Skip to main content
SpringerLink
Log in

Cell models for viscous flow past a swarm of Reiner–Rivlin liquid spherical drops

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

This paper presents an analytical study of Stokes flow of an incompressible viscous fluid through a swarm of immiscible Reiner–Rivlin liquid droplets-in-cell using the cell model technique. The stream function solution of Stokes equation is obtained for the flow in the fictitious envelope region, while for the inner flow field within the liquid drop, the solution is obtained by expanding the stream function in a power series of S. The proper boundary conditions are taken on the surface of the liquid sphere, while the appropriate conditions applied on the fictitious boundary of the fluid envelope vary depending on the kind of cell-model. The analytical solution of the problem for four models: Happel’s, Kuwabara’s, Kvashnin’s and Mehta–Morse’s model (usually referred to as Cunningham’s) is derived. The velocity profile and the pressure distribution outside of the droplet are shown in numerous graphs for different values of the parameters. Numerical results for the normalized hydrodynamic drag force \(W_{C}\) acting, in each case, on the spherical droplet-in-cell obtained for different values of the parameters characterizing volume fraction \(\gamma ,\) the relative viscosity \(\lambda\), and the cross-viscosity, i.e., S are presented in tabular and graphical forms as well. It is found that normalized hydrodynamic drag force \(W_{C}\) is a monotonic increasing function of particle volume fraction \(\gamma .\) It is also observed that solid sphere in-cell experiences greater drag force \(C_{D}\), whereas spherical bubble experiences smaller. One of the important findings of the present investigation is that the cross-viscosity \(\mu _{c}\) of Reiner–Rivlin fluid decreases \(W_{C}\) on the liquid droplet-in-cell. Further, the drag coefficient \(C_{D}\) get reduced to analytical results obtained earlier.

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

Similar content being viewed by others

References

  1. Abramowitz M, Stegun IA (1970) Handbook of mathematical functions. Dover, New York

    MATH  Google Scholar 

  2. Bart E (1968) The slow unsteady settling of a fluid sphere toward a flat fluid interface. Chem Eng Sci 23:193–210

    Article  Google Scholar 

  3. Brenner H (1957) Eng. Sc. D. thesis, New York University, New York

  4. Choudhuri D, Sri Padamavati B (2010) A study of an arbitrary Stokes flow past a fluid coated sphere in a fluid of a different viscosity. Z Angew Math Phys 61:317–328

    Article  MathSciNet  MATH  Google Scholar 

  5. Cunningham E (1910) On the velocity of steady fall of spherical particles through fluid medium. Proc R Soc Lond Ser A 83:357–369

    Article  ADS  MATH  Google Scholar 

  6. Dassios G, Hadjinicolaou M, Coutelieris FA, Payatakes AC (1995) Stokes flow in spheroidal particle-in-cell models with Happel and Kuwabara boundary conditions. Int J Eng Sci 33:1465–1490

    Article  MATH  Google Scholar 

  7. Datta S, Deo S (2002) Stokes flow with slip and Kuwabara boundary conditions. Proc Ind Acad Sci (Math Sci) 112:463–475

    Article  MathSciNet  MATH  Google Scholar 

  8. Deo S, Gupta BR (2009) Stokes flow past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition. Acta Mech 203:241–254

    Article  MATH  Google Scholar 

  9. Deo S, Shukla P (2009) Creeping flow past a swarm of porous spherical particles with Mehta–Morse boundary condition. Indian J Biomech 7–8:123–127

    Google Scholar 

  10. Deo S (2009) Stokes flow past a swarm of deformed porous spheroidal particles with Happel boundary condition. J Porous Media 12:347–359

    Article  Google Scholar 

  11. Datta S, Raturi S (2014) Cell model for slow viscous flow past spherical particles with surfactant layer coating. J Appl Fluid Mech 7:263–273

    Google Scholar 

  12. Faltas MS, Saad EI (2012) Slow motion of a porous eccentric spherical particle-in-cell models. Transp Porous Media 95:133–150

    Article  MathSciNet  Google Scholar 

  13. Gupta BR, Deo S (2013) Axisymmetric creeping flow of a micropolar fluid over a sphere coated with a thin fluid film. J Appl Fluid Mech 6:149–155

    Google Scholar 

  14. Happel J (1958) Viscous flow in multi particle system: slow motion of fluids relative to beds of spherical particles. AIChE J 4:197–201

    Article  MathSciNet  Google Scholar 

  15. Happel J (1959) Viscous flow relative to arrays of cylinders. AIChE J 5:174–177

    Article  Google Scholar 

  16. Happel J, Brenner H (1983) Low Reynolds number hydrodynamics. Martinus Nijhoff, The Hague

    MATH  Google Scholar 

  17. Hetsroni G, Haber S (1970) The flow in and around a droplet or bubble submerged in an unbounded arbitrary velocity field. Rheol Acta 9:488–496

    Article  MATH  Google Scholar 

  18. Jaiswal BR, Gupta BR (2014) Drag on Reiner–Rivlin liquid sphere placed in a micro-polar fluid with non-zero boundary condition for microrotations. Int J Appl Math Mech 10:90–103

    Google Scholar 

  19. Jaiswal BR, Gupta BR (2015) Brinkman flow of a viscous fluid past a Reiner–Rivlin liquid sphere immersed in a saturated porous medium. Transp Porous Media 7:907–925

    Article  MathSciNet  Google Scholar 

  20. Jaiswal BR, Gupta BR (2014) Wall effects on Reiner–Rivlin liquid spheroid. Appl Comput Mech 2:157–176

    Google Scholar 

  21. Jaiswal BR, Gupta BR (2015) Stokes flow of polar fluid past a non-Newtonian liquid spheroid. Int J Fluid Mech Res 42:170–189

    Article  Google Scholar 

  22. Jaiswal BR, Gupta BR (2014) Reiner–Rivlin liquid sphere in an approximate spherical container (communicated)

  23. Keh HJ, Lee TC (2010) Axisymmetric creeping motion of a slip spherical particle in a non-concentric spherical cavity. Theor Comput Fluid Dyn 24:497–510

    Article  MATH  Google Scholar 

  24. Keh MP, Keh HJ (2010) Slow motion of an assemblage of porous spherical shells relative to a fluid. Transp Porous Media 81:261–275

    Article  MathSciNet  Google Scholar 

  25. Kim S, Karrila SJ (1991) Micro hydrodynamics: principles and selected applications. Butterworth-Heinemann, Boston

    Google Scholar 

  26. Kuwabara S (1959) The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers. J Phys Soc Jpn 14:527–532

    Article  ADS  MathSciNet  Google Scholar 

  27. Kvashnin AG (1979) Cell model of suspension of spherical particles. Fluid Dyn 14:598–602

    Article  ADS  MATH  Google Scholar 

  28. Mehta GD, Morse TF (1975) Flow through charged membranes. J Chem Phys 63:1878–1889

    Article  ADS  Google Scholar 

  29. Ramkissoon H, Majumadar SR (1976) Drag on axially symmetric body in the Stokes’ flow of micropolar fluids. Phys Fluids 19:16–21

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Ramkissoon H (1989) Slow flow of a non-Newtonian liquid past a fluid sphere. Acta Mech 78:73–80

    Article  MATH  Google Scholar 

  31. Ramkissoon H (1989) Stokes flow past a Reiner–Rivlin liquid sphere. J Appl Math Mech (ZAMM) 69:259–261

    Article  MathSciNet  MATH  Google Scholar 

  32. Ramkissoon H (1998) Stokes flow past a non-Newtonian fluid spheroid. J Appl Math Mech (ZAMM) 78:61–66

    Article  MathSciNet  MATH  Google Scholar 

  33. Ramkissoon H (1999) Polar flow past a Reiner–Rivlin fluid sphere. J Math Sci 10:63–68

    MathSciNet  Google Scholar 

  34. Ramkissoon H, Rahaman K (2001) Non-Newtonian fluid sphere in a spherical container. Acta Mech 149:239–245

    Article  MATH  Google Scholar 

  35. Reiner M (1945) A mathematical theory of dilatancy. Am J Math 67:350–362

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Saad EI (2012) Cell models for micropolar flow past a viscous fluid sphere. Meccanica 47:2055–2068

    Article  MathSciNet  MATH  Google Scholar 

  37. Saad EI (2012) Stokes flow past an assemblage of axisymmetric porous spheroidal particle-in-cell models. J Porous Media 15:849–866

    Article  Google Scholar 

  38. Saad EI (2013) Stokes flow past an assemblage of axisymmetric porous spherical shell-in-cell models: effect of stress jump condition. Meccanica 48:1747–1759

    Article  MathSciNet  MATH  Google Scholar 

  39. Stokes GG (1851) On the effect of the internal friction of fluid on pendulums. Trans Camb Philos Soc 9:8–106

    ADS  Google Scholar 

  40. Uchida S (1954) Slow viscous flow through a mass of particles. Ind Eng Chem 46:1194–1195 (transl: T. Motai)

  41. Vasin SI, Filippov AN, Starov VM (2008) Hydrodynamic permeability of membranes built up by particles covered by porous shells: cell models. Adv Colloid Interface Sci 139:83–96

    Article  Google Scholar 

  42. Yadav PK, Tiwari A, Deo S, Filippov A, Vasin S (2010) Hydrodynamic permeability of membranes built up by spherical particles covered by porous shells: effect of stress jump condition. Acta Mech 215:193–209

    Article  MATH  Google Scholar 

  43. Zholkovskiy EK, Shilov VN, Masliyah JH, Bondarenko MP (2007) Hydrodynamic cell model: general formulation and comparative analysis of different approaches. Can J Chem Eng 85:701–725

    Article  Google Scholar 

Download references

Acknowledgments

Authors acknowledge with thanks to Department of Mathematics, JUET, Guna (M.P.) for providing necessary facilities during the preparation of this research paper. Further, the authors acknowledge their sincere thanks to the anonymous reviewers/referees of the paper for making their useful and valuable comments and giving fruitful suggestions which led to much significant improvement in the presentation of the manuscript.

Author information

Authors and Affiliations

  1. AKS University, Sherganj, Panna road, Satna, M.P., 485001, India

    B. R. Jaiswal

  2. Jaypee University of Engg. & Tech., AB road, Raghogarh, Guna, M.P., 473226, India

    B. R. Gupta

Authors
  1. B. R. Jaiswal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. R. Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. R. Jaiswal.

Appendix: Determination of arbitrary unknowns

Appendix: Determination of arbitrary unknowns

Applying the boundary conditions (36)–(39), and one condition from (40)–(43), we obtain the resulting equations for different models as follows.

1.1 For Happel model

$$\begin{aligned}&{ a_{2} +b_{2} +c_{2} +d_{2} =0,} \\&{ e_{2} +f_{2} +\frac{4}{63} S^{2} =0,} \\&{ b_{2} -2a_{2} -d_{2} -4c_{2} +2e_{2} +4f_{2} +4+\frac{8}{21} S^{2} =0,} \\&{2\lambda b_{2} -\lambda a_{2} -\lambda d_{2} +2\lambda c_{2} +e_{2} -2f_{2} -6-\frac{4}{7} S^{2} =0,} \\&{ b_{2} +\gamma ^{-1} a_{2} +\gamma ^{-2/3} d_{2} +\gamma ^{-5/3} c_{2} -\gamma ^{-1} =0,} \\&{ b_{2} -\gamma ^{-5/3} c_{2} =0.} \end{aligned}$$
(62)

1.2 For Kuwabara model

$$\begin{aligned}&{ a_{2} +b_{2} +c_{2} +d_{2} =0,} \\&{ e_{2} +f_{2} +\frac{4}{63} S^{2} =0,} \\&{ b_{2} -2a_{2} -d_{2} -4c_{2} +2e_{2} +4f_{2} +4+\frac{8}{21} S^{2} =0,} \\&{2\lambda b_{2} -\lambda a_{2} -\lambda d_{2} +2\lambda c_{2} +e_{2} -2f_{2} -6-\frac{4}{7} S^{2} =0,} \\&{ b_{2} +\gamma ^{-1} a_{2} +\gamma ^{-2/3} d_{2} +\gamma ^{-5/3} c_{2} -\gamma ^{-1} =0,} \\&{ d_{2} -5\gamma ^{-1} c_{2} =0.} \end{aligned}$$
(63)

1.3 For Kvashnin model

$$\begin{aligned}&{a_{2} +b_{2} +c_{2} +d_{2} =0,} \\&{ e_{2} +f_{2} +\frac{4}{63} S^{2} =0,} \\&{ b_{2} -2a_{2} -d_{2} -4c_{2} +2e_{2} +4f_{2} +4+\frac{8}{21} S^{2} =0,} \\&{2\lambda b_{2} -\lambda a_{2} -\lambda d_{2} +2\lambda c_{2} +e_{2} -2f_{2} -6-\frac{4}{7} S^{2} =0,} \\&{b_{2} +\gamma ^{-1} a_{2} +\gamma ^{-2/3} d_{2} +\gamma ^{-5/3} c_{2} -\gamma ^{-1} =0,} \\&{ 3b_{2} -\gamma ^{-2/3} d_{2} +8\gamma ^{-5/3} c_{2} =0.} \end{aligned}$$
(64)

1.4 For Cunningum/Mehta–Morse model

$$\begin{aligned}&{ a_{2} +b_{2} +c_{2} +d_{2} =0,} \\&{ e_{2} +f_{2} +\frac{4}{63} S^{2} =0,} \\&{ b_{2} -2a_{2} -d_{2} -4c_{2} +2e_{2} +4f_{2} +4+\frac{8}{21} S^{2} =0,} \\&{2\lambda b_{2} -\lambda a_{2} -\lambda d_{2} +2\lambda c_{2} +e_{2} -2f_{2} -6-\frac{4}{7} S^{2} =0,} \\&{b_{2} +\gamma ^{-1} a_{2} +\gamma ^{-2/3} d_{2} +\gamma ^{-5/3} c_{2} -\gamma ^{-1} =0,} \\&{b_{2} -2\gamma ^{-1} a_{2} -\gamma ^{-2/3} d_2-4\gamma ^{-5/3} c_{2} +2\gamma ^{-1} =0.} \end{aligned}$$
(65)

The solution of these systems of Eqs. (62)–(65), for different models are as follows.

1.5 For Happel model

$$\begin{aligned} a_{2} &=-(378+32S^{2} \gamma ^{1/3} -64S^{2} \gamma -567\gamma ^{5/3} \\ &\quad +\,32S^{2} \gamma ^{2} +378\lambda +378\gamma ^{5/3} \lambda) {\varDelta }_{1}, \\ b_{2} &=-(189+32S^{2} -32S^{2} \gamma ^{1/3} ){\varDelta }_{1}, \\ c_{2} &=(-189-32S^{2} +32S^{2} \gamma ^{1/3} ){\varDelta }_{1} \gamma ^{5/3}, \\ d_{2} &=-(-567-32S^{2} +64S^{2} \gamma +378\gamma ^{5/3}\\ &\quad -\,32S^{2} \gamma ^{5/3} -378\lambda -378\gamma ^{5/3} \lambda ){\varDelta }_{1}, \\ e_{2} &=\frac{2}{27} (27+2S^{2} )+(189+32S^{2} -32S^{2} \gamma ^{1/3} )\\&\quad\,\times\,(\lambda +\gamma ^{5/3} \lambda ){\varDelta }_{1}, \\ f_{2}& =-\frac{2}{189} (189+20S^{2} )+(189+32S^{2} -32S^{2} \gamma ^{1/3} )\\&\quad \times\,(-\lambda -\gamma ^{5/3} \lambda ){\varDelta }_{1}, \end{aligned}$$
(66)

where

$$\begin{aligned} \frac{1}{{\varDelta }_{1} } &=189(-1+\gamma ^{1/3} )(2-\gamma ^{1/3} -\gamma ^{2/3} +\gamma +\gamma ^{4/3} \\&\quad-\,2\gamma ^{5/3} +2\lambda +2\gamma ^{5/3} \lambda ). \end{aligned}$$

1.6 For Kuwabara model

$$\begin{aligned} a_{2} &=-(-1890-192S^{2} \gamma ^{1/3} -945\gamma +160S^{2} \gamma \\ &\quad +\,32S^{2} \gamma ^{2} -1890\lambda ){\varDelta }_{2}, \\ b_{2} &=-(-945-160S^{2} +192S^{2} \gamma ^{1/3} +378\gamma -32S^{2} \gamma \\ &\quad -\,378\gamma \lambda ){\varDelta }_{2}, \\ c_{2} &=\gamma (-567-32S^{2} +32S^{2} \gamma -378\lambda ){\varDelta } _{2}, \\ d_{2} &=5(-567-32S^{2} +32S^{2} \gamma -378\lambda ){\varDelta }_{2}, \\ e_{2} &=2+\frac{4S^{2} }{63} +\frac{3}{2(-1+\gamma )} \\ &\quad -\,(5-4\gamma ^{1/3} -4\gamma ^{2/3} +\gamma +\gamma ^{4/3} +\gamma ^{5/3} ) \\ &\quad \times\, (-567-32S^{2} +32S^{2} \gamma -378\lambda ){\varDelta }_{2}, \\ f_{2} &=-2-\frac{8S^{2} }{63} -\frac{3}{2(-1+\gamma )} \\ &\quad -\,(-5+4\gamma ^{1/3} +4\gamma ^{2/3} -\gamma -\gamma ^{4/3} -\gamma ^{5/3}) \\ &\quad \times \, (-567-32S^{2} +32S^{2} \gamma -378\lambda ){\varDelta }_{2}, \end{aligned}$$
(67)

where

$$\begin{aligned}\frac{1}{{\varDelta }_{2} } &=378(-1+\gamma ^{1/3} )^{2} (5+\gamma ^{1/3} -3\gamma ^{2/3} -2\gamma -\gamma ^{4/3} \\ &\quad+\,5\lambda +4\gamma ^{1/3} \lambda +3\gamma ^{2/3} \lambda +2\gamma \, \lambda +\gamma ^{4/3} \lambda ). \end{aligned}$$

1.7 For Kvashnin model

$$\begin{aligned} a_{2} &=-(-3024-288S^{2} \gamma ^{1/3} -945\gamma +160S^{2} \gamma \\&\quad -\,1701\gamma ^{5/3} +128S^{2} \gamma ^{2} -\,3024\lambda +1134\gamma ^{5/3} \lambda ){\varDelta }_{3}, \\ b_{2} &=2(756+128S^{2} -144S^{2} \gamma ^{1/3} -189\gamma \\& \quad +\,16S^{2} \gamma +189\gamma \, \lambda ){\varDelta }_{3}, \\ c_{2} &=-(567\gamma +32S^{2} \gamma +567\gamma ^{5/3} +96S^{2} \gamma ^{5/3}\\ &\quad -\,128S^{2} \gamma ^{2} +378\gamma \, \lambda ){\varDelta } _{3}, \\ d_{2} &=2(-2268-128S^{2} +80S^{2} \gamma -567\gamma ^{5/3}\\ &\quad +\,48S^{2} \gamma ^{5/3} -1512\lambda +567\gamma ^{5/3} \lambda ){\varDelta }_{3}, \\ e_{2} &=2+\frac{4S^{2} }{63} +\frac{8(-\frac{1}{2} +\frac{\gamma }{8} )}{8-9\gamma ^{1/3} +\gamma } \\ &\quad -\,(16-11\gamma ^{1/3} -11\gamma ^{2/3} -\gamma -\gamma ^{4/3} +8\gamma ^{5/3} )\\ &\quad\times\,(756+128S^{2} -144S^{2} \gamma ^{1/3} \\ & \quad -\,189\gamma +16S^{2} \gamma +189\gamma \, \lambda ){\varDelta }_{3}, \\ f_{2} &=-2-\frac{8S^{2} }{63} +\frac{8(\frac{1}{2} -\frac{\gamma }{8} )}{8-9\gamma ^{1/3} +\gamma } \\ &\quad -\,(-16+11\gamma ^{1/3} +11\gamma ^{2/3} +\gamma +\gamma ^{4/3} -8\gamma ^{5/3} )\\ &\quad\times\,(756+128S^{2} -144S^{2} \gamma ^{1/3} \\ & \quad -189\gamma +16S^{2} \gamma +189\gamma \, \lambda ){\varDelta }_{3}, \end{aligned}$$
(68)

where

$$\begin{aligned}\frac{1}{{\varDelta }_{3} } &=189(-1+\gamma ^{1/3} )^{2} (16+5\gamma ^{1/3} -6\gamma ^{2/3} -7\gamma -8\gamma ^{4/3} \\&\quad+\,16\lambda +14\gamma ^{1/3} \lambda +12\gamma ^{2/3} \lambda +10\gamma \, \lambda +8\gamma ^{4/3} \lambda ). \end{aligned}$$

1.8 For Cunningum/Mehta–Morse model

$$\begin{aligned} a_{2} &=-(756+96S^{2} \gamma ^{1/3} +945\gamma -160S^{2} \gamma -1701\gamma ^{5/3} \\ &\quad +\,64S^{2} \gamma ^{2} +\,756\lambda +1134\gamma ^{5/3} \lambda ){\varDelta }_{4}, \\ b_{2} &=-[32S^{2} (2-3\gamma ^{1/3} +\gamma )+378(1+\gamma (-1+\lambda ))]{\varDelta }_{4}, \\ c_{2} &=-(-567\gamma -32S^{2} \gamma +567\gamma ^{5/3} +96S^{2} \gamma ^{5/3} \\ &\quad -\,64S^{2} \gamma ^{2} -378\gamma \, \lambda ){\varDelta } _{4}, \\ d_{2} &=2(567+32S^{2} -80S^{2} \gamma -567\gamma ^{5/3} +48S^{2} \gamma ^{5/3} \\ & \quad +\,378\lambda +567\gamma ^{5/3} \lambda ){\varDelta }_{4}, \\ e_{2} &=[4S^{2} \{ 42\gamma ^{2/3} +\gamma ^{1/3} (7-6\lambda )+4\gamma ^{4/3} (-7+3\lambda )\\ &\quad-\,4(7+3\lambda )+\gamma (7+6\lambda )\} \, \\&\quad +\,189\{ -8+2\gamma ^{1/3} +8\gamma ^{4/3} (-1+\lambda )-6\lambda \\ &\quad +\,6\gamma ^{2/3} (2+\lambda )+\gamma (2+7\lambda )\} ]{\varDelta }_{4}, \\ f_{2}& =-[8S^{2} \{ 30\gamma ^{2/3} +\gamma ^{1/3} (5-6\lambda )+4\gamma ^{4/3} (-5+3\lambda )\\ &\quad -\,4(5+3\lambda )+\gamma (5+6\lambda )\} \\&\quad +\,189\{ -8+2\gamma ^{1/3} +8\gamma ^{4/3} (-1+\lambda )-6\lambda\\ &\quad +\,6\gamma ^{2/3} (2+\lambda )+\gamma (2+7\lambda )\} ]{\varDelta }_{4},\end{aligned}$$
(69)

where

$$\begin{aligned} \frac{1}{{\varDelta }_{4} } &=189(-1+\gamma ^{1/3} )^{3} [4\gamma (-1+\lambda )+4(1+\lambda )\\ &\quad+\,\gamma ^{2/3} (-3+6\lambda )+\gamma ^{1/3} (3+6\lambda )], \end{aligned}$$

and

$$\begin{aligned} \, \gamma =\eta ^{3}. \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jaiswal, B.R., Gupta, B.R. Cell models for viscous flow past a swarm of Reiner–Rivlin liquid spherical drops. Meccanica 52, 69–89 (2017). https://doi.org/10.1007/s11012-016-0385-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-016-0385-3

Keywords

Search

Navigation