Stream Function Technique Research Articles

The Kinematic Flow Structure for the Gvirtzman-Gorelick In Situ VOC Remediation System

Potential theory and Stokes' stream function techniques are used to investigate the flow structure around the recirculation system developed by Gvirtzman and Gorelick (1992, 1993), which consists of an extraction well and a gallery (trench) for the recharge of treated water to the aquifer. Analytical formulas are derived for the drawdown, velocity, and stream function for a model in which the extraction well is modeled as a uniformly distributed line sink and the gallery is modeled as a uniformly distributed ring source. Travel times are reported for water particles traveling along the streamlines containing 50 and 90% of the flow for various degrees of well penetration and various radii of the ring source. The travel times along the streamline resulting in the shortest travel time (not necessarily the shortest path) are also reported for various degrees of well penetration and various radii of the ring source. The method completely eliminates the use of numerical finite-difference or finite-element methods and can be used for optimization of technological parameters of this remediation system.

True energy-minimal and finite-size biplanar gradient coil design for MRI.

Finite-sized high-performance planar magnetic field gradient coils in today's open configuration magnetic resonance imaging (MRI) systems have always been desirable for ever demanding imaging applications. We present a Lagrange multiplier technique for designing a minimum-energy gradient coil under a finite-size planar geometry constraint in addition to a set of magnetic field constraints. In this new design methodology, the surface current density on a finite size plane is represented by a two-dimensional (2-D) Fourier series expansion. Following the standard approach, we construct a functional F in terms of the stored magnetic energy and a set of field constraint points which are chosen over the desired imaging volume. Minimizing F, we obtain the continuous current density distribution for the finite-size planar gradient coil. Applying the stream function technique to the resulting continuous current distribution, the discrete current pattern can be generated. Employing the Biot-Savart law to the discrete current loops, the gradient magnetic field has been re-evaluated in order to validate the theory. Using this approach, we have been able to design a finite-size biplanar z-gradient coil which is capable of generating a gradient field of 40 mT/m @ 266 A. The excellent agreement between the analytical and numerical results has been achieved.

Spherical gradient coil for ultrafast imaging

High performance magnetic field gradient coils have always been desirable in today’s ultrafast magnetic resonance imaging (MRI) applications, such as single-shot echo-planar imaging and fast spin echo imaging, as well as MR diffusion imaging and microscopy. We present a Lagrange multiplier technique of a minimum inductance gradient coil with spherical geometry. Based on this minimization approach, we construct a functional F in terms of the stored magnetic energy, the magnetic field and a set of field constraint points which are chosen over the desired imaging volume. Minimizing F, we obtain the continuous current density distribution for the spherical gradient coil. Applying the stream function technique to the continuous current distribution, the discrete current pattern can then be generated. Employing the Biot–Savart law to the discrete current loops, the gradient magnetic field has been re-evaluated in order to validate the theory. Using this approach, we have been able to design a spherical z-gradient coil which is capable of generating a gradient field of 176 mT/m with slew rate of 3422 T/m/s over a 30-cm-diam spherical volume if driven by a 350 V–220 A current amplifier. A prototype of the spherical z-gradient coil has been built. The agreement between the analytical and experimental results is excellent. Initial imaging experiments have been conducted. The results indicate the potential use of such a coil for in vivo and in vitro fast NMR applications.

EFFECTS OF STREAM WISE CONVERGENCE IN RADIUS ON THE LAMINAR FORCED CONVECTION IN AXISYMMETRIC DUCTS

A systematic study has been carried out on the effects of streamwise convergence in radius on the laminar forced convection in an axisymmetric duct. This transport circumstance is relevant to many practical processes such as injection molding, glass molding, fiber drawing, and extrusion, where large variations in the radius may occur downstream and where the flow rates are generally small enough to yield a laminar flow. A fairly uncommon transformation technique was used to transform the pseudo-transient conservation equations for the stream function, vorticity, and energy, and several new numerical techniques were developed. These include a nonuniform grid scheme, a second-order-accurate vorticity condition for an arbitrary surface, and a nominally second-order-accurate approximation for the derivatives on a nonuniform grid. The three geometries studied were those of the straight, periodic, and converging ducts, where the results for the first two were obtained mainly for validation purposes. However, new results were also obtained for the periodic duct, showing the attainment of a sinusoidal steady state with the local Nusselt number varying from 1.0 to 6.0. For the converging duct, the local Nusselt number was found, for the first lime, to increase with increasing convergence of the duct wall.

The use of dual-stream functions in the analysis of three-dimensional metal forming processes

Modelling metal forming processes with the dual stream function technique is considered. The dual stream function concept is reviewed. Its numerical implementation in a computer program is presented. The method enables the numerical evaluation of the process power and gives detailed information about other process parameters such as friction between the workpiece and the tool, workpiece geometry and material characteristics. Two selected examples concerning three-dimensional flat rolling and polygonal forging are analysed by this technique. The theoretical predictions are compared with results obtained by other authors using alternative methods. Computational advantages over other methods, namely the finite element method are the main feature of the present analysis.