Three-dimensional MHD duct flows with strong transverse magnetic fields. Part 3. Variable-area rectangular ducts with insulating walls

Abstract

The general analysis developed in Parts 1 and 2 of three-dimensional duct flows subject to a strong transverse magnetic field is used to examine the flow in diverging ducts of rectangular cross-section, the walls of which are electrically non-conducting. A dramatically different flow is found in this case from that studied in Part 2, where the side walls parallel to the magnetic field were highly conducting. Now it is found that the core velocity normalized with respect to the mean velocity is of O(M−½) while the velocity in the side-wall boundary layers is of O(M½), so that these boundary layers carry most of the flow. The problem of entry is solved by analysing the change from fully developed Hartmann flow in a rectangular duct to the flow in the diverging duct. It is found that the disturbance in the upstream duct decays exponentially. The analysis of the side-wall boundary layers is more difficult than that in Part 1 on account of the different boundary conditions and requires the solution of two coupled integro-differential equations. Numerical solutions are obtained for a duct whose width increases linearly in the flow direction.

MHD flow in an insulating rectangular duct under a non-uniform magnetic field

“…Although we intend to focus on the particular example where B is only x-dependent, it is worth keeping the full expression of S 0 in the derivation of our general model equations and to limit ourselves to B(x) later on. The expression of the charge conservation principle, which comes out from (14), is:…”

“…For the reference case C = 0 (insulated duct), we can further modify equation (32) by using expressions (14) for the x-and y-components of the core current density j 0⊥ , equation (31) for j 0z , and equations (29) and (30) for 0 and 1 , respectively. The first and second terms on the right-hand-side of this new equation contain a purely twodimensional term and a parabolic z-dependent term:…”

“…At the same time, providing that ω 0 ,ψ 0 , 0 and 1 are known, the current density given by (14), remains three-dimensional since it contains the z-dependent terms and j 0z ≠ 0.…”

“…Although this concept is limited to the asymptotic case of a perfectly conducting fluid, it allows for qualitative predictions of the main features of many MHD flows, including those in a fringing field. Subsequently, different research groups, around Walker [14][15][16][17][18][19][20][21][22][23] and more recently Molokov [24][25][26][27], investigated such flows, using simplifying assumptions, e.g. limiting the analysis to inertialess flows.…”

3D MHD free surface fluid flow simulation based on magnetic-field induction equations

“…Importantly, without using 3D formulations, numerical solutions can not capture the essential essence of the problems, and often provide providing poor information [1]. Furthermore, numerical codes based on the approach of solving the electrical potential equation [2] can only be applied to relatively weak magnetic fields (a few orders of magnitude weaker than those required for fusion plasma confinement). They also encounter considerable convergent difficulty if the Hartmann number becomes high (e.g.…”