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In this paper, we applied wavelet collocation method to study the second-order boundary value problem of non-linear differential equation of a fully wetted moving porous fin depending on different geometrical configuration of nanofluids, such as spherical, needle and disk. We compared the exact solution in a particular case using numerical method to validate the results and found good agreement. The novelty of the work is a highly nonlinear problem solved by a hybrid numerical method, i.e., the Legendre wavelet collocation method. This method gives a percentage error of (10^{-7}) with exact results, which demonstrates the method’s accuracy. We also observed that when sphere-shaped nanoparticles are present, the heat transfer rate in the fin is enhanced. Detailed investigations are done to determine the impact of various factors. The findings and error analysis are displayed in the form of figures and tables.

The rheological behaviour of motile micro-organisms with activation energy within the heat transport phenomena holds significant properties in various areas, including industrial processes, biomedical engineering and environmental science. The present investigation shows the influence of the motile micro-organisms with Arrhenius kinetics on the motion of Williamson fluid over a lubricated surface for the consideration of dissipative heat, thermal radiation and heat generation/absorption. The choice of Williamson fluid is the best approach to accurately depict the non-Newtonian characteristics, which is often used in biological and industrial fields. A suitable similarity approach is useful to transform the governing set of model problems into its non-dimensional form. Further, numerical approaches are utilised to explore the combined influences of the considered factors on the flow phenomena. The validation of the result in special cases is obtained exhibiting a good correlation. However, the important finding are: the velocity bounding surface thickness retards for the interaction of the magnetisation along with the non-Newtonian Weissenberg number and the heat transport phenomena of the fluid enhances significantly with the increasing thermal radiation.

The primary aim of this paper is to undertake an analytical investigation of the dispersion and damping behaviours of Love-type waves propagation in a porous piezoelectric layer sandwiched between a dissipative transversely isotropic poroelastic layer of finite thickness and a homogeneous transversely isotropic poroviscoelastic half-space. The acquisition of a dispersion equation for the propagation of Love-type waves has been accomplished by employing appropriate boundary conditions. Calculations for specific cases have been conducted, demonstrating the transformation of dispersion equation into the conventional Love wave equation in those particular situations. This affirmation validates the current mathematical model. Numerical analyses were performed for the parameters involved and the results were depicted through graphical representations. The effects of viscoelastic parameter, porosity parameters, thickness ratio, dielectric and piezoelectric parameters in the dispersion curves are highlighted. The current investigation could prove valuable in applications related to geophysics, material science, oil and gas exploration and earthquake engineering, aiding in the comprehension of seismic wave propagation characteristics in complex layered structures.

In this study, we identify two distinct families of solutions for the Vakhnenko–Parkes (VP) equation using a hybrid computational scheme. This methodology, referred to as the hybrid scheme, integrates B-spline functions with a finite-element approach. The notable advantage of this scheme is its unconditional stability, which enables the successful development of both topological and non-topological solutions. To demonstrate its efficacy, several test cases are examined. We visualise the dynamics of topological and non-topological solitary wave profiles by presenting results through figures and calculating the associated errors. Furthermore, we derive an analytical solution using the Khater II technique and compute conservation laws. In summary, our findings suggest that the presented scheme is highly effective and adaptable to various other nonlinear models.

Internal waves in a stratified liquid propagate to fit the dispersion relation, depending only on the frequency and the angle to the gravity. The phenomenon of self-focussing of such waves is known as wave attractor. To study this phenomenon numerically or in a laboratory, it is common to use a wave-maker providing the forcing through one of the boundaries. In this work, we study the influence of the wave-maker forcing on the wave attractor formation and its characteristics.

A novel segmented time-varying transformation stochastic resonance is proposed, and its use in extracting and enhancing weak misalignment fault features of the coupling under variable speed conditions is studied. At first, the theory of the segmented time-varying transformation stochastic resonance is described in detail. Numerical simulations are carried out to successfully extract the weak misalignment fault characteristic of a coupling. Then, based on the coupling misalignment fault test bench, the method is verified under different amounts of coupling misalignment and time-variable speed conditions. The results demonstrate that the method can effectively extract the fault feature under strong noise background and time-variable speed conditions. The proposed method has a certain engineering value in coupling misalignment fault diagnosis.

The current mathematical model investigates the unsteady flow of fluid–particle suspension and multiphase heat exchange in ferrofluid flow over a spinning and upward migrating disk under the constant Kelvin magnetisation force. The motion of the disk along the rotational axis slows its angular velocity, reducing the radial and circumferential motion of the fluid. Excluding the effect of suspended particles, vertical movement of the disk and the Kelvin magnetisation force, the current problem is reduced to the Von Karman rotating disk problem. Normalised nonlinear differential equations of the system are solved using the technique of finite elements. When a magnetic field is present, increasing the volume concentration of ferromagnetic nanoparticles in the flow increases velocity and temperature. The variation in both temperature and velocity of the suspended particles is greater than that of fluid. The magnetic fluid and magnetic particle radial velocities fall by approximately 7% and 6%, respectively, with an increase in the ferromagnetic interaction number (β₁). On the other hand, there was an approximate 4.5% increase in axial velocity. The radial and tangential components of particle velocities, and the particle temperature, are greatly decreased by increasing the vertical motion parameter. The particle's axial velocity does, however, increase.

We develop models of compact stars with pressure anisotropy on Finch–Skea space–time by applying a generalised equation of state (EoS), which includes specific cases such as linear, quadratic, polytropic, Chaplygin, and colour-flavour-locked (CFL) equations of state. The physical viability of these models is tested for the strange star candidate 4U 1820-30, with a mass ( M = 1.58M_odot ) and radius (R = 9.1) km. All models are found to be physically plausible.

Exploration of a non-linear stretching sheet has been conducted by utilising the Casson model, focussing on an exclusive mixture known as a ternary hybrid nanofluid. This particular nanofluid is a mixture of ethylene glycol hosting suspended graphene, carbon nanotubes and aluminium oxide. The nanoparticles scattered in the underlying liquid are anticipated to exhibit distinct formations like brick blades and lamina, ensuring an efficient connection between the expansive surface of the nanoparticles and the foundational fluid. As a result, this configuration allows for improved absorption of heat from the surface. Many fluid flow problems require the use of a similarity transformation to simplify the equations and make them more tractable. However, it is crucial to derive the correct similarity transformation for each specific problem. Unfortunately, many researchers overlook this step or employ inappropriate transformations, leading to erroneous results. This paper aims to address this issue by systematically deriving suitable similarity transformations for various flow problems, ensuring the accuracy and reliability of the solutions obtained. We derived an appropriate set of similarity transformations for a particular flow problem, in which the similarity variable is dimensionless and appears in the formulation as a function of all independent variables. The numerical solution of the modified equations was obtained using the Runge–Kutta Fehlberg 4–5th method. The study suggests that the temperature distribution is most significant with nanoparticles featuring brick–brick–lamina configurations, followed by brick–brick–blade and brick–brick–brick arrangements.

Graphical abstract

The present study examines the impact of heat source/sink, magnetic dipole and heterogeneous–homogeneous chemical reactions on the hybrid nanofluid flow via a stretching cylinder in the presence of porous media. Scientists and engineers can enhance the efficiency of heat transfer by optimising system flow and investigating the impact of chemical reactions on flow dynamics. Many chemical engineering activities, including absorption, leaching, drying, adsorption, evaporation and solvent extraction, can be used in the analysis of mass transfer to or from surfaces. The governing partial differential equations (PDEs) are modelled and presented. The use of similarity variables transforms the modelled PDEs of the present problem into non-dimensional ordinary differential equations (ODEs). The resultant ordinary differential equations (ODEs) are solved using the Runge–Kutta–Fehlberg fourth–fifth order (RKF-45) method and the obtained results are compared using the physics-informed neural network (PINN) approach. Graphical representations illustrate the effects of various parameters on temperature, concentration and velocity profiles. The thermal profile increases as the ferromagnetic interaction and heat source/sink parameters increase. As homogeneous and heterogeneous reaction parameters rise, the concentration profile decreases. The outcomes obtained by PINN are in good agreement with the solution obtained by RKF-45, indicating good convergence.