Skip to main content
SpringerLink
Log in

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

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

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.

Graphic abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

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

References

  1. K. Lu, E.E. Brodsky, H.P. Kavehpour, A thermodynamic unification of jamming. Nature 4, 404–407 (2008)

    Google Scholar 

  2. H. Jaeger, S. Nagel, Introduction to the focus issue on granular materials. Chaos 9, 509–510 (1999)

    Article  ADS  Google Scholar 

  3. I.S. Aranson, L.S. Tsimring, Patterns and collective behavior in granular media: Theoretical concepts. Rev. Mod. Phys. 78, 641–692 (2006)

    Article  ADS  Google Scholar 

  4. S. Douady, S. Fauve, C. Laroche, Subharmonic instabilities and defects in a granular layer under vertical vibrations. Europhys. Lett. 8, 621–627 (1989)

    Article  ADS  Google Scholar 

  5. J.M. Ottine, D.V. Khakhar, Mixing and segregation of granular materials. Ann. Rev. Fluid Mech. 32, 55–91 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  6. R. Ramirez, D. Risso, P. Cordero, Thermal convection in fluidized granular systems. Phys. Rev. Lett. 85, 1230–1233 (2000)

    Article  ADS  Google Scholar 

  7. M.G. Clerc, P. Cordero, J. Dunstan, K. Huff, N. Mujica, D. Risso, G. Varas, Liquid–solid-like transition in quasi-one-dimensional driven granular media. Nature 4, 249–254 (2008)

    Google Scholar 

  8. M. Argentina, M. Clerc, R. Soto, Van der Waals-like transition in fluidized granular matter. Phys. Rev. Lett. 89, 044301 (2002)

    Article  ADS  Google Scholar 

  9. C. Cartes, M.G. Clerc, R. Soto, van der Waals normal form for a one-dimensional hydrodynamic model. Phys. Rev. E 70, 031302 (2004)

    Article  ADS  Google Scholar 

  10. 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)

    Article  MathSciNet  Google Scholar 

  11. 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)

    Article  ADS  Google Scholar 

  12. P. Duru, E. Guazzelli, Experimental investigation on the secondary instability of liquid-fluidized beds and the formation of bubbles. J. Fluid Mech. 470, 359–382 (2002)

    Article  ADS  Google Scholar 

  13. E.W.C. Lim, Voidagewaves in hydraulic conveying through narrow pipes. Chem. Eng. Sci. 62, 4529–4543 (2007)

    Article  Google Scholar 

  14. A.M. Abourabia, T.S. El-Danaf, A.M. Morad, Exact solutions of the hierarchical Korteweg–de Vries equation of microstructured granular materials. Chaos Solitons Fractals 41, 716–726 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  15. 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)

    Article  ADS  Google Scholar 

  16. 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. Ruiz-Chavarría (Springer, New York, 2016), pp. 445–463

    Chapter  Google Scholar 

  17. M. Guzmán, R. Soto, Critical phenomena in quasi-two-dimensional vibrated granular systems. Phys. Rev. E. 97, 012907 (2018)

    Article  ADS  Google Scholar 

  18. R. Brito, R. Soto, V. Garzó, Energy nonequipartition in a collisional model of a confined quasi-two-dimensional granular mixture. Phys. Rev. E 102, 052904 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  19. R. Conte, M. Musette, The Painlevé Handbook (Springer Science Business Media B.V, Berlin, 2008).

    MATH  Google Scholar 

  20. 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)

    MathSciNet  MATH  Google Scholar 

  21. E.S. Selima, A. Seadawy, Y. Xiaohua, The nonlinear dispersive Davey-Stewartson system for surface waves propagation in shallow water and its stability. European Phys. J. Plus 131(425), 1–16 (2016)

    Google Scholar 

  22. 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)

    Article  ADS  MathSciNet  Google Scholar 

  23. E.S. Selima, Y. Mao, Y. Xiaohua, A.M. Morad, T. Abdelhamid, B.I. Selim, Applicable symbolic computations on dynamics of small-amplitude long waves and Davey-Stewartson equations in finite water depth. Appl. Math. Model. 57, 376–390 (2018)

    Article  MathSciNet  Google Scholar 

  24. S.A. Mohammadein, A.K. Abu-Nab, G.A. Shalaby, The behavior of vapour bubbles inside a vertical cylindrical tube under the effect of peristaltic motion with two-phase density flow and heat transfer. J Nanofluids 6, 1–6 (2017)

    Article  Google Scholar 

  25. A.K. Abu-Nab, E.S. Selima, A.M. Morad, Theoretical investigation of a single vapor bubble during Al2O3/H2O nanofluids in power-law fluid affected by a variable surface tension. Phys. Scr. 96(3), 035222 (2021)

    Article  Google Scholar 

  26. S.A. Mohammadein, G.A. Shalaby, A.F. Abu-Bakr, A.K. Abu-Nab, Analytical solution of gas bubble dynamics between two-phase flow. Results Phys. 7, 2396–3403 (2017)

    Article  ADS  Google Scholar 

  27. S. Chapman, T.G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, Cambridge, 1970).

    MATH  Google Scholar 

  28. J.J. Brey, F. Moreno, J.W. Dufty, Model kinetic equation for low-density granular flow. Phys. Rev. E 54(1), 445–456 (1996)

    Article  ADS  Google Scholar 

  29. A. Santos. From gases to glasses in granular matter: thermodynamics and hydrodynamic aspect, (Sphinx, 2005).

  30. J.J. Brey, P. Maynar, M.I. García de Soria, Kinetic model for a confined quasi-two-dimensional gas of inelastic hardspheres. J. Stat. Mech. 3, 034002 (2020)

    Article  Google Scholar 

  31. V. Grazó, A. Santos, J.M. Montanero, Modified sonine approximation for the Navier-Stokes transport coefficients of a granular gas. Phys. A 376, 94–107 (2007)

    Article  Google Scholar 

  32. A. Puglisi, Transport and Fluctuations in Granular Fluids: From Boltzmann Equation to Hydrodynamics (Springer International Publishing, Diffusion and Motor Effects, 2015).

    Book  Google Scholar 

  33. 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 non-uniform electric field. Results in Physics 19, 103554 (2020)

    Article  Google Scholar 

  34. 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)

    Article  Google Scholar 

  35. 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).

  36. M.G. Clerc, D. Escaff, Solitary waves in van der Waals-like transition in fluidized granular matter. Phys. A 371(1), 33–36 (2006)

    Article  Google Scholar 

Download references

Acknowledgements

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

Author information

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

Authors
  1. Adel M. Morad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ehab S. Selima
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ahmed K. Abu-Nab
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adel M. Morad.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-021-01277-3

Search

Navigation