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Burgers' equation is a nonlinear partial differential equation that appears in various areas of physics and engineering. Finding accurate and efficient numerical methods to solve this equation is crucial for understanding complex fluid flow phenomena. In this study, we propose a spline-based numerical technique for the numerical solution of Burgers' equation. The space derivative is discretized using cubic B-splines with new approximations for the second order. Typical finite differences are used to estimate the time derivative. Additionally, the scheme undergoes a stability study to ensure minimal error accumulation, and its convergence is investigated. The primary advantage of this scheme is that it generates an approximate solution as a smooth piecewise continuous function, enabling approximation at any point within the domain. The scheme is subjected to a numerical study, and the obtained results are compared to those previously reported in the literature to demonstrate the effectiveness of the proposed approach. Overall, this study aims to contribute to the development of efficient and accurate numerical methods for solving Burgers' equation. The spline-based approach presented herein has the potential to advance our understanding of complex fluid flow phenomena and facilitate more reliable predictions in a range of practical applications. [ABSTRACT FROM AUTHOR]

This work explores the influence of double diffusion over thermally radiative flow of thin film hybrid nanofluid and irreversibility generation through a stretching channel. The nanoparticles of silver and alumina have mixed in the Maxwell fluid (base fluid). Magnetic field influence has been employed to channel in normal direction. Equations that are going to administer the fluid flow have been converted to dimension-free notations by using appropriate variables. Homotopy analysis method is used for the solution of the resultant equations. In this investigation it has pointed out that motion of fluid has declined with growth in magnetic effects, thin film thickness, and unsteadiness factor. Temperature of fluid has grown up with upsurge in Brownian motion, radiation factor, and thermophoresis effects, while it has declined with greater values of thermal Maxwell factor and thickness factor of the thin film. Concentration distribution has grown up with higher values of thermophoresis effects and has declined for augmentation in Brownian motion. [ABSTRACT FROM AUTHOR]

In this work, the thermal analysis for bio-convective hybrid nanofluid flowing upon a thin horizontally moving needle is carried out. The chemical reaction and viscous dissipation has also considered for flow system in the presence of microorganism. The hybrid nanoparticles comprising of Copper Cu and Alumina A l 2 O 3 are considered for current flow problem. Mathematically the flow problem is formulated by employing the famous Buongiorno's model that will also investigate the consequences of thermophoretic forces and Brownian motion upon flow system. Group of similar variables is used to transform the model equations into dimensionless form and have then solved analytically by homotopy analysis method (HAM). It has established in this work that, flow of fluid declines due to increase in bioconvection Rayleigh number, buoyancy ratio and volume fractions of nanoparticles. Thermal flow grows due to rise in Eckert number, Brownian, thermophoresis parameters and volume fraction of nanoparticles. Concentration profiles increase due to growth in Brownian motion parameter and reduces due to increase in thermophoresis parameter and Lewis number. Motile microorganism profile declines due to augmentation in Peclet and bioconvection Lewis numbers. Moreover, the percentage enhancement in the drag force and rate of heat transfer using conventional nanofluid and hybrid nanofluid are observed and discussed. The hybrid nanofluid increases the skin friction and heat transfer rate more rapidly and efficiently as compared to other traditional fluids. A comparison of the present study with the existing literature is also conducted with a closed agreement between both results for variations in thickness of the needle. [ABSTRACT FROM AUTHOR]

This paper is concerned with the derivation of a well-balanced kinetic scheme to approximate a shallow flow model incorporating non-flat bottom topography and horizontal temperature gradients. The considered model equations, also called as Ripa system, are the non-homogeneous shallow water equations considering temperature gradients and non-uniform bottom topography. Due to the presence of temperature gradient terms, the steady state at rest is of primary interest from the physical point of view. However, capturing of this steady state is a challenging task for the applied numerical methods. The proposed well-balanced kinetic flux vector splitting (KFVS) scheme is non-oscillatory and second order accurate. The second order accuracy of the scheme is obtained by considering a MUSCL-type initial reconstruction and Runge-Kutta time stepping method. The scheme is applied to solve the model equations in one and two space dimensions. Several numerical case studies are carried out to validate the proposed numerical algorithm. The numerical results obtained are compared with those of staggered central NT scheme. The results obtained are also in good agreement with the recently published results in the literature, verifying the potential, efficiency, accuracy and robustness of the suggested numerical scheme. [ABSTRACT FROM AUTHOR]

• The current work aims to study the double diffusive MHD Jeffrey fluid flow through the space of disc and cone apparatus subject to impact of multiple rotations. • Growth in magnetic, Jeffery fluid, and Maxwell factors retards the radial velocity. • Augmentation in the thermophoresis factor results in in an expansion in concentration and thermal distributions. • Growth in the Brownian motion factor supports thermal panels and opposes concentration panels. • Thermal distribution retards with growth in thermal relaxation factor and Prandtl number. • It has also been revealed from this work that growth in Nb , N t , M , λ , and λ 1 the thermal transference, rate has experienced a respective variation of 30%, 35%, 20%, 8%, and 7% showing the highest growth in case of variations in the thermophoresis factor. Non-Fourier's and non-Fick's models enhance predictions of transient heat transfer and mass transport, especially in situations where local equilibrium assumptions break down. In such fluids, the deformation rate and shear stress display a natural parabolic structure. In this study, we explore the flow of a Jeffrey fluid with dual diffusions, incorporating both non-Fourier's and non-Fick's assumptions, through the gap of disk-cone devices. Four scenarios have been investigated for fluid flow and heat transformation in this work comprising of (i) both disc and cone are rotating in opposite directions (ii) both disc and cone are rotating in same direction, (iii) cone is rotating while disc remains stationary, (iv) disc is rotating while cone remains stationary. After deriving the basic equations, a set of appropriate variables is used to convert them into dimension from form. RK-4 (Runge-Kutta 4th order) technique has been employed for the solution of nonlinear equations. As the magnetic and Maxwell factors experience a notable increase, whether the disk and cone move in tandem, in opposite directions, or involve a static cone with a gyrating disk, or a static disk with a gyrating cone, the transverse velocity panels consistently exhibit retardation in all scenarios. Additionally, a noteworthy observation is made when evaluating changes in both the Nusselt number and the Sherwood number, with significantly more pronounced variations detected at the surface of the disk compared to that of the cone. There has been a 35% increase observed in the thermal flow rate with variations in the thermophoresis factor. [ABSTRACT FROM AUTHOR]

• Casson-Maxwell fluid flow in the gap of the disk and cone system is considered for heat and mass transfer applications. To the best of our knowledge and belief, it is the pioneer attempt to consider the combination of two fluids for this unique kind of geometry. The combination of non-Newtonian and viscoelastic nature properties plays a vital role in heat and mass transfer analysis. • Magnetic field in combination with the Casson-Maxwell fluid flow over the rotating con and disk is also a new addition. • The Artificial Neural Network (ANN) strategy is used to solve the problem. The results are validated through error estimation. • The various cone and disk rotations are considered to analyze the flow behavior. The Casson-Maxwell nanofluid is affected by the same and opposite rotations of the devices, which are also derived and discussed in detail. The Casson-Maxwell model is particularly useful for studying viscoplastic fluids or fluids with yield stress, making it applicable to various engineering applications, including extrusion processes, coating applications, and biomedical fluid dynamics. Casson-Maxwell fluid flow enhances mass transfer rates due to the combined effects of non-Newtonian viscosity and viscoelastic behavior. This is particularly useful in processes where mass transfer limitations play a significant role, such as in multiphase reactions or reactive distillation systems. In the context of the above applications the present model, is the combination of Casson- Maxwell fluids that flow through the varying gap of the spinning cone and disk system (CDS) for the heat transfer enhancement. The magnetic field is also imposed in the upright direction to the flow field. The solution of the transform equations has been obtained through artificial neural networks (ANN). The skin friction, heat transfer rate, and comparative analysis have been done. The Casson-Maxwell parameters that depend on the viscoelastic behavior and high viscosity term, causes the fluid to slow down and tends to store the temperature, for a long time as compared to traditional fluids. The radial component of velocity decreases due to the increase in magnetic field. The relative error of the reference and targeted dates is calculated to demonstrate the best precision efficiency of ANN, with a range of 10−3 to 10−4. [ABSTRACT FROM AUTHOR]

• This work explores Darcy-Forchheimer flow on a slandering stretchable sheet. • The flow is impacted by Dufour and Soret effects. • Velocity distribution has augmented for both scenarios h 1 = 0.0 and h 1 = 0.5 where the impacts are more significant for h 1 = 0.0. • Thermal distribution has declined with augmentation in Prandtl number, permeability factor and Dufour number h 2 = 0.0 and h 2 = 0.5. • Concentration distribution has diminished with escalation in Schmidt number, and has augmented with growth in Soret factor for h 3 = 0.0 and h 3 = 0.5 with more significant impacts for h 3 = 0.0. The flow of fluid on a slandering stretchable sheet with variable thickness can have several engineering applications like design and optimization of, aerodynamic structures such as wings and airfoils, wind turbine blade design, efficient cooling systems for electronic devices, power plants and industrial processes. Keeping in view these important applications, this work explores Darcy-Forchheimer flow of magnetohydrodynamics (MHD) fluid across a slandering stretchable sheet with variable thickness in the presence of chemically reactive effects, and heat source. The flow is impacted by the combined influences of diffusive-thermo and thermo-diffusive phenomena. The fluid is set into motion along x-axis due to the stretching behavior of the sheet with y-axis as normal to it. Similarity transformations are employed to convert the modeled equations into nonlinear ordinary differential equations. It has observed as outcomes of this study that velocity distribution has augmented with expansion in viscosity factor and has declined with growth in permeability factor, inertia factor, and magnetic factor for both scenarios h 1 = 0.0 and h 1 = 0.5 where the impacts are more significant for the scenario h 1 = 0.0. Thermal distribution has declined with augmentation in Prandtl number, permeability factor and Dufour number while has upsurge with escalation in thermal source factor for both scenarios h 2 = 0.0 and h 2 = 0.5. Concentration distribution has diminished with escalation in Schmidt number, chemical reactivity factor and has augmented with growth in Soret factor for both the scenarios h 3 = 0.0 and h 3 = 0.5 with more significant impacts in the scenario h 3 = 0.0. Stream lines and contour lines for fluid flow have also sketched for the scenarios h 1 = 0.0 and h 1 = 0.5 as well as for M = 0.0 and M = 2.0. Validation of the proposed model and the methodology has ensured through comparison with fine agreement amongst current results and previously published work. [ABSTRACT FROM AUTHOR]