Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T22:19:19.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Slow viscous flow of two porous spherical particles translating along the axis of a cylinder

Published online by Cambridge University Press:  28 December 2018

Xin Yao ,
Chyi Huey Ng ,
Jia Rui Amanda Teo ,
Marcos Open the ORCID record for Marcos [Opens in a new window]  and
Teck Neng Wong Open the ORCID record for Teck Neng Wong [Opens in a new window]
Xin Yao
Affiliation:
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore
Chyi Huey Ng
Affiliation:
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore
Jia Rui Amanda Teo
Affiliation:
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore
Marcos
Affiliation:
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore
Teck Neng Wong
Affiliation:
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe the motion of two freely moving porous spherical particles located along the axis of a cylindrical tube with background Poiseuille flow at low Reynolds number. The stream function and a framework based on cylindrical harmonics are adopted to solve the flow field around the particles and the flow within the tube, respectively. The two solutions are employed in an iterated framework using the method of reflections. We first consider the case of two identical particles, followed by two particles with different dimensions. In both cases, the drag force coefficients of the particles are solved as functions of the separation distance between the particles and the permeability of the particles. The detailed flow field in the vicinity of the two particles is investigated by plotting the streamlines and velocity contours. We find that the particle–particle interaction is dependent on the separation distance, particle sizes and permeability of the particles. Our analysis reveals that when the permeability of the particles is large, the streamlines are more parallel and the particle–particle interaction has less effect on the particle motion. We further show that a smaller permeability and bigger particle size generally tend to squeeze the streamlines and velocity contour towards the wall.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Porous media Multiphase and Particle-laden Flows: Particle/fluid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 861 , 25 February 2019 , pp. 643 - 678
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Auriault, J.-L. 2009 On the domain of validity of Brinkman’s equation. Trans. Porous Med. 79 (2), 215–223.Google Scholar
Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30 (1), 197–207.Google Scholar
Bhattacharya, S., Mishra, C. & Bhattacharya, S. 2010 Analysis of general creeping motion of a sphere inside a cylinder. J. Fluid Mech. 642, 295–328.Google Scholar
Bocquet, L. & Charlaix, E. 2010 Nanofluidics, from bulk to interfaces. Chem. Soc. Rev. 39 (3), 1073–1095.Google Scholar
Brenner, H. 1963 The Stokes resistance of an arbitrary particle. Chem. Engng Sci. 18 (1), 1–25.Google Scholar
Brenner, H. 1964 The Stokes resistance of an arbitrary particle. Part II. An extension. Chem. Engng Sci. 19 (9), 599–629.Google Scholar
Brenner, H. & Happel, J. 1958 Slow viscous flow past a sphere in a cylindrical tube. J. Fluid Mech. 4 (2), 195–213.Google Scholar
Brinkman, H. C. 1949 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1 (1), 27–34.Google Scholar
Buehler, K. L. & Anderson, J. L. 2002 Solvent effects on the permeability of membrane-supported gels. Ind. Engng Chem. Res. 41 (3), 464–472.Google Scholar
Chen, T. C. & Skalak, R. 1970 Stokes flow in a cylindrical tube containing a line of spheroidal particles. Appl. Sci. Res. 22 (1), 403–441.Google Scholar
Chudasama, N. A., Prasad, K. & Siddhanta, A. K. 2016 Agarose functionalization: synthesis of PEG-agarose amino acid nano-conjugate – its structural ramifications and interactions with BSA in a varying pH regime. Carbohyd. Polym. 151, 735–742.Google Scholar
Daiguji, H., Yang, P., Szeri, A. J. & Majumdar, A. 2004 Electrochemomechanical energy conversion in nanofluidic channels. Nano Lett. 4 (12), 2315–2321.Google Scholar
Feng, H., Wong, T. N., Che, Z. & Marcos 2016 Chaotic micromixer utilizing electro-osmosis and induced charge electro-osmosis in eccentric annulus. Phys. Fluids 28 (6), 062003.Google Scholar
Foquet, M., Korlach, J., Zipfel, W., Webb, W. W. & Craighead, H. G. 2002 DNA fragment sizing by single molecule detection in submicrometer-sized closed fluidic channels. Anal. Chem. 74 (6), 1415–1422.Google Scholar
Greenstein, T. & Happel, J. 1968 Theoretical study of the slow motion of a sphere and a fluid in a cylindrical tube. J. Fluid Mech. 34 (4), 705–710.Google Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media. Springer.Google Scholar
Hasimoto, H. 1976 Slow motion of a small sphere in a cylindrical domain. J. Phys. Soc. Japan 41 (6), 2143–2144.Google Scholar
Higdon, J. J. L. & Kojima, M. 1981 On the calculation of Stokes’ flow past porous particles. Intl J. Multiphase Flow 7 (6), 719–727.Google Scholar
Ho, C.-M. & Tai, Y.-C. 1998 Micro-electro-mechanical-systems (MEMS) and fluid flows. Annu. Rev. Fluid Mech. 30, 579–612.Google Scholar
Huang, Y., Chen, J., Wong, T. & Liow, J.-L. 2016 Experimental and theoretical investigations of non-Newtonian electro-osmotic driven flow in rectangular microchannels. Soft Matt. 12 (29), 6206–6213.Google Scholar
Huang, Y., Wang, Y. L. & Wong, T. N. 2017 AC electric field controlled non-Newtonian filament thinning and droplet formation on the microscale. Lab on a Chip 17 (17), 2969–2981.Google Scholar
Jeffrey, D. J. 1982 Low-Reynolds-number flow between converging spheres. Mathematika 29 (1), 58–66.Google Scholar
Kapur, V., Charkoudian, J. C., Kessler, S. B. & Anderson, J. L. 1996 Hydrodynamic permeability of hydrogels stabilized within porous membranes. Ind. Engng Chem. Res. 35 (9), 3179–3185.Google Scholar
Kim, S. 1987 Stokes flow past three spheres: an analytic solution. Phys. Fluids 30 (8), 2309–2314.Google Scholar
Kim, S. & Karrila, S. J. 2005 Microhydrodynamics: Principles and Selected Applications. Dover.Google Scholar
Kim, S.-K., Koo, H.-J., Liu, J. & Braun, P. V. 2017 Flexible and wearable fiber microsupercapacitors based on carbon nanotube-agarose gel composite electrodes. ACS Appl. Mater. Interfaces 9 (23), 19925–19933.Google Scholar
Koh, J. B. Y. & Marcos 2014 Effect of dielectrophoresis on spermatozoa. Microfluid. Nanofluid. 17 (4), 613–622.Google Scholar
Krainov, V. P., Smirnov, B. M. & Tereshonok, D. V. 2014 Permeability of porous materials for liquid and gases. Europhys. Lett. 108 (3), 34002.Google Scholar
Lamb, H. & Caflisch, R. 1993 Hydrodynamics. Cambridge University Press.Google Scholar
Lee, C., Yang, E.-H., Myung, N. V. & George, T. 2003 A nanochannel fabrication technique without nanolithography. Nano Lett. 3 (10), 1339–1340.Google Scholar
Leichtberg, S., Pfeffer, R. & Weinbaum, S. 1976 Stokes flow past finite coaxial clusters of spheres in a circular cylinder. Intl J. Multiphase Flow 3 (2), 147–169.Google Scholar
Marcos, Kang, Y. J., Ooi, K. T., Yang, C. & Wong, T. N. 2005a Frequency-dependent velocity and vorticity fields of electro-osmotic flow in a closed-end cylindrical microchannel. J. Micromech. Microengng 15 (2), 301–312.Google Scholar
Marcos, Ooi, K. T., Yang, C., Chai, J. C. & Wong, T. N. 2005b Developing electro-osmotic flow in closed-end micro-channels. Intl J. Engng Sci. 43 (17–18), 1349–1362.Google Scholar
Marcos, Tran, N. P., Saini, A. R., Ong, K. C. H. & Chia, W. J. 2014 Analysis of a swimming sperm in a shear flow. Microfluid. Nanofluid. 17 (5), 809–819.Google Scholar
Mazur, P. & van Saarloos, W. 1982 Many-sphere hydrodynamic interactions and mobilities in a suspension. Physica A 115 (1–2), 21–57.Google Scholar
Mika, A. M. & Childs, R. F. 2001 Calculation of the hydrodynamic permeability of gels and gel-filled microporous membranes. Ind. Engng Chem. Res. 40 (7), 1694–1705.Google Scholar
Mo, G. & Sangani, A. S. 1994 A method for computing Stokes flow interactions among spherical objects and its application to suspensions of drops and porous particles. Phys. Fluids 6 (5), 1637–1652.Google Scholar
Mokadam, R. G. 1961 Thermodynamic analysis of the Darcy law. Trans. ASME J. Appl. Mech. 28 (2), 208–212.Google Scholar
Navardi, S. & Bhattacharya, S. 2013 General methodology to evaluate two-particle hydrodynamic friction inside cylinder-bound viscous fluid. Comput. Fluids 76, 149–169.Google Scholar
Navardi, S., Bhattacharya, S. & Wu, H. 2015 Stokesian simulation of two unequal spheres in a pressure-driven creeping flow through a cylinder. Comput. Fluids 121, 145–163.Google Scholar
Neale, G., Epstein, N. & Nader, W. 1973 Creeping flow relative to permeable spheres. Chem. Engng Sci. 28 (10), 1865–1874.Google Scholar
Neale, G. & Nader, W. 1974 Practical significance of Brinkman’s extension of Darcy’s law: coupled parallel flows within a channel and a bounding porous medium. Can. J. Chem. Engng 52 (4), 475–478.Google Scholar
Nield, D. A. & Bejan, A. 2006 Convection in Porous Media. Springer.Google Scholar
O’Neill, M. E. 1968 A sphere in contact with a plane wall in a slow linear shear flow. Chem. Engng Sci. 23 (11), 1293–1298.Google Scholar
O’Neill, M. E. 1969 Exact solutions of the equations of slow viscous flow generated by the asymmetrical motion of two equal spheres. Appl. Sci. Res. 21 (1), 452–466.Google Scholar
Padmavathi, B. S., Amaranath, T. & Nigam, S. D. 1993 Stokes flow past a porous sphere using Brinkman’s model. Z. Angew. Math. Phys. 44 (5), 929–939.Google Scholar
Prakash, J. & Sekhar, G. P. R. 2012 Arbitrary oscillatory Stokes flow past a porous sphere using Brinkman model. Meccanica 47 (5), 1079–1095.Google Scholar
Purcel, E. A. 1977 Life at low Reynolds number. Am. J. Phys. 45 (1), 3–11.Google Scholar
Reuland, P., Felderhof, B. U. & Jones, R. B. 1978 Hydrodynamic interaction of two spherically symmetric polymers. Physica A 93 (3–4), 465–475.Google Scholar
Saad, E. I. & Faltas, M. S. 2014 Slow motion of a porous sphere translating along the axis of a circular cylindrical pore subject to a stress jump condition. Trans. Porous Med. 102 (1), 91–109.Google Scholar
Schatzman, M. & Schatzman, M. 2002 Numerical Analysis: A Mathematical Introduction. Clarendon Press.Google Scholar
Shen, X., Marcos & Fu, H. C. 2017 Traction reveals mechanisms of wall effects for microswimmers near boundaries. Phys. Rev. E 95 (3), 033105.Google Scholar
Sinha, P. M., Valco, G., Sharma, S., Liu, X. & Ferrari, M. 2004 Nanoengineered device for drug delivery application. Nanotechnology 15 (10), S585–S589.Google Scholar
Song, Y., Kim, M.-O., Kwon, D.-S., Kim, Y.-J. & Kim, J. 2012 Facile fabrication of sub-20-nm nanochannels based on crystallinity-dependent anisotropic etching of silicon. Microelectron. Engng 98, 309–312.Google Scholar
Sonshine, R. M. & Brenner, H. 1966 The Stokes translation of two or more particles along the axis of an infinitely long circular cylinder. Appl. Sci. Res. 16 (1), 425–454.Google Scholar
Urbakh, M., Daikhin, L. & Klafter, J. 1995 Velocity profiles and the Brinkman equation in nanoscale confined liquids. Europhys. Lett. 32 (2), 125–130.Google Scholar
Vainshtein, P., Shapiro, M. & Gutfinger, C. 2002 Creeping flow past and within a permeable spheroid. Intl J. Multiphase Flow 28 (12), 1945–1963.Google Scholar
van der Heyden, F. H. J., Bonthuis, D. J., Stein, D., Meyer, C. & Dekker, C. 2006 Electrokinetic energy conversion efficiency in nanofluidic channels. Nano Lett. 6 (10), 2232–2237.Google Scholar
Wang, H. & Skalak, R. 1969 Viscous flow in a cylindrical tube containing a line of spherical particles. J. Fluid Mech. 38 (1), 75–96.Google Scholar
Wilson, H. J. 2013 Stokes flow past three spheres. J. Comput. Phys. 245, 302–316.Google Scholar
Yao, X., Marcos & Wong, T. N. 2017 Slow viscous flow around two particles in a cylinder. Microfluid. Nanofluid. 21 (10), 161.Google Scholar
Zhong, X. & Duan, F. 2016 Flow regime and deposition pattern of evaporating binary mixture droplet suspended with particles. Eur. Phys. J. E 39 (2), 1–6.Google Scholar
Zhong, X., Xie, H. & Duan, F. 2017 Deposition patterns from evaporating sessile droplets with suspended mixtures of multi-sized and multi-species hydrophilic and non-adsorbing nanoparticles. Appl. Therm. Engng 111, 1565–1572.Google Scholar