This paper deals with the thermal stability of completely confined fluids subject to a body force and a temperature gradient which are parallel and oriented in the same direction. It describes a method to obtain upper bounds to the instability criterion (the critical Rayleigh number) for piecewise cylindrical configurations, and demonstrates the use of this method treating some particular practical configurations. These upper bounds are shown to coincide with the critical Rayleigh number under some conditions. An account of experimental investigation of three of the particular configurations is presented and the experimental results compare favorably with the computed upper bounds.