#import "@preview/charged-ieee:0.1.3": ieee

#show: ieee.with(
  title: [Stokes flow - Particle interaction in low Reynolds Number environments],
  abstract: [
  ],
  authors: (
    (
      name: "Claudius Holeksa",
      department: [CSSR],
      organization: [NORCE Research AS],
      location: [Bergen, Norway],
      email: "clho@norceresearch.no"
    ),
  ),
  index-terms: ("Fluid Mechanics", "Review"),
  bibliography: bibliography("refs.bib"),
  figure-supplement: [Fig.],
)

= Introduction

Flow in porous subsurface structures often is dominated by low velocity and high viscous fluids with low Reynolds numbers,
often referred to as Stokes flow or creeping flow.
It occurs when the viscous forces are significantly larger compared to inertial forces.
For the understanding of porous media, where particle transport, bridging and clogging phenomena are critical to the use of subsurface
reservoirs. Be it either for carbohydrate resource recovery or storage of CO2. Understanding the behaviour of particles in Stokes flow
is key to analyzing more complex phenomena emerging over time such as permeability reduction over periods of time.\


The study of the Stokes Flow has a long history, beginning with Stokes' solution flow along a sphere@stokes1851.\
Subsequent work expanded to explore a multitude of shapes and the interaction of multiple particles.
More recently, research was performed in the understanding of creeping flow which are essential in water treatment, carbohydrate recovery
and CO2 storage. // CITE

This report delves into the central aspects governing the known behaviour past single and multiple spheres.
Explores the effect of it in porous media and examines the more recent particle bridging behaviour found in porous structures.


== Stokes Flow

// Write about Navier
The incompressible Navier Stokes equations are derived from Newton are a set of equations

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

While we are often interested in Navier-Stokes flows, a generalized analytical solution does not exist,
most analytical approaches rely on a set of restrictions or assumptions.
Luckily we decided to work on a fluid which tends $"Re" arrow 0$ where due to the viscous forces
the inertial term is negligable and is assumed to be zero.

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

== Solid-Fluid interaction with Stokes flow

For a solid sphere of radius a moving with velocity $U$ in an unbounded creeping flow,
we define the 

// Insert analytical solution



Since 
// Boundary
//


// Different boundary examples ?

== Multiple particles

== Clogging and Bridging


// References
== References