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This paper presents an investigation of magnetohydrodynamics (MHD) Casson nanofluid flow along a stretchable surface through a permeable medium. The modeling of the physical phenomena is considered with impact of thermal radiation, heat generation, slip conditions and suction. Transformations of the governing set of mathematical equations for the physical model are carried out into nonlinear ordinary differential equations (ODEs) with appropriate similarity variables. The nonlinear ODE solutions are carried out using the optimal homotopy analysis technique (OHAM), and the findings are presented for determining the influences of the emerging important parameters. The results indicate that velocity field increases in respect of porosity parameter, Casson fluid parameter and magnetic parameter while it declines for enhancing velocity slip and suction parameters. The temperature profile shows rising behavior for heat source, Prandtl number, thermophoresis, radiation and Brownian motion parameters while it declines for enhancing thermal slip parameter. Moreover, the concentration profile enhances for rise in Brownian motion parameter while it reduces for Schmidt number and nanoparticle parameter. We also showed the accuracy of the present results by indicating that skin friction values for varied magnetic parameters agree with earlier findings in the literature.

Studying the dynamics of beams subjected to a moving mass is important due to their wide applications, including railways, machining processes, and microelectromechanical systems (MEMS). Various numerical and analytical approaches have been used for modeling such structures. In this analytical study, we have used a combination of the Optimal homotopy analysis method (Optimal HAM) and enriched multiple scales (MS) to analytically study the dynamics of a simply supported Euler–Bernoulli beam traversed by a moving mass and resting on a viscoelastic foundation. The viscoelastic foundation contributes to the modeling by adding a linear and nonlinear term to the formulation. Further, we have considered a fifth-order nonlinear term to account for the bending vibration of the flexible beam. Using the Galerkin method, we have formed the corresponding ordinary differential equation (ODE). Then, we used the enriched MS Optimal HAM to calculate the dynamic response of the beam. After validating our method by comparing our results with the dynamic results of the beam obtained from finite element analysis (FEA), we investigated the effects of the determining parameters on the beam dynamic response. The effects of the foundation nonlinear and linear terms, the moving load weight, and its velocity have been investigated by studying the variation of the normalized beam lateral deflection versus the normalized moving mass instantaneous position in each case. We showed that the difference between linear and nonlinear modeling results is pronounced, and it becomes more pronounced for faster and heavier moving loads.

This paper deals with approximate homotopy series solution of imbibition phenomenon occurring in multiphase flow during the secondary oil recovery process. In heterogeneous porous media, the geometry of pores is irregular while in homogeneous porous media, the geometry of pores is uniformly same. The comparative study of counter-current imbibition phenomenon in heterogeneous and homogeneous porous medium has been also discussed. The governing partial differential equation obtained by mathematical formation of imbibition phenomenon has been solved by the optimal homotopy analysis method. The numerical as well as graphical interpretation of the solution have been given.