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

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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.

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Data Availability

The data used to support the findings of this study are included within the article.

Abbreviations

\(s\) :

Dissipation coefficient

\(R\) :

Restitution coefficient

\(\hat{e}\) :

Unite vector in the direction joining of two colliding particles

\(\gamma\) :

Diameter of sphere (\(m\))

\(v_{i}\) :

Velocities of sphere pre-collision (\(m/s\))

\(v^{\prime}_{1} ,v^{\prime}_{2}\) :

Velocities of the sphere after collision (\(m/s\))

\(m\) :

Mass of grains (\(kg\))

\(\Xi\) :

Total collision term

\(\omega\) :

Angular frequencies (\(Hz\))

\(A\) :

Amplitude (\(m\))

\(N\) :

Total number of grains

\(\ell\) :

Aspect ratio

\(h_{x} ,h_{y}\) :

Dimensions of the box containing grains in the direction \(x\) and \(y\) (\(m\))

\(n_{0}\) :

Number density

\(n\left( {r,t} \right)\) :

Local number density (\(m\))

\(u\left( {r,t} \right)\) :

Local average velocity (velocity of flow) (\(m/s\))

\(\psi \left( {r,t} \right)\) :

Test function

\(V_{p}\) :

Phase velocity (\(m/s\))

\(V_{g}\) :

Group velocity (\(m/s\))

\(\beta\) :

\(\left( {k_{\beta } T} \right)^{ - 1}\) (\(J^{ - 1}\))

\(k_{\beta }\) :

Boltzmann constant (\(J/k\))

\(\kappa\) :

Thermal conductivity (\(w/mk\))

\(x^{\prime}\) :

Dimensionless coordinates

\(x\) :

Spatial coordinate in the horizontal direction of the FGM (\(m\))

\(\rho_{0}\) :

Density at the Maxwell point (\(kg/m^{3}\))

\(\rho\) :

Density (fraction of area that occupied by the grains) (\(kg/m^{2}\))

\(\varphi\) :

Distribution function

\(\eta\) :

Effective viscosity (\(N.s/m^{2}\))

\(\nu\) :

Effective shear viscosity (Pa)

\(\chi\) :

Bifurcation parameter

\(C\) :

Peculiar velocity

\(\overline{P}\) :

Averaged pressure

\(M\) :

Horizontal momentum (\(kg.m/s\))

\(\mu\) :

Transport coefficient (\(kg.m/s\))

\(p_{i}\) :

Momentum quantity (\(kg.m/s\))

\(Kn\) :

Standard Knudsen number

\(\tilde{u}\) :

The complex amplitude of the wave (\(m\))

\(u\) :

Critical average vertical density (\(kg/m^{3}\))

\(q\) :

Heat flux (\(kg/s^{3}\))

\(\lambda\) :

Friction coefficient

\(k\) :

Wavenumber (\(m^{ - 1}\))

\(\lambda^{*}\) :

Wavelength (m)

\(D_{t} = \frac{\partial }{{\partial {\text{t}}}} + u.\nabla\) :

Material derivative

\(P_{ij} \left[ {r,\left. t \right|\varphi } \right]\) :

Pressure tensor

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Acknowledgements

This project is supported financially by the Academy of Scientific Research and Technology (ASRT), Egypt, Grant No. 6567.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El-Koom, 32511, Egypt

    Adel M. Morad, Ehab S. Selima & Ahmed K. Abu-Nab

  2. Department of Computational Mathematics and Mathematical Physics, Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Rostov on Don, 344090, Russia

    Adel M. Morad

  3. Fluid Dynamics and Seismics Lab, Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Moscow region, Russia

    Ahmed K. Abu-Nab

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  1. Adel M. Morad
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Correspondence to Adel M. Morad.

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Morad, A.M., Selima, E.S. & Abu-Nab, A.K. Bubbles interactions in fluidized granular medium for the van der Waals hydrodynamic regime. Eur. Phys. J. Plus 136, 306 (2021). https://doi.org/10.1140/epjp/s13360-021-01277-3

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