Laminar flow velocity profile measurement from magnetic resonance spin echoes at incomplete polarization

Abstract

Laminar flow velocity profiles are directly related to the rheological properties of the flowing fluids. Magnetic resonance spin echo measurements at complete polarization, with a flow-oriented magnetic field gradient, can be utilized to determine the velocity profile of laminar flow in a circular pipe. However, fluids with a long spin-lattice relaxation time will not have time to completely polarize before signal acquisition in typical applications. This will restrict applications of the method, and modification of the original methodology is required to work with the general case of incomplete polarization. In this paper, magnetic resonance spin echo measurements at incomplete polarization with a flow-oriented magnetic field gradient are employed to determine the velocity profile of laminar flow in a circular pipe. The governing equations describing phase shifts and magnitude changes of odd echoes for laminar flows were derived, at incomplete polarization, based on the flow behavior index, an effective polarization length, spin-lattice relaxation time, and the average velocity. The objective function for least squares minimization was constructed, based on the first odd echo phase shifts and magnitude changes at different echo times, to solve for the flow behavior index and average velocity. The Nelder–Mead algorithm was employed to minimize the objective function. Discrete simulations for three kinds of laminar flows in a circular pipe, that is, shear-thickening flow, Poiseuille flow, and shear-shinning flow, were employed to validate the proposed method. Magnetic resonance experiments for Poiseuille flow were undertaken for further verification.