DOI: 10.1017/s0022112096008531. The transverse shear-induced liquid and particle tracer diffusivities of a dilute suspension of spheres undergoing a simple shear flow

The transverse shear-induced liquid and particle tracer diffusivities of a dilute suspension of spheres undergoing a simple shear flow

10.1017/s0022112096008531

Crossref journal-article

Cambridge University Press (CUP)

Journal of Fluid Mechanics (56)

Abstract

We study the shear-induced self-diffusion of both a liquid tracer and a tagged spherical particle along the directions perpendicular to the ambient flow in a dilute suspension of neutrally buoyant spheres undergoing a simple shearing motion in the absence of inertia and Brownian motion effects. The calculation of the liquid diffusivity requires the velocity of a fluid point under the influence of two spheres, which was determined via Lamb's series expansion; conversely, the calculation of the particle diffusivity involves the trajectories of three spheres, which were determined using far-field and near-field asymptotic expressions. The displacements of the liquid tracer and of the tagged sphere were then computed analytically when the spheres and the tracer are all far apart, and numerically for close encounters. After summing over all possible encounters, the leading terms of the lateral liquid diffusion coefficients, both within and normal to the plane of shear, were thereby found to be 0.12γac and 0.004γac, respectively, where γ is the applied shear rate, a the radius of the spheres and c their volume fraction. The analogous coefficients of the lateral particle diffusivity were found to be 0.11γac, and 0.005γac, respectively. Also, liquid and particle diffusivities in a monolayer, with the liquid tracer and all the particle centres lying on the same plane of shear, were found to be 0.067γyac, and 0.032γac, respectively, with c denoting the areal fraction occupied by the spheres on the plane.

Bibliography

Wang, Y., Mauri, R., & Acrivos, A. (1996). The transverse shear-induced liquid and particle tracer diffusivities of a dilute suspension of spheres undergoing a simple shear flow. Journal of Fluid Mechanics, 327, 255â272.

Authors 3

Y. Wang (first)
R. Mauri (additional)
A. Acrivos (additional)

References 18 Referenced 51

Bossis, G. & Brady, J. F. 1987 Self-diffusion of Brownian particles in concentrated suspensions under shear.J. Chem. Phys. 87,5437.
Chang, C. & Powell, R. L. 1994 Self-diffusion of bimodal suspensions of hydrodynamically interacting spherical particles in shearing flow.J. Fluid Mech. 281,51.
Cichocki, B. , Felderhof, B. U. , Hinsen, K. , Wajnryb, E. & Blawzdziewicz, J. 1994 Friction and mobility of many spheres in Stokes flow.J. Chem. Phys. 100,3780.
Van Saarloos, W. & Mazur, P. 1983 Many-sphere hydrodynamic interactions H. Mobilities at finite frequencies.Physica A120,77.
Eckstein, E. C. , Bailey, D. G. & Shapiro, A. H. 1977 Self-diffusion of particles in shear flow of a suspension.J. Fluid Mech. 79,191.
Yoon, B. J. & Kim, S. 1987 Note on the direct calculation of mobility functions for two equal-sized spheres in Stokes flow.J. Fluid Mech. 185,437.
Biemfohr, S. , Looby, T. , Biemfohr, S. & Leighton, D. 1996 Measurement of the shear-induced coefficient of self-diffusion in dilute suspensions.J. Fluid Mech. (submitted).
Batchelor, G. K. & Green, J. T. 1972b The determination of the bulk stress in a suspension of spherical particles to order c 2 .J. Fluid Mech. 56,400.
Hassonjee, Q. , Pfeffer, R. & Ganatos, P. 1992 Behavior of multiple spheres in shear and Poiseuille flow fields at low Reynolds number.Int. J. Multiphase Flow 18,353.
Kim, S. 1987 Stokes flow past three spheres: An analytic solution.Phys. Fluids 30,2309.
Cunha, F. R. da & Hinch, E. J. 1996 Shear-induced dispersion in a dilute suspension of rough spheres.J. Fluid Mech. 309,211.
Acrivos, A. , Batchelor, G. K. , Hinch, E. J. , Koch, D. L. & Mauri, R. 1992 Longitudinal shear-induced diffusion of spheres in a dilute suspension.J. Fluid Mech. 240,651.
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications .Butterworth-Heinemann. (10.1016/B978-0-7506-9173-4.50007-4)
Leighton, D. & Acrivos, A. 1987b The shear-induced migration of particles in concentrated suspensions.J. Fluid Mech. 181,415.
Batchelor, G. K. & Green, J. T. 1972a The hydrodynamic interaction of two small freely-moving spheres in a linear flow field.J. Fluid Mech. 56,375.
Mazur, P. & Van Saarloos, W. 1982 Many-sphere hydrodynamic interactions and mobilities in a suspension.Physica A115,21.
Phan, S. E. & Leighton, D. 1996 Measurement of the shear-induced tracer diffusivity in concentrated suspensions.J. Fluid Mech. (submitted).
Leighton, D. & Acrivos, A. 1987a Measurement of shear-induced self-diffusion in concentrated suspensions of spheres.J. Fluid Mech. 177,109.

Dates

Type	When
Created	19 years, 3 months ago (May 3, 2006, 9:47 p.m.)
Deposited	6 years, 2 months ago (June 7, 2019, 4:22 p.m.)
Indexed	1 year, 6 months ago (Feb. 10, 2024, 8:40 p.m.)
Issued	28 years, 8 months ago (Nov. 25, 1996)
Published	28 years, 8 months ago (Nov. 25, 1996)
Published Online	19 years, 3 months ago (April 26, 2006)
Published Print	28 years, 8 months ago (Nov. 25, 1996)

Funders 0

None

BibTeX

@article{Wang_1996, title={The transverse shear-induced liquid and particle tracer diffusivities of a dilute suspension of spheres undergoing a simple shear flow}, volume={327}, ISSN={1469-7645}, url={http://dx.doi.org/10.1017/s0022112096008531}, DOI={10.1017/s0022112096008531}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Wang, Y. and Mauri, R. and Acrivos, A.}, year={1996}, month=nov, pages={255–272} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2024,
        2,
        11
      ]
    ],
    "date-time": "2024-02-11T01:40:39Z",
    "timestamp": 1707615639578
  },
  "reference-count": 18,
  "publisher": "Cambridge University Press (CUP)",
  "license": [
    {
      "start": {
        "date-parts": [
          [
            2006,
            4,
            26
          ]
        ],
        "date-time": "2006-04-26T00:00:00Z",
        "timestamp": 1146009600000
      },
      "content-version": "unspecified",
      "delay-in-days": 3439,
      "URL": "https://www.cambridge.org/core/terms"
    }
  ],
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "published-print": {
    "date-parts": [
      [
        1996,
        11,
        25
      ]
    ]
  },
  "abstract": "<jats:p>We study the shear-induced self-diffusion of both a liquid tracer and a tagged spherical particle along the directions perpendicular to the ambient flow in a dilute suspension of neutrally buoyant spheres undergoing a simple shearing motion in the absence of inertia and Brownian motion effects. The calculation of the liquid diffusivity requires the velocity of a fluid point under the influence of two spheres, which was determined via Lamb's series expansion; conversely, the calculation of the particle diffusivity involves the trajectories of three spheres, which were determined using far-field and near-field asymptotic expressions. The displacements of the liquid tracer and of the tagged sphere were then computed analytically when the spheres and the tracer are all far apart, and numerically for close encounters. After summing over all possible encounters, the leading terms of the lateral liquid diffusion coefficients, both within and normal to the plane of shear, were thereby found to be 0.12\u03b3<jats:italic>a</jats:italic><jats:italic>c</jats:italic> and 0.004\u03b3<jats:italic>a</jats:italic><jats:italic>c</jats:italic>, respectively, where \u03b3 is the applied shear rate, <jats:italic>a</jats:italic> the radius of the spheres and <jats:italic>c</jats:italic> their volume fraction. The analogous coefficients of the lateral particle diffusivity were found to be 0.11\u03b3<jats:italic>a</jats:italic><jats:italic>c</jats:italic>, and 0.005\u03b3<jats:italic>a</jats:italic><jats:italic>c</jats:italic>, respectively. Also, liquid and particle diffusivities in a monolayer, with the liquid tracer and all the particle centres lying on the same plane of shear, were found to be 0.067\u03b3y<jats:italic>a</jats:italic><jats:italic>c</jats:italic>, and 0.032\u03b3<jats:italic>a</jats:italic><jats:italic>c</jats:italic>, respectively, with c denoting the areal fraction occupied by the spheres on the plane.</jats:p>",
  "DOI": "10.1017/s0022112096008531",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2006,
        5,
        4
      ]
    ],
    "date-time": "2006-05-04T01:47:10Z",
    "timestamp": 1146707230000
  },
  "page": "255-272",
  "source": "Crossref",
  "is-referenced-by-count": 51,
  "title": "The transverse shear-induced liquid and particle tracer diffusivities of a dilute suspension of spheres undergoing a simple shear flow",
  "prefix": "10.1017",
  "volume": "327",
  "author": [
    {
      "given": "Y.",
      "family": "Wang",
      "sequence": "first",
      "affiliation": []
    },
    {
      "given": "R.",
      "family": "Mauri",
      "sequence": "additional",
      "affiliation": []
    },
    {
      "given": "A.",
      "family": "Acrivos",
      "sequence": "additional",
      "affiliation": []
    }
  ],
  "member": "56",
  "published-online": {
    "date-parts": [
      [
        2006,
        4,
        26
      ]
    ]
  },
  "reference": [
    {
      "key": "S0022112096008531_ref005",
      "unstructured": "Bossis, G. & Brady, J. F. 1987 Self-diffusion of Brownian particles in concentrated suspensions under shear.J. Chem. Phys. 87,5437."
    },
    {
      "key": "S0022112096008531_ref006",
      "unstructured": "Chang, C. & Powell, R. L. 1994 Self-diffusion of bimodal suspensions of hydrodynamically interacting spherical particles in shearing flow.J. Fluid Mech. 281,51."
    },
    {
      "key": "S0022112096008531_ref007",
      "unstructured": "Cichocki, B. , Felderhof, B. U. , Hinsen, K. , Wajnryb, E. & Blawzdziewicz, J. 1994 Friction and mobility of many spheres in Stokes flow.J. Chem. Phys. 100,3780."
    },
    {
      "key": "S0022112096008531_ref017",
      "unstructured": "Van Saarloos, W. & Mazur, P. 1983 Many-sphere hydrodynamic interactions H. Mobilities at finite frequencies.Physica A120,77."
    },
    {
      "key": "S0022112096008531_ref009",
      "unstructured": "Eckstein, E. C. , Bailey, D. G. & Shapiro, A. H. 1977 Self-diffusion of particles in shear flow of a suspension.J. Fluid Mech. 79,191."
    },
    {
      "key": "S0022112096008531_ref018",
      "unstructured": "Yoon, B. J. & Kim, S. 1987 Note on the direct calculation of mobility functions for two equal-sized spheres in Stokes flow.J. Fluid Mech. 185,437."
    },
    {
      "key": "S0022112096008531_ref004",
      "unstructured": "Biemfohr, S. , Looby, T. , Biemfohr, S. & Leighton, D. 1996 Measurement of the shear-induced coefficient of self-diffusion in dilute suspensions.J. Fluid Mech. (submitted)."
    },
    {
      "key": "S0022112096008531_ref003",
      "unstructured": "Batchelor, G. K. & Green, J. T. 1972b The determination of the bulk stress in a suspension of spherical particles to order c 2 .J. Fluid Mech. 56,400."
    },
    {
      "key": "S0022112096008531_ref010",
      "unstructured": "Hassonjee, Q. , Pfeffer, R. & Ganatos, P. 1992 Behavior of multiple spheres in shear and Poiseuille flow fields at low Reynolds number.Int. J. Multiphase Flow 18,353."
    },
    {
      "key": "S0022112096008531_ref011",
      "unstructured": "Kim, S. 1987 Stokes flow past three spheres: An analytic solution.Phys. Fluids 30,2309."
    },
    {
      "key": "S0022112096008531_ref008",
      "unstructured": "Cunha, F. R. da & Hinch, E. J. 1996 Shear-induced dispersion in a dilute suspension of rough spheres.J. Fluid Mech. 309,211."
    },
    {
      "key": "S0022112096008531_ref001",
      "unstructured": "Acrivos, A. , Batchelor, G. K. , Hinch, E. J. , Koch, D. L. & Mauri, R. 1992 Longitudinal shear-induced diffusion of spheres in a dilute suspension.J. Fluid Mech. 240,651."
    },
    {
      "key": "S0022112096008531_ref012",
      "doi-asserted-by": "crossref",
      "unstructured": "Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications .Butterworth-Heinemann.",
      "DOI": "10.1016/B978-0-7506-9173-4.50007-4"
    },
    {
      "key": "S0022112096008531_ref014",
      "unstructured": "Leighton, D. & Acrivos, A. 1987b The shear-induced migration of particles in concentrated suspensions.J. Fluid Mech. 181,415."
    },
    {
      "key": "S0022112096008531_ref002",
      "unstructured": "Batchelor, G. K. & Green, J. T. 1972a The hydrodynamic interaction of two small freely-moving spheres in a linear flow field.J. Fluid Mech. 56,375."
    },
    {
      "key": "S0022112096008531_ref015",
      "unstructured": "Mazur, P. & Van Saarloos, W. 1982 Many-sphere hydrodynamic interactions and mobilities in a suspension.Physica A115,21."
    },
    {
      "key": "S0022112096008531_ref016",
      "unstructured": "Phan, S. E. & Leighton, D. 1996 Measurement of the shear-induced tracer diffusivity in concentrated suspensions.J. Fluid Mech. (submitted)."
    },
    {
      "key": "S0022112096008531_ref013",
      "unstructured": "Leighton, D. & Acrivos, A. 1987a Measurement of shear-induced self-diffusion in concentrated suspensions of spheres.J. Fluid Mech. 177,109."
    }
  ],
  "container-title": "Journal of Fluid Mechanics",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112096008531",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2019,
        6,
        7
      ]
    ],
    "date-time": "2019-06-07T20:22:30Z",
    "timestamp": 1559938950000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://www.cambridge.org/core/product/identifier/S0022112096008531/type/journal_article"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1996,
        11,
        25
      ]
    ]
  },
  "references-count": 18,
  "alternative-id": [
    "S0022112096008531"
  ],
  "URL": "http://dx.doi.org/10.1017/S0022112096008531",
  "relation": {},
  "ISSN": [
    "0022-1120",
    "1469-7645"
  ],
  "subject": [],
  "container-title-short": "J. Fluid Mech.",
  "published": {
    "date-parts": [
      [
        1996,
        11,
        25
      ]
    ]
  }
}