A linear stability analysis examining the stability of a fully-developed, two-dimensional Hagen-Poiseuille resuspension flow is presented. Whilst the analysis includes the non-uniformity of the particle concentration distribution, the inclusion of any concentration fluctuations is ignored. Numerical solutions to the relevant Orr-Sommerfeld equations for temporal disturbance modes are obtained with the aid of ortho-normalization for a variety of different parameters by means of a classical shooting technique. It is found that interfacial instabilities result from long wavelength disturbances even in a small Reynolds number range. The growth rates of disturbances are shown to increase with decreasing initial feed concentration, whilst increasing density stratification is shown to stabilize the resuspension flow. It is also shown that the neglect of fluctuations in the particle concentration severely limits the validity of the stability analysis performed, especially for flows in which the particle concentration in the interfacial region varies rapidly.