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This paper aims to present dual solutions for the two-dimension copper oxide with silver (CuO–Ag) and zinc oxide with silver (ZnO–Ag) hybrid nanofluid flow past a permeable shrinking sheet in a dusty fluid with velocity slip.

The governing partial differential equations for the two dust particle phases are reduced to the pertinent ordinary differential equations using a similarity transformation. Closed-form analytical solutions for the reduced skin friction and reduced Nusselt number, as well as for the velocity and temperature profiles, were presented, both graphically and in tables, under specific non-dimensional physical parameters such as the suction parameter, Prandtl number, slip parameter and shrinking parameter, which are also presented in both figures and tables.

The results indicate that for the shrinking flow, the wall skin friction is higher in the dusty fluid when compared with the clear (viscous) fluid. In addition, the effect of the fluid–particle interaction parameter to the fluid phase can be seen more clearly in the shrinking flow. Furthermore, multiple (dual, upper and lower branch solutions) are found for the governing similarity equations and the upper branch solution expanded with higher values of the suction parameter. It can be confirmed that the lower branch solution is unstable.

In practice, the study of the stretching/shrinking flow is crucially important and useful. Both the problems of steady and unsteady flow of a dusty fluid have a wide range of possible applications in practice, such as in the centrifugal separation of particles, sedimentation and underground disposal of radioactive waste materials.

Even though the problem of dusty fluid has been broadly investigated, very limited results can be found for a shrinking sheet. Indeed, this paper has succeeded to obtain analytically dual solutions. The stability analysis can be performed by following many published papers on stretching/shrinking sheets. Finally, the critical values and plotting curves for obtaining single or dual solution are successfully presented.

This study aims to investigate the dual solutions for axisymmetric flow and heat transfer due to a permeable radially shrinking disk in copper oxide (CuO) and silver (Ag) hybrid nanofluids with radiation effect.

The partial differential equations that governed the problem will undergo a transformation into a set of similarity equations. Following this transformation, a numerical solution will be obtained using the boundary value problem solver, bvp4c, built in the MATLAB software. Later, analysis and discussion are conducted to specifically examine how various physical parameters affect both the flow characteristics and the thermal properties of the hybrid nanofluid.

Dual solutions are discovered to occur for the case of shrinking disk (λ < 0). Stronger suction triggers the critical values’ expansion and delays the boundary layer separation. Through stability analysis, it is determined that one of the solutions is stable, whereas the other solution exhibits instability, over time. Moreover, volume fraction upsurge enhances skin friction and heat transfer in hybrid nanofluid. The hybrid nanofluid’s heat transfer also heightened with the influence of radiation.

Flow over a shrinking disk has received limited research focus, in contrast to the extensively studied axisymmetric flow problem over a diverse set of geometries such as flat surfaces, curved surfaces and cylinder. Hence, this study highlights the axisymmetric flow due to a shrinking disk under radiation influence, using hybrid nanofluids containing CuO and Ag. Upon additional analysis, it is evidently shows that only one of the solutions exhibits stability, making it a physically dependable choice in practical applications. The authors are very confident that the findings of this study are novel, with several practical uses of hybrid nanofluids in modern industry.

The purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves.

By means of the Cole–Hopf transform, the bilinear form of the studied equation is extracted. Then the ansatz function method combined with the symbolic computation is implemented to construct the breather, multiwave and the interaction wave solutions. In addition, the subequation method tis also used to search for the diverse travelling wave solutions.

The breather, multiwave and the interaction wave solutions and other wave solutions like the singular periodic wave structure and dark wave structure are obtained. To the author’s knowledge, the solutions obtained are all new and have never been reported before.

The solutions obtained in this work have never appeared in other literature and can be regarded as an extension of the solutions for the new (3 + 1)-dimensional integrable fourth-order nonlinear equation.

The objective of this paper is to analyze the relationship between innovation and the development of artificial intelligence (AI) and digital technologies in Europe. The use of digital technologies among European companies is studied through a composite index, while the relationship between innovation and AI is studied through a log-linear regression model. The results of the model have made possible to develop interesting indications for economic and industrial policy.

The use of digital technologies among European companies is studied through a composite index of AI and information technology (ICT) (using the Fair and Sustainable Welfare methodology) with the aim of measuring territorial gaps and to know which European countries are more or less inclined to its use, while the relationship between innovation and AI is studied through a log-linear regression model.

In the paper, two different methodologies were used to analyze the relationship between innovation and the development of digital technologies in Europe. The synthetic indicator made possible to develop a taxonomy between the different countries, the log-linear model made possible to identify and explain the determinants of innovation.

The description of the biunivocal relationship between innovation and AI is a topical and relevant issue that is treated in the paper in an original way using a synthetic indicator and a log-linear model.

本文旨在探討在歐洲、創新與人工智能和數字技術的發展之間的關係。研究人員透過一個綜合指數、去探討歐洲公司之間數字技術的使用狀況。至於創新與人工智能之間的關係, 則以對數線性回歸模型來進行研究。從模型所得的結果, 為我們提供了建議、去訂定適切的經濟和產業政策。

研究人員透過一個人工智能和資訊科技的綜合指數, 去探討歐洲企業之間數字技術的使用狀況 (研究人員使用了公平和可持續福利方法論), 其目標為測量領土差距, 以及確定哪些歐洲國家、大體上傾向於使用數字技術;至於創新與人工智能之間的關係, 則以對數性回歸模型來進行研究。

本文使用了兩個不同的方法、去探討在歐洲、創新與數字技術發展之間的關係。有關的合成指標, 使研究人員可製定一個不同國家間的分類法;而有關的對數線性模型, 則讓研究人員可確立並說明創新的決定因素。

本文使用了合成指標和對數線性模型、去探討創新與人工智能之間的一對一的關係, 這是時下受到關注和適宜的課題;就研究法而言, 本研究確是新穎獨創的。

This paper aims to study a non-Newtonian micropolar fluid flow over a bidirectional flexible surface for multiple exact solutions of momentum boundary layer and thermal transport phenomenon subject to wall mass flux, second-order slip and thermal jump conditions.

The coupled equations are transformed into ordinary differential equations using similarity variables. Analytical and numerical techniques are used to solve the coupled equations for single, dual or multiple solutions.

The results show that the stretching flow, shrinking flow, the wall drag, thermal profile and temperature gradient manifest large changes when treated for special effects of the standard parameters. The role of critical numbers is definitive in locating the domains for the existence of exact solutions. The nondimensional parameters, such as mass transfer parameter, bidirectional moving parameter, plate deformation strength parameter, velocity slips, material parameter, thermal jump and Prandtl number, are considered, and their physical effects are presented graphically. The presence of governing parameters exhibits special effects on the flow, microrotation and temperature distributions, and various exact solutions are obtained for the special parametric cases.

The originality and value of this work lie in its exploration of non-Newtonian micropolar fluid flow over a bidirectional flexible surface, highlighting the multiple exact solutions for momentum boundary layers and thermal transport under various physical conditions. The study provides insights into the effects of key parameters on flow and thermal behavior, contributing to the understanding of complex fluid dynamics.

This study aims to improve the tribological properties of hydrodynamic journal bearing via surface texture, as well as the wear and antifriction mechanisms of textured bearing were represented. It provides a design direction for solving the tribological problem of rotor-bearing system.

In this paper, the variation of surface texture parameters (e.g. texture diameter, d; area density, sp; and depth, hp) were analyzed based on finite difference method. The optimal surface texture parameters were obtained by designing orthogonal experiments, and the relationship between friction and wear properties and microstructure was studied via combining electron probe microanalyzer, scanning electron microscope, X-ray diffractometer and friction and wear testing machine.

Dimensionless film pressure P increased as the d increased, whereas P first increased and then decreased as the sp and hp increased, and the maximum P was got as sp = 15% and hp = 25 µm, respectively. The friction coefficient of textured surface with suitable parameters was effectively reduced and the textured surface with the best antifriction effect was 5#. Orthogonal experimental design analysis showed that the influence order of factors on friction coefficient was as follows: sp > sp × d > d > d × hp > hp > sp × hp and the friction coefficient first decreased and then increased as the sp, d and hp increased. In addition, the friction and wear mechanism of textured bearing were three body friction and abrasive wear as the matrix structure and hard phase were a single β phase and Mn5Si3, respectively. While the antifriction mechanism of textured surface was able to store abrasive particles and secondary hydrodynamic lubrication was formed.

The sample with reasonable texture parameter design can effectively reduce friction and wear of hydrodynamic journal bearing without reducing the service life, which can provide a reference for improving the lubrication performance and mechanical efficiency of rotor-bearing system.

This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm.

The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations.

Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software.

The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement.

There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively.

This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution.

To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

The purpose of this study is to analyze the impact of inclined magnetohydrodynamics (MHD) and thermal radiation on the flow of a ternary micropolar nanofluid on a sheet that is expanding and contracting while applying mass transpiration and velocity slip conditions to the flow. The nanofluid, which is composed of Au, Ag and Cu nanoparticles dispersed in water as the base fluid, possesses critical properties for increasing the heat transfer rate and is frequently used in manufacturing and industrial establishments.

The set of governing nonlinear partial differential equations is transformed into a set of nonlinear ordinary differential equations. The outcome of this differential equation is solved and obtained the closed-form solution and energy equation in the form of hypergeometric functions.

The velocity, micro-rotation and temperature field are investigated versus a parametric variation. The physical domains of mass suction or injection and micropolar characteristics play an important role in specifying the presence, singleness and multiplanes of exact solutions. In addition, many nondimensional characteristics of the profiles of temperature, angular velocity and velocity profiles are graphically shown with substantial consequences. Furthermore, adding nanoparticles increases the heat transfer rate of the fluid used in manufacturing and industrial establishments. The current findings may be used for better oil recovery procedures, smart materials such as magnetorheological fluids, targeted medicine administration and increased heat transmission. Concerning environmental cleanup, nanomaterial fabrication and biomedical devices, demonstrate their potential influence in a variety of disciplines.

The originality of this paper is to analyze the impact of inclined MHD at an angle with the ternary nanofluid on a micropolar fluid over an expanding and contracting sheet with thermal radiation effect.

An integrated queueing network focused on container storage/retrieval operations occurring on the yard of a transshipment hub is proposed. The purpose of the network is to support decisions related to the organization of the yard area, while also accounting for operations policies and times on the quay.

A discrete-event simulation model is used to reproduce container handling on both the quay and yard areas, along with the transfer operations between the two. The resulting times, properly estimated by the simulation output, are fed to a simpler queueing network amenable to solution via algorithms based on mean value analysis (MVA) for product-form networks.

Numerical results justify the proposed approach for getting a fast, yet accurate analytical solution that allows carrying out performance evaluation with respect to both organizational policies and operations management on the yard area.

Practically, the expected performance measures on the yard subsystem can be obtained avoiding additional time-expensive simulation experiments on the entire detailed model.

As a major takeaway, deepening the MVA for generally distributed service times has proven to produce reliable estimations on expected values for both user- and system-oriented performance metrics.

The determination of mode-I fracture toughness of brittle structural materials by means of the notched Brazilian disc configuration is studied. Advantage is taken of a recently introduced analytical solution and, also, of data provided by an experimental protocol with notched marble specimens under diametral compression using the loading device suggested by International Society for Rock Mechanics (ISRM) and also the three-dimensional digital image correlation (3D-DIC) technique.

The analytical solution highlighted the role of geometrical factors, like, for example, the width of the notch, which are usually disregarded. The data of the experimental protocol were comparatively considered with those concerning the response of the specific material under uniaxial tensile load.

This combined study provided interesting data concerning some open issues, as it is the exact crack initiation point and the level of the critical load causing crack initiation. It was definitely indicated that the crack initiation point is not a priori known (even for notched specimens) and, also, that the maximum recorded load does not correspond by default to the critical load responsible for the onset of catastrophic macroscopic fracture.

It was suggested that the load considered critical one for the determination of mode-I fracture toughness K_IC is erroneous. At a load equal to about 70% of the maximum one, a process zone is formed (zone of non-reversible phenomena) around the notch's crown, designating termination of the validity of any linear elastic solution used to determine the normalized stress intensity factors (SIFs). Moreover, at a load level equal to about 95% of the macroscopically observed fracture load, crack propagation has already begun. Therefore, the experimental procedure must be monitored with additional equipment, providing an overview of the displacement field developed during loading.