Two-dimensional PFG NMR for encoding correlations of position, velocity, and acceleration in fluid transport

A generalized approach to obtain two-dimensional maps of spatial particle c oordinates and their derivatives with respect to time by PFG-NMR employing multiple gradient pulses is presented. A sequence of n magnetic field gradi ent 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 gra dient pulses. In particular, two gradient pulses of effective area k(1) and k(2) separated by a time interval Delta lead to a plot of the combined two -time probability density, W-2(r(1), 0; r(2), Delta), to find a particle at a coordinate r(1) at t = 0 and at r(2) at t = Delta. A conventional experi ment for measuring transport properties by simultaneous stepping of the gra dients 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 represen ts an average position between the two timepoints t = 0 and t = Delta, so t hat a rotation of the coordinate plot by an angle of 45 degrees 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 ti me interval Delta can be defined as (v) over bar = R/Delta, an extension to ward acceleration and higher order derivatives is straightforward by modifi cation of the pulse sequence. We discuss this concept by application to how through a circular and a narrowing pipe (confusor), respectively, the expe rimental results of which are compared to numerical simulations. (C) 2000 A cademic Press.