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 threedimensional graphics for seeking model solutions under the influence of friction and viscosity, qualitative agreements with previous related works are shown.
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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 precollision (\(m/s\))
 \(v^{\prime}_{1} ,v^{\prime}_{2}\) :

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
References
K. Lu, E.E. Brodsky, H.P. Kavehpour, A thermodynamic unification of jamming. Nature 4, 404–407 (2008)
H. Jaeger, S. Nagel, Introduction to the focus issue on granular materials. Chaos 9, 509–510 (1999)
I.S. Aranson, L.S. Tsimring, Patterns and collective behavior in granular media: Theoretical concepts. Rev. Mod. Phys. 78, 641–692 (2006)
S. Douady, S. Fauve, C. Laroche, Subharmonic instabilities and defects in a granular layer under vertical vibrations. Europhys. Lett. 8, 621–627 (1989)
J.M. Ottine, D.V. Khakhar, Mixing and segregation of granular materials. Ann. Rev. Fluid Mech. 32, 55–91 (2000)
R. Ramirez, D. Risso, P. Cordero, Thermal convection in fluidized granular systems. Phys. Rev. Lett. 85, 1230–1233 (2000)
M.G. Clerc, P. Cordero, J. Dunstan, K. Huff, N. Mujica, D. Risso, G. Varas, Liquid–solidlike transition in quasionedimensional driven granular media. Nature 4, 249–254 (2008)
M. Argentina, M. Clerc, R. Soto, Van der Waalslike transition in fluidized granular matter. Phys. Rev. Lett. 89, 044301 (2002)
C. Cartes, M.G. Clerc, R. Soto, van der Waals normal form for a onedimensional hydrodynamic model. Phys. Rev. E 70, 031302 (2004)
A.M. Abourabia, A.M. Morad, Exact travelling wave solutions of the van der Waals normal form for fluidized granular matter. Phys. A 437, 333–350 (2015)
D. Blair, I.S. Aranson, G.W. Crabtree, V. Vinokur, L.S. Tsimring, C. Josserand, Patterns in thin vibrated granular layers: Interfaces, hexagons, and superoscillons. Phys. Rev. E. 61(5), 5600–5610 (2000)
P. Duru, E. Guazzelli, Experimental investigation on the secondary instability of liquidfluidized beds and the formation of bubbles. J. Fluid Mech. 470, 359–382 (2002)
E.W.C. Lim, Voidagewaves in hydraulic conveying through narrow pipes. Chem. Eng. Sci. 62, 4529–4543 (2007)
A.M. Abourabia, T.S. ElDanaf, A.M. Morad, Exact solutions of the hierarchical Korteweg–de Vries equation of microstructured granular materials. Chaos Solitons Fractals 41, 716–726 (2009)
A.M. Abourabia, K.M. Hassan, A.M. Morad, Analytical solutions of the Magma equations for molten rocks in a granular matrix. Chaos Solitons Fractals 42, 1170–1180 (2009)
N. Mujica, R. Soto, Dynamics of noncohesive confined granular medi, in Recent Advances in Fluid Dynamics with Environmental Applications. ed. by J. Klapp, L. Sigalotti, A. Medina, A. López, G. RuizChavarría (Springer, New York, 2016), pp. 445–463
M. Guzmán, R. Soto, Critical phenomena in quasitwodimensional vibrated granular systems. Phys. Rev. E. 97, 012907 (2018)
R. Brito, R. Soto, V. Garzó, Energy nonequipartition in a collisional model of a confined quasitwodimensional granular mixture. Phys. Rev. E 102, 052904 (2020)
R. Conte, M. Musette, The Painlevé Handbook (Springer Science Business Media B.V, Berlin, 2008).
A.M. Abourabia, K.M. Hassan, E.S. Selima, The derivation and study of the nonlinear schrdinger equation for long waves in shallow water using the reductive perturbation and complex ansatz methods. Int. J. Nonlinear Sci. 9(4), 430–443 (2010)
E.S. Selima, A. Seadawy, Y. Xiaohua, The nonlinear dispersive DaveyStewartson system for surface waves propagation in shallow water and its stability. European Phys. J. Plus 131(425), 1–16 (2016)
E.S. Selima, X. Yao, A. Wazwaz, Multiple and exact soliton solutions of the perturbed Korteweg–de Vries equation of long surface waves in a convective fluid via Painlevé analysis, factorization, and simplest equation methods. Phys. Rev. E 95(6), 062211 (2017)
E.S. Selima, Y. Mao, Y. Xiaohua, A.M. Morad, T. Abdelhamid, B.I. Selim, Applicable symbolic computations on dynamics of smallamplitude long waves and DaveyStewartson equations in finite water depth. Appl. Math. Model. 57, 376–390 (2018)
S.A. Mohammadein, A.K. AbuNab, G.A. Shalaby, The behavior of vapour bubbles inside a vertical cylindrical tube under the effect of peristaltic motion with twophase density flow and heat transfer. J Nanofluids 6, 1–6 (2017)
A.K. AbuNab, E.S. Selima, A.M. Morad, Theoretical investigation of a single vapor bubble during Al_{2}O_{3}/H_{2}O nanofluids in powerlaw fluid affected by a variable surface tension. Phys. Scr. 96(3), 035222 (2021)
S.A. Mohammadein, G.A. Shalaby, A.F. AbuBakr, A.K. AbuNab, Analytical solution of gas bubble dynamics between twophase flow. Results Phys. 7, 2396–3403 (2017)
S. Chapman, T.G. Cowling, The Mathematical Theory of NonUniform Gases (Cambridge University Press, Cambridge, 1970).
J.J. Brey, F. Moreno, J.W. Dufty, Model kinetic equation for lowdensity granular flow. Phys. Rev. E 54(1), 445–456 (1996)
A. Santos. From gases to glasses in granular matter: thermodynamics and hydrodynamic aspect, (Sphinx, 2005).
J.J. Brey, P. Maynar, M.I. García de Soria, Kinetic model for a confined quasitwodimensional gas of inelastic hardspheres. J. Stat. Mech. 3, 034002 (2020)
V. Grazó, A. Santos, J.M. Montanero, Modified sonine approximation for the NavierStokes transport coefficients of a granular gas. Phys. A 376, 94–107 (2007)
A. Puglisi, Transport and Fluctuations in Granular Fluids: From Boltzmann Equation to Hydrodynamics (Springer International Publishing, Diffusion and Motor Effects, 2015).
E.M. Elsaid, T.Z. Abdel Wahid, A.M. Morad, Exact solutions of plasma flow on a rigid oscillating plate under the effect of an external nonuniform electric field. Results in Physics 19, 103554 (2020)
T.Z. Abdel Wahid, A.M. Morad, Unsteady plasma flow near an oscillating rigid plane plate under the influence of an unsteady nonlinear external magnetic field. IEEE Access 8, 76423–76432 (2020)
T. Z. Abdel Wahid and A. M. Morad. On Analytical Solution of a Plasma Flow over a Moving Plate under the Effect of an Applied Magnetic Field. Adv. Math. Phys. Volume 2020, Article ID 1289316 (2020).
M.G. Clerc, D. Escaff, Solitary waves in van der Waalslike transition in fluidized granular matter. Phys. A 371(1), 33–36 (2006)
Acknowledgements
This project is supported financially by the Academy of Scientific Research and Technology (ASRT), Egypt, Grant No. 6567.
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Morad, A.M., Selima, E.S. & AbuNab, 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/s13360021012773
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DOI: https://doi.org/10.1140/epjp/s13360021012773