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On Einstein’s Effective Viscosity Formula
 
Mitia Duerinckx Université Libre de Bruxelles, Belgium and Université Paris-Saclay, France
Antoine Gloria Sorbonne Université, France and Institut Universitaire de France and Université Libre de Bruxelles, Belgium
A publication of European Mathematical Society
Softcover ISBN:  978-3-98547-055-6
Product Code:  EMSMEM/7
List Price: $75.00
AMS Member Price: $60.00
This item is temporarily out of stock. Order now and your item will ship as soon as stock becomes available.
Expected availability date: August 05, 2024
Please note AMS points can not be used for this product
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On Einstein’s Effective Viscosity Formula
Mitia Duerinckx Université Libre de Bruxelles, Belgium and Université Paris-Saclay, France
Antoine Gloria Sorbonne Université, France and Institut Universitaire de France and Université Libre de Bruxelles, Belgium
A publication of European Mathematical Society
Softcover ISBN:  978-3-98547-055-6
Product Code:  EMSMEM/7
List Price: $75.00
AMS Member Price: $60.00
This item is temporarily out of stock. Order now and your item will ship as soon as stock becomes available.
Expected availability date: August 05, 2024
Please note AMS points can not be used for this product
  • Book Details
     
     
    Memoirs of the European Mathematical Society
    Volume: 72023; 196 pp
    MSC: Primary 76; Secondary 35

    In his Ph.D. thesis, Einstein derived an explicit first-order expansion for the effective viscosity of a Stokes fluid with a suspension of small rigid particles at low density. His formal derivation relied on two implicit assumptions:

    (i) There is a scale separation between the size of the particles and the observation scale.

    (ii) At first order, dilute particles do not interact with one another.

    In mathematical terms, the first assumption amounts to the validity of a homogenization result defining the effective viscosity tensor, which is now well understood. The second assumption allowed Einstein to approximate this effective viscosity at low density by considering particles as being isolated. The rigorous justification is, in fact, quite subtle as the effective viscosity is a nonlinear nonlocal function of the ensemble of particles and as hydrodynamic interactions have borderline integrability.

    In this memoir, the authors establish Einstein's effective viscosity formula in the most general setting. In addition, they pursue the low-density expansion to arbitrary order in form of a cluster expansion, where the summation of hydrodynamic interactions crucially requires suitable renormalizations. In particular, they justify a celebrated result by Batchelor and Green on the second-order correction and explicitly describe all higher-order renormalizations for the first time.

    In some specific settings, the authors further address the summability of the whole cluster expansion. The authors' approach relies on a combination of combinatorial arguments, variational analysis, elliptic regularity, probability theory, and diagrammatic integration methods.

    A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

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    Review Copy – for publishers of book reviews
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Volume: 72023; 196 pp
MSC: Primary 76; Secondary 35

In his Ph.D. thesis, Einstein derived an explicit first-order expansion for the effective viscosity of a Stokes fluid with a suspension of small rigid particles at low density. His formal derivation relied on two implicit assumptions:

(i) There is a scale separation between the size of the particles and the observation scale.

(ii) At first order, dilute particles do not interact with one another.

In mathematical terms, the first assumption amounts to the validity of a homogenization result defining the effective viscosity tensor, which is now well understood. The second assumption allowed Einstein to approximate this effective viscosity at low density by considering particles as being isolated. The rigorous justification is, in fact, quite subtle as the effective viscosity is a nonlinear nonlocal function of the ensemble of particles and as hydrodynamic interactions have borderline integrability.

In this memoir, the authors establish Einstein's effective viscosity formula in the most general setting. In addition, they pursue the low-density expansion to arbitrary order in form of a cluster expansion, where the summation of hydrodynamic interactions crucially requires suitable renormalizations. In particular, they justify a celebrated result by Batchelor and Green on the second-order correction and explicitly describe all higher-order renormalizations for the first time.

In some specific settings, the authors further address the summability of the whole cluster expansion. The authors' approach relies on a combination of combinatorial arguments, variational analysis, elliptic regularity, probability theory, and diagrammatic integration methods.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.