// Title Page
= Solid particle interaction in Stokes flow

Author: "Claudius Holeksa"
Date: 2025-11-09

// Abstract
// == Abstract
// This is a brief summary of the report. It gives an overview of the objectives, methods, and conclusions.

// Table of Contents
[toc]

== Introduction

For the understanding of near-well injections multiple elements such as multiphase behaviour, particle-solid interaction and
the geometry of the porous structure is required.  
While at heart most numerical modelling approaches such as the Lattice-Boltzmann-Method are based on the Navier-Stokes Equations,
here we will take a glance on a more special case.  
Since we are interested in very viscous cases with our Reynolds number (Re \< \< 1) our base equation reduces itself to the Stokes Equations,
which are well understand in terms of the description by Stokes [dummy, I mean the old paper from 1851].
So the Navier-Stokes equations

todo insert NS eqs here

are reduced to

todo insert S eqs here
// Rather move this to the lower chapters and use book citations I guess. Well, maybe also Stokes paper. The initial rant is quite fun


== Navier

The Navier Stokes equations are governed as follows

// we don't really need the external force here? do we?

#math.equation(
block: true,
$ ρ((∂)/(∂t)u + (u · ∇)u) = −∇p + μ∇²u + f\
∇ · u = 0 $
)

// Introduction
== Stokes flow

While we are often interested in Navier-Stokes flows, on higher viscous fluids the viscosity dominates.


== Solid-Fluid interaction with Stokes flow

So while

// References
== References