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 the Stokes regime have been addressed theoretically. Practical expressions, useful to investigate the transport of particles in complex geometries, have been provided for force, torque and higher order moments on the particle and for 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 limits its application in confined problems. The formalism proposed permits naturally to distinguish between pole and field points of tensorial singular fields and to clearly define each singularity from its associated Stokes problem. 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 of 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 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 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 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 distances 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 ("lb" being the characteristic size of the body and "ld" the characteristic distance from the confinement boundaries) 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 geometries, 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. Specifically, the general setting of the overdamped approximation in confined geometries is provided, by explicitly expressing the thermal contributions associated with the rigid rototranslational motion of a body. In passing, the extension of fluctuation-dissipation results to non-equilibrium conditions, such as those arising in thermophoretic flows in the presence of a steady temperature profile is developed. The influence of boundary conditions on the fluctuational form of the force acting on a rigid particle near a solid wall is addressed, showing that the classical Stokesian paradox of infinite touching time in the presence of no-slip boundary conditions can be resolved by considering the arbitrarily small slippage effects on both surfaces, leading to an integrable logarithmic singularity. Finally, a preliminary extension of fluid-particle interactions either in a time-dependent Stokes regime or in the presence of complex (viscoelastic) flows is addressed, focusing on the modal representation of the dissipative and fluid inertial memory kernels, and on the fluctional form of the latter. Specifically, it is shown that for a viscoelastic fluid, characterized by a finite and non-vanishing relaxation rate, the generalized Basset kernel is a regular function of time, also close to t=0, which is not the case of a Newtonian fluid for which the Basset kernel scales inversely with the square root of t.