We have investigated the theoretical and experimental linear dependence of the reciprocal of the apparent longitudinal relaxation time [(T*1)-1] of the NMR signal from spins in a flowing fluid on the volume flow rate, Fv, the so-called inflow effect. We refer to the coefficient of this dependence as the longitudinal flow relaxivity, r1F. A very simple model predicts that, under a range of conditions pertinent to modern flow studies and perfusion imaging experiments, r1F is controlled by the volume of the fluid in which the magnetization is perturbed by pulsed RF inversion or saturation, not the detection volume, and that it can be approximated as the reciprocal of half of the inversion volume. Phantom sample experiments, using a new, quantitative approach that we call flow relaxography, confirm the general predictions of this simple model. There are two intriguing implications of these findings for general NMR flow studies as well as for medical applications. It should be possible to vary the value of r1F by simply (noninvasively) adjusting the inversion slice thickness, and thus measure the value of (blood 1H2O, for example) Fv in a vessel without changing Fv, from the resultant varying T*1 values. Also, it should be possible to extrapolate to the intrinsic T1 value of the fluid signal (as if it were stationary), without altering or stopping the flow. Again, these are quite successful in phantom sample studies. Imaging versions of the flow relaxographic experiments are also possible. The twin goals of flow studies in medical MRI are the quantitative discrimination of the signals from flowing and nonflowing spins, and the accurate measurement of the flow rate of the former.