Innovation in biotechnology research has resulted in a number of fungi being identified for diverse industrial applications. Much research has been done in developing optimised membrane bioreactor (MBR) systems for the cultivation of these fungi as a consequence of their potent industrial applications. This research has been hampered by the lack of a thorough understanding of the fluid mechanics through these devices. In this article, analytical and numerical solutions to the Navier-Stokes equations were developed to describe the hydrostatic pressure and velocity profiles in a single fiber membrane gradostat reactor (SFMGR). A generic equation for low wall Reynolds number (Re-w = rho nu(w)r(H)/mu) flows was developed and solved for the case of negligible angular variations of the flow profiles. The mathematical expressions were proposed as solutions to transient state, laminar, incompressible, viscous and isothermal flow through a membrane with a variable hydraulic permeability. These profiles were developed for the lumen and shell sides, taking into account the osmotic pressure and gel formation that occurs when solute particles are rejected on the membrane. The models developed are applicable to different orientations and configurations. A numerical scheme, with a complete stability analysis, was developed to complement the analytical models. The models were tested on a vertically orientated MGR, operated in the dead-end mode. The model solutions gave profiles that are in agreement with the experimental results. (c) 2007 Elsevier B.V. All rights reserved.