In this paper, the effect of electrons and positrons trapping on positive and negative ions are analysed in collision-less magnetised system. Relying on the widely used reductive perturbation technique (RPT), a three-dimensional (3D) modified trapped Korteweg-de Varies Burgers’ (mTKdVB) equation is governed due to the presence of trapped positrons, trapped electrons, and both kinematic as well as bulk viscosity. This study examines the influence of different parameters, such as negative ion number densities, electron ion densities, and positron ion density on dynamical and structural properties. Also, the impact of the viscosity coefficient and magnetised parameter of the small-amplitude shock waves has been examined. The current theoretical research may be helpful to understand the shock wave structure in space environments related to astrophysics such as white dwarfs, neutron stars etc.

Natural convection, driven by buoyancy, is utilized for the heat transport in various applications including thermal exchangers, cooling of heat sources, solar collectors, geothermal power systems, electronic devices, microelectronics, and nuclear industries. This research focuses on the free convection of a hybrid nanoliquid containing Ag–MgO nanoparticles in an enclosure having partially active borders. The hybrid nanosuspension utilized is a mixture of MgO and Ag nanoparticles in equal proportions, suspended in water as the base liquid. The square enclosure is subject to the Lorentz force impact. The study examines two cases. In Case 1, the left wall experiences heat dissipation via a heat sink at a fixed temperature T c , whilst the right wall is partly affected by the active chamber borders with a heater at temperature T h (where T h > T c ). The rest sections of vertical borders are adiabatic. In addition, the cavity is thermally insulated on both the upper and lower surfaces. In Case 2, the chamber's vertical sides are heated to a certain extent ( T h ), whereas the bottom wall is somewhat cold ( T c ) and has some level of activity. The remaining inactive sections of the cavity are adiabatic. The control flow equations were resolved with the help of COMSOL Multiphysics, which is complex modelling software for computational fluid dynamics (CFD). The computational study has been performed with the following parameters, Rayleigh number ( Ra ) = 10 3 –10 6 , Hartmann number ( Ha ) = 0–80, and nanoparticles volume fraction (ϕ) = 0.01, 0.02. The effect of important variables, such as Hartmann and Rayleigh numbers, in conjunction with the concentration of nano additives has been examined by analyzing streamlines and isotherms to understand their effect on thermal convection. It is found from the isotherms within the cavity in Case 2, that increment in Ha leads to slight rise the temperature within the cavity. Further, in Case 1, Nu avg is decreasing function of Ha and Q . While in Case 2, the average Nu is decreasing function of Q and increasing function of Ra and ϕ.

In this study, the effects of pressure anisotropy and viscosity on the propagation of shock waves in spin-polarized degenerate quantum magnetoplasma are studied under the influence of the streaming energy of ion beams. The effects of different suitable plasma parameters on the shock wave’s potential profile are studied using the steady state solution of the Zakharov–Kuznetsov–Burgers (Z–K–B) equation, as well as the numerical simulation of the governing non-linear Z–K–B equation. First-order analysis of the non-linear wave propagation depicted a new beam-induced stable mode whose Mach number may be subsonic or supersonic depending on the anisotropic pressure combination in the presence of different spin density polarization ratios. This is the first observation of this new beam-induced stable mode in ion beam plasma, apart from the other existing modes of ion beam plasma systems, namely, the fast beam mode, the slow beam mode, the inherent ion acoustic mode, and the coupled mode, which also has unique propagation characteristics compared to the other modes. The spin density polarization ratio of spin-up and spin-down electrons have an unprecedented effect on the polarity and the direction of propagation of different shock wave modes in such plasma systems. Apart from the spin effect, anisotropic pressure combinations, as well as the viscosity of ions and ion beams, also play an outstanding role in controlling the nature of propagation of shock waves, especially in the newly detected beam-induced stable mode, and depending on the viscosity parameters of ions and ion beams, both oscillatory and monotonic shock waves can propagate in such plasma.

In this work, we have studied the propagation of nonlinear electrostatic shock waves in an anisotropically pressured magnetized plasma containing positive and negative ions, trapped electrons, and positrons. A reductive perturbation technique is used to generate the trapped Zakharov-Kuznetsov Burgers’ equation is generated for nonlinear analysis. We study and debate the effects of positron and electron trapping on shock structures. In addition, the influence of an external magnetic field and an anisotropic pressure on the shock waves for both positive and negative ions is thoroughly discussed. It is shown that the amplitude and magnitude of the shock wave are affected by changes in the bulk viscosity and kinematic viscosity for both ions. The paper examines the unique characteristics of rarefactive shock-like formations under various plasma conditions, including ion bulk and kinematic viscosities, electrons and positron trapping with the influence of magnetic fields. This result of this study will provide a better understanding of the nonlinear transmission of ion acoustic shock waves in astrophysical environments, such as neutron stars and pulsar magnetospheres.

An efficient mathematical model of electroosmotic blood flow in a non-uniform wavy microvessel is investigated. In the present study, the microvessel is considered as an impermeable microchannel in which the Herschel–Bulkley (H–B) model of shear-thinning character is chosen to represent the complex flow of blood. An external electric field is applied along the channel length. Due to the negative charge of the glycocalyx layer located at the inner surface of the microchannel, an electric double layer is formed. As a result, an electric potential developed, which is described by the Poisson–Boltzmann equation. Eventually, the study analytically solves a boundary value problem to determine the axial velocity of H–B fluid flow by employing a long wavelength and low Reynolds number. Additionally, the analysis derives the volumetric flow rate in the microchannel across a single wavelength and stream function for the flow field. Using Mathematica symbolic software, graphs are plotted to visualize the impact of rheological features on the axial velocity, streamlines, and volumetric flow rate concerning various physical parameters such as H–B shear-thinning flow index, plug radius, Debye length, and Helmholtz–Smoluchowski velocity. It is found that the flow of blood becomes smoother as blood behaves more shear-thinning in nature, which is the key innovation of this work. Also, an increment in Debye length helps in increasing the size of fluid bolus remarkably, which adds the novelty of physics to this study. Such a model can have applications in canalicular flow, transport in human skin, fluid dialysis, and separation methods.

Our research demonstrates that positive and negative ions, along with trapped electrons and positrons, propagate as small-amplitude electrostatic shock waves in a magnetized collisions-less plasma. The trapped Korteweg-de Vries Burger's (T-KdVB) equation was derived by the widely recognized standard reductive perturbation technique (RPT), employing two separate scales of standard coordinates. The study examines the impact of temperature ratio on the shock wave characteristics of positive and negative ions, as well as positrons with electrons. The analysis and discussion focus on the impacts of negative ion number densities, as well as the number densities of electrons and positrons. The T-KdVB equations are responsible for governing rarefactive shock waves. Additionally, it is important to acknowledge that nonlinearity and dissipation play a key role in controlling the amplitude of the shock.

In this investigation, the exact solutions of variable coefficients of generalized Zakharov-Kuznetsov (ZK) equation and the Gardner equation are studied with the help of an extended generalizedG′/Gexpansion method. The main objective of this study is to establish the closed-form solutions and dynamics of analytical solutions to the generalized ZK equation and the Gardner equation. The generalized ZK equation and the Gardner equation govern the behavior of nonlinear wave phenomena in the presence of magnetic field in plasma dynamics, turbulence, bottom topography, and quantum field theory. We construct innovative solutions to the models under consideration using various computing tools and a recently developed extended generalizedG′/Gexpansion technique. The extended generalizedG′/Gexpansion technique is a well-defined and simple technique which is based on the initial assumed solutions of the polynomial ofG′/G. The derived solutions for both the equations are the hyperbolic, trigonometric, and rational functions. The obtained solutions have shock/kink waves and multisoliton, which depict the dynamical representations of the acquired solutions through the three-dimensional surface plots and the contour plots.

In this paper 2+1–dimensional Kadomtsev–Petviashvili (KP) equation with variable coefficients is investigated through the extended generalized G′G–expansion technique. One of the most universal model is KP equation, which is used to explain the ion acoustic waves in plasma physics, to model two dimensional shallow water waves, and in ferromagnetic, Bose–Einstein condensation and string theory. The obtained exact solutions of KP equation are in the form of hyperbolic function, trigonometric function, and rational function. With the aid of symbolic computational software Mathematica, the three dimensional surface plots with corresponding contour plots are provided for the obtained closed from solutions, which are of the form of solitary waves, multi solitons and periodic solitary wave like dynamical structures.

In this investigation, a different form of extended generalized G′G2-expansion technique with variable coefficient is proposed. The advantage of the extended generalized G′G2–expansion technique is that it can be used to solve nonlinear evolution equations with both constant and variable coefficients, whereas the basic G′G2 method can only be used with constant coefficients. The proposed technique is used to solve the Bogoyavlensky–Konopelchenko (BK) equation with variable coefficients, which depicts the interaction of a Riemann wave propagating along the y-axis and a long wave propagating along the x-axis in a fluid. Further, the BK equation with variable coefficients is applied for stratified internal waves, shallow-water waves, ion-acoustic waves, and water propagation in a liquid. The hyperbolic, trigonometric, and rational form solutions of the BK equation are dynamically represented as the annihilation of three-dimensional kink waves, multi-soliton waves, single solitons, and so on. Furthermore, the derived solutions of the considered equation, which comprise arbitrary functional parameters and other constant parameters, can be used to enhance the advanced behaviours of physical situations.

The Schamel Burgers’ equation is a nonlinear partial differential equation which produces the shock type traveling waves in extraordinary physical situations. Previously, this sort of equation is solved with well known tan−hyperbolicmethod. In this paper, we present the exact solutions of the Schamel Burgers’ equation using generalized-improved G′Gand generalized G′Gexpansion methods with the assistance of Mathematica-12 and observe the behavior of the solutions both analytically and numerically with requisite graphs. The solutions of mentioned equation admits the shock like dynamical structures.

The extended generalized G′G–expansion is well defined and an efficient technique, which is used to obtain the exact traveling wave solutions to the governing nonlinear equations with constant coefficients as well as variable coefficients. In this paper, the modified Korteweg–de Vries (mKdV) equation and Burgers equation with variable coefficients are investigated through the extended generalized G′G–expansion method, which are exceptional cases of the nonlinear evolution equations widely used in a two-layer fluid system, in fluid-filled elastic tubes, in an atmospheric and oceanic dynamical system, traffic flow, turbulence in fluid dynamics, dusty plasma, ion-acoustic waves in a plasma system. New families of exact closed-form solutions are obtained in hyperbolic, trigonometric, and rational function solutions with the available free constants. With the help of computerized symbolic computation work, the newly formed closed-form solutions are validated by back substituting them into the equations using the computational mathematical software. Furthermore, the graphical representations of all these obtained solutions are discussed and demonstrated by giving the suitable best values of arbitrary functions and constants via three-dimensional surface and density plots. The dynamics of the solution profiles demonstrate the annihilation of 3D kink-type soliton waves, shock waves, double solitons, and multi-soliton wave structures.

The nonlinear solitary wave in a magnetized plasma with the combined effect of ion collision and trapped electrons distribution is investigated. Using the reductive perturbation technique, the damped modified Schamel–Zakarov–Kuznetsov nonlinear wave equation is derived and analyzed numerically. Solitary wave propagation through collisional plasma medium is analyzed for different parameters such as free to trapped electrons temperature ratio, collisional term, positive and negative ion density ratio and ion mass ratio. It is observed that the increase in free to trapped electron temperature ratio enhances the speed of the solitons toward its direction of propagation where we observed the formation of fast wave mode, and the whole analysis has been carried out for fast wave mode. While the collisional effect due to ions and neutral particles significantly reduces the amplitude. The findings of this work will be helpful in understanding trapped electrons effect in collisional plasmas, containing negative ions.