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Introduction to Noninvasive Therapies
Published in Robert B. Northrop, Non-Invasive Instrumentation and Measurement in Medical Diagnosis, 2017
When a moving conductive fluid intersects a magnetic field, an EMF, Ef, is induced given by the Faraday's law of induction equation (Northrop 2014): EF=−∫0d[(vXB)⋅dL]volts, where X is the vector cross-product operation (see Glossary) and ⋅ denotes the vector dot product operation. The net result in the brackets is a scalar.
Edge-Based Finite Element Formulation of Magnetohydrodynamics at High Mach Numbers
Published in International Journal of Computational Fluid Dynamics, 2021
Wenbo Zhang, Wagdi G. Habashi, Guido S. Baruzzi, Nizar Ben Salah
As part of a sustained effort to create HALO3D, an all-Mach number CFD code that can simulate the conditions of the entire trajectory of a re-entry vehicle, this article addresses hypersonic flows under the influence of an externally imposed magnetic field. This field can be used to increase stability and control of the spacecraft, as well as increase the bow shock stand-off distance and reduce surface heat loads. The modeling of the problem takes advantage of the low-magnetic Reynolds number approximation, thus simplifying the problem by eliminating the need to solve the magnetic induction equation. The current-continuity equation, cast in terms of a scalar electric potential, is solved to obtain the induced electric field. The flow can then be modeled through the RANS equations with added source terms accounting for the electromagnetic Lorentz forces and Joule heating.
A geometric look at MHD and the Braginsky dynamo
Published in Geophysical & Astrophysical Fluid Dynamics, 2021
Andrew D. Gilbert, Jacques Vanneste
Perhaps a more familiar route for some readers is to express the momentum and induction equations in terms of the covariant derivative. Here we write the components of u as and those of ν as , etc. Using standard results, in particular that (Arnold and Khesin 1998), the momentum equation (43) may be written as In this formulation, it is interesting to see the magnetic pressure term emerge, with no corresponding term involving . For magnetic field, equation (11) can be reexpressed as a familiar form of the induction equation, for , and similarly for equations such as (13), (15). We have considered the compressible case, as it does allow a clear emergence of the magnetic pressure term. For the incompressible case, one drops the internal energy and in its place uses a Lagrange multiplier to enforce that the flow map conserves volumes, . The theory can be developed further from Lagrangian to Hamiltonian mechanics, using a non-canonical Poisson bracket as discussed in Morrison (1982, 1998), Morrison et al. (2020) and reviewed in Webb (1994), with extensions to relativistic MHD (D'Avignon et al.2015) and two-fluid MHD (Lingam et al.2016).
Advances in geodynamo modelling
Published in Geophysical & Astrophysical Fluid Dynamics, 2019
Johannes Wicht, Sabrina Sanchez
The magnetic Reynolds number measures the ratio of induction and Ohmic dissipation in the induction equation. The Elsasser number is often used to estimate the relative importance of the Lorentz force to the Coriolis force. However, Soderlund et al. (2015) show that the dynamic Elsasser number provides a more direct and better measure for the true force ratio when faster variations in are considered.