Eur. Phys. J. Plus (2021) 136: 306
https://doi.org/10.1140/epjp/s13360-021-01277-3

Regular Article

Bubbles interactions in fluidized granular medium for the van der Waals hydrodynamic regime

¹ Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, 32511, Shebin El-Koom, Egypt
² Department of Computational Mathematics and Mathematical Physics, Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, 344090, Rostov on Don, Russia
³ Fluid Dynamics and Seismics Lab, Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Moscow region, Russia

^a dr_adel_morad@yahoo.com

Received: 29 January 2021
Accepted: 24 February 2021
Published online: 11 March 2021

Abstract

This paper investigates the fluidized granular materials (FGM) with the van der Waals normal form (VDWF) under the effects of friction and viscosity. The system of macroscopic balance is presented, including the mass, momentum, and energy equations of local densities. For two different types of collisions, elastic and inelastic collisions, analytical solutions of the nonlinear PDEs governing the granular model are investigated using the hydrodynamic equations for granular matter motion. The integrability of the proposed model is analyzed by applying the Painlevé analysis. Moreover, the Bäcklund transformation (BT) is established using the Painlevé truncation expansion. New traveling wave solutions of the VDWF within FGM are obtained by using the BT, tanh function, Jacobi elliptic function methods to study the phase separation phenomenon. As two pairs of rarefaction and shock waves emerge and travel away giving the appearance of bubbles, the resulting solutions of the proposed model show a behavior similar to those found in the molecular dynamic simulations. The dispersion relation and their properties to the model equation are investigated. Besides, stability analysis of the VDWF in its ODE form is demonstrated using the phase portrait classifications. Finally, using two- and three-dimensional graphics for seeking model solutions under the influence of friction and viscosity, qualitative agreements with previous related works are shown.