The paper aims at a detailed description of the physical background, for the so-called Coriolis mass flow meter. It presents essentially an analysis of the (free) vibration modes of a fluid conveying straight pipe segment. Due to the inertial effects of the flowing fluid, mainly the Coriolis force, these modes deviate in shape (and in frequency) from those appearing in the absence of fluid motion. The effect of fluid inertia may, therefore, be exploited for the purpose of flow measurement. The analysis is performed under a simplifying approximation: The pipe is considered as a beam, the fluid as a moving string. This approximation leaves the fluid with only one degree of freedom, connected with its mean velocity, and eliminates an infinity of degrees of freedom of the pipe. Yet it keeps, the essential features of the phenomenon. The equations describing the vibrations are derived variationally, with the constraint of a common vibration amplitude of both fluid and pipe. The Lagrange multiplier associated with the constraint gives the interaction force between pipe and fluid. The modes are determined by a perturbation procedure, wherein the small (perturbation) parameter is related to the fluid velocity. The analysis shows, as main result, how the time delay between the vibrations of two appropriately chosen points of the pipe may serve to determine the mass flow rate of the fluid. Other aspects of the problem, like the precise role of the Coriolis force, are considered. The possible improvement of the used approximation is discussed.