The soliton waves’ physical behavior on the pseudo spherical surfaces is studied through the analytical solutions of the nonlinear (1+1)–dimensional Kaup–Kupershmidt $\left(\mathcal{KK}\right)$ equation. This model is named after Boris Abram Kupershmidt and David J. Kaup. This model has been used in various branches such as fluid dynamics, nonlinear optics, and plasma physics. The model’s computational solutions are obtained by employing two recent analytical methods. Additionally, the solutions’ accuracy is checked by comparing the analytical and approximate solutions. The soliton waves’ characterizations are illustrated by some sketches such as polar, spherical, contour, two, and three-dimensional plots. The paper’s novelty is shown by comparing our obtained solutions with those previously published of the considered model.

The purpose of this article is to investigate the mass and heat transport phenomena associated with micropolar fluid flow created by a vertically stretched Riga surface. This is constructed using an array of irregular electrodes and permanent magnets that are oriented spanwise. Additionally, we investigate the particles’ micro rotational impacts. Furthermore, the flow behaviour of the modeled problem has been numerically calculated with bvp4c solver and the obtained results are presented graphically. Numerical data are used to illustrate physical parameters such as skin friction, Nusselt, and Sherwood numbers. For precise values of different flow parameters, the characteristics of fluid velocity, angular velocity, temperature, and concentration gradients are investigated graphically. The flowing parallel to the Riga plate in a positive $x$–path is aided by Lorentz forces introduced into the flowing simulation by the electro-magnetic poles of the Riga plate, which produces a rapidity greater than the inner speed. It is confirmed that the numerical calculations fit well with the results of earlier published investigations. Due to the participation of the Riga plate, the updated Hartmann number has a considerable effect on flow profiles.

The suspension of tiny solid particles inside the energy transport liquids could enhance their thermal conductivity as well as provide an efficient and inventive approach to significantly improve their properties of heat transport. Therefore, our aim is to explore the radiative two-dimensional unsteady flow of a viscous nanofluid about an aligned magnetic field that includes the joint effect of suction, velocity slip, and heat source across a porous convective stretching/shrinking surface. Initially, using non-dimensional variables, the nonlinear governing partial differential equations (PDEs) were transformed into ordinary differential equations (ODEs) which were subsequently solved with the help of bvp4c built-in package in MATLAB. The results declare that escalating the values of the unsteadiness parameter escalates the friction drag whereas it reduces with the escalation of the slip parameter. Furthermore, the heat transfer rate escalates with the escalation of radiation and concentration parameter, and the escalation of the heat source parameter causes to reduce the heat transfer rate. Finally, it is found that the rate of heat transfer and friction drag continuously improve and decline against the rising rates of stretching, respectively.

This particular research was conducted with the aim of describing the impact of a rotating nanoliquid on an elasting surface. This specific study was carried out using a two-phase nanoliquid model. In this study engine oil is used as the base fluid, and two forms of nanoparticles are used, namely, titanium oxide and zinc oxide (TiO₂ and ZnO). Using appropriate similarity transformations, the arising system of partial differential equations and the related boundary conditions are presented and then converted into a system of ordinary differential equations. These equations are numerically tackled using powerful techniques. Graphs for nanoparticle rotation parameter and volume fraction for both types of nanoparticles present the results for the velocity and heat transfer features. Quantities of physical significance are measured and evaluated, such as local heat flux intensity and local skin friction coefficients at the linear stretching surface. Numerical values for skin friction and local heat flux amplitude are determined in the presence of slip factor.

The current analysis deals with radiative aspects of magnetohydrodynamic boundary layer flow with heat mass transfer features on electrically conductive Williamson nanofluid by a stretching surface. The impact of variable thickness and thermal conductivity characteristics in view of melting heat flow are examined. The mathematical formulation of Williamson nanofluid flow is based on boundary layer theory pioneered by Prandtl. The boundary layer nanofluid flow idea yields a constitutive flow laws of partial differential equations (PDEs) are made dimensionless and then reduce to ordinary nonlinear differential equations (ODEs) versus transformation technique. A built-in numerical algorithm bvp4c in Mathematica software is employed for nonlinear systems computation. Considerable features of dimensionless parameters are reviewed via graphical description. A comparison with another homotopic approach (HAM) as a limiting case and an excellent agreement perceived.