In this paper, an analytical and numerical technique are examined in order to analyse the Stokes flow determination problem due to a viscous sphere droplet moving at a concentric instantaneous position inside a spherical interface separating finite and semi-infinite immiscible fluid phases. Here, when only one of the three phases of the fluid (micropolar fluid) has a microstructure, attention is focused on this case. The motion is considered when Reynolds- and capillary-numbers are low, and the droplet surface and the fluid-fluid interface have insignificant deformation. A general solution is obtained in a spherical coordinate system based on a concentric position to analyse the slow axisymmetric movement of the micropolar fluid, considering microrotation and velocity components. Boundary conditions are initially fulfilled at the fluid-fluid interface and subsequently at the droplet surface. The normalised hydrodynamic drag force applying to a moving viscous droplet appears to be a function of the droplet-to-interface radius ratio, which increases monotonically and becomes unbounded when the droplet surface touches the fluid-fluid interface. The numerical outcomes of the normalised drag force acting on the viscous droplet are derived for different values of the parameters, and are presented in a tabular and graphical framework. A comparison was made between our numerical outcomes for the drag force and the pertinent data for the special cases found in the literature.

The aim of the present study is to investigate the damping of slow sausage MHD waves propagating in a gravitationally-stratified magnetic cylindrical structure when the plasma is strongly partially ionised. The problem is treated as an initial value problem and the analysis deals with the temporal evolution of waves in an asymptotic sense, i.e., large values of time compared to the period of waves. The plasma is assumed to be collision-dominated, i.e., we employ a two-fluid approximation. The set of equations describing the plasma dynamics is reduced to a coupled partial differential equations. Our findings show that the slow wave of charged species is affected by the presence of a cut-off. The mode associated with the neutral fluid propagates without any cut-off and decay very quickly due to collisions between particles.

We utilize a fluid kinetic hybrid approach to analyze dust acoustic wave propagation in collisionless, unmagnetized dusty plasmas, specifically investigating how linear Landau damping effects in the ion population affect weakly nonlinear and weakly dispersive wave behavior. An electron-depleted plasma is considered, consisting of a cold fluid of negatively charged dust particles and two types of ions at different temperatures, modelled by a kappa-type distribution. Anticipating nonlinear solitary waves, a reductive perturbation technique is employed, leading to a nonlinear partial differential equation for the electrostatic potential in the form of a modified Korteweg–de Vries (mKdV) equation, featuring an additional term to account for linear Landau damping of the ions. The solitary wave's amplitude is found to decay with time. A parametric analysis is carried out of the impact of the plasma configuration on the Landau damping rate under the influence of this latter (Landau damping-related) term. The results of this work are highlighted in space plasma, such as that around Enceladus and Saturn's E ring, where the occurrence of Landau damping in combination with a nonthermal ion distribution may affect wave propagation significantly.

This study investigates magnetohydrodynamic (MHD) boundary-layer flow, heat transfer, and entropy generation in a ternary hybrid nanofluid composed of $Cu–Mo{S}_2-GO$ nanoparticles dispersed in an ethylene glycol-water mixture flowing over an expanding porous sheet. Using similarity transformations, the governing partial differential equations are reduced to a coupled system of nonlinear ordinary differential equations and solved numerically. The effects of key physical parameters-including magnetic field strength, nanoparticle volume fraction, particle shape, porous expansion rate, and unsteadiness-on velocity, temperature, skin friction, Nusselt number, and entropy generation are examined in detail. In addition, an artificial neural network (ANN) model is developed to accurately predict the flow and thermal characteristics, demonstrating excellent agreement with numerical results. The findings reveal that nanoparticle shape and concentration significantly influence heat transfer and entropy production, while magnetic and slip effects suppress momentum transport. The combined numerical and ANN-based framework provides an efficient and reliable tool for analyzing complex MHD nanofluid systems.

This article presents an analytical and numerical investigation on the quasi-steady, slow flow generated by the movement of a micropolar fluid drop sphere of at a concentrical position within another immiscible viscous fluid inside a spherical slip cavity. Additionally, the effect of a cavity with slip friction along with the change in the micropolarity parameter on the movement of the fluid sphere is introduced. When Reynolds numbers are low, the droplet moves along a diameter that connects their centres. The governing and constitutive differential equations are reduced to a computationally convenient form using appropriate transformations. By using the resulting linear partial differential equations for the stream functions and using the method of separation variables, we can obtain their solutions. General solutions for velocity fields are found using spherical coordinate systems, which are based on the concentric point of the cavity; this allows to obtain solutions to the Navier-Stokes equations internal and external to the spherical droplet. The vorticity-microrotation boundary condition is used in regard to the micropolar droplet case in a viscous fluid. The normalised drag forces acted upon the micropolar drop are illustrated via graphs and tables for diverse values of the viscosity ratio and drop-to-wall radius ratio, with the change of the spin parameter that attaches the microrotation to vorticity. The correction wall factor is shown to increase with an increase in the drop-to-wall radius ratio, when moving from the gas bubble case to the solid sphere case, with an increase in the micropolarity parameter, and with an increase in the slip frictional resistance. This study is relevant due to its potential uses in a variety of biological, natural, and industrial processes, including the creation of raindrops, the investigation of blood flow, fluid-fluid extraction, the forecasting of weather conditions, the rheology of emulsions, and sedimentation phenomena.

This paper presents a comprehensive mathematical model for analyzing the dynamics of illicit drug consumption within a population. Building upon and extending classical epidemiological models such as the susceptible, infected, and recovered (SIR) framework, the proposed model categorizes individuals into four compartments: Non-susceptible, susceptible, addicted, and rehabilitated. The model incorporates key social factors such as peer influence, intervention efforts, and the probability of relapse. A nonlinear system of differential equations was developed to describe the transitions between these states. The basic reproduction number $R_0$ was derived to assess the potential spread of addiction and stability analysis of the equilibrium points was carried out. Numerical simulations explore the impact of various intervention levels, highlighting how increased societal and family involvement can reduce the prevalence of addiction. Furthermore, an optimal control problem was formulated and solved using Pontryagin's maximum principle to determine the most cost-effective intervention strategy over time. The results demonstrate that adaptive, well-balanced policies can significantly reduce both susceptibility and addiction rates while minimizing social and economic costs.