Two-Dimensional PFG NMR for Encoding Correlations of Position, Velocity, and Acceleration in Fluid Transport

摘要

A generalized approach to obtain two-dimensional maps of ipatial particle coordinates and their derivatives with respect to time by PFG-NMR employing multiple gradient pulses is presented. A sequence of n magnetic field gradient pulses makes it possible, after independent stepping of each pulse and subsequent Fourier transformation, to plot the spin density distribution in coordinate space at n times and along the respective directions of the gradient pulses. In particular, two gradient pulses of effective area k_(1) and k_(2) separated by a time interval △ lead to a plot of the combined two-time probability density, W_(2)(r_(1), 0; r_(2), △), to find a particle at a coordinate r_(1) at t=0 and at r_(2) at t=△. A conventional experiment for measuring transport properties by simultaneous stepping of the gradients under the condition k_(1)=-k_(2) is equivalent to a projection onto the secondary diagonal in the [r_(1), r_(2)] plot. The main diagonal represents an average position between the two timepoints t=0 and t=△, so that a rotation of the coordinate plot by an angle of 45° allows one to correlate the displacement R=r_(2)-r_(1) with the averaged position r parallel to the gradient direction. While an average velocity during the time interval △ can be defined as v=R/△, an extension toward acceleration and higher order derivatives is straightforward by modification of the pulse sequence. We discuss this concept by application to flow through a circular and a narrowing pipe (confusor), respectively, the experimental results of which are compared to numerical simulations.