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Journal articles on the topic 'Euler's pump equation'

Author: Grafiati

Published: 28 June 2021

Last updated: 29 July 2025

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1

Meddah, Medjahed Driss, Medjadji Nassira, Boukhadia Karima, Abdelmajid Kaddour, Yahiaoui Mostapha, and Ariss Abdelkader. "A visual basic implementation of the design and analysis of centrifugal pump rotors using Euler's theory and similarity laws." STUDIES IN ENGINEERING AND EXACT SCIENCES 5, no. 3 (2024): e12476. https://doi.org/10.54021/seesv5n3-032.

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The continuous evolution of hydraulic machinery demands innovative tools for improving the design and performance of centrifugal pumps. This study presents the development and validation of a Visual Basic application for designing centrifugal pump rotors, leveraging Euler's theory and similarity laws. The mathematical modeling includes essential parameters such as pump head, flow rate, and efficiency, which are computed based on theoretical equations and user-provided inputs. The software's ability to automate calculations provides engineers with a quick and reliable solution for optimizing pu

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2

Liu, Yong, Dezhong Wang, Hongjuan Ran, Rui Xu, Yu Song, and Bo Gong. "RANS CFD Analysis of Hump Formation Mechanism in Double-Suction Centrifugal Pump under Part Load Condition." Energies 14, no. 20 (2021): 6815. http://dx.doi.org/10.3390/en14206815.

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The RANS (Reynolds-averaged Navier–Stokes equations) with CFD (Computational Fluid Dynamics) simulation method is used to analyze the head hump formation mechanism in the double-suction centrifugal pump under a part load condition. The purpose is to establish a clear connection between the head hump and the microcosmic flow field structure, and reveal the influence mechanism between them. It is found that the diffuser stall causes a change in the impeller capacity for work, and this is the most critical reason for hump formation. The change in the hydraulic loss of volute is also a reason for

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3

Sheth, K. K., G. L. Morrison, and W. W. Peng. "Slip Factors of Centrifugal Slurry Pumps." Journal of Fluids Engineering 109, no. 3 (1987): 313–18. http://dx.doi.org/10.1115/1.3242666.

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Experiments have been carried out in order to determine the effects on slip factor due to the various parameters affecting the performance characteristics of a centrifugal slurry pump. The experiments were conducted with water, sand slurry, and a glass bead slurry at three different pump speeds. Measurements of power, flow rate, head developed by the pump and the density of the slurry were made in order to obtain the characteristic curves of the pump. Using Euler’s equation, equations were derived for calculating the slip and friction factors of the flow. The deduced slip factors for centrifug

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4

Sun, Datong, and Mauricio G. Prado. "Single-Phase Model for Electric Submersible Pump (ESP) Head Performance." SPE Journal 11, no. 01 (2006): 80–88. http://dx.doi.org/10.2118/80925-pa.

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Summary This paper presents a new incompressible single-phase model for ESP head performance. Sachdeva (1988, 1994) and Cooper and Bosch (1966) developed models for ESP channels and for inducers, respectively. The model presented in this paper is based on 1D approximation along an ESP channel. The new derived pressure ordinary differential equation (ODE) for frictionless incompressible flow (Bird 1960) is consistent with the pump Euler equation. New models for pump frictional and shock losses have been proposed. Finally, a comparison between the predicted pump performance and the pump performa

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Yedidiah, S. "An overview of methods for calculating the head of a rotodynamic impeller and their practical significance." Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering 217, no. 3 (2003): 221–32. http://dx.doi.org/10.1243/095440803322328872.

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This paper explains why Euler's equation and the airfoil theory, while analytically correct, sometimes produce disappointing results. It also emphasizes the merits of a recently developed approach and demonstrates its usefulness in solving problems encountered in practice. The subject matter relates, directly, only to rotodynamic pumps. However, with proper modifications, it can be easily expanded to other fluids machines.

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6

Lee, Y. T., C. Hah, and J. Loellbach. "Flow Analyses in a Single-Stage Propulsion Pump." Journal of Turbomachinery 118, no. 2 (1996): 240–48. http://dx.doi.org/10.1115/1.2836631.

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Steady-state analyses of the incompressible flow past a single-stage stator/rotor propulsion pump are presented and compared to experimental data. The purpose of the current study is to validate a numerical method for the design application of a typical propulsion pump and for the acoustic analysis based on predicted flowfields. A steady multiple-blade-row approach is used to calculate the flowfields of the stator and the rotor. The numerical method is based on a fully conservative control-volume technique. The Reynolds-averaged Navier–Stokes equations are solved along with the standard two-eq

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7

Churakov, Evgeny O., Vladimir N. Makarov, Nikolay V. Makarov, and Anna M. Belskikh. "Stages of Technical Improvement of Mine Drainage Centrifugal Pumps." Transbaikal State University Journal 30, no. 1 (2024): 81–89. http://dx.doi.org/10.21209/2227-9245-2024-30-1-81-89.

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The article shows the timeliness of an in-depth analysis of changes in the main hydrodynamic parameters of centrifugal pumps for the main drainage of mines, depending on their depth and time history, as well as for processing plants of various technological processes. The requirements for the characteristics of drainage pumping equipment, determined by modern needs for mining in the conditions of technical and economic possibilities for mining, taking into account the geological features of the most accessible deposits explored and accepted for development are determined up by the authors. The

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8

Li, Huokun, Hongkang Lin, Wei Huang, et al. "A New Prediction Method for the Complete Characteristic Curves of Centrifugal Pumps." Energies 14, no. 24 (2021): 8580. http://dx.doi.org/10.3390/en14248580.

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Complete pump characteristics (CPCs) are the key for establishing pump boundary conditions and simulating hydraulic transients. However, they are not normally available from manufacturers, making pump station design difficult to carry out. To solve this issue, a novel method considering the inherent operating characteristics of the centrifugal pump is therefore proposed to predict the CPCs. First, depending on the Euler equations and the velocity triangles at the pump impeller, a mathematical model describing the complete characteristics of a centrifugal pump is deduced. Then, based on multipl

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9

Chami, Francis A. "Use of Simplified Mathematical Formulations in Multi-phase Thermal Pump (MPTP) for Theoretical Prediction of Working Conditions." Tanzania Journal of Engineering and Technology 30, no. 2 (2007): 130–38. http://dx.doi.org/10.52339/tjet.v30i2.406.

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In this work mathematical expressions have been formed and solved to get predictions of working parameters of multiphase thermal pump (MPTP). MPTP is a simple pump which uses steam to pump water. The results have been comparedwith data found experimentally. Experimental and theoretical values for a range of pressures versus velocity differed byapproximately 8.7% up 12% Through dimensional analysis dimensionless parameters were found Re, Eu, Fo and h/d pmp .These helped to further elucidate the pump’s pumping phenomenon. It was experimentally shown that the Reynoldsnumber found theoretically ga

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10

Hatami, Hamed, Ahmad Bagheri, and Reza Ansari. "An Analytical Investigation for Vibration Characteristics of a Beam-Type Liquid Micro-Pump." International Journal of Applied Mechanics 12, no. 02 (2020): 2050016. http://dx.doi.org/10.1142/s1758825120500167.

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This paper studies the characteristics of a micro-beam interacting with an incompressible fluid in a fluid chamber with an opening in its bottom face for fluid flow. The Euler–Bernoulli equation for transverse deformation of an elastic beam is coupled with the fundamental hydrodynamic equation, which is solved by Galerkin and separation of variables method. The 2D fluid flow assumption in Cartesian coordinate has been used. Natural frequencies and mode shapes of wet beam are calculated and compared with the dry beam. The effects of geometrical parameter changes are also computed as a benchmark

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11

Zhang, Heng, and Yan Chao Yin. "Turbulence Numerical Simulation and Particle Track Analysis of Slurry Pump Impeller." Advanced Materials Research 655-657 (January 2013): 336–39. http://dx.doi.org/10.4028/www.scientific.net/amr.655-657.336.

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In order to improve the pump wear performance and optimize pump design, the Euler-Lagrange approach is chosen to simulate the flow in the impeller. The fluid phase is treated as a continuum by solving the N-S equations, while the dispersed phase is solved by using Lagrange Method through the calculated flow field. The results show that the reflex appears in the blade inlet of pressure sides in high flow conditions, but appears in the blade inlet of suction sides in low flow conditions, which possibly induce the cavitation erosion. In low flow conditions, reflex and eddies appear in the blade o

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12

Mata, V., S. Provenzano, J. L. Cuadrado, and F. Valero. "Inverse dynamic problem in robots using Gibbs-Appell equations." Robotica 20, no. 1 (2002): 59–67. http://dx.doi.org/10.1017/s0263574701003502.

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In this paper, two algorithms for solving the Inverse Dynamic Problem based on the Gibbs-Appell equations are proposed and verified. Both are developed using mainly vectorial variables, and the equations are expressed in a recursive form. The first algorithm has a computational complexity of O(n2) and is the least efficient of the two; the second algorithm has a computational complexity of O(n). This algorithm will be compared with one based on Newton-Euler equations of motion, formulated in a similar way, and using mainly vectors in its recursive formulation. The O(n) proposed algorithm will

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13

Liu, Ya Jun, Shu Yan Zhan, Jia Kun Ye, and Wen Hua Xie. "New Designs in Fuel Dispensing System to Control Maximum Flow of Volatile Liquid." Applied Mechanics and Materials 868 (July 2017): 75–80. http://dx.doi.org/10.4028/www.scientific.net/amm.868.75.

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The dispenser is a fuel pumping and measurement device used in the service station. During the refueling process of volatile liquid, the cavitation phenomenon occur easily due to the large flow rate. The serious cavitation will not only reduce the pumping efficiency, produce loud work noise, but also aggravate the pollution of oil and gas and the energy consumption of the system. Therefore, it is necessary to control the maximum flow rate of the pump. Based on this problem, this paper firstly designs a new flow control valve, and a method of mathematical modeling is proposed to analyze the flo

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14

Valera, A., V. Mata, M. Vallés, F. Valero, N. Rosillo, and F. Benimeli. "Solving the inverse dynamic control for low cost real-time industrial robot control applications." Robotica 21, no. 3 (2003): 261–69. http://dx.doi.org/10.1017/s0263574702004769.

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This work deals with the real-time robot control implementation. In this paper, an algorithm for solving Inverse Dynamic Problem based on the Gibbs-Appell equations is proposed and verified. It is developed using mainly vectorial variables, and the equations are expressed in a recursive form, it has a computational complexity of O(n). This algorithm will be compared with one based on Newton-Euler equations of motion, formulated in a similar way, and using mainly vectors in their recursive formulation. This algorithm was implemented in an industrial PUMA robot. For the robot control a new and o

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15

Quan, Hui, Baiheng Fu, Rennian Li, Guangxian Li, Zhengjie Zhang, and Jin Li. "Mathematical model of energy conversion mechanism in screw centrifugal pump based on load criteria of blade airfoil." Engineering Computations 34, no. 7 (2017): 2168–88. http://dx.doi.org/10.1108/ec-04-2017-0155.

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Purpose To analyze the work principle and capacity of energy conversion in each segment of profile lines, the energy transfer from impeller to transmission medium is separated into head coefficient and load coefficient to analyze the energy transfer process. The concepts of airfoil lift coefficient and drag coefficient are used; the third manifestation of the Euler equations is used as well. Design/methodology/approach The numerical simulation of energy conversion mechanism based on load criteria of vane airfoil has been established in screw centrifugal pump to explain its energy conversion me

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16

Chaban, Andriy, Marek Lis, and Andrzej Szafraniec. "Voltage Stabilisation of a Drive System Including a Power Transformer and Asynchronous and Synchronous Motors of Susceptible Motion Transmission." Energies 15, no. 3 (2022): 811. http://dx.doi.org/10.3390/en15030811.

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A mathematical model is developed of the master circuit of an electric driver system including a power transformer and susceptible motion transmission of asynchronous and synchronous drives. Electric motors drive water pumps by means of motion transmission that comprises two elastic couplings of lumped mechanical parameters and a long shaft of distributed mechanical parameters. Differential equations for oscillatory processes for the long shaft and the elastic couplings are different. The shaft is described with partial derivative Euler–Poisson equations, which, combined with the boundary cond

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17

Sazonov, Yuri Appolonievich, Mikhail Albertovich Mokhov, Inna Vladimirovna Gryaznova, et al. "Prototyping and Study of Mesh Turbomachinery Based on the Euler Turbine." Energies 14, no. 17 (2021): 5292. http://dx.doi.org/10.3390/en14175292.

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This paper presents a scientific development aimed at improving the efficiency of turbomachines through the joint use of rotary-vane and vortex workflows. In the well-known Euler turbine, the rotor flow channels represent a set of curved pipes. The authors propose to consider in more detail the possibilities of using such rotating pipes in the implementation of an ejection (vortex) workflow. A hybrid pump was considered with the conclusion that its workflow can be described using two Euler equations. The results of computer simulation indicate that hybrid turbomachines are promising. The use o

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18

Namangan, Institute of Textile Industry. "THE PROCESS CLEANING ZONE'S THEORETICAL THE FEATURES BETWEEN RAW COTTON PARTICULAR AREAS FROM TRASH CONTAMINANTS." SCIENTIFIC-TECHNICAL JOURNAL OF NAMANGAN INSTITUTE OF TEXTILE INDUSTRY 1, no. 1 (2025): 124–31. https://doi.org/10.5281/zenodo.15244968.

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The article proposes to use the Euler equations to describe the motion of a stationary stream in the cleaning zones, which allows us to determine the laws of the distribution of pressure, density and speed along the arc of contact of the moving layer of raw cotton with the surface of the mesh during impact with spikes on the pulp. It was found that the pressure, density and flow velocity along the cleaning arc as a result of the hammer spikes change stepwise with decreasing pressure and density and increasing flow velocity along this arc. 

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19

Simon, Frank, R. Roncen, P. Vuillemin, P. Klotz, Fabien Méry, and E. Piot. "Design and optimization of acoustic liners with a shear grazing flow: OPAL software platform description." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 263, no. 6 (2021): 508–18. http://dx.doi.org/10.3397/in-2021-1496.

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In the context of aircraft noise reduction in varied applications where a cold or hot shear grazing flow is present (i.e., engine nacelle, combustion chamber, jet pump, landing gear), improved acoustic liner solutions are being sought. This is particularly true in the low-frequency regime, where space constraints limit the efficiency of conventional liner technology. Therefore, liner design must take into account the dimensional and phenomenological characteristics of constituent materials, assembly specifications and industrial requirements involving multiphysical phenomena. To perform the si

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20

Roncen, Remi, Pierre Vuillemin, Patricia Klotz, et al. "Design and optimization of acoustic liners with a shear grazing flow: OPAL software platform applications." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 263, no. 6 (2021): 152–63. http://dx.doi.org/10.3397/in-2021-1308.

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In the context of noise reduction in diverse applications where a shear grazing flow is present (i.e., engine nacelle, jet pump, landing gear), improved acoustic liner solutions are being sought. This is particularly true in the low-frequency regime, where space constraints currently limit the efficiency of classic liner technology. To perform the required multi-objective optimization of complex meta-surface liner candidates, a software platform called OPAL was developed. Its first goal is to allow the user to assemble a large panel of parallel/serial assembly of unit acoustic elements, includ

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21

Zhu, Jianjun, Haiwen Zhu, Guangqiang Cao, et al. "A New Mechanistic Model To Predict Boosting Pressure of Electrical Submersible Pumps Under High-Viscosity Fluid Flow with Validations by Experimental Data." SPE Journal 25, no. 02 (2019): 744–58. http://dx.doi.org/10.2118/194384-pa.

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Summary As the second most widely used artificial-lift method in petroleum production (and first in accumulative production), electrical submersible pumps (ESPs) increase flow rates by converting kinetic energy to hydraulic pressure. ESPs are routinely characterized with water flow, and water performance curves are provided by the manufacturers (catalog curves) for designing ESP-based artificial-lift systems. However, the properties of hydrocarbon fluids are very different from those of water, especially the dynamic viscosities, which can significantly alter the ESP performance. Most of the ex

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22

Mazurenko, A., O. Klymchuk, G. Luzhanska, P. Ivanov, and I. Sergeiev. "ENSURING INCREASED RELIABILITY AND EFFICIENCY OF HEAT SUPPLY SYSTEMS DUE TO THE USE OF MICROTURBINES IN CONDITIONS OF UNSTABLE POWER SUPPLY." Odes’kyi Politechnichnyi Universytet Pratsi 2, no. 66 (2022): 58–63. http://dx.doi.org/10.15276/opu.2.66.2022.07.

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Research on the effectiveness of using microturbines for reliable energy supply during the operation of boilers, heat pumps and other small and non-traditional energy systems. In view of the total global economy of fuel and energy resources, the difficult energy situation in Ukraine, the issues of energy saving and reliable operation of heat supply systems, including autonomous ones, come first. The development and implementation of new technologies in the field of small energy are becoming more and more relevant. The compact size of energy generators, and, accordingly, their mobility, arouse

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23

"Velocity Slip in a Deep-sea Slurry Pump and Its Effect on Particle Transportation." Journal of Applied Fluid Mechanics 16, no. 8 (2023). http://dx.doi.org/10.47176/jafm.16.08.1661.

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The slurry pump, which forms the core equipment of the deep-sea mining (DSM) system, provides lifting power for the ore from the seabed to the sea level, which is crucial for the safety of coarse ore particle transportation. Velocity slip plays a significant role in revealing the migration of the pump particles. Therefore, this study analyzes the velocity slip in a slurry pump using the computational fluid dynamics–discrete element method (CFD-DEM) for the first time. The relationship between the pump head and velocity slip was proposed and verified in this study based on the velocity triangle

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24

Yan, Peng, Ning Chu, Dazhuan Wu, Linlin Cao, Shuai Yang, and Peng Wu. "Computational Fluid Dynamics-Based Pump Redesign to Improve Efficiency and Decrease Unsteady Radial Forces." Journal of Fluids Engineering 139, no. 1 (2016). http://dx.doi.org/10.1115/1.4034365.

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In this study, a double volute centrifugal pump with relative low efficiency and high vibration is redesigned to improve the efficiency and reduce the unsteady radial forces with the aid of unsteady computational fluid dynamics (CFD) analysis. The concept of entropy generation rate is applied to evaluate the magnitude and distribution of the loss generation in pumps and it is proved to be a useful technique for loss identification and subsequent redesign process. The local Euler head distribution (LEHD) can represent the energy growth from the blade leading edge (LE) to its trailing edge (TE)

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25

Куликов, А. А., А. Ф. Смоляков, И. В. Иванова, and И. Н. Дюкова. "Thermodynamic analysis of dynamic processes in a centrifugal pump." Известия СПбЛТА, no. 221() (December 28, 2017). http://dx.doi.org/10.21266/2079-4304.2017.221.197-217.

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Предложена термодинамическая и математическая модель процессов энер- гообмена в центробежном насосе. Для процесса повышения в насосе давления несжимаемой жидкости получено выражение. Из первого закона термодинамики выведена формула для КПД центробежного насоса. Построена физическая и математическая модель, позволяющие получить уравнение Эйлера для лопастных машин из первого закона термодинамики. Показано, что термодинамическая и математическая модели, примененные для лопастных машин, могут быть использованы для анализа процесса дросселирования. The proposed thermodynamic and mathematical model

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26

Mutuku, Muvengei, and Kihiu John. "Bond Graph Modeling of Mechanical Dynamics of an Excavator for Hydraulic System Analysis and Design." March 22, 2009. https://doi.org/10.5281/zenodo.1071484.

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This paper focuses on the development of bond graph dynamic model of the mechanical dynamics of an excavating mechanism previously designed to be used with small tractors, which are fabricated in the Engineering Workshops of Jomo Kenyatta University of Agriculture and Technology. To develop a mechanical dynamics model of the manipulator, forward recursive equations similar to those applied in iterative Newton-Euler method were used to obtain kinematic relationships between the time rates of joint variables and the generalized cartesian velocities for the centroids of the links. Representing th

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27

Al-Safran, Eissa, Ahmed Aql, and Tan Nguyen. "Analysis and Prediction of Fluid Flow Behavior in Progressing Cavity Pumps." Journal of Fluids Engineering 139, no. 12 (2017). http://dx.doi.org/10.1115/1.4037057.

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A progressing cavity pump (PCP) is a positive displacement pump with an eccentric screw movement, which is used as an artificial lift method in oil wells. Downhole PCP systems provide an efficient lifting method for heavy oil wells producing under cold production, with or without sand. Newer PCP designs are also being used to produce wells operating under thermal recovery. The objective of this study is to develop a set of theoretical operational, fluid property, and pump geometry dimensionless groups that govern fluid flow behavior in a PCP. A further objective is to correlate these dimension

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28

Ferrari, Alessandro, Michele Manno, and Antonio Mittica. "Cavitation Analogy to Gasdynamic Shocks: Model Conservativeness Effects on the Simulation of Transient Flows in High-Pressure Pipelines." Journal of Fluids Engineering 130, no. 3 (2008). http://dx.doi.org/10.1115/1.2842226.

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A comparison between conservative and nonconservative models has been carried out for evaluating the influence of conservativeness on the prediction of transient flows in high-pressure pipelines. For the numerical tests, a pump-line-nozzle Diesel injection system was considered because the pipe flow presented interesting cases of cavitation. The validity of a conservative model in the simulation of cavitating transient flows was substantiated by the comparison between computed pressure time histories and experimental results at two pipeline locations in the injection system. Although nonconser

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29

Quan, Hui, Xueling Yang, Xiaoyi Liu, Chenxi Sun, and Jianying Lan. "Research on energy conversion characteristics in impeller of Helico-Axial Multiphase Pump during crude transportation." Physics of Fluids, January 31, 2023. http://dx.doi.org/10.1063/5.0140739.

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In order to explore the influence of different crude viscosity on the energy conversion characteristics of impeller of Helical-Axial Multiphase Pump, it was carried out numerical simulation which based on the standard k-ε equation and Euler multiphase flow model under different viscosity. At the same time, Helical-Axial Multiphase Pump was analyzed internal flow structure of by experiments when transported water. The changes of pressure, GVF（gas volume fraction）, flow field and pressurization are obtained in impeller with different viscosity and IGVF（inlet gas volume fraction）.The results show

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30

Akbarovich, Abdurakhmonov Akmalzhon, and Gaybnazarov Egamnazar Eryigitovich. "Theoretical study of the process of interaction of particles of weed impurities in the flow of raw cotton with the working bodies of the cleaning machine." Journal of Textile Engineering & Fashion Technology 6, no. 4 (2020). http://dx.doi.org/10.15406/jteft.2020.06.00244.

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The article proposes to use the Euler equations to describe the motion of a stationary flow in the cleaning zones, which allows determining the laws of pressure, density and velocity distribution along the arc of contact of a moving layer of raw cotton with a mesh surface in the process of impact impact with spikes on the pulp. It has been established that the pressure, densities and flow rates along the cleaning arc as a result of hammer spikes vary in steps, with a decrease in pressure and density and an increase in the flow velocity along this arc.

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31

Zhang, Tao, Haoran Gou, Kefan Mu, Jianchun Guo, Ruoyu Yang, and Ming Li. "Numerical Study on Particle Transport and Placement Behaviors of Ultralow Density Particles in Fracture-Vuggy Reservoirs." SPE Production & Operations, February 1, 2022, 1–14. http://dx.doi.org/10.2118/209193-pa.

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Summary A solid/liquid two-phase flow numerical model based on the computational fluid dynamics-discrete element method (CFD-DEM) model was established to study the transport and settlement law of ultralow-density (ULD) particles during the waterdrive channel adjustment of the Tahe carbonate fractured-vuggy reservoir. The mass, momentum, and turbulence equations of the fluid phase were established in Euler coordinates, whereas the particle motion equations were established based on Newton’s second law. The interaction between the ULD particles was described using a soft sphere model, and the w

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d’Agostino, Luca, Lucio Torre, Angelo Pasini, and Angelo Cervone. "On the Preliminary Design and Noncavitating Performance Prediction of Tapered Axial Inducers." Journal of Fluids Engineering 130, no. 11 (2008). http://dx.doi.org/10.1115/1.2979007.

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A reduced order model for preliminary design and noncavitating performance prediction of tapered axial inducers is illustrated. In the incompressible, inviscid, irrotational flow approximation, the model expresses the 3D flow field in the blade channels by superposing a 2D cross-sectional vorticity correction to a fully guided axisymmetric flow with radially uniform axial velocity. Suitable redefinition of the diffusion factor for bladings with non-negligible radial flow allows for the control of the blade loading and the estimate of the boundary layer blockage at the specified design flow coe

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Böhl, P., B. King, and H. Ruhl. "Vacuum high-harmonic generation and electromagnetic shock." Journal of Plasma Physics 82, no. 2 (2016). http://dx.doi.org/10.1017/s0022377816000210.

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When one takes into account the presence of virtual charged states in the quantum vacuum, a nonlinear self-interaction can arise in the propagation of electromagnetic fields. This self-interaction is often referred to as ‘real photon–photon scattering’. When the centre-of-mass energy of colliding photons is much lower than the rest energy of an electron–positron pair, this quantum effect can be included in the classical field equations of motion as a vacuum current and charge density using the Heisenberg–Euler Lagrangian. Using analytical and numerical methods for subcritical fields, the intri

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