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In the present investigation, we investigated the chemically reactive, laminar flow of a micropolar fluid over a stretching sheet in the presence of a transverse magnetic field, thermal radiation and the mutual effect of Soret–Dufour and viscous dissipation. The governing partial differential equations are converted into a system of nonlinear ordinary differential equations for finding the mathematical solutions and the transformed system is solved using the bvp4c algorithm in the MATLAB environment. Effects of different physical factors on dimensionless velocity, microrotation, temperature and concentration profiles are addressed and illustrated graphically. When the Schmidt number is low, it indicates that the momentum diffusivity dominates over the mass diffusivity.

The study aims at different families of analytical solutions and their dynamics for the ((2+1))-dimensional extended Boiti–Leon–Manna–Pempinelli (eBLMP) problem, which is widely used in the fields of physics such as non-linear optics, fluid dynamics, mathematical physics, plasma physics and quantum mechanics. The paper utilises two recently developed efficient mathematical methods: the generalised exponential rational function (GERF) method and the generalised Kudryashov (gK) method. These two methods are versatile, simply applicable to enlighten the new non-linear waveforms. Consequently, these discoveries enhance our understanding of complex systems like ((2+1))-dimensional eBLMP in the realm of non-linear science.

The multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) is used in a suddenly expanded channel to demonstrate the flow of viscoplastic Bingham nanofluid with Al(_2)O(_3) nanoparticles. The geometry has two sections namely, inlet and outlet, and the corresponding heights are denoted by h and H, respectively. The length of the entire channel is 20H, and the expanded channel has a height of 16H. The purpose of the MRT-LBM simulation is to investigate the impact of changing the Bingham number (( 0 le Bn le 200)), keeping the Reynolds number (Re) fixed for different volume fractions ((phi =) 0.00 and 0.04). In addition, the consequences of variations in the Reynolds number (( 50 le Re le 1000 )) at constant Bingham number (Bn) are also studied for those two different volume fractions. The results demonstrate that with fixed (Bn=2), (Re=400) is the point where the flow pattern and recirculation regions are exactly the same for both volume fractions. An increase in Re causes the recirculation regions to grow for a fixed Bn for both volume fractions as Re’s rise increases the velocity and decreases the viscous force. Bn’s increment with Re and volume fraction unchanged lowers the recirculation region’s size due to a rise in viscous force. Higher Re and lower Bn cause the more significant recirculation regions to break down into smaller areas. Incrementing the volume fraction lowers size of the recirculation region. An unstable flow was observed for higher Bn (e.g., (Bn ge 100)) and lower Bn (e.g., (0 le Bn le 10)) when (Re ge 500) for both volume fractions in maximum cases. Unstable flow for lower Bn makes the recirculation regions asymmetric, and when Re is high, the recirculation regions break down for the base fluid ((phi =0.00)). When (Re=300) and (Bn=2), the length of the recirculation region of the upper wall decreases by (28.58%), and the length of the lower wall falls a bit less by (26.37%) when (phi ) is increased from 0.00 to 0.04. For (x/h=2), the nanoparticle mixed fluid’s velocity ((phi =0.04)) never gets a negative magnitude till the final position for (Re=700). In most situations, an increased volume fraction increases the skin-friction effect on both walls.

This paper explores the effectiveness of the Lie derivative discretisation scheme applied to two particular types of nonlinear dynamical equations, both of which have the characteristic of time variables in the denominator position. The discrete structure of non-autonomous systems is established. In particular, we exclude time variables as state variables to prevent non-autonomous systems from becoming autonomous systems. Using this method, we compute the numerical solution of the system above and compare it with the precise solution and the numerical findings of Runge–Kutta, demonstrating the broad applicability of the Lie derivative numerical algorithm. Finally, we determine the CPU consumption time of two numerical algorithms, thus providing evidence of the high efficiency of the Lie derivative numerical algorithm.

The interaction between a Riemann wave propagating along the y-axis and a long wave along the x-axis results in a generalised breaking soliton (gBS) equation. Lie symmetries of the equation are generated in this article to derive some rarely available classes of invariant solutions. The presence of arbitrary functions in each solution opens up a broad class of solution profiles. 3D profiles are used to explore more properties of the solutions to the gBS equation. The profiles describe doubly solitons, annihilation of parabolic, periodic solitons, line solitons and solitons on curved surface types. Solution profiles are useful in optical fibre, acoustic waves in a crystal lattice, long waves in stratified oceans, long-distance transmission and shallow water waves. The Lie symmetry approach has future scope to provide more variety in solutions due to the capability of solutions to include functions and arbitrary constants. This research effectively demonstrates the uniqueness of the solutions when compared with the previously published result. Moreover, the adjoint equation and conserved vectors are determined using Noether’s theorem.

Fluid cavities have important integral roles in different engineering systems. However, a significant challenge is created by the natural convection (NC) within these cavities. Hence, the present work aimed to assess the heat transfer (HT) and fluid flow within a porous medium. For this purpose, a base fluid (f) was chosen comprising a 50–50 mixture of water–ethylene glycol. Moreover, by incorporating TiO₂–Al₂O₃ hybrid nanoparticles (HNP) into the base fluid, their effect on the HT processes and flow was explored. Primarily, the governing equations were derived by considering momentum, continuity and energy equations. Then, similarity solutions were utilised to convert partial differential equations (PDEs) for the flow and energy functions into ordinary differential equations (ODEs). Then, the problem was solved by considering the boundary conditions. To solve the ODEs, the non-commercial software Flex PDE was used through the numerical solution and finite element discretisation methods. Moreover, in the present work, optimal values were used to determine the response surface method (RSM). According to the results, an upward trend was presented by the temperature (T) profile with a decrementation in the electromagnetic intensity and porosity level. Moreover, the temperature profile was not significantly affected by increasing the radiation parameter (Rd).

The self-energy in Born approximation including exchange of interacting one-dimensional systems is expressed in terms of a single integral about the potential which allows a fast and precise calculation for any potential analytically. The imaginary part of the self-energy as damping of single-particle excitations shows a rich structure of different areas limited by single-particle and collective excitation lines. The corresponding spectral function reveals a pseudogap, a splitting of excitations into holons and antiholons as well as bound states.

The current study investigates the three-dimensional radiative and convective Casson hybrid nanofluid flow and heat transfer with the Cattaneo–Christov heat flux model over an inclined spinning and extending disk subjected to an applied magnetic field. Additionally, the study considers the impacts of Joule’s heating and viscous dissipation. Mathematical modelling of the nanofluid flow problem containing Ag and multiwalled carbon nanotubes (MWCNT) nanoparticles with water as the base fluid in a Darcy medium is done using a cylindrical coordinate system. The simplified system of equations is subjected to the spectral quasilinearisation method (SQLM) approach for the graphical and tabular representations. Examining key parameters, such as magnetic field, Bejan number, angle of inclination, disk movement parameter and disk rotation reveals interesting results on velocity and temperature profiles. The research concludes that the Bejan number increases with higher values of temperature ratio, radiation and magnetic parameters, while it decreases with increasing Casson parameter and Brinkman number. Radial wall friction decreases with improved magnetic field, temperature ratio, stretching and porosity parameters, but tangential wall friction increases. The present results are compared with the one already existing in literature to validate the numerical scheme and the results are found to agree well with the previously published work. The application of hybrid nanofluid flow over rotating and stretching disks is widespread in various fields, including rotating machinery, electronic devices, patient treatment instruments, crystal growth method, etc.

This article describes the bilinear form, biliear Bäcklund transformation and Lax pair of the geophysical Boussinesq equation using the Bell polynomial approach. The integrability of the said equation is asserted in the sense of the Lax pair. The Darboux transformation is employed to produce periodic solutions as well as single and N-complexiton-type solutions. Several steps of the Hirota bilinear technique are used to illustrate the flow characteristics of multi-solitons, higher-order breathers and rogue waves. Every sort of wave dynamics response to the Coriolis force is carefully studied.

This study investigates the influence of a magnetic field on thermal convection within a PIP confined between distinct combinations of bounding surfaces. Nonlinear stability analysis (employing the energy method) and linear instability analysis (utilising the normal mode analysis method) are conducted and eigenvalue problems are formulated for both analyses. Numerical analysis is executed using the Galerkin-weighted residual method. Additionally, the study examines the effects of magnetic field, collisional frequency and compressibility on the onset of thermal convection. Collisional frequency plays a significant role in the decay of energy. The principle of exchange of stability is validated for linear analysis, indicating the absence of oscillatory modes of convection. The findings show that the Rayleigh number for both nonlinear and linear analyses is the same. Hence, the subcritical region is not possible, which affirms the existence of global stability. It is noteworthy that both magnetic field and compressibility contribute to the delay in the onset of thermal convection. Also, the PIP confined in rigid–rigid bounding surfaces is thermally more stable than the other configurations of the bounding surfaces.