No abstract

The increasing number of ways in which carbon nanotubes (CNTs) may be used in business and technology has led to an explosion of interest in these tiny tubes. As a result, the Yamada–Ota model is used to investigate the unsteady, non-axisymmetric MHD Homann stagnation point of carbon nanotubes passing over a convective surface with nonlinear radiation. Consisting of single-walled and multi-walled carbon nanotubes that are suspended in water (H2O). The length of the nanomaterial is between [Formula: see text] nanometers, while its radius is between [Formula: see text]. The method of similarity transformation is altered so that it may be used to get the dimensionless system of differential equations from the mathematical model that is envisioned for PDEs. After that, approximate solutions are obtained using MATHEMATICA and the Shooting with RK-IV technique. In this paper, we provide a graphical discussion and a physical interpretation of the results of measures of practical significance as a function of key factors. The results indicated that an increase in the volume fraction led to a corresponding rise in the heat transfer rate. However, it is reduced by the magnetic energy that is supplied to it. Carbon nanoliquids with a single wall have a greater melting point than nanoliquids with multiple walls. Industrial and technological uses of the issue under examination span several fields, including aviation and health. The results of the interface velocity and heat transfer rate at the surface, as well as the solution of each profile, are shown graphically, along with an analysis of the effects of MHD on the flow and heat transfer characteristics of the fluid under the influence of radiation.

Oceanic internal waves often have curvilinear fronts and propagate over vertically sheared currents. We present the first study of long weakly-nonlinear internal ring waves in a three-layer fluid in the presence of a horizontally uniform background current with a constant vertical shear. The leading order of this theory leads to the angular adjustment equation—a nonlinear first-order ordinary differential equation describing the dependence of the linear long-wave speed on its angle to the direction of the current. The compact ring waves, well studied in the absence of a current, correspond to the singular solution (envelope of the general solution) of this equation, and they can exist only under certain conditions. The constructed solutions reveal qualitative differences in the shapes of the wavefronts of the two baroclinic modes: the wavefront of the faster mode is elongated in the direction of the current, while the wavefront of the slower mode is squeezed. Moreover, depending on the vorticity strength, several different regimes have been identified. When the vertical shear is weak, part of the wavefront is able to propagate upstream, while when the shear is strong enough, the whole wavefront propagates downstream. A richer pattern of behaviour is observed for the slower mode. As the shear increases, singularities of the swallowtail-type may arise and, eventually, solutions with compact wavefronts crossing the downstream axis cease to exist. We show that the latter is related to the long-wave instability of the base flow. We obtain the cKdV-type amplitude equation and examine analytical expressions for its coefficients. Using this cKdV-type equation we numerically model the evolution of the waves for both modes. The initial stage of the evolution is in agreement with the leading-order predictions for the deformations of the wavefronts. Then, as the wavefronts expand, strong dispersive effects occur in the upstream direction. Moreover, when nonlinearity is enhanced, fission of waves is observed in the upstream part of the ring waves.

The non-linear propagation of different kinds of ion-acoustic waves (IAWs) in multi-component plasma involving proton beam, positive ions and isothermal electrons has been studied. Using the derivative expansion method of basic equations, namely the hydrodynamic and Poisson equations, they are reduced to a single evolution equation of the non-linear Schrödinger (NLS) equation. By applying this model to plasma formed in Earth’s magnetosphere, different waves can be predicted that express the properties of the plasma. Using the separating variables method and the G /G-expansion method, we derived the exact analytical solutions to the evolution equation by using different solution regions in which non-linear waves are defined. A comparison has been made between the solutions describing the differential equation in each region in which the solution can appear using the data on the Earth’s magnetosphere in the study by Alotaibi et al. (2021).