In this paper we present a time accurate computational model based on the slip-flow theory to simulate momentum and heat transport phenomena in complex micro-geometries, encountered in typical components of micro-devices such as micro-capillaries, micro-valves, micro-rotors, and micro-bearings. In the first part, we present extensions to the classical Maxwell/Smoluchowski slip conditions to include high-order Knudsen number effects as well as to take into account the coupling of momentum and heat transfer through thermal creep and viscous heating effects. The numerical method is based on the spectral element technique; validation of the method is obtained by comparison of the numerical simulation results in simple prototype flows (e.g. channel slip -flows) with analytical results. Reduction of pressure drop in micro-channels, reported in similar experimental studies, is investigated using slip-flow theory and simulations. In the second part, we consider model inlet flows and a slip-flow past a micro-cylinder. The effect of slip-flow on skin friction reduction and associated increase in mass flow rate as well as the variation of normal stresses is investigated as a function of Knudsen number. Finally, the effect of compressibility is examined and possible extensions of the current model to take into account such effect are discussed.