Analysis of transport processes in random media

The focus of this dissertation has revolved around a number of issues related to the effect of disorder on the transport of momentum and heat through heterogeneous media. Four problems are addressed: (1) The influence of hydrodynamic dispersion on natural convection through random porous media. We use weakly nonlinear analysis to study bifurcation of conduction into cellular convection in an internally-heated fluid-saturated porous medium. The resulting bifurcation structure is unlike any pitchfork bifurcation type. The model is an attempt to account for the scatter of observed critical values for the first bifurcation. (2) The effect of structural disorder of mammalian blood vessels on pulsatile blood flow. A microvascular dentritic network with random vessel dimensions is built on the basis of statistical analysis of conjuctival beds. Our results show that the assumed statistical variation of vessel lengths results to flow rate deviations as high as 50% of the mean, while the corresponding effect of vessel diameter variation is much smaller. (3) The effect of random variations of conductivity on steady and unsteady heat conduction in disordered packed beds. A hybrid numerical-symbolic manipulation scheme is developed and applied in the study of one-dimensional heat conduction in fully-saturated packed beds in order to examine the effects of packing disorder on the medium effective conductivity. The steady-state solution is used to obtain both the mean value and the standard deviation of the effective conductivity. The unsteady heat conduction equation is discretized on a finite spatial grid and an explicit integration in time is carried out symbolically for each time step. (4) Lattice gas simulation of complex flows. The lattice gas method is introduced. The results are compared with these of a conventional finite element method in two problems; the plane Poiseuille flow and flow over a single cylinder. Good agreement has been achieved. The algorithm is then extended to study flow in a porous medium modelled as a disordered array of parallel cylinders. The relationship between pressure gradient and mean flow is determined in both linear (Darcy) and non-linear (Forchheimer) regimes.; These problems necessitate the use of applied mathematics tools including asymptotic expansions, symbolic algebra, finite-difference and finite-element methods. In addition to demonstrating the flexibility of these mathematical tools, used independently or in combination, this work contains new methodologies for the study of transport in random systems.