## GIUSEPPE PROCOPIO

PhD Graduate**PhD program:**: XXXV

**supervisor**: prof. Massimiliano Giona

**Thesis title:**Hydrodynamic characterization of finite-sized particle transport in confined microfluidic systems, Brownian motion and stochastic modeling of particle transport at microscale.

In this thesis the peculiar effects of the hydrodynamic confinement on the dynamic
of a colloid in Stokes regime has been addressed theoretically. Practical expressions,
useful to investigate transport of particles in complex geometries, have been provided
for force, torque and higher order moments on the particle and the disturbed velocity
field of the fluid.
To begin with, a new formulation of the Stokesian singularity method is developed
by introducing a bitensorial distributional formalism. This formalism overcomes the
ambiguities of the classical hydrodynamic formulation of the singularity method
that limit its application in confined problems. The formalism proposed permits
naturally to distinguish between poles and field points of tensorial singular fields
and to clearly define each singularity from its associated Stokes problem, connecting
singularities each other.
As a consequence of this approach an explicit expression for the singularity
operator is provided, giving the disturbance field due to a body once applied to
an ambient flow of the fluid. The operator is expressed in terms of the volume
moments and its expression is valid regardless the boundary conditions applied to
the surface of the body. The dualism between the singularity operator giving the
disturbance flow of a n-th order ambient flow and the n-th order Faxén operator has
been investigated. It has been found that this dualism, referred to as the Hinch-Kim
dualism, holds only if the boundary conditions satisfy a property that is referred to
as the Boundary-Condition reciprocity (BC-reciprocity, for short). If this property is
fulfilled, the Faxén operators can be expressed in terms of (m, n)-th order geometrical
moments of volume forces (defined in the Chapter 3). In addition, it is shown that in
these cases, the hydromechanics of the fluid-body system is completely determined
by the entire system of the Faxén operators. Classical boundary conditions of
hydrodynamic practice (involving slippage, fluid-fluid interfaces, porous materials,
etc.) are investigated in the light of this property. It is found the analytical expression
for the 0-th, 1-st and 2-nd Faxén operators for a sphere with Navier-slip boundary
conditions.
These results are applied in order to express the hydrodynamics of particles in
confined fluids in terms of quantities related to the geometry of the particle and the
geometry of the confinement separately using the reflection method. Specifically,
closed-form results and practical expressions for the velocity field of the fluid and
for the functional form of force and torque acting on a particle are derived in terms
of: (i) the Faxén operators of the body of the particle (given by its unbounded
geometrical moments) and (ii) the multi-poles in the domain of the confinement. The
convergence of the reflection method is examined and it is found that the expressions
obtained are also valid for distance between particle and walls of the confinement
of the same magnitude order, failing only in the limit case of the lubrication range.
The reflection
solutions obtained with the present theory, approximated to the order
O(lb /ld )^5 , are compared with the exact solution of a sphere near a planar wall,
and the expressions for forces and torques considering the more general situation of
Navier-slip boundary conditions on the body are provided.
A general formulation of the fluctuation-dissipation relations in confined geome-
tries, the paradoxes associated with no-slip boundary conditions close to a solid
boundary, and the modal representation of the inertial kernels for complex fluids
complete the present dissertation.

**Research products**