfus. "Reconsidering the mechanisms of laser sputtering with Knudsen-layer formation taken into account." Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 32, no. 1-4 (1988): 341–48. http://dx.doi.org/10.1016/0168-583x(88)90235-2.

Full text

APA, Harvard, Vancouver, ISO, and other styles

24

Hori, Tomoki, Takuya Mabuchi, Ikuya Kinefuchi, and Takashi Tokumasu. "Molecular Dynamics Simulation of Scattering and Surface Diffusion of Oxygen Molecules on Ionomers in Catalyst Layers of PEFCs." ECS Meeting Abstracts MA2022-02, no. 39 (2022): 1390. http://dx.doi.org/10.1149/ma2022-02391390mtgabs.

Full text

Abstract:

Polymer electrolyte fuel cells (PEFCs) are used in fuel cell vehicles and households stationary power sources because of their low operating temperatures, small size, and light weight. For further development of PEFCs, it is necessary to reduce the amount of platinum used in the catalyst layers. If the amount of platinum used in the catalyst layers is reduced, the current density per mass of platinum increases. The effect of diffusion polarization then becomes dominant in the overall voltage drop. One factor that causes diffusion polarization is the transport resistance of oxygen molecules in the catalyst layers. Therefore, an understanding of the transport of oxygen molecules in the catalyst layers is necessary to improve the performance of PEFCs under high current density conditions. The catalyst layers comprise porous structures by aggregation of ionomer-coated platinum-supported carbon. The Knudsen number is important for gas diffusion phenomena in porous structures. The Knudsen number is the ratio of the representative length of the system to the mean free path of gas molecules, and when this value is sufficiently small, the gas diffusion phenomenon can be regarded as molecular diffusion. When the Knudsen number is sufficiently large, the gas diffusion phenomenon becomes Knudsen diffusion, in which collisions with solid walls are more dominant than collisions between gas molecules. The Knudsen number is sufficiently large in the porous structure of the catalyst layer. Therefore, the effect of collision with solid surfaces is important in the oxygen transport in the catalyst layer. In previous studies, it has been clarified that the behavior of oxygen molecules incident on an ionomer thin film includes surface scattering, in which diffuse reflection occurs on the surface, and surface diffusion, in which oxygen molecules trapped on the surface move across the surface in a series of multiple collisions. However, the effect of surface diffusion on oxygen transport across the catalyst layer is not clear. The purpose of this study is to analyze the effect of oxygen molecules and ionomer conditions on surface diffusion using the MD simulation. We analyze the behavior of oxygen molecules impinging on ionomers in a computational system that reproduces the pores of the catalyst layer. The simulation system consists of polymer chains, solvent molecules (water molecules and hydronium ions), and oxygen molecules. Periodic boundary conditions were applied in the x-, y-, and z-directions. The behavior of oxygen molecules after they impact the ionomer wall such as their residence time during surface diffusion and the surface diffusion coefficient will be analyzed. Acknowledgments This work was supported by the New Energy and Industrial Technology Development Organization (NEDO) of Japan, Grant number JPNP20003. It was performed on the Supercomputer system “AFI-NITY” at the Advanced Fluid Information Research Center, Institute of Fluid Science, Tohoku University. Figure 1

APA, Harvard, Vancouver, ISO, and other styles

25

Besse, Christophe, Saja Borghol, Thierry Goudon, Ingrid Lacroix-Violet, and Jean-Paul Dudon. "Hydrodynamic Regimes, Knudsen Layer, Numerical Schemes: Definition of Boundary Fluxes." Advances in Applied Mathematics and Mechanics 3, no. 5 (2011): 519–61. http://dx.doi.org/10.4208/aamm.10-m1041.

Full text

Abstract:

AbstractWe propose a numerical solution to incorporate in the simulation of a system of conservation laws boundary conditions that come from a microscopic modeling in the small mean free path regime. The typical example we discuss is the derivation of the Euler system from the BGK equation. The boundary condition relies on the analysis of boundary layers formation that accounts from the fact that the incoming kinetic flux might be far from the thermodynamic equilibrium.

APA, Harvard, Vancouver, ISO, and other styles

26

Charles, Fréderique, Nicolas Vauchelet, Christophe Besse, et al. "Numerical approximation of Knudsen layer for the Euler-Poisson system." ESAIM: Proceedings 32 (October 2011): 177–94. http://dx.doi.org/10.1051/proc/2011020.

Full text

APA, Harvard, Vancouver, ISO, and other styles

27

Erofeev, A. I., and O. G. Friedlender. "Nonlinear Knudsen boundary layer on an infinitely thin permeable membrane." Fluid Dynamics 45, no. 6 (2010): 965–74. http://dx.doi.org/10.1134/s0015462810060141.

Full text

APA, Harvard, Vancouver, ISO, and other styles

28

Tang, G. H., Y. H. Zhang, X. J. Gu, and D. R. Emerson. "Lattice Boltzmann modelling Knudsen layer effect in non-equilibrium flows." EPL (Europhysics Letters) 83, no. 4 (2008): 40008. http://dx.doi.org/10.1209/0295-5075/83/40008.

Full text

APA, Harvard, Vancouver, ISO, and other styles

29

Skorov, Yu V., and H. Rickman. "Gas flow and dust acceleration in a cometary Knudsen layer." Planetary and Space Science 47, no. 8-9 (1999): 935–49. http://dx.doi.org/10.1016/s0032-0633(99)00008-2.

Full text

APA, Harvard, Vancouver, ISO, and other styles

30

Boutilier, Michael S. H., Nicolas G. Hadjiconstantinou, and Rohit Karnik. "Knudsen effusion through polymer-coated three-layer porous graphene membranes." Nanotechnology 28, no. 18 (2017): 184003. http://dx.doi.org/10.1088/1361-6528/aa680f.

Full text

APA, Harvard, Vancouver, ISO, and other styles

31

Cheng, Zhi Lin, Zan Liu, and Sheng Wang. "Preparation of MFI Membrane on Mesoporous-Layer-Modified Macroporous Al₂O₃ Substrate by Secondary Growth Method and its Permeation Property." Applied Mechanics and Materials 633-634 (September 2014): 417–21. http://dx.doi.org/10.4028/www.scientific.net/amm.633-634.417.

Full text

Abstract:

The compact MFI membrane supported on a mesoporous-layer-modified macroporous Al2O3 substrate has been prepared by the secondary growth method. SEM images indicate that the MFI membranes with the thickness of ca. 15 μm consist of highly intergrown crystals with remarkable integrity. The H2 permeance of MFI membranes on mesoporous/macroporous substrate is 8-18 times larger than that on the macroporous substrate, and further the permselectivity of H2/C3H8 maintains at the level of 20, which is 4 times higher than that of the corresponding Knudsen diffusion. More importantly, the real selectivity of the mixture of H2/C3H8 on these membranes is 2-4 times higher than that of the corresponding Knudsen diffusion. The method is attractive for the practical application.

APA, Harvard, Vancouver, ISO, and other styles

32

Zhang, Yudong, Aiguo Xu, Feng Chen, Chuandong Lin, and Zon-Han Wei. "Non-equilibrium characteristics of mass and heat transfers in the slip flow." AIP Advances 12, no. 3 (2022): 035347. http://dx.doi.org/10.1063/5.0086400.

Full text

Abstract:

Slip flow is a common phenomenon in micro-/nano-electromechanical systems. It is well known that the mass and heat transfers in slip flow show many unique behaviors, such as the velocity slip and temperature jump near the wall. However, the kinetic understanding of slip flow is still an open problem. This paper first clarifies that the Thermodynamic Non-Equilibrium (TNE) flows can be roughly classified into two categories: near-wall TNE flows and TNE flows away from the wall. The origins of TNE in the two cases are significantly different. For the former, the TNE mainly results from the fluid–wall interaction; for the latter, the TNE is primarily due to the considerable (local) thermodynamic relaxation time. Therefore, the kinetic modeling methods for the two kinds of TNE flows are significantly different. Based on the Discrete Boltzmann Modeling (DBM) method, the non-equilibrium characteristics of mass and heat transfers in slip flow are demonstrated and investigated. The method is solidly verified by comparing with analytic solutions and experimental data. In pressure-driven flow, the DBM results are consistent with experimental data for the Knudsen number up to 0.5. It is verified that, in the slip flow regime, the linear constitutive relations with standard viscous or heat conduction coefficients are no longer applicable near the wall. For the Knudsen layer problem, it is interesting to find that a heat flux (viscous stress) component in the velocity (temperature) Knudsen layer approximates a hyperbolic sinusoidal distribution. The findings enrich the insights into the non-equilibrium characteristics of mass and heat transfers at micro-/nano-scales.

APA, Harvard, Vancouver, ISO, and other styles

33

TIAN, ZHI-WEI, SHENG CHEN, and CHU-GUANG ZHENG. "LATTICE BOLTZMANN SIMULATION OF GASEOUS FINITE-KNUDSEN MICROFLOWS." International Journal of Modern Physics C 21, no. 06 (2010): 769–83. http://dx.doi.org/10.1142/s0129183110015464.

Full text

Abstract:

In this study, microscale gaseous flows in the transitional regime have been investigated by lattice Boltzmann method (LBM). In the existing microflows LBM models, the Knudsen layer correction function has been introduced into the models. According to the kinetic theory rigorously, we choose a proper expression of correction function, and then determine its adjustable parameter. A substitute high-order boundary conditions treatment is adopted to capture the velocity slip, without any difficulties in computing the high-order velocity derivatives. The numerical results of two typical microflows show that: the present results agree with the analytical solutions better than the existing LBM simulations. Evident improvements can also be found, especially for finite Kn microflows.

APA, Harvard, Vancouver, ISO, and other styles

34

SZALMÁS, LAJOS. "LATTICE BOLTZMANN METHOD WITH OPTIMIZED BOUNDARY LAYER AT FINITE KNUDSEN NUMBERS." International Journal of Modern Physics C 19, no. 02 (2008): 249–57. http://dx.doi.org/10.1142/s0129183108012078.

Full text

Abstract:

We present an optimization procedure in high-order lattice Boltzmann models in order to fine-tune the method for micro-channel flows in the transition region. Both the first and second slip coefficients are tunable, and the hydrodynamic and Knudsen layer solutions can be tailored. Very good results are obtained in comparison with the continuous solution for hard sphere molecules. For the first time, we provide an accurate description of Poiseuille flow in the transition region.

APA, Harvard, Vancouver, ISO, and other styles

35

Akhlaghi, Hassan, and Ehsan Roohi. "Generalized description of the Knudsen layer thickness in rarefied gas flows." Physics of Fluids 33, no. 6 (2021): 061701. http://dx.doi.org/10.1063/5.0052263.

Full text

APA, Harvard, Vancouver, ISO, and other styles

36

Pekker, L., M. Keidar, and J. L. Cambier. "Effect of thermal conductivity on the Knudsen layer at ablative surfaces." Journal of Applied Physics 103, no. 3 (2008): 034906. http://dx.doi.org/10.1063/1.2838210.

Full text

APA, Harvard, Vancouver, ISO, and other styles

37

Bhagat, Apurva, Harshal Gijare, and Nishanth Dongari. "Implementation of Knudsen Layer Phenomena in Rarefied High-Speed Gas Flows." Journal of Aerospace Engineering 32, no. 6 (2019): 04019100. http://dx.doi.org/10.1061/(asce)as.1943-5525.0001097.

Full text

APA, Harvard, Vancouver, ISO, and other styles

38

Gupta, Roop N., and Carl D. Scott. "Comment on 'Corrective term in wall slip equations for Knudsen layer'." Journal of Thermophysics and Heat Transfer 10, no. 1 (1996): 190. http://dx.doi.org/10.2514/3.773.

Full text

APA, Harvard, Vancouver, ISO, and other styles

39

Lilley, Charles R., and John E. Sader. "Velocity profile in the Knudsen layer according to the Boltzmann equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2096 (2008): 2015–35. http://dx.doi.org/10.1098/rspa.2008.0071.

Full text

Abstract:

Flow of a dilute gas near a solid surface exhibits non-continuum effects that are manifested in the Knudsen layer. The non-Newtonian nature of the flow in this region has been the subject of a number of recent studies suggesting that the so-called ‘effective viscosity’ at a solid surface is half that of the standard dynamic viscosity. Using the Boltzmann equation with a diffusely reflecting surface and hard sphere molecules, Lilley & Sader discovered that the flow exhibits a striking power-law dependence on distance from the solid surface where the velocity gradient is singular. Importantly, these findings (i) contradict these recent claims and (ii) are not predicted by existing high-order hydrodynamic flow models. Here, we examine the applicability of these findings to surfaces with arbitrary thermal accommodation and molecules that are more realistic than hard spheres. This study demonstrates that the velocity gradient singularity and power-law dependence arise naturally from the Boltzmann equation, regardless of the degree of thermal accommodation. These results are expected to be of particular value in the development of hydrodynamic models beyond the Boltzmann equation and in the design and characterization of nanoscale flows.

APA, Harvard, Vancouver, ISO, and other styles

40

Tidecks, Reinhard. "Evidence for electron-tunneling-limited Knudsen diffusion of mercury in phosphor layers and coatings of fluorescent lamps." European Physical Journal Applied Physics 91, no. 3 (2020): 30801. http://dx.doi.org/10.1051/epjap/2020190286.

Full text

Abstract:

Recent experimental studies and modelling of the mercury loss (i.e. the mercury consumption) in fluorescent lamps yield diffusion coefficients of mobile mercury in phosphor layers and coatings, which are several orders of magnitude smaller than expected for a gas diffusion in a situation in which the mean free path of the diffusing particles is restricted by the pore radius in those materials (Knudsen diffusion). In the present work it is shown that the transition of mercury ions from the plasma to the Knudsen diffusion regime may be one reason for this observation. Another possibility is that only discharged ions from the plasma form the mercury oxide as which mercury is deposited in the phosphor layer and coating, from the investigation of which the diffusion coefficient of mobile mercury is concluded by fitting the model to the experiment.

APA, Harvard, Vancouver, ISO, and other styles

41

Nabapure, D., A. Singh, and R. C. M. Kalluri. "Investigation of rarefied flow over backward-facing step in different rarefaction regimes using direct simulation Monte Carlo." Aeronautical Journal 126, no. 1298 (2021): 617–44. http://dx.doi.org/10.1017/aer.2021.88.

Full text

Abstract:

AbstractHypersonic aerothermodynamics for a re-entry vehicle approaching the earth’s atmosphere is critical in the exploration of space. These vehicles often encounter various flow regimes due to the density variations and have surface abnormalities. The backward-facing step (BFS) is one such simplified configuration for modeling anomalies around such space vehicles. The present work examines rarefied hypersonic flow over a BFS using the direct simulation Monte Carlo (DSMC) method. The purpose of this research is focused on exploring the various loads encountered by a re-entry vehicle passing through different altitudes covering different rarefaction regimes. The fluid considered was non-reacting air, with the free-stream Mach number as 25, and the Knudsen number considered ranged from 0.05-21.10. The influence of the Knudsen number on flow characteristics has been elucidated graphically in various streamwise directions. The normalised flow properties such as velocity, pressure, temperature and density showed an increasing trend with the Knudsen number due to compressibility and viscous heating effects. In all flow regimes, there was an appearance of flow recirculation. With rarefaction, the recirculation lengths decreased, whereas the boundary layer thickness showed an increase. The aerodynamic surface properties such as pressure coefficient, skin friction, and heat transfer coefficient, by and large, showed an increase with the Knudsen number. When the chemical reactions were accounted for and compared against the non-reacting flows, the velocity, pressure, and density field showed no marked variation; however, considerable variations were observed in the temperature field. Furthermore, the present study also depicts the compressibility factor contour, showing the flow regions that diverge from the ideal gas behaviour.

APA, Harvard, Vancouver, ISO, and other styles

42

Chun, Hyunsoo, Youngseop Lee, Jiwoong Kim, Jung Hyo Chang, Jin Young Kim, and Kyoungdoug Min. "Method for Precise Oxygen Resistance Analysis in Polymer Electrolyte Membrane Fuel Cells Under Various Operating Conditions." ECS Meeting Abstracts MA2024-02, no. 44 (2024): 3068. https://doi.org/10.1149/ma2024-02443068mtgabs.

Full text

Abstract:

The simultaneous changes occurring in the internal materials of Polymer Electrolyte Membrane Fuel Cells (PEMFCs) under dynamic operating conditions make characterizing the inherent resistance of each material under different operating conditions highly complex. Therefore, the isolation of the internal resistance of components and the identification of the resistance elements that significantly affect PEMFC performance are essential, but comprehensive studies analyzing the internal resistance under different operating conditions have been scarce. This study proposes a complex resistance analysis methodology capable of isolating the oxygen resistance of a PEMFC single cell under different operating conditions (relative humidity, temperature, supply gas). The Knudsen diffusion within the catalyst layer is calculated according to the operating conditions using a catalyst agglomerate model. This calculation is combined with the dissection of the oxygen transport resistance to distinguish between molecular diffusion, Knudsen diffusion and ionomer film resistance under limiting current density conditions. In addition, an equation has been developed that accurately predicts the ionomer film resistance as a function of changes in temperature and RH, proposing a new approach to mapping ionomer film resistance. Then, by inputting three forms of resistance from the dissection of the oxygen transport resistance analysis as percentages into the distribution of relaxation time analysis, the newly developed methodology gives insight into the contributions of seven different resistance elements at different current densities. This provides an indicator for determining ionomer resistance without the need for temperature and RH experiments. In addition, this research identifies the design of the ionomer within the catalyst layer as a key determinant of PEMFC cell performance, highlighting the significant impact that molecular or Knudsen diffusion can have depending on operating conditions. Consequently, through an in-depth analysis of oxygen stability under different operating conditions, this study will provide essential insights for future PEMFC cell design.

APA, Harvard, Vancouver, ISO, and other styles

43

Lockerby, Duncan A., Jason M. Reese, and Michael A. Gallis. "Capturing the Knudsen Layer in Continuum-Fluid Models of Nonequilibrium Gas Flows." AIAA Journal 43, no. 6 (2005): 1391–93. http://dx.doi.org/10.2514/1.13530.

Full text

APA, Harvard, Vancouver, ISO, and other styles

44

Cohen, Bruce I., Andris M. Dimits, George B. Zimmerman, and Scott C. Wilks. "One-dimensional particle simulations of Knudsen-layer effects on D-T fusion." Physics of Plasmas 21, no. 12 (2014): 122701. http://dx.doi.org/10.1063/1.4903323.

Full text

APA, Harvard, Vancouver, ISO, and other styles

45

Loyalka, S. K., and K. C. Lea. "Density profiles in the knudsen layer of vapor condensing on a drop." Journal of Colloid and Interface Science 107, no. 2 (1985): 581–83. http://dx.doi.org/10.1016/0021-9797(85)90217-6.

Full text

APA, Harvard, Vancouver, ISO, and other styles

46

Lockerby, Duncan A., Jason M. Reese, and Michael A. Gallis. "The usefulness of higher-order constitutive relations for describing the Knudsen layer." Physics of Fluids 17, no. 10 (2005): 100609. http://dx.doi.org/10.1063/1.1897005.

Full text

APA, Harvard, Vancouver, ISO, and other styles

47

Yadav, Upendra, Anirudh Jonnalagadda, and Amit Agrawal. "Third-order accurate 13-moment equations for non-continuum transport phenomenon." AIP Advances 13, no. 4 (2023): 045311. http://dx.doi.org/10.1063/5.0143420.

Full text

Abstract:

The derivation of analytical equations of non-continuum macroscopic transport phenomena is underpinned by approximate descriptions of the particle distribution function and is required due to the inability of the Navier–Stokes equations to describe flows at high Knudsen number ( Kn ∼ 1). In this paper, we present a compact representation of the second-order correction to the Maxwellian distribution function and 13-moment transport equations that contain fewer terms compared to available moment-based representations. The intrinsic inviscid and isentropic assumptions of the second-order accurate distribution function are then relaxed to present a third-order accurate representation of the distribution function, using which corresponding third-order accurate moment transport equations are derived. Validation studies performed for Grad’s second problem and the force-driven plane Poiseuille flow problem at non-zero Knudsen numbers for Maxwell molecules highlight an improvement over results obtained by using the Navier–Stokes equations and Grad’s 13-moment (G13) equations. To establish the ability of the proposed equations to accurately capture the bulk behavior of the fluid, the results of Grad’s second problem have been validated against the analytical solution of the Boltzmann equation. For the planar Poiseuille flow problem, validations against the direct simulation Monte Carlo method data reveal that, in contrast to G13 equations, the proposed equations are capable of accurately capturing the Knudsen boundary layer.

APA, Harvard, Vancouver, ISO, and other styles

48

Berger, Anne, та Hubert Andreas Gasteiger. "Determination of the τ/ε-Ratio for Gas Diffusion Substrates and Microporous Layers in an Operating Fuel Cell". ECS Meeting Abstracts MA2022-01, № 35 (2022): 1456. http://dx.doi.org/10.1149/ma2022-01351456mtgabs.

Full text

Abstract:

Proton-exchange-membrane fuel cells (PEMFCs) are expected to play a major role in the electrification of the transportation sector,[1] with recent focus shifting to the heavy-duty market. One essential aspect for increasing the power density of a PEMFC is optimizing the mass transport in the gas diffusion layer (GDL) on the cathode side of the cell. A careful choice of the commonly used carbon fiber based GDL is therefore necessary to optimize the performance at different relative humidity conditions. Among other important GDL properties such as the pore size distribution and the thermal conductivity, the porosity (ε) and the tortuosity (τ) are important descriptors. The τ/ε-ratio has been characterized for diffusion media using in-situ or ex-situ techniques.[2-3] This study is based on the theory developed by Baker et al.[2] In addition to their work, which solely considers molecular diffusion, we will present a method to include Knudsen diffusion occurring in smaller pores for the in-situ evaluation of the τ/ε-ratio. This adaptation makes it possible to evaluate GDL substrates to which carbon black (creating small pores) has been added to increase the electrical conductivity,[4] and also to extend the theory to the microporous layer (MPL). In this study, we characterize the τ/ε-ratio for two GDL substrates (without the addition of an MPL), one from Toray and one from Freudenberg, using limiting current measurements in an operating fuel cell. The τ/ε-ratio describes the deviation of the effective diffusivity compared to ideal molecular diffusion. However, depending of the range of pore diameters in the gas diffusion medium, a mixture of molecular diffusion and Knudsen diffusion has to be taken into account. Fig. 1 shows the theoretical contribution ratio of Knudsen diffusion and molecular diffusion (left y-axis) versus the relevant range of pore diameters for different pressures. The graphic illustrates that Knudsen diffusion dominates at lower pore sizes, while molecular diffusion is mostly present at larger pore sizes. An increase in pressure shifts the regime of molecular diffusion to smaller pores. In addition to the theoretical contribution ratio, Fig. 1 depicts the pore size distribution of a Toray and a Freudenberg GDL substrate determined by mercury intrusion porosimetry (MIP). While the Toray paper has a narrow pore size distribution at pores of 30-50 µm, where only molecular diffusion occurs, the Freudenberg GDL contains a broader range of larger pores at 10-40 µm (molecular diffusion) together with the presence of small pores at ca. 70-80 nm that derive from the addition of carbon black and that present a medium where Knudsen and molecular diffusion occur. After the validation of the principle, the method is transferred to determine the τ/ε-number of a microporous layer based on vapor-grown carbon-fibers (VGCF), whose transport properties in a PEMFC have been described previously.[5] References [1] O. Gröger, H. A. Gasteiger, J.-P. Suchsland, J. Electrochem. Soc. 2015, 162, A2605-A2622. [2] D. R. Baker, D. A. Caulk, K. C. Neyerlin, M. W. Murphy, J. Electrochem. Soc. 2009, 156, B991. [3] D. Kramer, S. A. Freunberger, R. Flückiger, I. A. Schneider, A. Wokaun, F. N. Büchi, G. G. Scherer, Journal of Electroanalytical Chemistry 2008, 612, 63-77. [4] K.-D. Wagner, A. Bock, K. Salama, A. Weller, Vol. US 2010/0219069 A1, Carl Freudenberg KG 2010. [5] C. Simon, J. Endres, B. Nefzger-Loders, F. Wilhelm, H. A. Gasteiger, J. Electrochem. Soc. 2019, 166, F1022-F1035. Acknowledgements We gratefully acknowledge funding from the Swiss National Foundation under the funding scheme Sinergia (project grant number 180335). We also thank Michael Striednig and Christoph Simon for initial work on the topic. Figure 1: Left y-axis: theoretical contribution ratio of Knudsen (green) and molecular diffusion (orange) versus pore diameter for different absolute pressures of 115 kPaabs (solid lines), 150 kPaabs (dashed lines), 200 kPaabs (dotted lines), and 300 kPaabs (dash-dotted lines). With smaller pores and lower pressures, more contributions from Knudsen diffusion can be expected. Right y-axis: log. differential intrusion measured by MIP analysis for the Toray (blue) and the Freudenberg (grey) GDL substrates, whereby the Freudenberg GDL shows pores at ~70-80 nm that are caused by the addition of carbon black. Figure 1

APA, Harvard, Vancouver, ISO, and other styles

49

BATISHCHEV, O. V., M. M. SHOUCRI, A. A. BATISHCHEVA, and I. P. SHKAROFSKY. "Fully kinetic simulation of coupled plasma and neutral particles in scrape-off layer plasmas of fusion devices." Journal of Plasma Physics 61, no. 2 (1999): 347–64. http://dx.doi.org/10.1017/s0022377898007375.

Full text

Abstract:

Fluid descriptions of plasmas, which are usually applied to a collisional plasma, can only be justified for very small Coulomb Knudsen numbers. However, the scrape-off layer (SOL) plasmas of experimental magnetic confinement fusion devices tend to have operational regimes characterized by a Coulomb Knudsen number around 0.1. In interesting detached regimes of an SOL plasma in a tokamak, when the plasma detaches from the limiters or divertors, this number may increase along with the local plasma gradients. Plasma gradients are also known to increase (and thus drive non-local effects) in inertial confinement fusion. Neutrals, which are being produced owing to plasma recombination at the plasma–divertor interface, may be in a mixed collisional regime as well. Thus simultaneous kinetic treatments of plasma and neutral particles with self-consistent evaluation of boundary conditions at the material walls are required. We present a physical model and a numerical scheme, and discuss results of purely kinetic simulations of plasmas and neutrals for actual conditions in the Alcator C-Mod and Tokamak-de-Varennes experimental tokamaks. Results for both steady-state and transient regimes of SOL plasma flow are presented. Our approach, unlike particle-in-cell and Monte Carlo methods, is free from statistical noise.

APA, Harvard, Vancouver, ISO, and other styles

50

Grechnikov, Alexander A., Alexey S. Borodkov, Yaroslav O. Simanovsky, and Sergey M. Nikiforov. "Silicon surface assisted laser desorption ionization mass spectrometry for quantitative analysis." European Journal of Mass Spectrometry 27, no. 2-4 (2021): 84–93. http://dx.doi.org/10.1177/14690667211006017.

Full text

Abstract:

The approach to quantitative analysis by silicon Surface Assisted Laser Desorption Ionization Mass Spectrometry (Si-SALDI) is proposed. The approach is based on the new method for forming an active surface layer on a silicon substrate by exposing to laser radiation directly in the ion source of a mass spectrometer. The method can be used repeatedly on the same substrate, providing high reproducibility of its surface ionization properties and high ionization efficiency of organic compounds. Within the proposed approach, the methods of improvement of signal reproducibility are also considered, including continuous monitoring of the silicon surface ionization properties using a Knudsen effusion cell; scanning the surface of a silicon substrate with a laser beam; selecting the optimal value of laser fluence and using a reproducible sample introduction technique. It is demonstrated that this approach can be successfully applied to quantify clinically relevant concentrations of pharmaceutical drugs in extracts of blood.

APA, Harvard, Vancouver, ISO, and other styles

We offer discounts on all premium plans for authors whose works are included in thematic literature selections. Contact us to get a unique promo code!