Separation Point Research Articles

Эластичные хоны для полирования профилей зубьев термообработанных цилиндрических колес специального назначения

Introduction. The most important component of the technological process of manufacturing of gear wheels of critical products is the operation of teeth honing. Special requirements are imposed on the surface quality of special-purpose gears, where imported abrasive tools were used, the supply of which in modern economic conditions is impossible. Purpose of work: development of formulation, technological equipment and technology of manufacturing of elastic diamond gear hones instead of imported ones for teeth honing of gear wheels of special purpose. Research methods. Subject of research are samples of imported elastic gear hones and created domestic analogs. The mechanical properties, morphology and chemical composition of the abrasive (diamond) layer of the working surface of the teeth and the annular gear were determined. The content of chemical elements was controlled in separate points of the surface and by scanning over the area on a scanning electron microscope. The formulation and technology of production of annular gears were determined. Results and Discussion. Designs of molds for forming the abrasive layer and the hub of the gear hone are developed. The peculiarities of morphology of the material of the working layer and the annular gear of the elastic diamond gear hone are revealed. On the basis of the conducted research, domestic analogs of materials of constituent elements of the gear hone are determined. Two manufacturing technologies were considered: pressing and injection molding. Two molds were made to test the technology: a simplified model consisting of two teeth and a round mold. Several methods of manufacturing hone teeth were analyzed: manufacturing of an abrasive layer with different degree of pre-vulcanization, subsequent introduction of gear material and final vulcanization of the whole product. The mechanical properties of the materials of the working abrasive layer and the annular gear were determined. The chemical composition of the components of the hone and the boundary zone are studied. As a result of the conducted research, recommendations on the formulation of the abrasive layer and the annular gear, technology of manufacturing of the gear hone intended for final treatment of teeth of heat-treated spur wheels of special purpose are given.

Optimizing the Performance of different Airfoils at Various Angles of Attack through CFD Simulation

This study specifically examines the NACA 0012, NACA 4412, and NACA 2412 airfoil profiles using ANSYS FLUENT. By simulating the flow over these airfoils, we can comprehensively explore the impact of the angle of attack on lift and drag coefficients. Notably, the study reveals that the angle of attack directly influences lift force, with a critical angle beyond which the aircraft may stall. Thus, the research underscores the importance of maintaining an optimal angle of attack to avoid turbulence and optimize aircraft performance. The aerodynamics of airfoil shapes play a crucial role in the performance and safety of aircraft. Understanding airflow characteristics over airfoils, particularly concerning the critical angle of attack, is paramount in achieving optimal lift while avoiding stalling. This paper delves into the shift of the separation point on the upper surface of most airfoil shapes, emphasizing the shift from the trailing edge to the leading edge as the angle of attack increases. Stalling becomes a critical concern beyond the critical angle of attack, necessitating comprehensive research to enhance aircraft performance and safety.

Wake of two tandem square cylinders

The wake of two tandem square cylinders of identical width (d) is experimentally studied, with a view to understanding the dependence of the flow structure, aerodynamics forces and Strouhal number on the centre-to-centre spacing ratio L/d and Reynolds number Re, where L is the distance between the cylinder centres. Extensive measurements are carried out, using hot-wire, particle imaging velocimetry, laser-induced fluorescence flow visualization, surface-oil-flow visualization and surface pressure scanning techniques, for L/d = 1.0 ~ 5.0 and Re ≡ U∞d/ν = 2.8 × (103 ~ 104), where U∞ is the free-stream velocity and ν is the kinematic viscosity of the fluid. The flow is classified into four regimes, i.e. the extended-body (L/d ≤ 1.5–2.0), reattachment (1.5–2.0 < L/d < 2.7–3.2), co-shedding (L/d ≥ 3.0–3.4) and transition (2.7 ≤ L/d ≤ 3.3) where both reattachment and co-shedding phenomena may take place. The mean drag and fluctuating drag and lift exhibit distinct features for different flow regimes, which is fully consistent with the proposed flow classification. Comparison is made between this flow and the wake of two tandem circular cylinders, which provides valuable insight into the profound effect of the flow separation point and the presence of sharp corners on the flow development and classification.

ПОЛУЧЕНИЕ И ИССЛЕДОВАНИЕ ПОЛИМЕРНЫХ НАНОЧАСТИЦ

Currently, there is an increasing interest in the study of polymers in a nanodisperse state. Since polymer materials, as a rule, are heterogeneous systems with a sufficiently variable surface, a correct assessment of their chemical structure, physical properties and morphology is of great scientific and practical importance. Polymer nanoparticles (PNPs) have been successfully studied for a long time as nanoscale colloidal particles obtained from polymers. Research on HDPE is most intensively conducted in the field of nanomedicine, and the possible application of HDPE in the field of optoelectronics, photonics, gadgets, sensors, sensors, and new nanocomposites is of great interest to researchers. An independent task is to compare the properties of the HDPE with the properties of a bulk polymer, including the features of its supramolecular structure. Previous studies of the supramolecular structure of PDFs have suggested that charge transfer in the polymer during resistive switching occurs along the grids of intergranular boundaries. This paper presents the results of an experimental study of polymer nanoparticles on different substrates by atomic force microscopy. The main metrological parameters of polymer nanoparticles were determined, the results were compared with the available data on the supramolecular structure of polymer films. The possibility of using atomic force microscopy techniques to visualize polymer nanoparticles on dif- ferent substrates has been demonstrated. Using the AFM method with a conductive probe, it was found that in some cases, during the aggregation of nanoparticles on a metal substrate, charge transfer is realized along separate nanometer-sized sections. The conducting channels in the current image have the form of separate points with a height corresponding to the locally flowing current. It was found that the arrangement of the observed channels correlates well with the model of conductivity along the grain boundaries of the supramolecular structure of the polymer. The issues of interpretation of current images of polymer nanoparticles and the physical model of charge transfer in polymer nanostructures are discussed. The work was carried out with the support of the Mirror Labora- tories project of the National Research University Higher School of Economics and an intra-university grant from the Bashkir State Pedagogical University named after M. Akmulla.

Derived energy storage systems from Brayton cycle

Various energy storage systems (ESS) can be derived from the Brayton cycle, with the most representative being compressed air energy storage and pumped thermal electricity storage systems. Although some important studies on above ESS are reported, the topological structure behind those systems (i.e., derivations of the Brayton cycle) has not been studied, and the underlying thermodynamic ideas still need to be further explored. This paper first introduces the topological structure and the symmetry of ESS and their based Brayton cycles. The formation method of ESS based on paths and separation points is specified. It is found that round-trip path can form ESS directly. Then various ESS formed are compared. Finally, the synergistic effect and gain principle of thermal cycle and ESS are revealed. This work helps to reveal the intrinsic relationship between thermal cycles and ESS, understand the general laws behind ESS, and guide the combination of thermal cycles and ESS.

Absencja usprawiedliwiona? O kilku wątkach nieobecnych na wystawie Ukraina. Wzajemne spojrzenia

The aim of this article is to analyse the content of the exhibition Ukraine. A different angle on neighbourhood, presented in Cracow from September 2021 to January 2022, aimed at identifying exemplary themes that would have significantly enriched the educational and informational dimension of this exhibition, but for various reasons did not make it into it. Exemplary absences included the regional differentiation of Ukraine, the historical-independence status of Galicia, the demographic decline of Poland’s neighbours from across the river Bug, the importance of the Cossacks in the context of the 17th-century problems of the Polish-Lithuanian Commonwealth and the place of the “Vistula” action in the consciousness of Poles. A separate point is devoted to the absence of the motif of the Ukrainian Insurgent Army (UPA) and the Volhynian massacre from the exhibition, indicating – also through references to the declarations of people involved in the creation of the exhibition – why this motif was to be expected, and proposing a way in which this motif could have been creatively presented. Although the essence of the considerations is to indicate points which would enrich the school knowledge of the primarily Polish audience of the exhibition in relation to the framework of Ukrainian identity (identities) and the Polish-Ukrainian neighbourhood, in its essence the article does not prejudge the necessity to take up the aforementioned motifs, nor does it constitute a closed list of topoi which would be crucial in determining such a framework.

On the problem of determining the separation point of the laminar boundary layer by the example of the Howart–Tani flow

A new approach is proposed how to calculate the laminar boundary layer in slow flows. It is based on describing the velocity profile using a polynomial of indefinite degree and on introducing two additional coordinate-dependent parameters, one of which defines the separation of the boundary layer from a wall once this parameter reaches zero. The approach based on three integral relations and reducing the problem to the system of three ordinary differential equations was further developed. A numerical analysis performed for the Howart–Tani flow showed that the separation point of a laminar boundary layer is determined highly exactly using this approach. It was shown that introducing into consideration certain restrictions for the outer surface of a boundary layer allows one to find the problem solutions which would adequately define and fairly exactly determine the flow velocity distribution within this layer, and at any point up to the point of its separation. The proposed numerical-analytical calculation method based on three integral relations and two additional parameters and involving the definition of the flow velocity profile by a polynomial of indefinite degree can be extended to other slow flows past smooth two-dimensional surfaces.

A low-cost open-source imaging platform reveals spatiotemporal insight into leaf elongation and movement.

Plant organs move throughout the diurnal cycle, changing leaf and petiole positions to balance light capture, leaf temperature, and water loss under dynamic environmental conditions. Upward movement of the petiole, called hyponasty, is one of several traits of the shade avoidance syndrome (SAS). SAS traits are elicited upon perception of vegetation shade signals such as far-red light (FR) and improve light capture in dense vegetation. Monitoring plant movement at a high temporal resolution allows studying functionality and molecular regulation of hyponasty. However, high temporal resolution imaging solutions are often very expensive, making this unavailable to many researchers. Here, we present a modular and low-cost imaging set-up, based on small Raspberry Pi computers, that can track leaf movements and elongation growth with high temporal resolution. We also developed an open-source, semi-automated image analysis pipeline. Using this setup we followed responses to FR enrichment, light intensity, and their interactions. Tracking both elongation and the angle of the petiole, lamina, and entire leaf in Arabidopsis (Arabidopsis thaliana) revealed insight into R:FR sensitivities of leaf growth and movement dynamics and the interactions of R:FR with background light intensity. The detailed imaging options of this system allowed us to identify spatially separate bending points for petiole and lamina positioning of the leaf.

Electromagnetic response analysis of high temperature superconducting coated conductor tape with local debonding: Case of transporting current in an external magnetic field

In this article, the electromagnetic mechanical behavior of a long superconducting coating tape within a local internal cracking was analyzed when it transport current in a perpendicular magnetic field. Based on the superconducting critical state model, the distribution of current and magnetic field within the structure were calculated. Then, the control equation for the correlation between the flux pinning force inside the superconducting thin film and the shear force at the substrate interface was given using the plane strain method of linear elasticity theory. Combined with the Chebyshev polynomial solution, the distribution patterns of normal stress inside the superconducting thin film and shear stress at the substrate interface were finally obtained. The results indicate that, unlike the phenomenon observed under a single field excitation, the stress distribution in the structure exhibits an asymmetric distribution when both transport current and external magnetic field are applied. The normal stress inside the superconducting thin film is significantly larger on the right side of the structure than that on the left side, and the same distribution law is found in the shear stress at the substrate interface. The stress concentration occurs at the separation point between the film and the substrate. When the Young's modulus of the substrate material is high, the normal stress inside the superconducting film is low, while the shear stress at the interface is high, and vice versa. All these results provide theoretical guidance for the application of superconducting tapes.

Computational study of shock diffraction over convex edges

Numerical analysis of normal shock diffraction over various step corners is presented using an implicit second-order upwind least squares cell-based method. Slipstream angle calculation for various corner angles and Mach numbers, variation of pressure load and specific heat flux with incident shock Mach numbers and different step corners along the step wall, separation point movement, and reattachment of the streamlines are some of the key features of the present study. Detailed studies have been conducted about the perturbed region enclosed within the diffracted shock wave and the last running expansion wave with the increase in the incident shock Mach numbers. The finite Volume Method is used to solve the governing equations numerically for the simulation of the moving shock. Various flow characteristics such as secondary shock, contact surface, expansion wave, slipstream, vortex, etc are well captured. Apart from isopycnics; Mach contours, isobars, and isotherms are plotted as well. The reattachment region formed after the flow separation near the step edge is mentioned and corresponding lengths over various steps are presented. Separation point movement along the step wall is highlighted and a relevant increase in reattachment length is reported. The slipstream angles get increased with incident shock Mach numbers and with corner angles. However, the rate of increment of slipstream angle decreases gradually. The core of the vortices generated and the reattachment of the separated streamlines are indicated by the two downfalls in the heat flux values.

Mitigation of Shock-Induced Separation Using Square-Shaped Micro-Serrations—A Preliminary Study

Suppressing shock-induced flow separation has been a long-standing problem in the design of supersonic vehicles. To reduce the structural and design complexity of control devices, a passive control technique based on micro-serrations is proposed and its controlling effects are preliminarily investigated under test conditions in which the Mach number is 2.5 and the ramp creating an incident shock is 15 deg. Meanwhile, a vorticity-based criterion for assessing separation scales is developed to resolve the inapplicability of the zero skin friction criterion caused by wall unevenness. The simulations demonstrate that the height of the first stair significantly influences the separation length. Generally, the separation length is shorter at higher stairs, but when the height is greater than half of the thickness of the incoming boundary layer, the corresponding separation point moves upstream. A stair with a height of only 0.4 times the thickness of the boundary layer reduces the separation length by 2.69%. Further parametric analysis reveals that while the remaining serrations have limited effects on the flow separation, an optimization of their shape (depth and width) can create more favorable spanwise vortices and offer a modest improvement of the overall controlling performance. Compared to the plate case, a 9.13% reduction in the separation length can be achieved using a slightly serrated design in which the leading stair is 0.1 high and the subsequent serrations are 0.2 deep and 0.05 wide (nondimensionalized, with the thickness of the incoming boundary layer). Meanwhile, the micro-serration structure even brings less drag. Considering the minor modification to the structure, the proposed method has the potential for use in conjunction with other techniques to exert enhanced control on separations.

An Experimental Anatomic CBCT Study on the Correlations Between MB1 and MB2 of the Mesio-Vestibular Root of the Upper First Molars.

This experimental anatomic study aimed to investigate the correlations between the mesio-vestibular canal (MB1) and the second mesio-vestibular canal (MB2) of the mesio-vestibular root of the upper sexts using cone-beam computed tomography (CBCT) imaging. A total of 24 extracted maxillary first molars were collected and subjected to CBCT imaging. The presence, location, and morphology of MB1 and MB2 canals were evaluated using axial, coronal, and sagittal CBCT sections. The interrelation between MB1 and MB2 canals was assessed, including their separate canals, merging, and division points. Among the 24 maxillary first molars examined, 86.5% demonstrated the presence of an MB2 canal in addition to MB1. The MB2 and MB1 were confluent in 80% of the cases with a confluent height of 4,16mm. The mean distance between MB1 and MB2 canals was 1.85mm. This study provides detailed information on the anatomic correlations between MB1 and MB2 canals of the mesio-vestibular root in the upper sexts using CBCT imaging. The high prevalence of MB2 canals suggests their significance in endodontic procedures and emphasizes the importance of thorough exploration and identification during root canal treatment. The knowledge of the location and morphology of these canals can aid in successful endodontic therapy and enhance treatment outcomes.

Motility-induced phase separation of soft active Brownian particles

Motility-induced phase separation (MIPS) is the hallmark of non-equilibrium phase transition in active matter. Here, by means of Brownian dynamics simulations, we determine the phase behavior and the critical point for phase separation induced by motility of a two-dimensional system of soft active Brownian particles, whose interaction is modeled by the generalized purely repulsive Weeks–Chandler–Andersen potential. We refer to this model as soft active Brownian particles. We determine and analyze the influence of particle softness on the MIPS and show that the liquid–gas coexistence region is wider, the softer the interparticle interactions becomes. Moreover, the critical value of the self-propulsion velocity at which diluted and dense phases start to coexist also increases; as a consequence, the softer the particle interaction is, the bigger self-propulsion velocities are needed in order to observe a MIPS.

Shape optimization to enhance energy harvesting from vortex-induced vibration of a circular cylinder

In previous research on energy harvesting from vortex-induced vibrations (VIVs), the cross section of the structure commonly utilizes basic geometric shapes like circular, ellipse, square, and semicircle. Nevertheless, exploring optimized shapes for energy harvesting from VIV remains an understudied area. To address this gap, this paper employs adjoint-based unsteady shape optimization to increase the efficiency of energy harvesting from VIV of a circular cylinder at low Reynolds numbers. The goal of the optimization is to maximize the plunge-damping derivative of a single-degree-of-freedom transversely vibrating cylinder, which represents the rate of energy injected into the structure by the flow. To facilitate this process, an efficient method to evaluate the gradient of the objective function with respect to shape parameters is provided via the proposed unsteady discrete adjoint method. Results show that, through optimization, the low-pressure region behind the cylinder is significantly enlarged and the separation points move forward, resulting in the faster development of separation vortex and reduced stability of the fluid–structure coupling system. As a consequence, the intensity of VIV as well as the corresponding power generation efficiency is remarkably enhanced, accompanied by a notable expansion in the energy harvesting region.

Effect of Vortex Generations on Aerodynamic Performance of Rough Wind Turbine Airfoil

Experimental investigations were carried out in wind tunnel facilities to assess the effect of vortex generators on the aerodynamic characteristics of a NACA4418 airfoil, particularly in scenarios where surface roughness is precipitated by environmental elements like dust, sand, and insects. These elements are known to adversely affect the efficiency of power generation. These studies meticulously analyzed the airfoil’s pressure distribution and its aerodynamic coefficients, taking into account the roughness positioned at varying locations along the chord. Special attention was given to the roughness located at 0.2c due to its pronounced impact on aerodynamic efficiency. The research highlighted that vortex generators play a crucial role in mitigating boundary layer separation, thereby enhancing the lift-to-drag ratio, especially at higher angles of attack. Notably, the deployment of vortex generators on airfoils affected by surface roughness resulted in a more pronounced improvement, elevating the maximum lift coefficient by 11.6% compared to their effect on smooth airfoils. Furthermore, the study revealed that an increase in the Reynolds number significantly augments the benefits provided by vortex generators, improving the lift coefficient and delaying the stall angle for airfoils with surface roughness, while also shifting the boundary layer separation point towards the airfoil’s trailing edge.

Evaluation and Measurement Properties of a Patient-Reported Experience Measure for Home Dialysis.

No previously validated patient-reported experience measures exist for use among patients undergoing home dialysis. We tested the Home Dialysis Care Experience survey, a newly developed 26-item experience measure, among patients from 30 dialysis facilities in the United States. Using mail and telephone survey modalities, we approached 1372 patients treated with peritoneal dialysis or home hemodialysis for participation. Using the results from completed surveys, we evaluated item calibration by assessing item floor and ceiling effects. We tested three sets of composite scores and used factor analysis to assess model fit for each. We evaluated associations of composite scores with global ratings and separately with patient and dialysis facility characteristics. Finally, we measured test-retest reliability in patients who completed the survey at two separate time points. Overall, 495 eligible patients completed at least one survey (response rate 36%). Of these, 49 completed the survey in Spanish and 61 completed a second survey within 30 days. We did not detect significant floor or ceiling effects, except for one item that demonstrated >90% responses at the top response option. Analyses supported one 12-item composite scale with high internal consistency reliability: Quality of Home Dialysis Care and Operations (Cronbach alpha=0.85). This scale strongly correlated with overall staff rating ( r =0.73) and overall center rating ( r =0.70). Patient demographic and dialysis facility characteristics were not consistently associated with composite scale scores or overall staff or center ratings. Intraclass correlation coefficients in the test-retest population were 0.74 for the Quality scale, 0.88 for overall staff rating, and 0.90 for overall center rating. The Home Dialysis Care Experience survey is a 26-item measure that includes one composite scale and two global rating scores and is an informative tool to evaluate patient experience of care for home dialysis.

Assessing the influence of political factors on the dynamics of exports of Russian oil and petroleum products based on the use of a combination of PESTE and SWOT analysis methods

The presented work opens a series of articles devoted to assessing the influence of external factors on the dynamics of Russian exports. In this and subsequent related works, the authors proposed to consider the external environment, which had a significant impact on the dynamics of Russian exports and the relative diffi culty of revising its structure, the basis of commodity exports of which in value terms are hydrocarbon energy carriers. This work represents a conceptual basis for the use of instrumental methods for assessing the infl uence of external factors, in this case, political ones, on sectoral aspects of Russian exports and, in particular, on the export of hydrocarbon energy resources. In the second part of this article, it is proposed to use a set of environmental analysis methods, the basis of which is the real estate market research presented in this article, which tests the possibility of combining PESTEL analysis methods with the results of SWOT analysis, which will allow, based on the use of point estimates, to describe the cumulative impact external factors on the dynamics of the main sectors of Russian exports. The novelty of the presented approach lies in the absence of a methodology for a comprehensive assessment of the infl uence of external factors on the dynamics of Russian exports using the selected tools, which seems useful and timely in light of the emergence of new political and economic factors that have a signifi cant impact on Russian foreign trade in general and on the dynamics of the carbohydrate energy market, in particular. The parameters for assessing external factors proposed by the authors were chosen by the author independently, taking into account the modern realities of the development of the world economy, which can also represent a separate point of novelty of the research conducted. The merit of the authors of this work lies in the fact that this work proposed their own vision of a comprehensive assessment of the influence of external factors on the dynamics of the country’s exports.

Vortex-induced vibration response of the cylinder inspired by Terebridae

Vortex-induced Vibration (VIV) responses of the three-dimensional cylinders inspired by Terebridae are numerically investigated at the Reynolds number ranges of 0.8 × 104 ≤ Re ≤ 8.0 × 104. By Terebridae, we mean that inspired by its slender structural shape, and whether it can be used to improve the VIV suppression performance of the traditional structures. Three different cylinders are considered, including the smooth cylinder (T0), the Terebridae-inspired cylinder with four-start helical ribs (T1), and the Terebridae-inspired cylinder with six-start helical ribs (T2). The VIV responses of T1 and T2 is effectively suppressed, and T1 shows better suppression performance. The results show that the maximum cross-flow amplitude ratio of T1 is reduced by 70.7 % compared with that of T0, and the maximum in-line amplitude ratio of T1 is reduced by 85.7 % compared with that of T0. Compared with the maximum mean-drag coefficient of T0, it is reduced by 44.0 % for T1. In the high Reynolds number ranges, the galloping phenomenon does not occur for T1 and T2, and the relationship between the mean-drag coefficients of T0, T1 and T2 is Cd‾ (T2) > Cd‾ (T1) > Cd‾ (T0). It is found that the changes of Q-criterion vortex structure and wake flow play an important role in the VIV response. The Q-criterion vortex of T0 is a slightly curved linear shaped, and the Q-criterion vortices of T1 and T2 are break up. The three-dimensional geometry of the Terebridae-inspired cylinder undermines the correlation of vortex-shedding along the span. The wake flow of T0 can develop normally. The wake flow of T1 quickly rolls up after passing through the top separation point, and the reattachment of the wake vortex destroys the development of the original wake flow. The continuous development of the boundary layer is destroyed by multiple Terebridae-inspired ribs of T2, resulting in the broken wake. Through the comparison of vortex force, it can be seen that the vortex force of T1 is significantly weakened compared to that of T0, and this also indicates that the three-dimensional geometry of T1 can effectively reduce the intensity of the wake flow.

An Investigation on the Steady and Unsteady Wake Flow Regimes and Aerodynamic Characteristics of the Trapezoidal Cylinders Using Immersed Boundary-Lattice Boltzmann Method

Abstract In this study, the flow physics of the forward-facing (FF) and backward-facing (BF) trapezoidal cylinders (TC) subjected to two-dimensional, incompressible, and laminar flow is investigated using an in-house developed flexible forcing immersed boundary-lattice Boltzmann solver. The Reynolds number (Re) is defined based on the cylinder's characteristic length D. For the steady and unsteady flow regimes, Re is varied in the ranges of 10–40 and 75–125, respectively. The TCs shape is varied by modifying its nondimensional axial H/D and transverse Y/D length scales, between 0.5 to 2 and 0 to 1, respectively. Here, TCs horizontal central axis is always aligned along the incoming flow direction. It is observed that the flow separation points on the FF-TC and BF-TC are strongly influenced by the geometric (H/D and Y/D) and flow parameters (Re). Based on the boundary layer separation point, we have categorized the wake flow regimes behind the FF-TC and BF-TC into four types. In addition, the effect of the geometric and flow parameters on the drag coefficient (Cd), vortex shedding frequency, and steady and unsteady wake characteristics are thoroughly investigated here. Furthermore, by performing nonlinear regression analysis, we have proposed a set of correlation equations for the Cd and Strouhal number (St), using which the aerodynamic characteristics of differently shaped TC can be derived in the considered Re range without performing rigorous numerical simulations or experiments.

Separation of the two-dimensional steady MHD boundary layer

In this paper, we study the separation of the two-dimensional steady MHD boundary layer. When the outer flow satisfies appropriate conditions, we find that there exists a separation point on the boundary. Moreover, the singularity of the solution and the separation point occur simultaneously. Our result shows that the magnetic field has a certain positive effect on preventing the occurrence of separation. This gives a rigorous mathematical verification to previous study (Buckmaster, 1969) on the MHD boundary layer separation.