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Spherical particle sedimenting in weakly viscoelastic shear flow

J. Einarsson, B. Mehlig

TL;DR

This work analyzes a small sphere translating through an unbounded viscoelastic shear flow under an external force, using a second-order perturbation in $De$ and $Wi$ to uncover nontrivial viscoelastic effects on mobility and resistance. A key methodological advance is the introduction of a Cartesian $T$-tensor basis, enabling coordinate-free, algebraic solutions to the inhomogeneous Stokes equations and seamless use of the Lorentz reciprocal theorem. The authors derive explicit expressions for the particle velocity and rotation (mobility problem) and for the hydrodynamic force and angular velocity (resistance problem), revealing a shear-induced lift at $O(Wi)$, drag corrections at $O(De^2)$ and $O(Wi^2)$, and a second lift at $O(Wi^2)$; these corrections depend sensitively on the forcing orientation relative to the shear. The results explain reduced settling speeds in cross-shear configurations and predict additional lateral drifts when gravity is inclined with respect to the vorticity, while providing a versatile tensorial perturbation framework amenable to computer implementation for other geometries and constitutive models.

Abstract

We consider the dynamics of a small spherical particle driven through an unbounded viscoelastic shear flow by an external force. We give analytical solutions to both the mobility problem (velocity of forced particle) and the resistance problem (force on fixed particle), valid to second order in the dimensionless Deborah and Weissenberg numbers, which represent the elastic relaxation time of the fluid relative to the rate of translation and the imposed shear rate. We find a shear-induced lift at $O({\rm Wi})$, a modified drag at $O({\rm De}^2)$ and $O({\rm Wi}^2)$, and a second lift that is orthogonal to the first, at $O({\rm Wi}^2)$. The relative importance of these effects depends strongly on the orientation of the forcing relative to the shear. We discuss how these forces affect the terminal settling velocity in an inclined shear flow. We also describe a new basis set of symmetric Cartesian tensors, and demonstrate how they enable general tensorial perturbation calculations such as the present theory. In particular this scheme allows us to write down a solution to the inhomogenous Stokes equations, required by the perturbation expansion, by a sequence of algebraic manipulations well suited to computer implementation.

Spherical particle sedimenting in weakly viscoelastic shear flow

TL;DR

This work analyzes a small sphere translating through an unbounded viscoelastic shear flow under an external force, using a second-order perturbation in and to uncover nontrivial viscoelastic effects on mobility and resistance. A key methodological advance is the introduction of a Cartesian -tensor basis, enabling coordinate-free, algebraic solutions to the inhomogeneous Stokes equations and seamless use of the Lorentz reciprocal theorem. The authors derive explicit expressions for the particle velocity and rotation (mobility problem) and for the hydrodynamic force and angular velocity (resistance problem), revealing a shear-induced lift at , drag corrections at and , and a second lift at ; these corrections depend sensitively on the forcing orientation relative to the shear. The results explain reduced settling speeds in cross-shear configurations and predict additional lateral drifts when gravity is inclined with respect to the vorticity, while providing a versatile tensorial perturbation framework amenable to computer implementation for other geometries and constitutive models.

Abstract

We consider the dynamics of a small spherical particle driven through an unbounded viscoelastic shear flow by an external force. We give analytical solutions to both the mobility problem (velocity of forced particle) and the resistance problem (force on fixed particle), valid to second order in the dimensionless Deborah and Weissenberg numbers, which represent the elastic relaxation time of the fluid relative to the rate of translation and the imposed shear rate. We find a shear-induced lift at , a modified drag at and , and a second lift that is orthogonal to the first, at . The relative importance of these effects depends strongly on the orientation of the forcing relative to the shear. We discuss how these forces affect the terminal settling velocity in an inclined shear flow. We also describe a new basis set of symmetric Cartesian tensors, and demonstrate how they enable general tensorial perturbation calculations such as the present theory. In particular this scheme allows us to write down a solution to the inhomogenous Stokes equations, required by the perturbation expansion, by a sequence of algebraic manipulations well suited to computer implementation.

Paper Structure

This paper contains 13 sections, 71 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Equation of motion and dimensionless parameters
  4. Reciprocal theorem
  5. Method: T-tensors
  6. Definition
  7. Properties of the $T$-tensors
  8. Particular solution for the inhomogenous Stokes equation
  9. Results
  10. The mobility problem
  11. The resistance problem
  12. Summary and Discussion
  13. Multiplication of $T$-tensors

Figures (3)

  • Figure 1: The inclined shear flow geometry discussed in Section \ref{['sec:discussion']}. In this example the external force lies in the plane spanned by the vorticity axis and the flow direction. This situation corresponds to settling between two far-apart shearing walls, parallel to the walls, but where the shearing is at an angle to gravity. We denote by $\varphi$ the angle between $\boldsymbol{F}^\mathrm{ext}$ and vorticity.
  • Figure 2: Shear-dependent velocity of a sphere forced by an external force in the flow-vorticity plane, for different angle of attack $\varphi$ between the forcing $\boldsymbol{F}^\mathrm{ext}$ and the vorticity axis. The cross-shear flow corresponds to $\varphi=0$, whereas the force is along the flow direction when $\varphi=90^\circ$. (a) Velocity along $\boldsymbol{F}^\mathrm{ext}$. (b) Lateral drift in the shear direction $\boldsymbol{\hat{y}}$. (c) Lateral drift perpendicular to the shear direction. Parameters: $\mu_r = 0.3$, $\textrm{De}=0.1$.
  • Figure 3: The trace of the first order viscoelastic stress $\boldsymbol{\Pi}^{(1)}$ around a particle driven by an external force through a shear flow. All panels show a center cross-section of the particle, viewed along the direction of external forcing. In these panels $y$ indicates the shear direction, and $x$ indicates the direction perpendicular to both $\boldsymbol{F}^\mathrm{ext}$ and $\boldsymbol{\hat{y}}$, see Fig. \ref{['fig:inclinedshear']}. Left column shows full stress field, right column shows its asymmetric part under inversion of $\boldsymbol{r}$. Top row shows stress field when the external forcing is aligned with vorticity, bottom row shows same when the external forcing is at angle $\varphi = 45^\circ$ to vorticity. Parameters: $\alpha=1$ (implying $\textrm{Wi}=\textrm{De}$). The trace of $\boldsymbol{\Pi}^{(1)}$ is independent of $\mu_r$, which follows from Eq. (\ref{['eq:eqPi1']}) because the flow field $\boldsymbol{u}$ is incompressible.