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#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 a set of equations defined with the density $rho$, the dynamic viscosity $mu$ and the
velocity field $u(x,t)$
#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.
With the Reynolds number defined by
$ "Re" = (rho U L ) / mu
$
Luckily we decided to work on a fluid which tends $"Re" arrow 0$ or at least $"Re" << 1$ where due to the viscous forces
the inertial term is negligable and is assumed to be zero.
Which is why we arrive at this term
// Stokes
#math.equation(
block: true,
$ μ∇²u + f = ∇p,\
∇ · u = 0 $
)
An additional restriction by the creeping flow equations is the selection of boundary criteria where it depends on
an infinite space with a set solid boundary interface.
#math.equation(
block: true,
$ v = U "at" a = r,\
v arrow 0 "as" a arrow infinity
$
)
== Solid-Fluid single sphere interaction with Stokes flow
With the previous assumptions a solid sphere of radius $r$ moving with a relative velocity $v$ in an unbounded creeping flow,
we receive the drag force provided by Stokes
#math.equation(
block: true,
$ F_S = 6 pi mu r v $
)
where according to Proudman and Pearson@proudman_pearson_1957 at a Reynolds number of 0.05 the predicted drag is two percent lower
than the possibly more correct value by Proudman and Pearson@proudman_pearson_1957.
//#math.equation(
//block: true,
//$ F = F_S ( 1 + (3/8)"Re" + (9/40)("Re")^2 * (log "Re" + gamma + (5/3) log 2 - (323/360)) + (27/80)*"Re"^3 log "Re" ) $
//)
//Using spherical coordinates $(r,theta,phi.alt)$ and no external force we receive the velocity components
//#math.equation(
//block: true,
//$ v_r = U cos theta (1- (3r)/(2a) + r^3 / (2a^3)) $
//$ v_theta = - U cos theta (1- (3r)/(4a) + r^3 / (4a^3)) $
//)
== Lubrication Forces for Near Contact
When spheres or a sphere near a straight wall approach each other with a small separation $h<<r$ the flow in the gap is dominated by
lubrication forces. For a sphere approaching a straight wall with velocity $U$ normal to the wall we
have
#math.equation(
block: true,
$ F_(L) tilde.eq (6 pi mu r^2 U) / h (1-(9h)/(16r) + ...) $
)
With two spheres of equal radius r and otherwise the same conditions approaching
#math.equation(
block: true,
$ F_(L) tilde.eq (6 pi mu r^2 U) / h (1-(h)/(5r) + ...) $
)
== Clogging and Bridging
== Lattice-Boltzmann-Method
=== Immersed Boundary Method
=== Homogenized Lattice-Boltzmann
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