Theoretically a Couette flow in a narrow channel can be utilized to simulate microgravity conditions experienced by a surface flame due to the linear velocity profile. Hence, the Couette channel is a potential apparatus for the study of flame spread in an environment that recreated microgravity flow conditions. Simulated microgravity conditions were achieved by limiting the vertical extent over and under the flame to suppress buoyancy. This numerical study was done for a 2-D channel using Fire Dynamics Simulator (FDS). This thesis is divided into two sections; the first is the study of Couette flow with a non-reacting cold flow in a finite length channel, a subject with surprisingly little past research, despite the ubiquity of "infinite" Couette channels in text books. The channel was placed in a room to allow for a better representation of a realistic channel and allow the flow and pressure field to develop without forcing them at the inlet and outlet. The plate's velocities, channel's gap and the channel's length were varied and the results of the u-velocity profile, w-velocity profile and pressure were investigated. The entrance length relationship with Reynolds number for a finite Couette Channel was determined for the first time - as far as the author knows - in order to ensure the flame occurs in a fully developed flow. In contrast to an infinite channel, the u-velocity was found to be nonlinear due to an adverse pressure differential created along the channel attributed to the pull force along the entrance of the channel created by the top plate a well as the pressure differential created by the flow exiting the channel. The linearity constant was derived for the one moving plate case. The domain consisted of a rectangular region with the top plate moving and the bottom plate fixed except for a few cases in which the bottom plate also moved and were compared with only one moving plate. The second section describes the combustion of a thin cellulose sample placed 3.5 mm from the bottom using a combined pseudo-Hagen-Poiseuille-Couette narrow channel. The gap on top of the paper sample as well as the top plate's velocities were varied and the effect on the flame spread rate was investigated. The results were compared between no gravity and normal gravity and it was determined that some of the cases run the 0g created a flame tail that was longer than that of the 1g flame while others that had narrower gaps both 0g and 1g flame tail are similar. The channel is not placed in a room since the entrance velocity profile of the fluid is in fact set at the inlet (hence pseudo-Hagen-Poiseuille). While varying the plate velocity while keeping the height equal, the 0g flames spread faster than the 1g flames. Varying the height and keeping the plate velocity constant presented a crossover phenomenon of the flame spread rate.