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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.

The purpose of this paper is to address unsteady heat conduction in two subsets of ordinary bodies. One subset consists of a large plane wall, a long cylinder and a sphere in one dimension. The other subset consists of a short cylinder and a large rectangular bar in two dimensions. The prevalent assumptions in the two subsets are: constant initial temperature, uniform surface heat flux and thermo-physical properties invariant with temperature. The engineering applications of the unsteady heat conduction deal with the determination of temperature–time histories in the two subsets using electric resistance heating, radiative heating and fire pool heating.

To this end, a novel numerical procedure named the enhanced method of discretization in time (EMDT) transforms the linear one-dimensional unsteady, heat conduction equations with non-homogeneous boundary conditions into equivalent nonlinear “quasi–steady” heat conduction equations having the time variable embedded as a time parameter. The equivalent nonlinear “quasi–steady” heat conduction equations are solved with a finite difference method.

Based on the numerical computations, it is demonstrated that the approximate temperature–time histories in the simple subset of ordinary bodies (large plane wall, long cylinder and sphere) exhibit a perfect matching over the entire time domain 0 < t < ∞ when compared against the rigorous exact temperature–time histories expressed by classical infinite series. Furthermore, using the method of superposition of solutions in the convoluted subset (short cylinder and large rectangular crossbar), the same level of agreement in the approximate temperature–time histories in the simple subset of ordinary bodies is evident.

The performance of the proposed EMDT coupled with a finite difference method is exhaustively assessed in the solution of the unsteady, one-dimensional heat conduction equations with prescribed surface heat flux for: a subset of one-dimensional bodies (plane wall, long cylinder and spheres) and a subset of two-dimensional bodies (short cylinder and large rectangular bar).

This work aims to investigates exact solutions of the classical Glauert’s laminar wall jet mass and heat transfer under wall suction, wall contraction or dilation, and two thermal transport boundary conditions; prescribed constant surface temperature and prescribed constant surface flux in nanofluidic environment.

The flow system arranged in terms of partial dif- ferential equations is non-dimensionalized with suitable dimensionless transformation variables, and this new set of equations is reduced into ordinary differential equations via a set of similarity transformations, where they are treated analytically for closed form solutions.

Exact solutions of nanofluid flow for velocity distributions, momentum flux, wall shear stress and heat transfer boundary layers for commonly studied nanoparticles; namely copper, alumina, silver, and titanium oxide are presented. The flow behavior of alumina and titanium oxide is identical, and a similar behavior is seen for copper and silver, making two pairs of identical traits. The mathematical expressions as well as visual analysis of wall shear drag and temperature gradient which are of practical interest are analyzed. It is shown that wall stretching or shrinking, wall transpiration and velocity slip together influences the jet flow mechanism and extends the original Glauert’s jet solutions. The exact solutions for the two temperature boundary layer conditions and temperature gradients are analyzed analytically. It is found that the effect of nanopar- ticles concentration on thermal boundary layer is intense, causing temperature uplift, whereas the wall transpiration causes a decrease in thermal layers.

The analysis carried out in nanofluid environment is genuinely new and unique, as our work generalizes the Glauert’s classical regular wall jet fluid problem.

The purpose of this study is to solve the Reynolds equation for finite journal bearings by using the physics-informed neural networks (PINNs) method. As a meshless method, it is unnecessary to use big data to train the neural networks, but to satisfy the Reynolds equation and the corresponding boundary conditions by using the known physics information.

Here, the boundary conditions are enforced through the loss function firstly, i.e. the soft constrain method. After this, an equation was constructed to build a surrogate model for satisfying the corresponding boundary conditions naturally, i.e. the hard constrain method.

For the soft one, in brief, the pressure results agree well with existing results, apart from the ones on the boundaries. While for the hard one, it can be noted that the discrepancies on the boundaries are reduced significantly.

The PINNs method is used to solve the Reynolds equation for finite journal bearings, and the error values on the boundaries for the results of the soft constrain method are improved by using the hard constrain method. Therefore, the hard constraint maybe also a good option when the pressure results on the boundaries are emphasized.

The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-02-2023-0045/

A higher-order implicit shock-capturing scheme is presented for the Euler equations based on time linearization of the implicit flux vector rather than the residual vector.

The flux vector is linearized through a truncated Taylor-series expansion whose leading-order implicit term is an inner product of the flux Jacobian and the vector of differences between the current and previous time step values of conserved variables. The implicit conserved-variable difference vector is evaluated at cell faces by using the reconstructed states at the left and right sides of a cell face and projecting the difference between the left and right states onto the right eigenvectors. Flux linearization also facilitates the construction of implicit schemes with higher-order spatial accuracy (up to third order in the present study). To enhance the diagonal dominance of the coefficient matrix and thereby increase the implicitness of the scheme, wave strengths at cell faces are expressed as the inner product of the inverse of the right eigenvector matrix and the difference in the right and left reconstructed states at a cell face.

The accuracy of the implicit algorithm at Courant–Friedrichs–Lewy (CFL) numbers greater than unity is demonstrated for a number of test cases comprising one-dimensional (1-D) Sod’s shock tube, quasi 1-D steady flow through a converging-diverging nozzle, and two-dimensional (2-D) supersonic flow over a compression corner and an expansion corner.

The algorithm has the advantage that it does not entail spatial derivatives of flux Jacobian so that the implicit flux can be readily evaluated using Roe’s approximate Jacobian. As a result, this approach readily facilitates the construction of implicit schemes with high-order spatial accuracy such as Roe-MUSCL.

A novel finite-volume-based higher-order implicit shock-capturing scheme was developed that uses time linearization of fluxes at cell interfaces.

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.

This study describes the applicability of the a priori estimate method on a nonlocal nonlinear fractional differential equation for which the weak solution's existence and uniqueness are proved. The authors divide the proof into two sections for the linear associated problem; the authors derive the a priori bound and demonstrate the operator range density that is generated. The authors solve the nonlinear problem by introducing an iterative process depending on the preceding results.

The functional analysis method is the a priori estimate method or energy inequality method.

The results show the efficiency of a priori estimate method in the case of time-fractional order differential equations with nonlocal conditions. Our results also illustrate the existence and uniqueness of the continuous dependence of solutions on fractional order differential equations with nonlocal conditions.

The authors’ work can be considered a contribution to the development of the functional analysis method that is used to prove well-positioned problems with fractional order.

The authors confirm that this work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere.

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.