Skip to main content
Springer Nature Link
Log in

Inviscid Suspension Flow along a Flat Boundary

  • MATHEMATICAL PHYSICS
  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

A previously developed self-consistent field method is used to study an arbitrary finite set of identical spherical particles of arbitrary density moving in a uniform inviscid incompressible flow specified at infinity in the presence of a flat wall. For a given initial particle distribution in space, expressions for the particle and fluid velocities are derived taking into account the collective hydrodynamic interaction of the particles with each other and the wall. For a statistically uniform particle distribution in a semibounded inviscid fluid, analytical averaged particle and fluid velocity profiles are obtained in the first approximation with respect to the particle volume fraction in the suspension.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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.

Similar content being viewed by others

REFERENCES

  1. W. M. Hicks, “On the motion of two spheres in a fluid,” Philos. Trans. 171, 455–492 (1880).

    Article  Google Scholar 

  2. A. B. Basset, “On the motion of two spheres in a liquid, and allied problems,” Proc. London Math. Soc. 18, 369–377 (1886).

    Article  MathSciNet  Google Scholar 

  3. E. Cunningham, “On the velocity of steady fall of spherical particles through fluid medium,” Proc. R. Soc. London A 83, 357–365 (1910).

    Article  Google Scholar 

  4. N. Zuber, “On the dispersed two-phase flow in the laminar flow regime,” Chem. Eng. Sci. 19, 897–917 (1964).

    Article  Google Scholar 

  5. L. Wijngaarden and D. J. Jeffrey, “Hydrodynamic interaction between gas bubbles in liquid,” J. Fluid Mech. 77 (1), 27–44 (1976).

    Article  Google Scholar 

  6. B. U. Felderhof, “Virtual mass and drag in two-phase flow,” J. Fluid Mech. 225, 177–196 (1991).

    Article  MathSciNet  Google Scholar 

  7. V. V. Struminskii, O. B. Gus’kov, and G. A. Korol’kov, “Hydrodynamic interaction of particles in potential flow of an ideal fluid,” Sov. Phys. Dokl. 31 (10), 787–789 (1986).

    MATH  Google Scholar 

  8. O. B. Gus’kov and B. V. Boshenyatov, “Hydrodynamic interaction of spherical particles in an inviscid-fluid flow,” Dokl. Phys. 56 (6), 352–354 (2011).

    Article  Google Scholar 

  9. O. B. Gus’kov and B. V. Boshenyatov, “Interaction of phases and virtual mass of dispersed particles in potential flows of fluid,” Vestn. Nizhegorod. Univ. im. Lobachevskogo, No. 4–3, 740–741 (2011).

  10. O. B. Gus’kov, “Virtual mass of a solid moving through a suspension of spherical particles,” Dokl. Phys. 57 (1), 29–32 (2012).

    Article  Google Scholar 

  11. O. B. Gus’kov, “The virtual mass of a sphere in a suspension of spherical particles,” J. Appl. Math. Mech. 76 (1), 93–97 (2012).

    Article  MathSciNet  Google Scholar 

  12. O. B. Gus’kov, “The motion of a cluster of spherical particles in an ideal fluid,” J. Appl. Math. Mech. 78 (2), 126–131 (2014).

    Article  MathSciNet  Google Scholar 

  13. O. B. Gus’kov, “The concept of a self-consistent field as applied to the dynamics of inviscid suspensions,” Proceedings of the 10th International Conference on Nonequilibrium Processes in Nozzles and Jets (NPNJ'2014), May 22–31, 2014, Alushta (Mosk. Aviats. Inst., Moscow, 2014), pp. 87–89.

  14. O. B. Gus’kov, “On the virtual mass of a rough sphere,” J. Appl. Math. Mech. 81 (4), 325–333 (2017).

    Article  MathSciNet  Google Scholar 

  15. S. K. Zaripov, M. V. Vanyunina, A. N. Osiptsov, and E. V. Skvortsov, “Calculation of concentration of aerosol particles around a slot sampler,” Atm. Environ. 41 (23), 4773–4780 (2007).

    Article  Google Scholar 

  16. Shuyan Wang, Jin Sun, Qian Yang, Yueqi Zhao, Jinsen Gao, and Yang Liu, “Numerical simulation of flow behavior of particles in an inverse liquid-solid fluidized bed,” Powder Technol. 261, 14–21 (2014).

    Article  Google Scholar 

  17. Shuai Wang, Huilin Lu, Qinghong Zhang, Guodong Liu, Feixiang Zhao, and Liyan Sun, “Modeling of bubble-structure-dependent drag for bubbling fluidized beds,” Ind. Eng. Chem. Res. 53 (40), 15776–15785 (2014).

    Article  Google Scholar 

  18. Shuyan Wang, Xiaoxue Jiang, Ruichen Wang, Xu Wang, Shanwen Yang, Jian Zhao, and Yang Liu, “Numerical simulation of flow behavior of particles in a liquid-solid stirred vessel with baffles,” Adv. Powder Technol. 28, 1611–1624 (2017).

    Article  Google Scholar 

  19. Wen-rui Wang, Zhao Li, Jia-ming Zhang, and Han-lin Li, “Simulation study of particle-fluid two-phase coupling flow field and its influencing factors of crystallization process,” Chem. Pap. 72 (12), 3105–3117 (2018).

    Article  Google Scholar 

  20. P. van Beek, “A counterpart of Faxen’s formula in potential flow,” Int. J. Multiphase Flow 11 (6), 873–879 (1985).

    Article  Google Scholar 

  21. G. K. Batchelor, “Sedimentation in a dilute dispersion of spheres,” J. Fluid Mech. 52 (2), 245–268 (1972).

    Article  Google Scholar 

  22. C. W. J. Beenakker and P. Mazur, “Is sedimentation container-shape dependent?” Phys. Fluids 28 (11), 3203–3206 (1985).

    Article  Google Scholar 

  23. O. B. Gus’kov and A. V. Zolotov, “Sedimentation of a suspension of spherical particles in a cylinder,” J. Appl. Math. Mech. 51 (6), 745–748 (1987).

    Article  Google Scholar 

  24. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Nauka, Moscow, 1986; Butterworth-Heinemann, Oxford, 1987).

  25. B. V. Boshenyatov, “The theory of electric and thermal conductivity in bubble gas-liquid media,” Dokl. Phys. 59 (12), 601–603 (2014).

    Article  Google Scholar 

  26. B. V. Boshenyatov, “The contribution of interactions of spherical inclusions into electrical and thermal conductivity of composite materials,” Compos. Mech. Comput. Appl. Int. J. 7 (2), 95–104 (2016).

    Article  Google Scholar 

  27. B. V. Boshenyatov, “The role of particle interaction in the cluster model of heat conductivity of nanofluids,” Tech. Phys. Lett. 44 (2), 94–97 (2018).

    Article  Google Scholar 

Download references

Funding

This work was performed in the framework of the State assignment, state number registration AAAA-A19-119012290136-7.

Author information

Authors and Affiliations

  1. Institute of Applied Mechanics, Russian Academy of Sciences, 125040, Moscow, Russia

    O. B. Gus’kov

Authors
  1. O. B. Gus’kov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. B. Gus’kov.

Additional information

Translated by I. Ruzanova

APPENDIX

APPENDIX

The functions \({{q}_{p}}({{z}_{i}})\), \({{k}_{p}}({{z}_{i}})\), \({{q}_{f}}(z)\), and \({{k}_{f}}(z)\) involved in formulas (2.1) for the averaged particle and fluid velocities are given by

$${{q}_{p}}({{z}_{i}}) = \left\{ \begin{gathered} {{q}_{{p1}}}({{z}_{i}}),\quad 1 \ll {{z}_{i}} < 3, \hfill \\ {{q}_{{p2}}}({{z}_{i}}),\quad 3 \ll {{z}_{i}} < \infty , \hfill \\ \end{gathered} \right.\quad {{k}_{p}}\left( {{{z}_{i}}} \right) = \left\{ \begin{gathered} {{k}_{{p1}}}({{z}_{i}}),\quad 1 \ll {{z}_{i}} < 3, \hfill \\ {{k}_{{p2}}}({{z}_{i}}),\quad 3 \ll {{z}_{i}} < \infty , \hfill \\ \end{gathered} \right.$$
$${{q}_{{p1}}}({{z}_{i}}) = \frac{{406\,933}}{{573\,440}} + \frac{{{{z}_{i}}(547 + 189{{z}_{i}} - 63z_{i}^{2})}}{{2048}} - \frac{{19 + 27{{z}_{i}} + 15z_{i}^{2} + 3z_{i}^{3}}}{{1024{{{({{z}_{i}} + 1)}}^{5}}}} + \frac{1}{{3\,440\,640z_{i}^{8}}}(107\,520 $$
$$ - \;1\;843\,200{{z}_{i}} - 1\;106\,700z_{i}^{2} + 703\,878z_{i}^{3} - 1\;117\,200z_{i}^{4} + 1\;702\,365z_{i}^{5} - 362\,600z_{i}^{6} + 17\,535z_{i}^{7}) $$
$$ - \;\frac{{\sqrt {{{z}_{i}} + 1} }}{{215\,040z_{i}^{8}}}(6720 - 118\,560{{z}_{i}} - 7080z_{i}^{2} + 31\,308z_{i}^{3} - 43\,884z_{i}^{4}$$
$$ + \;113\;148z_{i}^{5} - 72\,625z_{i}^{6} + 44\,030z_{i}^{7} + 49\,875z_{i}^{8} - 840z_{i}^{9}) + \frac{{(75 + 1301z_{i}^{2})}}{{8192z_{i}^{7}}}\ln ({{z}_{i}} + 1),$$
$${{q}_{{p2}}}({{z}_{i}}) = \frac{{15}}{{16}} + \frac{5}{{16z_{i}^{5}}}\ln ({{z}_{i}} - 1) + \frac{{3(25 + 7z_{i}^{2})}}{{4096z_{i}^{7}}}\ln \left( {\frac{{{{z}_{i}} + 1}}{{{{z}_{i}} - 1}}} \right) $$
$$ + \;\frac{{4480 - 76\,800{{z}_{i}} - 47\,425z_{i}^{2} + 28\,672z_{i}^{3} - 1610z_{i}^{4} + 71\;680z_{i}^{5} - 770z_{i}^{6}}}{{71\;680z_{i}^{8}}} $$
$$ + \;\frac{{311 + 2632z_{i}^{2} + 1302z_{i}^{4} - 32z_{i}^{6} + 11z_{i}^{8}}}{{1024{{{(z_{i}^{2} - 1)}}^{5}}}},$$
$${{k}_{{p1}}}({{z}_{i}}) = \frac{{267}}{{512}} + \frac{{3(3 + 2{{z}_{i}})}}{{128{{{({{z}_{i}} + 1)}}^{3}}}} - \frac{{{{z}_{i}}(18 + 33{{z}_{i}} - 43z_{i}^{2})}}{{1024}} - \frac{{64 + 20z_{i}^{3} + 30z_{i}^{4} + 15z_{i}^{5}}}{{1024z_{i}^{6}}} $$
$$ + \;\frac{{\sqrt {{{z}_{i}} + 1} }}{{1024z_{i}^{6}}}(64 - 32{{z}_{i}} + 24z_{i}^{2} - 40z_{i}^{3} + 32z_{i}^{4} - 24z_{i}^{5} - 336z_{i}^{6} + 108z_{i}^{7} + 30z_{i}^{8} - 45z_{i}^{9}) $$
$$ - \;\frac{{9z_{i}^{2}(12 - 5z_{i}^{2})}}{{2048}}\ln \left( {\frac{{{{{(1 + \sqrt {{{z}_{i}} + 1} )}}^{2}}}}{{{{z}_{i}} + 2}}} \right) - \frac{3}{{512z_{i}^{3}}}\ln ({{z}_{i}} + 1) + \frac{3}{{128}}\ln (1 + \sqrt {{{z}_{i}} + 1} ),$$
$${{k}_{{p2}}}({{z}_{i}}) = \frac{{73}}{{256}} - \frac{3}{{64({{z}_{i}} - 2)}} - \frac{{3(27 + 44z_{i}^{2} + z_{i}^{4})}}{{128{{{(z_{i}^{2} - 1)}}^{3}}}} - \frac{{64 - 12z_{i}^{4} + 45z_{i}^{8}}}{{512z_{i}^{6}}} $$
$$ + \;\frac{3}{{128}}\ln \left( {\frac{{{{z}_{i}}}}{{{{z}_{i}} - 2}}} \right) - \frac{3}{{256z_{i}^{3}}}\ln \left( {\frac{{{{z}_{i}} + 1}}{{{{z}_{i}} - 1}}} \right) - \frac{{9z_{i}^{2}(12 - 5z_{i}^{2})}}{{2048}}\ln \left( {\frac{{z_{i}^{2}}}{{z_{i}^{2} - 4}}} \right),$$
$${{q}_{f}}(z) = \left\{ {\begin{array}{*{20}{c}} {{{q}_{{f1}}}(z),\quad 0 \ll z < 2,} \\ {{{q}_{{f2}}}(z),\quad 2 \ll z < \infty ,} \end{array}} \right.\quad {{k}_{f}}(z) = \left\{ \begin{gathered} {{k}_{{f1}}}(z),\quad 0 \ll z < 2, \hfill \\ {{k}_{{f2}}}(z),\quad 2 \ll z < \infty , \hfill \\ \end{gathered} \right.$$
$${{q}_{{f1}}}(z) = \frac{{13 + 138z + 768{{z}^{2}} - 176{{z}^{3}} - 3349{{z}^{4}} - 1274{{z}^{5}} + 3780{{z}^{6}} + 2520{{z}^{7}}}}{{64{{{(z + 1)}}^{2}}}} $$
$$ + \;\frac{{67 + 182z - 1071{{z}^{2}} - 2672{{z}^{3}} + 5618{{z}^{4}} + 16548{{z}^{5}} - 6720{{z}^{6}} - 20160{{z}^{7}}}}{{64{{{(4z + 1)}}^{{3/2}}}}} $$
$$ - \;\frac{3}{{32}}\ln (z + 1) - \frac{3}{{32}}(1 - 60{{z}^{2}} + 350{{z}^{4}} - 420{{z}^{6}})\ln \left( {\frac{{2z + 1 + \sqrt {4z + 1} }}{{2(z + 1)}}} \right) + \frac{{(1 - (1 - 2z + 6{{z}^{2}})\sqrt {4z + 1} )}}{{16{{z}^{3}}}},$$
$${{q}_{{f2}}}(z) = 1 + \frac{1}{{8{{z}^{3}}}} - \frac{3}{{16}}\ln \left( {\frac{z}{{z - 1}}} \right) + \frac{{15}}{{16}}{{z}^{2}}(6 - 35{{z}^{2}} + 42{{z}^{4}})\ln \left( {\frac{{{{z}^{2}}}}{{{{z}^{2}} - 1}}} \right) $$
$$ + \;\frac{{19 + 29z - 489{{z}^{2}} - 495{{z}^{3}} + 1680{{z}^{4}} + 1680{{z}^{5}} - 1260{{z}^{6}} - 1260{{z}^{7}}}}{{32(z - 1){{{(z + 1)}}^{2}}}},$$
$${{k}_{{f1}}}(z) = \frac{3}{{64}}(5 - 20z - 29{{z}^{2}} + 60{{z}^{3}}) - \frac{{3(3z + 1)}}{{64\sqrt {4z + 1} }}(5 - 21z - 10{{z}^{2}} + 40{{z}^{3}}) $$
$$ + \;\frac{3}{{32}}\ln (z + 1) + \frac{3}{{32}}(1 - 18{{z}^{2}} + 30{{z}^{4}})\ln \left( {\frac{{2z + 1 + \sqrt {4z + 1} }}{{2(z + 1)}}} \right),$$
$${{k}_{{f2}}}(z) = \frac{{3( - 5 + 3z + 30{{z}^{2}} - 30{{z}^{3}})}}{{32(z - 1)}} + \frac{3}{{16}}\ln \left( {\frac{z}{{z - 1}}} \right) - \frac{9}{{16}}{{z}^{2}}(3 - 5{{z}^{2}})\ln \left( {\frac{{{{z}^{2}}}}{{{{z}^{2}} - 1}}} \right).$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gus’kov, O.B. Inviscid Suspension Flow along a Flat Boundary. Comput. Math. and Math. Phys. 61, 470–479 (2021). https://doi.org/10.1134/S0965542521030088

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542521030088

Keywords:

Search

Navigation