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Modeling of macroscopic stresses in a dilute suspension of small weakly inertial particles

Alexander Vibe, Nicole Marheineke

Abstract

In this paper we derive asymptotically the macroscopic bulk stress of a suspension of small inertial particles in an incompressible Newtonian fluid. We apply the general asymptotic framework to the special case of ellipsoidal particles and show the resulting modification due to inertia on the well-known particle-stresses based on the theory by Batchelor and Jeffery.

Modeling of macroscopic stresses in a dilute suspension of small weakly inertial particles

Abstract

In this paper we derive asymptotically the macroscopic bulk stress of a suspension of small inertial particles in an incompressible Newtonian fluid. We apply the general asymptotic framework to the special case of ellipsoidal particles and show the resulting modification due to inertia on the well-known particle-stresses based on the theory by Batchelor and Jeffery.

Paper Structure

This paper contains 17 sections, 7 theorems, 100 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical model of a single particle in a fluid
  3. Three-dimensional interface problem
  4. Tracer particles of different inertial type
  5. Asymptotic analysis
  6. Extension of Stokes solutions to the particle domain
  7. Kinetic model for a particle suspension
  8. General framework
  9. Macroscopic stresses and forces
  10. Asymptotical suspension model
  11. Special case of a suspension with ellipsoidal particles
  12. General properties
  13. Suspension model of weakly inertial ellipsoidal particles
  14. Conclusion
  15. Analytical statements for ellipsoidal geometry
  16. ...and 2 more sections

Key Result

Lemma 4

Let the following four requirements be fulfilled, then $\bm{\mathsf{u}}^e$, $p^e$, $\bm{\mathsf{c}}^e$, $\bm{\mathsf{v}}^e$, $\bm{\mathsf{\omega}}^e$ and $\bm{\mathsf{R}}^e$ defined in eq_asymAn_expanAll are a solution of the complete system eq_mathMod_dimFreeSystem up to an order of $\mathcal{O}(\e

Figures (1)

  • Figure 2.1: Lagrangian description, bijective mapping between reference state and the actual time-dependent state.

Theorems & Definitions (18)

  • Lemma 4: Asymptotic one-particle model
  • proof
  • Lemma 5: Solvability conditions of Stokes problem
  • proof
  • Remark 6
  • Remark 7: Density ratio scaling
  • Lemma 8: Asymptotic model of stresses in particle domain
  • proof
  • Remark 9: Local velocity fields and forces in particle domain
  • Theorem 11: Asymptotical model of a suspension with weakly inertial tracer particles
  • ...and 8 more