The falling needle viscometer is a particularly useful instrument for measuring the shear viscosity when small sample volumes are necessitated. Under some conditions this instrument can exhibit bimodal terminal velocity observations which precludes a straightforward analysis to obtain the viscosity. This phenomenon is explained via an analysis for the fluid mechanics and associated entropy production of a needle falling asymmetrically with respect to the centerline of the instrument. This analysis provides a rationale for using only the smaller terminal velocities, thereby permitting an unambiguous data analysis.