Narrow Layer Research Articles

Internal boundary layer in a singularly perturbed problem of fractional derivative

This paper is devoted to the study of internal boundary layer. Such motions are often associated with effect of boundary layer, i.e. low flow viscosity affects only in a narrow parietal layer of a streamlined body, and outside this zone the flow is as if there is no viscosity - the so-called ideal flow. Number of exponentials in the boundary layer is determined by the number of non-zero points of the limit operator spectrum. In the paper we consider the case when spectrum of the limit operator vanishes at the point To study the problem the Lomov regularization method is used. The original problem is regularized and the main term of asymptotics of the problem solution is constructed as the low viscosity tends to zero. Numerical results of solutions are obtained for different values of low viscosity.

Transient flow-driven distortion of a nematic liquid crystal in channel flow with dissipative weak planar anchoring

Motivated by the one-drop-filling (ODF) method for the industrial manufacturing of liquid crystal displays, we analyze the pressure-driven flow of a nematic in a channel with dissipative weak planar anchoring at the boundaries of the channel. We obtain quasisteady asymptotic solutions for the director angle and the velocity in the limit of small Leslie angle, in which case the key parameters are the Ericksen number and the anchoring strength parameter. In the limit of large Ericksen number, the solution for the director angle has narrow reorientational boundary layers and a narrow reorientational internal layer separated by two outer regions in which the director is aligned at the positive Leslie angle in the lower half of the channel and the negative Leslie angle in the upper half of the channel. On the other hand, in the limit of small Ericksen number, the solution for the director angle is dominated by splay elastic effects with viscous effects appearing at first order. As the Ericksen number varies, there is a continuous transition between these asymptotic behaviors, and in fact the two asymptotic solutions capture the behavior rather well for all values of the Ericksen number. The steady-state value of the director angle at the boundaries and the timescale of the evolution toward this steady-state value in the asymptotic limits of large and small Ericksen number are determined. In particular, using estimated parameter values for the ODF method, it is found that the boundary director rotation timescale is substantially shorter than the timescale of the ODF method, suggesting that there is sufficient time for significant transient flow-driven distortion of the nematic molecules at the substrates from their required orientation to occur.

Endoscopic submucosal dissection for superficial pharyngeal carcinoma using transnasal endoscope.

Endoscopic submucosal dissection for superficial pharyngeal carcinoma using transnasal endoscope.

Dynamics of a gas bubble during fluid decompression at a constant rate

The bubble growth process in a magmatic melt during decompression at a constant rate under various conditions is numerically studied. The effect of the formation of a narrow boundary layer with a high viscosity around the bubble, which significantly changes the dynamics of the bubble, is taken into account. It is shown that, at a low initial pressure in the medium, in the bubble high pressure is maintained throughout the calculation time. In the case of long decompression time, the final bubble radius is bigger than in the case with small decompression time. This effect can be associated with different dynamics of the medium viscosity in a narrow boundary layer around the bubble.

Laser breakdown model in the absorption mode behind a light supported detonation wave

The light supported detonation wave (LSDW) propagation with the laser radiation absorption in a narrow layer behind wave’s front is considered in the paper alternatively to the volumetric energy source model usually used for the energy deposition simulation [1-7]. The next laser plasma flow features were revealed in [8] by methods of numerical simulation:1) laser plasma forms high-speed jet flow behind the LSDW front along the light beam propagation direction,2) jet parameters are close to an isentropic flow mode at significant distances from the LSDW front, therefor can be determined with a good approximation using the unsteady pressure solution for the point explosion model with a kinematic x-t transformation. These features allow one to determine the plasma momentum value (in the direction of a laser beam) of the optical breakdown as an additional factor of effect on gas flow, first indicated in [9]. For an argon flow, we used a comparative analysis of the results of numerical simulations obtained both accounting the absorbed energy only and the breakdown plasma momentum additionally acquired at the absorption of radiation behind the LSDW front.The experimental results of an optical discharge plasma in a subsonic argon flow are also presented. Pulse-periodic radiation with a frequency of 40 kHz and average power of 1.6 kW (peak power more than 30 kW) was generated by CO2-laser created in ITAM SB RAS. In order to obtain the LSDW mode by increasing a length of the breakdown plasma a focusing system f / d ≈ 9 was applied. A high-speed video camera with the exposure time of 1.0 μs, and the shooting speed of 200,000 frames / sec was used to record process. It has been established that the plasma dynamics has two successive stages: from the initial high-speed (of the order of 1 μs) propagation of the optical discharge to the subsequent lower-velocity gas-dynamic stage.

Broadband and Compact TE-Pass Polarizer Based on Hybrid Plasmonic Grating on LNOI Platform

A broadband and compact TE-pass/TM-stop polarizer is presented based on hybrid plasmonic grating (HPG) on an x-cut Lithium-Niobate-on-isolator (LNOI) platform. By comprehensively analyzing the effects of metal width on mode effective index, mode similarity, and mode conversion, we demonstrate the structure with a narrow metal layer. The simulation results indicate that the polarizer with a compact length of 9 μm achieves an extinction ratio over 20 dB within the wavelength range from 1470 nm to 1700 nm. The insertion loss is below 2.3 dB in C-band. Furthermore, the polarizer exhibits large fabrication tolerance to current fabrication technology.

Numerical simulation of highly efficient lead-free perovskite layers for the application of all-perovskite multi-junction solar cell

The numerical simulation of lead-free perovskite cesium tin germanium halide (CsSnGeI3) and methyl ammonium germanium halide (CH3NH3GeI3) solar cells have been performed. It has been elucidated from the numerical simulation that as compare to the CH3NH3GeI3 based structure, CsSnGeI3 based structure depicts better photovoltaic performance under constant illumination condition. To enhance the device photovoltaic performance, the effects of different hole transport layer (HTL), defect density of the absorber layer, metal work function and the effect of temperature has been studied. In the present investigation, the effect of hole transport layers are analyzed by correlating the built-in voltage (Vbi) with the open circuit voltage (VOC). It has been obtained from the simulation results, that Vbi has a significant impact on VOC as the higher Vbi corresponds to the highest VOC. Moreover, the effect of defect density and metal work function is also studied. It is obtained from the simulation results, that the optimum defect density is found to be 1 × 1014 cm−3 and metal work function should be greater than or equal to 5 eV for both lead-free CH3NH3GeI3 and CsSnGeI3 based perovskite layer. Furthermore, to suggest possible lead-free alternatives for all-perovskite two-terminal multi-junction solar cell, an attempt is made to match the current of bottom perovskite layer i.e., CsSnGeI3 based perovskite layer to the top perovskite layer i.e., CH3NH3GeI3 based perovskite layer. To attain the current matching for bottom perovskite layer (CsSnGeI3 based perovskite layer) to top perovskite layer (CH3NH3GeI3 based perovskite layer), the bottom perovskite sub cell is fed with the filtered spectrum which is obtained by transmission from top electrode. After, the realization of current matching the short circuit current density of CsSnGeI3 based perovskite layer drops from 25.75 mA/cm2 to 15.32 mA/cm2 which is similar to CH3NH3GeI3 based perovskite absorber layer. In addition, it has been obtained from literature, that both the perovskite layers can be fabricated using solution–processing low-temperature technology. Hence, the numerical study suggests the possible alternatives for wide and narrow bandgap perovskite layers for the application to obtain highly efficient lead-free all-perovskite multi-junction solar cells.

Study of spontaneous mobility and imbibition of a liquid droplet in contact with fibrous porous media considering wettability effects

In this paper, droplet mobility and penetration into a fibrous porous medium are studied considering different physical and geometrical properties for the fibers. An in-depth insight into the droplet imbibition into the fibrous medium is beneficial for improving membrane products in different applications. Herein, a multiphase lattice Boltzmann method is employed as an efficient numerical algorithm for predicting the multiphase flow characteristics and the interfacial dynamics affected by the interaction between the droplet and fibrous substrates considered. This computational technique is validated by comparison of the present results obtained for different benchmark two-phase flow problems with those reported in the literature, which shows good agreement and confirms its accuracy and efficiency. Droplet spreading and penetration into the fibrous porous geometries are then studied considering various porous topologies, intrinsic contact angles, and fiber sizes. This study shows that the intrinsic contact angle has a great influence on the capillary pressure and, consequently, on the droplet imbibition into the porous medium. The droplet easily penetrates the porous substrate by decreasing the intrinsic contact angle of the fibers, and vice versa. It is also concluded that by coating the fibrous porous medium with a narrow layer of thin fibers, the surface resistance to liquid penetration significantly increases. The present results illustrate that the droplet size impacts the directional wicking ability of the fibrous porous structure used in this study. This property should be considered to produce appropriate two-layer membranes for different applications.

Miniature Dual-Band Substrate Integrated Waveguide Slotted Antenna Array for Millimeter-Wave 5G Applications

A compact substrate integrated waveguide (SIW) antenna array that operates at 28 GHz and 38 GHz is proposed for fifth generation (5G) applications. The proposed array consists of four SIW cavities fabricated on one single layer of substrate. Each cavity implements a rhombic slot and a triangular-split-ring slot, resonating on TE101 and TE102 modes at 28 GHz and 38 GHz, respectively. In comparison with dual-band SIW antennas in the literature, the proposed configuration depicts a miniature footprint (28.7 × 30.8 mm2) without stacking substrates. To excite the four cavities with equal power, a broadband power divider that supports the propagation of TE10 mode is designed. Accordingly, the impedance bandwidths are 26.6–28.3 GHz and 36.8–38.9 GHz. The measured realized peak gain over the lower and higher bands is 9.3–10.9 dBi and 8.7–12.1 dBi, respectively. The measured half-power beam widths (HPBWs) at 28 GHz and 38 GHz are 20.7° and 15.0°, respectively. Considering these characteristics, including dual bands, high gain, narrow beam widths, miniaturization, and single layer, the proposed antenna array is a suitable candidate for millimeter-wave 5G communication systems with the flexibility in switching operating frequency bands against channel quality variations.

The technical optimization of Na-K lidar and to measure mesospheric Na and K over Brazil

This paper reports that the sodium–potassium (Na–K) lidar was completed in November 2016 at São José Dos Compose, Brazil (23°S, 45°W), by the joint effort of the National Space Science Center, Chinese Academy of Sciences (NSSC) and Instituto Nacional de Pesquisas Espaciais (INPE). This system realized the Na and K metal layers simultaneously observe in Brazil, and this is the first instance of K layer detection in South America. Some of the key parameters and technologies have been optimized based on the Na and K layer dual-wave lidar in Beijing Yanqing station, such as improve technical parameters for receiving telescope, the narrow linewidth, efficient laser frequency doubling, the wavelength automatic locking techniques. By adopting these technologies, the output were 589 nm and 770 nm lasers, with high emission powers of 75 mJ and 83 mJ, respectively, and backscattered signals of Na and K layers with high signal quality were obtained. Observation data showed that the original echo photon count of the Na layer was approximately 42,486 (time resolution: 200 s, spatial resolution: 96 m) and the number of noise photons was 286 in a single data acquisition. The signal-to-noise ratio was up to 205:1. At the same spatiotemporal resolution, the original echo photon count of the K layer was approximately 1633, the noise photons were 38, and the signal-to-noise ratio was up to 40:1. The initial photocounts received has demonstrated that the Brazil K lidar has produced high quality signal with signal-to-noise level required by intended science studies. Moreover, the simultaneous phenomena of sporadic Na (Nas) and sporadic K (Ks), and the highly concentrated layers of atomic K have been observed, the K density in these narrow layers exceeds 1019 cm−3.

Assessment of the Potential for Genomic Selection To Improve Husk Traits in Maize.

Husk has multiple functions such as protecting ears from diseases, infection, and dehydration during development. Additionally, husks comprised of fewer, shorter, thinner, and narrower layers allow faster moisture evaporation of kernels prior to harvest. Intensive studies have been conducted to identify appropriate husk architecture by understanding the genetic basis of related traits, including husk length, husk layer number, husk thickness, and husk width. However, marker-assisted selection is inefficient because the identified quantitative trait loci and associated genetic loci could only explain a small proportion of total phenotypic variation. Genomic selection (GS) has been used successfully on many species including maize on other traits. Thus, the potential of using GS for husk traits to directly identify superior inbred lines, without knowing the specific underlying genetic loci, is well worth exploring. In this study, we compared four GS models on a maize association population with 498 inbred lines belonging to four subpopulations, including 27 lines in stiff stalk, 67 lines in non-stiff stalk, 193 lines in tropical-subtropical, and 211 lines in mixture subpopulations. Genomic Best Linear Unbiased Prediction with principal components as cofactor, performed the best and was selected to examine the impact of interaction between sampling proportions and subpopulations. We found that predictions on inbred lines in a subpopulation were benefited from excluding individuals from other subpopulations for training if the training population within the subpopulation was large enough. Husk thickness exhibited the highest prediction accuracy among all husk traits. These results gave strategic insight to improve husk architecture.

Buckling-induced interaction between circular inclusions in an infinite thin plate.

Design of slender artificial materials and morphogenesis of thin biological tissues typically involve stimulation of isolated regions (inclusions) in the growing body. These inclusions apply internal stresses on their surrounding areas that are ultimately relaxed by out-of-plane deformation (buckling). We utilize the Föppl-von Kármán model to analyze the interaction between two circular inclusions in an infinite plate that their centers are separated a distance of 2ℓ. In particular, we investigate a region in phase space where buckling occurs at a narrow transition layer of length ℓ_{D} around the radius of the inclusion, R (ℓ_{D}≪R). We show that the latter length scale defines two regions within the system, the close separation region, ℓ-R∼ℓ_{D}, where the transition layers of the two inclusions approximately coalesce, and the far separation region, ℓ-R≫ℓ_{D}. While the interaction energy decays exponentially in the latter region, E_{int}∝e^{-(ℓ-R)/ℓ_{D}}, it presents nonmonotonic behavior in the former region. While this exponential decay is predicted by our analytical analysis and agrees with the numerical observations, the close separation region is treated only numerically. In particular, we utilize the numerical investigation to explore two different scenarios within the final configuration: The first where the two inclusions buckle in the same direction (up-up solution) and the second where the two inclusions buckle in opposite directions (up-down solution). We show that the up-down solution is always energetically favorable over the up-up solution. In addition, we point to a curious symmetry breaking within the up-down scenario; we show that this solution becomes asymmetric in the close separation region.

Resistance Spot Welding of Aluminum to Aluminum Clad Steel Sheet: Experimental and Theoretical Analysis

Abstract In order to increase the cross-tension to the tensile-shear peak load ratio, aluminum clad steel sheet produced by roll bonding, was used for resistance spot welding to Al-Mg aluminum alloy. The results showed that the weld nugget formed between the aluminum clad layer and Al-Mg base sheet was the load bearing area. Presence of the metallurgical bond between the steel sheet and the aluminum clad layer led to the formation of the Al/Fe intermetallic compound with the thickness less than ∼3.5 μm. This narrow intermetallic layer did not play a detrimental role in the weld strength and therefore, the cross-tension to the tensile-shear peak load ratio reached up to ∼0.5 at the maximum tensile-shear peak load which was an advance compared to the results in the literature. Theoretical analysis was relatively successful for prediction of the critical nugget diameter and the tensile-shear peak failure loads. While the tensile-shear peak load increased with the welding current up to ∼4 kN, the peak load in the cross-tension test remained relatively constant at ∼2 kN.

Study on dry band discharge of wet polluted surface with float potential metal by electro-thermal synchronous observation

Severe partial arc occurred near metal bolts on the composite tower under wet pollution conditions, which had a close relationship with the temperature distribution on the surface. Artificial contamination tests on the small-sized composite plate were carried out. Temperature distribution and leakage current variation of the specimen were acquired synchronously. The effects of float potential metal, pollution layer width, metal shape, and moisture on temperature distribution, dry band formation, and discharge phenomenon were studied by a series of comparative experiments. The results show that the float potential metal makes the temperature distribution of the pollution layer more concentrated. The dry band begins to form in the region with the highest temperature. The formation of a dry band on the narrow pollution layer is faster. The temperature of the dry band endpoint is very high and increases with the extension of the dry band. The formation and passing through of dry band blocks leakage current leading to the occurrence of partial arc. Temperature distribution near metals with sharp shapes is more inhomogeneous. In a humid environment, dry band arc can exist stably and develop to float potential metal. Owing to the rewetting of the dry band, dry band formation, and dry band arc will occur repeatedly.

Mechanical Behavior Characterization of a Stainless Steel Dissimilar Metal Weld Interface : In-situ Micro-Tensile Testing on Carburized Martensite and Austenite

Stainless Steel Dissimilar Metal Welds (SS DMW) between low-alloy steel 18MND5 and austenitic 316L stainless steel are critical junctions in the currently operating reactors because of their heterogeneous microstructure and mechanical properties. The presence of a narrow hard layer of carburized martensite and austenite in the ferritic-austenitic interface creates an important hardness gradient which affects the crack behavior of the SS DMW. In order to evaluate the plastic properties of this hard layer, a micro tensile testing method was developed. Tensile specimens of 15 x 80 x 6 μm were extracted from the martensitic and carburized austenitic layers by focused ion beam (FIB) micro-processing and tested using an in-situ tensile testing device. A platinum FIB deposition was used to measure local strain in the specimen during the test through digital image correlation (DIC). Isotropic elasto-plastic constitutive laws for the martensite and carburized austenite were obtained from the true strain-stress curves calculated from the micro-tensile tests. It was found that the corresponding plastic properties were in a good agreement with nanoindentation measurements and with values obtained from macroscopic tensile tests on crossweld specimens machined perpendicularly to the ferritic-austenitic interface and characterized using laser beam local diameter measurements. In-situ tensile testing is a promising technique for plastic behavior characterization of small scale materials and local hard layers in dissimilar metal welds.

Quasi-Heteroface Perovskite Solar Cells.

Perovskite solar cells (PSCs) have attracted unprecedented attention due to their rapidly rising photoelectric conversion efficiency (PCE). In order to further improve the PCE of PSCs, new possible optimization path needs to be found. Here, quasi-heteroface PSCs (QHF-PSCs) is designed by a double-layer perovskite film. Such brand new PSCs have good carrier separation capabilities, effectively suppress the nonradiative recombination of the PSCs, and thus greatly improve the open-circuit voltage and PCE. The root cause of the performance improvement is the benefit from the additional built-in electric field, which is confirmed by measuring the external quantum efficiency under applied electric field and Kelvin probe force microscope. Meanwhile, an intermediate band gap perovskite layer can be obtained simply by combining a wide band gap perovskite layer with a narrow band gap perovskite layer. Tunability of the band gap is obtained by varying the film thicknesses of the narrow and wide band gap layers. This phenomenon is quite different from traditional inorganic solar cells, whose band gap is determined only by the narrowest band gap layer. It is believed that these QHF-PSCs will be an effective strategy to further enhance PCE in PSCs and provide basis to further understand and develop the perovskite materials platform.

Nanophase iron carbides in fine‐grained rims in CM2 carbonaceous chondrites: Formation of organic material by Fischer–Tropsch catalysis in the solar nebula

AbstractTransmission electron microscope studies of fine‐grained rims in three CM2 carbonaceous chondrites, Y‐791198, Murchison, and ALH 81002, have revealed the presence of widespread nanoparticles with a distinctive core–shell structure, invariably associated with carbonaceous material. These nanoparticles vary in size from ~20 nm up to 50 nm in diameter and consist of a core of Fe,Ni carbide surrounded by a continuous layer of polycrystalline magnetite. These magnetite shells are 5–7 nm in thickness irrespective of the diameter of the core Fe,Ni carbide grains. A narrow layer of amorphous carbon a few nanometers in thickness is present separating the carbide core from the magnetite shell in all the nanoparticles observed. The Fe,Ni carbide phases that constitute the core are consistent with both haxonite and cohenite, based on electron diffraction data, energy dispersive X‐ray analysis, and electron energy loss spectroscopy. Z‐contrast scanning transmission electron microscopy shows that these core–shell magnetite‐carbide nanoparticles can occur as individual isolated grains, but more commonly occur in clusters of multiple particles. In addition, energy‐filtered transmission electron microscopy (EFTEM) images show that in all cases, the nanoparticles are embedded within regions of carbonaceous material or are coated with carbonaceous material. The observed nanostructures of the carbides and their association with carbonaceous material can be interpreted as being indicative of Fischer–Tropsch‐type (FTT) reactions catalyzed by nanophase Fe,Ni metal grains that were carburized during the catalysis reaction. The most likely environment for these FTT reactions appears to be the solar nebula consistent with the high thermal stability of haxonite and cohenite, compared with other carbides and the evidence of localized catalytic graphitization of the carbonaceous material. However, the possibility that such reactions occurred within the CM parent body cannot be excluded, although this scenario seems unlikely, because the kinetics of the reaction would be extremely slow at the temperatures inferred for CM asteroidal parent bodies. In addition, carbides are unlikely to be stable under the oxidizing conditions of alteration experienced by CM chondrites. Instead, it is most probable that the magnetite rims on all the carbide particles are the product of parent body oxidation of Fe,Ni carbides, but this oxidation was incomplete, because of the buildup of an impermeable layer of amorphous carbon at the interface between the magnetite and the carbide phase that arrested the reaction before it went to completion. These observations suggest that although FTT catalysis reactions may not have been the major mechanism of organic material formation within the solar nebula, they nevertheless contributed to the inventory of complex insoluble organic matter that is present in carbonaceous chondrites.

A NOVEL BIDIRECTIONAL QUASI Z SOURCE INVERTER FOR ELECTRIC VEHICLE APPLICATIONS

The fixed frequency sliding mode control method for this project is a single-phase grid-connected LCL filtered quasi-Z-source inverter (QZSI). In the proposed method, smoothing is applied to the function of the sliding surface, thereby eliminating jitter and achieving a fixed switching frequency. The smoothing process is performed by passing the function of the sliding surface through a narrow boundary layer. To mitigate steady state errors due to smoothing, a PIC controller is used to handle current grid errors and determine the capacitor voltage reference required in the SMC. The DC output of the battery is converted to AC using a DC AC inverter after withstanding or increasing demand. The inverter function must change the DC input voltage to the desired voltage and voltage as the asymmetric AC output voltage achieved through the control branches of Quasi by using simple boost control technology.

Influence of white etching layer on rolling contact behavior at wheel-rail interface

The existence of narrow and brittle white etching layers (WELs) on the rail surface is often linked with the formation of rail defects such as squats and studs, which play the key roles in rail surface degradation and tribological performance. In the present study, a systematic investigation on stress/strain distribution and fatigue life of the WEL during wheel-rail rolling contact was conducted based on a numerical model considering the realistic wheel geometry. This is the first study considering the influence of rail materials, loading pressure, frictional condition, WEL geometry (a/b), and slip ratio (Sr) in the practical service conditions at the same time. The results revealed much higher residual stress in WEL than in rail matrix. Stress changes along the rail depth matched with the previously reported microstructure evolutions. The current work revealed that the maximum difference in contact stress between the wheel passages of rail matrix and the WEL region (noted as stress variation) rises with the increase of loading pressure, the value of a/b, and Sr; but drops with the friction coefficient (μ). In addition, a critical length-depth ratio of 5 for a/b has been found. The fatigue parameter, FP, of the WEL decreased quickly with the length-depth ratio when it was less than 5 and then increased slightly when it was larger than 5. This study also revealed that the fatigue life of the WEL was reduced for high strength head hardened (HH) rail compared with standard carbon (SC) rail.

Collisional effects on resonant particles in quasilinear theory

A careful examination of the effects of collisions on resonant wave–particle interactions leads to an alternate interpretation and deeper understanding of the quasilinear operator originally formulated by Kennel & Engelmann (Phys. Fluids, vol. 9, 1966, pp. 2377–2388) for collisionless, magnetized plasmas, and widely used to model radio frequency heating and current drive. The resonant and nearly resonant particles are particularly sensitive to collisions that scatter them out of and into resonance, as for Landau damping as shown by Johnston (Phys. Fluids, vol. 14, 1971, pp. 2719–2726) and Auerbach (Phys. Fluids, vol. 20, 1977, pp. 1836–1844). As a result, the resonant particle–wave interactions occur in the centre of a narrow collisional boundary when the collision frequency $\unicode[STIX]{x1D708}$ is very small compared to the wave frequency $\unicode[STIX]{x1D714}$ . The diffusive nature of the pitch angle scattering combined with the wave–particle resonance condition enhances the collision frequency by $(\unicode[STIX]{x1D714}/\unicode[STIX]{x1D708})^{2/3}\gg 1$ , resulting in an effective resonant particle collisional interaction time of $\unicode[STIX]{x1D70F}_{\text{int}}\sim (\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714})^{2/3}/\unicode[STIX]{x1D708}\ll 1/\unicode[STIX]{x1D708}$ . A collisional boundary layer analysis generalizes the standard quasilinear operator to a form that is fully consistent with Kennel–Englemann, but allows replacing the delta function appearing in the diffusivity with a simple integral (having the appropriate delta function limit) retaining the new physics associated with the narrow boundary layer, while preserving the entropy production principle. The limitations of the collisional boundary layer treatment are also estimated, and indicate that substantial departures from Maxwellian are not permitted.