Nonlinear Flow Research Articles

On the nonlinear moonpool responses in a drillship under regular heading waves

In this study, the nonlinear and viscous damping effects on the free-surface elevations of the recess-type moonpool inside a drillship are investigated. Based on a three-dimensional nonlinear potential flow (NPF3D) model, the nonlinear moonpool responses excited by regular heading waves are simulated in the time domain. To consider the vortex-shedding damping effects, induced by nonlinear moonpool responses, the pressure drop model of Chu et al. [Chu et al., “Effects of nonlinearity and viscous damping on the resonant responses in two-dimensional moonpools with a recess,” Appl. Ocean Res. 127, 103295 (2022)] is extended to three-dimensional and combined with NPF3D to form a viscous modified nonlinear potential flow model (referred to as NPF3D_V). The pressure drop model is composed of two parts in order to account for the energy loss from the first harmonic (piston-mode motions) and higher harmonics (sloshing-mode motions), respectively. The investigation focuses on the piston-mode resonance and secondary resonances of the first and second longitudinal sloshing modes. The response amplitude operators of the higher harmonics, by which the nonlinear effects are evaluated, are computed by the NPF3D_V model. It is found that the higher harmonics are noticeable at the excitation frequencies ωn0/m, where secondary resonances of the nth longitudinal sloshing mode are triggered. In addition, it is found that increasing the length of the recess can promote the nonlinear response of the moonpool significantly. For the moonpool with a long recess, the higher harmonics at secondary resonance are comparable to the first harmonics.

Variational principles for the hydrodynamics of the classical one-component plasma

Hydrodynamic equations for a one-component plasma are derived as a unification of the Euler equations with long-range Coulomb interaction. By using a variational principle, these equations self-consistently unify thermodynamics, dispersion laws, nonlinear motion, and conservation laws. In the moderate and strong coupling limits, it is argued that these equations work down to the length scale of the interparticle spacing. The use of a variational principle also ensures that closure is achieved self-consistently. Hydrodynamic equations are evaluated in both the Eulerian frame, where the fluid variables depend on the position in the laboratory, and the Lagrangian frame, where they depend on the position in some reference state, such as the initial position. Each frame has its advantages and our final theory combines elements of both. The properties of longitudinal and transverse dispersion laws are calculated for the hydrodynamic equations. A simple step function approximation for the pair distribution function enables simple calculations that reveal the structure of the equations of motion. The obtained dispersion laws are compared to molecular dynamics simulations and the theory of quasilocalized charge approximation. The action, which gives excellent agreement for both longitudinal and transverse dispersion laws for a wide range of coupling strengths, is elucidated. Agreement with numerical experiments shows that such a hydrodynamic approach can be used to accurately describe a one-component plasma at very small length scales comparable to the average interparticle spacing. The validity of this approach suggests considering nonlinear flows and other systems with long-range interactions in the future.

Online Decentralized Tracking for Nonlinear Time-Varying Optimal Power Flow of Coupled Transmission–Distribution Grids

Online Decentralized Tracking for Nonlinear Time-Varying Optimal Power Flow of Coupled Transmission–Distribution Grids

An adaptive harmonic polynomial cell method for three-dimensional fully nonlinear wave-structure interaction with immersed boundaries

To accurately simulate wave-structure interaction based on fully nonlinear potential flow theory, a three-dimensional (3 D) high-order immersed-boundary adaptive harmonic polynomial cell (IB-AHPC) method is proposed. Both the free surface and body surface are immersed in background octree cells that are adaptively refined near the boundaries of interest, thereby dramatically reducing computational costs without loss of accuracy. We also propose an easy-to-implement IB strategy to deal with possible instabilities in the time-domain solution arising from the intersection of Dirichlet–Neumann boundaries. For a linearized problem of wave-wall interaction, a matrix-based stability analysis is performed, providing mathematical support for the robustness of the proposed IB strategy. In contrast to the two-dimensional HPC method, compressed cells are found to offer superior stability compared to stretched cells in the vertical direction, while equal mesh aspect ratio in the horizontal plane is superior. Cubic octree cells are, however, still preferred in practice. The free surface is primarily described by a set of massless background wave markers; however, to address the challenges of IB methods in tracking the free surface evolution near the structure, additional body-fitted wave markers are introduced close to the waterline. The information exchange between these two sets of wave markers is realized by radial basis function (RBF) interpolation. While standard RBF schemes have grid-size-dependent filtering performance, we propose a normalized RBF scheme, which is then optimized in terms of the number of neighboring nodes, a smoothing coefficient and the basis functions. Excellent accuracy properties of the proposed 3 D IB-AHPC method are demonstrated by studying fully nonlinear wave propagation. The method is further applied to study relevant fully nonlinear wave-structure interaction problems, including sloshing in 3 D rectangular tanks and wave diffraction of a bottom-mounted cylinder in regular waves. Satisfactory agreement is demonstrated with existing experimental and numerical results, suggesting that the proposed 3 D IB-AHPC method is a promising potential-flow method in marine hydrodynamics.

Experimental study on nonlinear flow behavior in smooth and rough sandstone fractures subjected to various injection pressure and normal compressive stress

Experimental study on nonlinear flow behavior in smooth and rough sandstone fractures subjected to various injection pressure and normal compressive stress

Seepage disturbance mechanism and interface force of cylindrical barrier in fracture

The fluid flow in the crack is disturbed by the contact area. The parallel-plate model with a circular cylindrical contact area is regarded as an idealization of the real contact fracture, and a computational fluid dynamics simulation is performed on it to analyze the interaction between the contact area and the flow around it. The size of the contact area controls the disturbance intensity to the flow velocity magnitude and the disturbance range to the velocity direction. Due to the existence of viscosity, the flow near the wall has a larger shear rate. The intermediate transition region is located between the contact area and the fracture lateral wall and does not contain the region with strong viscous friction caused by the wall. The flow velocity magnitude in the intermediate transition zone changes exponentially in space and has a peak value near the contact area. After the flow bypasses the contact area, the inertial effect caused by the increase in flow velocity magnitude in the local flow channel controls the generation and development of the low-velocity region and the asymmetric degree of the velocity direction distribution. Both the mechanical aperture and the inlet flow velocity affect the stress distribution on the surface of the contact area. The occurrence of the nonlinear flow behavior of fracture seepage and nonlinear change trend of logarithmic drag coefficient curve can be predicted with the same critical Reynolds number. These results provide a useful guide for further exploring the local flow and the surface stress distribution of the local interior geometric property in a single rough-walled fracture on the mesoscopic scale.

Coupling dynamics of capsule interior defects and its impact on hydrodynamic instabilities at ablation fronts for inertial confinement fusion implosions

The micrometer-scale internal defect in the capsule is one of the most important factors that limit implosion performance in inertial confinement fusion (ICF) experiments, which creates instability seeds as shocks propagate through the capsule shell. Here, we report the generation mechanism of vortex pairs resulting from the interaction of shock waves with multiple bubbles, as well as the origin of more intricate perturbation waves than those observed in the case of single defects. Based on the subsequent evolution of hydrodynamic instability, it is evident that the vortex pairs induce the emergence of low-density (light-bubble case) or high-density (referred to as heavy-bubble case) jets on the ablative front. The presence of multiple side-by-side defects can rapidly amplify the dimensions of the jet. These jets could be responsible for the “meteor shower” observed in implosion experiments. Converging disturbed waves between vertically aligned defects lead to a more complex nonlinear flow field evolution compared to the scenario with a single defect. A systematic study of localized perturbation growth as a function of defect placement is presented. We investigate the dependence of circulation in the flow field on the locations of the defects. The scanning results of defect scenes with different sizes revealed the reason why the depth of fluid penetration is affected by the position and size, and found that the effects of the position and size on the perturbation expansion width can be equivalent to a certain extent. The extension of the perturbation width when the defect is off-axis limits the degree of penetration of the perturbation depth. The results contribute to a more comprehensive understanding of physical processes, such as the seeding mechanism, shell integrity, and mass injection into the central region, which may be applied to inform the development of more effective strategies to mitigate implosion degradation in ICF implosion experiments.

The Transition from Darcy to Nonlinear Flow in Heterogeneous Porous Media: I—Single-Phase Flow

Using extensive numerical simulation of the Navier–Stokes equations, we study the transition from the Darcy’s law for slow flow of fluids through a disordered porous medium to the nonlinear flow regime in which the effect of inertia cannot be neglected. The porous medium is represented by two-dimensional slices of a three-dimensional image of a sandstone. We study the problem over wide ranges of porosity and the Reynolds number, as well as two types of boundary conditions, and compute essential features of fluid flow, namely, the strength of the vorticity, the effective permeability of the pore space, the frictional drag, and the relationship between the macroscopic pressure gradient ∇P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\ abla }}P$$\\end{document} and the fluid velocity v. The results indicate that when the Reynolds number Re is low enough that the Darcy’s law holds, the magnitude ωz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _z$$\\end{document} of the vorticity is nearly zero. As Re increases, however, so also does ωz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _z$$\\end{document}, and its rise from nearly zero begins at the same Re at which the Darcy’s law breaks down. We also show that a nonlinear relation between the macroscopic pressure gradient and the fluid velocity v, given by, -∇P=(μ/Ke)v+βnρ|v|2v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-{\\varvec{\ abla }}P=(\\mu /K_e)\ extbf{v}+\\beta _n\\rho |\ extbf{v}|^2\ extbf{v}$$\\end{document}, provides accurate representation of the numerical data, where μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu$$\\end{document} and ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho$$\\end{document} are the fluid’s viscosity and density, Ke\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_e$$\\end{document} is the effective Darcy permeability in the linear regime, and βn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _n$$\\end{document} is a generalized nonlinear resistance. Theoretical justification for the relation is presented, and its predictions are also compared with those of the Forchheimer’s equation.

Soret and dufour effects on unsteady non-linear mixed convection flow past a stretching sheet influenced by non-linear thermal radiation

This article presents a novel mathematical model to analyze the behavior of a time-dependent, incompressible, non-linear (quadratic) mixed convection flow across a stretching surface heated by convection. The model takes into account the influence of non-linear thermal radiation, along with the Soret and Dufour effects. The equations governing the flow characteristics in the boundary layer approximation are derived and transformed into ordinary differential equations (ODEs) using similarity variables. The second-order ODEs are transformed into an initial value problem, which is then numerically solved using the bvp4c solver in MATLAB. The novel findings of the study have been presented using graphs and tables. The intensification of non-linear convection parameters has been shown to lead to the development of velocity near the sheet while simultaneously reducing velocity toward the free stream. The non-linear convection parameters reduce the temperature distribution, while the temperature ratio and the non-linear thermal radiation parameters enhance it. The non-linear convection parameters ( 0.5 ≤ α 1 , α 2 ≤ 2.0 ) , temperature ratio ( 1.2 ≤ θ w ≤ 1.5 ) , and non-linear thermal radiation ( 1.0 ≤ Rd ≤ 1.5 ) , parameters are influential factors in decreasing the surface drag force (by about 15.4%, 28.5%, 4.4%, and 4.3%) and increasing the cooling rate (by about 0.4%, 0.5%, 19.5%, and 23%), respectively. The Biot ( 0.5 ≤ Bi ≤ 1.0 ) number is also found to significantly contribute toward enhancing the cooling rate of the system by about 51.6%.

A Novel Spatial–Temporal Deep Learning Method for Metro Flow Prediction Considering External Factors and Periodicity

Predicting metro traffic flow is crucial for efficient urban planning and transit management. It enables cities to optimize resource allocation, reduce congestion, and enhance the overall commuter experience in rapidly urbanizing environments. Nevertheless, metro flow prediction is challenging due to the intricate spatial–temporal relationships inherent in the data and the varying influence of external factors. To model spatial–temporal correlations considering external factors, a novel spatial–temporal deep learning framework is proposed in this study. Firstly, mutual information is utilized to select the highly corrected stations of the examined station. Compared with the traditional correlation calculation methods, mutual information is particularly advantageous for analyzing nonlinear metro flow data. Secondly, metro flow data reflecting the historical trends from different time granularities are incorporated. Additionally, the external factor data that influence the metro flow are also considered. Finally, these multiple sources and dimensions of data are combined and fed into the deep neural network to capture the complex correlations of multi-dimensional data. Sufficient experiments are designed and conducted on the real dataset collected from Xi’an subway to verify the effectiveness of the proposed model. Experimental results are comprehensively analyzed according to the POI information around the subway station.

Modeling of a Jet Pipe Servovalve Considering Nonlinear Flow Forces Acting on the Spool

The design and analysis of jet pipe servovalves are mostly based on linear models. However, there are some nonlinear factors in this kind of electromechanical–hydraulic structure. The article deduces a linear rotation equation for the armature assembly and a linear flow equation for the control cavity. With the consideration of nonlinear hydraulic reaction forces in the second stage, the nonlinear dynamic equation of the main spool in an ideal jet pipe servovalve is derived. Based on the MATLAB (R2016a) software, the nonlinear model of a certain type of jet pipe servovalve is numerically investigated. The equilibrium points of the nonlinear system are calculated, the phase portraits are plotted, and the Hopf bifurcations caused by the flow-pressure coefficient as the control parameter and the period-doubling bifurcations caused by the variation of the input signal are analyzed. The vibration frequency of the time-domain response of the fifth-order system with a cosine signal as input is 242 Hz, which is similar to the experimental value of 233 Hz. The relative error between the two is 3.9%, verifying the validity of the nonlinear system model.

Analysis of heat and mass transfer of a non-linear convective heat generating fluid flow in a porous medium with variable viscosity

The present study addresses the significance of analyzing the heat and mass transfer characteristics of a non-linear convective fluid flow featuring variable viscosity within a porous medium. Our focus lies on understanding the impact of varying viscosity on the hydromagnetic coupling stress, particularly in the presence of internal heat generation. The fluid is confined within a porous medium, and convection cooling takes place at both wall of the fluid. A key assumption in our investigation is the nonlinearity nature of the buoyancy force, specifically when there is a substantial difference in temperature and concentration within the fluid and its surroundings, the buoyancy force becomes quadratic. The exploration of couple stress fluids, alongside considerations of temperature-dependent internal heat generation and mass transfer, contributes to advancing our understanding of thermal system performance. By accounting for nonlinearity in density, temperature, and concentration, the research aims to provide more accurate results than previous linear models, crucial for optimizing energy consumption in practical applications such as nuclear reactors and heat exchangers. Dealing with this type of problem often pose significant challenges, as they often lack analytical solutions, necessitating the use of more advanced numerical methods or approximation techniques. We employ Spectral Quasilinearisation Method (SQLM) to solve the coupled non-linear boundary value problems (PDE) arising from the model, momentum equation, energy equation, concentration equation, and entropy generation. The obtained results are then validated using the Runge–Kutta method and a good agreement is achieved. The findings for the dimensionless velocity, temperature, mass concentration and entropy generation are presented in graphics and in table manner, and effects of controlling parameters are extensively explained. The results highlight that an increase in variable viscosity leads to an enhancement in the velocity profile, while simultaneously reducing the temperature in the system. Furthermore, increments in the non-linear convection parameter and coupling stress parameter result in an appreciation of the velocity profile, temperature profile, and entropy generation rate.

Spline Model: A Hydrostatic/Non-Hydrostatic Dynamic Core with Space-Time Second-Order Precision and Its Exact Tests

We present a new explicit quasi-Lagrangian integration scheme with the three-dimensional cubic spline function transform (transform = fitting + interpolation, referred to as the “spline format”) on a spherical quasi-uniform longitude–latitude grid. It is a consistent longitude–latitude grid, and to verify the feasibility, accuracy, convergence, and stability of the spline format interpolation scheme for the upstream point on the longitude–latitude grid, which may map a quasi-uniform longitude–latitude grid, a set of ideal, exact test schemes is adopted, which are recognized and proven to be effective internationally. The equilibrium flow test, cross-polar flow test, and Rossby–Haurwitz wave test are used to illustrate the spline scheme uniformity to the linear scheme and to overcome the over-dense grid in the polar region and the non-singularity of the poles. The cross-polar flow test demonstrates that the geostrophic wind crosses the polar area correctly, including the South Pole and North Pole. A non-hydrostatic, fully compressible dynamic core is used to complete the density flow test, demonstrating the existence of a time-varying reference atmosphere and that the spline format can simulate highly nonlinear fine-scale transient flows. It can be compared for the two results of the density flow test between the solution with the spline format and the benchmark reference solution with the linear format. Based on the findings, the non-hydrostatic dynamic core with the spline format is recommended for adoption. When simulated for the flow over an ideal mountain, through the “topographic gravity wave test”, the bicubic surface terrain and terrain-following height coordinates, time-split integration, and vector discrete decomposition can be derived successfully. These may serve as the foundations for a global, unified spline-format numerical model in the future.

Investigation of the chloride ion transport mechanism in unsaturated concrete considering the nonlinear seepage effect

In this paper, the mechanism of chloride ion transport in unsaturated concrete considering the nonlinear seepage effect is investigated by indoor experiments and analytical modelling. The concrete sheet imbibition test is used to reveal the nonlinear characters of seepage including water convection and moisture capillary within the unsaturated concrete in case of various water-binder ratios. For modelling purpose, an exponential function is introduced to mathematically depict the nonlinear relation between the specific discharge and hydraulic gradient during both water convection and moisture capillary processes. Based on such nonlinear seepage models, new mathematical models are proposed for the problem of chloride invasion within the reinforced concrete structure. The solution for the mathematical model of chloride transport is resolved by Laplace transform and finite difference method, of which acceptability is verified by the comparison with the results from chloride ion invasion test. The result indicates that, compared with linear seepage law, the use of the proposed nonlinear seepage solution is preferred to depict the saturation-relative humidity curves from concrete sheet imbibition test. The consideration of nonlinear seepage can result in a slower chloride transport rate, which further makes a later flow start-up time and a smaller free chloride iron concentration within the concrete. Such effect can be enhanced with the increase of nonlinear flow degree. The paper provides an effective solution to evaluate the chloride invasion within the unsaturated concrete.

Aerodynamic and Aeroacoustic Effects of Different Transition Mechanisms on an Airfoil

The paper delves into the detailed sound generation and propagation mechanisms associated with tripping-induced flow perturbations and trailing-edge scattering using wall-resolved large-eddy simulations. Two distinct boundary-layer tripping techniques, namely a geometrically resolved stair strip and an artificially modeled trip using suction and blowing, are investigated. To facilitate comparison, the natural boundary-layer transition is also simulated as a baseline scenario. The analysis takes into account a Reynolds number of 4×105, a Mach number of 0.058, and a nonzero angle of attack of 6.25° over a NACA 0012 airfoil configuration. The mechanisms for sound generation and propagation related to trailing-edge noise remain consistent across the three transition scenarios. However, boundary-layer tripping notably leads to intricate, scenario-specific noise generation: there is an interaction between the laminar separation bubble and tripping for the stair strip case, whereas laminar boundary-layer instability is evident for the suction and blowing scenario. Aerodynamic flowfields involving acoustic noise sources, their propagating natures near the wall, and far-field acoustics are cross-examined in detail. The comprehensive analysis of observed phenomena provides valuable insights for understanding nonlinear flow and acoustic interactions relevant to airfoil noise and designing new types of trips under adverse pressure gradient flows.

A Novel Higher-Order Numerical Scheme for System of Nonlinear Load Flow Equations

Power flow problems can be solved in a variety of ways by using the Newton–Raphson approach. The nonlinear power flow equations depend upon voltages Vi and phase angle δ. An electrical power system is obtained by taking the partial derivatives of load flow equations which contain active and reactive powers. In this paper, we present an efficient seventh-order iterative scheme to obtain the solutions of nonlinear system of equations, with only three steps in its formulation. Then, we illustrate the computational cost for different operations such as matrix–matrix multiplication, matrix–vector multiplication, and LU-decomposition, which is then used to calculate the cost of our proposed method and is compared with the cost of already seventh-order methods. Furthermore, we elucidate the applicability of our newly developed scheme in an electrical power system. The two-bus, three-bus, and four-bus power flow problems are then solved by using load flow equations that describe the applicability of the new schemes.

Entropy generation analysis on zero mass flux effects of nonlinear mixed convective Williamson nanofluid flow with christov-Cattaneo heat flux

Entropy generation analysis on zero mass flux effects of nonlinear mixed convective Williamson nanofluid flow with christov-Cattaneo heat flux

Multi-Objective Stochastic Paint Optimizer for Solving Dynamic Economic Emission Dispatch with Transmission Loss Prediction Using Random Forest Machine Learning Model

Dynamic economic emission dispatch problems are complex optimization tasks in power systems that aim to simultaneously minimize both fuel costs and pollutant emissions while satisfying various system constraints. Traditional methods often involve solving intricate nonlinear load flow equations or employing approximate loss formulas to account for transmission losses. These methods can be computationally expensive and may not accurately represent the actual transmission losses, affecting the overall optimization results. To address these limitations, this study proposes a novel approach that integrates transmission loss prediction into the dynamic economic emission dispatch (DEED) problem. A Random Forest machine learning model was offline-trained to predict transmission losses accurately, eliminating the need for repeated calculations during each iteration of the optimization process. This significantly reduced the computational burden of the algorithm and improved its efficiency. The proposed method utilizes a powerful multi-objective stochastic paint optimizer to solve the highly constrained and complex dynamic economic emission dispatch problem integrated with random forest-based loss prediction. A fuzzy membership-based approach was employed to determine the best compromise Pareto-optimal solution. The proposed algorithm integrated with loss prediction was validated on widely used five and ten-unit power systems with B-loss coefficients. The results obtained using the proposed algorithm were compared with seventeen algorithms available in the literature, demonstrating that the multi-objective stochastic paint optimizer (MOSPO) outperforms most existing algorithms. Notably, for the Institute of Electrical and Electronics Engineers (IEEE) thirty bus system, the proposed algorithm achieves yearly fuel cost savings of USD 37,339.5 and USD 3423.7 compared to the existing group search optimizer algorithm with multiple producers (GSOMP) and multi-objective multi-verse optimization (MOMVO) algorithms.

Recurrence Rate spectrograms for the classification of nonlinear and noisy signals

Time series analysis of real-world measurements is fundamental in natural sciences and engineering, and machine learning has been recently of great assistance especially for classification of signals and their understanding. Yet, the underlying system’s nonlinear response behaviour is often neglected. Recurrence Plot (RP) based Fourier-spectra constructed through τ-Recurrence Rate (RR τ ) have shown the potential to reveal nonlinear traits otherwise hidden from conventional data processing. We report a so far disregarded eligibility for signal classification of nonlinear time series by training RESnet-50 on spectrogram images, which allows recurrence-spectra to outcompete conventional Fourier analysis. To exemplify its functioning, we employ a simple nonlinear physical flow of a continuous stirred tank reactor, able to exhibit exothermic, first order, irreversible, cubic autocatalytic chemical reactions, and a plethora of fast-slow dynamics. For dynamics with noise being ten times stronger than the signal, the classification accuracy was up to ≈ 75% compared to ≈ 17% for the periodogram. We show that an increase in entropy only detected by the RR τ allows differentiation. This shows that RP power spectra, combined with off-the-shelf machine learning techniques, have the potential to significantly improve the detection of nonlinear and noise contaminated signals.

Darcy–Forchheimer flow of Carreau fluid over a curved stretching surface with slip condition and variable thermal conductivity

The aim of this study is to perform analysis of nonlinear mixed convection flow of Carreau fluid due to curved stretched surface with slip condition. Furthermore, the effect of Darcy–Forchheimer and MHD is taken into consideration. The energy equation is modified by considering the variable thermal conductivity with Joule heating effect. The governing equations of the problem are highly nonlinear partial differential equations subjected to boundary conditions. Similarity transformations are used to transform nonlinear PDEs into ODEs. The numerical technique is used to solve this problem via bvp4c. The main outcomes of this research are that the impact of Forchheimer and porosity parameter tends to decrease the fluid velocity for shear thickening case. The velocity slip condition increases the fluid velocity only near the surface but the fluid velocity away from the surface does not get affected. The changing thermal conductivity raises the temperature of the fluid near the boundary. Furthermore, the impacts of various parameters on the velocity and temperature profiles are demonstrated. The values of local Nusselt number and skin friction for a different parameter are provided.