Simplemodels of heterogenousmagnetic fields for liquid metal flow simulations

A physically consistent approach is considered for defining an external magnetic field as needed in computational fluid dynamics problems involving magnetohydrodynamics (MHD). The approach results in simple analytical formulae that can be used in numerical studies where an inhomogeneous magnetic field influences a liquid metal flow. The resulting magnetic field is divergence and curl-free, and contains two components and parameters to vary. As an illustration, the following examples are considered: peakwise, stepwise, shelfwise inhomogeneous magnetic field, and the field induced by a solenoid. Finally, the impact of the streamwise magnetic field component is shown qualitatively to be significant for rapidly changing fields. There are few examples in recent history,when things being obvious to particular specialists, remained unexploited by people working in conjugated fields. These are, for instance, the Fast Fourier Transform (FFT), which was originally used by Gauss in 1805 and then several times rediscovered by Lanczos and Danielson in 1942, and Cooley and Tukey in mid-1960s. Another and more specific example is the so-called Savitzky-Golay smoothing filter. The least square method being the base of the filter has been formulated hundred years before experimentalists working with spectra started to use it to treat their data. The paper that popularized this method for the experimentalists is one of the most widely cited papers in the journal Analytical Chemistry. What is considered in this letter is not as far reaching as the two aforementioned examples, nevertheless, we believe it will help people studying numerically the flow of liquid metal under the influence of an inhomogeneous magnetic field Also, this letter is complementary to the discussion about magnetic field models given earlier in Votyakov et al. (2008). In reality, one always deals with an inhomogeneous magnetic field, moreover, it is a hard practical problem to create a strong and homogeneous magnetic field for experimental needs. Despite this fact, starting from the pioneeringwork of Hartmann & Lazarus (1937)many theoreticians love to workmostly with a homogeneous magnetic field. It is clear, that a constant magnetic field is already responsible for main phenomena such as the formation of the Hartmann and parallel layers. Nevertheless, a constant field could not produce awell developedM-shaped velocity profile frequently used for the electromagnetic brake. Due to Kulikovskii (1968), who showed that a flow under strong and slowly varying magnetic field 1 Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ”Fast Fourier Transform.” Ch. 12.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 3rd ed. Cambridge, England: Cambridge University Press, Page 498, 2007; see also http://mathworld.wolfram.com/FastFourierTransform.html 2 A. Savitzky and Marcel J.E. Golay (1964). Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Analytical Chemistry, 36: 1627-1639; see also http://en.wikipedia.org/wiki/Savitzky-Golay smoothing filter 2 E. V. Votyakov and S. C. Kassinos can be subdivided in a core and boundary layer, people started to exploit theoretically inhomogeneous magnetic fields. It was demonstrated that the flow goes parallel to characteristic surfaces, but all the calculations, including the numerical ones, were carried out either by neglecting the second magnetic field component, see, e.g. Sterl (1990), Molokov & Reed (2003b), Molokov & Reed (2003a), Alboussiere (2004), Kumamaru et al. (2004), Kumamaru et al. (2007) or by employing an especially curvilinear channel Todd (1968) to match boundary conditions. In the case of numerical simulations there is no particular technical problem preventing the inclusion of the second component of the inhomogeneous magnetic field; nevertheless this has not been done. A probable explanation for this is the absence of suitable and analytically simple models to define an inhomogeneous magnetic field. Therefore, there is the need for convenient formulae that could allow us to do so, and the goal of the paper is to provide a simple method that can be used in numerical studies so that both components can be varied in a consistent way. At the end of the paper we discuss briefly the legitimacy of omitting the streamwise component of the magnetic field, and show that in some cases this might have been done due to large aspect ratios. The necessity to have at least two nonzero components of the inhomogeneousmagnetic field (hereafter denoted as MF) follows directly from the requirement that, in the flow region, an externally applied field must be simultaneously divergence-free ∇ ·B = 0 and curl-free ∇×B = 0. Thus, if the transverse MF component varies along the streamwise coordinate, the streamwiseMF componentmust vary consistently along the transverse coordinate. (The spanwise component can be neglected without violation of the physical correctness, when the magnetic field is two dimensional.) A vector field that is simultaneously divergence-free and curl-free is known as a Laplace vector field and it can be defined in terms of the gradient of any function η which is harmonic in the flow region, B = −∇η, ∆η = 0. This function η is called the magnetic scalar potential. Although this is the most general approach, see e.g. McCaig (1977), it is not quite convenient because boundary conditions for η must be defined for each specific case. However, what we need is a magnetic field which either vanishes or goes monotonically onto a constant level far away from the central point, where the intensity of the field is maximal. So, among all the possible harmonic functions we may select those which do not vary at far distances, and this forms the basis for the proposed methodology. A simple, flexible, and physically consistent way to define an inhomogeneous MF for parametric numerical needs is based on the magnetic field induced by a single magnetic dipole. This field is local in space and its magnetic scalar potential belongs to harmonic functions. Therefore, the dipole can be taken as an elementary unit in the appropriate spatial distribution of magnetic sources. The field B(r, r) created at r = (x, y, z) by a single dipole m = (0, 0,m) located at r = (x, y, z) is given by Jackson (1999): B′(r, r′) = ∇× (