Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic limits of the axisymmetric solution of the Brinkman equation for a point force near a no-slip wall

Abdallah Daddi-Moussa-Ider, Andrej Vilfan

TL;DR

The paper addresses how a point force normal to a no-slip wall behaves in a Brinkman medium by deriving a Green's function that is explicit up to a single integral and presenting asymptotic expansions valid in both far-field and near-field regimes. The solution is decomposed into free-space and wall-induced components, yielding a near-field structure that includes Blake-like multipoles and a far-field series of wall-corrected source singularities; the authors also provide closed-form far-field and near-field expressions without requiring full numerical integration. The results recover the classical Blake tensor in the Stokes limit and extend the framework to Brinkman media, with direct relevance to near-wall flows induced by beating cilia or microswimmers and to the time-dependent cross-correlations of Brownian particles near boundaries. This work offers a robust analytical toolkit for modeling hydrodynamic interactions near boundaries in porous or partially permeable environments and can serve as a foundation for more complex force distributions and unsteady/inertial extensions.

Abstract

We derive the far-field and near-field solutions for the Green's function of a point force acting perpendicular to a no-slip wall in a Brinkman fluid, focusing on the regime where the distance between the force and the wall is much smaller than the screening length. The general solution is obtained in closed form up to a single integral and can be systematically expanded in a Taylor series in both the far-field and near-field limits. The flow can then be expressed as a series of source-multipole singularities with an additional, analytically known, correction in the proximity of the wall. Comparisons with numerical integration demonstrate the accuracy and reliability of the asymptotic expansions. The results are also applicable to the unsteady Stokes flow driven by a localized assembly of forces, such as a beating cilium protruding from a flat surface.

Asymptotic limits of the axisymmetric solution of the Brinkman equation for a point force near a no-slip wall

TL;DR

The paper addresses how a point force normal to a no-slip wall behaves in a Brinkman medium by deriving a Green's function that is explicit up to a single integral and presenting asymptotic expansions valid in both far-field and near-field regimes. The solution is decomposed into free-space and wall-induced components, yielding a near-field structure that includes Blake-like multipoles and a far-field series of wall-corrected source singularities; the authors also provide closed-form far-field and near-field expressions without requiring full numerical integration. The results recover the classical Blake tensor in the Stokes limit and extend the framework to Brinkman media, with direct relevance to near-wall flows induced by beating cilia or microswimmers and to the time-dependent cross-correlations of Brownian particles near boundaries. This work offers a robust analytical toolkit for modeling hydrodynamic interactions near boundaries in porous or partially permeable environments and can serve as a foundation for more complex force distributions and unsteady/inertial extensions.

Abstract

We derive the far-field and near-field solutions for the Green's function of a point force acting perpendicular to a no-slip wall in a Brinkman fluid, focusing on the regime where the distance between the force and the wall is much smaller than the screening length. The general solution is obtained in closed form up to a single integral and can be systematically expanded in a Taylor series in both the far-field and near-field limits. The flow can then be expressed as a series of source-multipole singularities with an additional, analytically known, correction in the proximity of the wall. Comparisons with numerical integration demonstrate the accuracy and reliability of the asymptotic expansions. The results are also applicable to the unsteady Stokes flow driven by a localized assembly of forces, such as a beating cilium protruding from a flat surface.

Paper Structure

This paper contains 10 sections, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Hydrodynamic fields
  4. Limit of small $h$
  5. Pressure field
  6. Shear-driven flow
  7. Velocity field
  8. Far field
  9. Near field
  10. Discussion

Figures (2)

  • Figure 1: Comparison of the numerically evaluated $P^{(2)}_2$ (symbols) with the asymptotic approximations (solid lines) from Eq.\ref{['eq:P2_far-field']} in the far-field limit $\alpha r \gg 1$ and Eq.\ref{['eq:P2_near-field']} (up to the order $\mathcal{O}(\alpha^2)$) in the near-field limit $\alpha r \ll 1$.
  • Figure 2: Streamline plot of the axisymmetric flow induced by a Brinkmanlet near a no-slip wall, obtained using (a) the far-field solution from Eqs. \ref{['eq:W']} and \ref{['eq:H']} and (b) numerical integration for $\alpha h = 0.1$.