Keywords: magnetohydrodynamics, curved duct, numerical analysis, complementary variational principle method.

The fully developed magnetohydrodynamic (MHD) flow of a liquid metal in a curved circular duct is studied in this paper. A transverse external magnetic field is applied. MHD liquid metal flows in curved circular ducts are met in many practical applications related to fusion reactors, power generation and engineering applications of magnetic fluids. The present analysis is unique as no computational and little theoretical and experimental research on curved MHD flows has been published.

The governing equations of the electromagnetic quantities, derived from the Maxwell equations, can be formulated in different ways. In this paper the electromagnetic variables are implemented from a hybrid formulation [1].

The MHD flow is governed by a set of non-dimensional equations that include the Navier-Stokes equations and the Maxwell equations which will be solved numerically using the complementary variational principle (CVP) numerical method [2]. CVP method has been validated and tested in several hydrodynamic and MHD channel flows. The governing equations are expressed in a toroidal-poloidal coordinate system. Non-uniform stretched meshes will be used in order to compute accurately the MHD boundary layers near the walls.

The results reveal the basic mechanism of the flow and some basic conclusions are obtained: As the curvature increases the centrifugal force tends to shift the axial velocity to the right region of the cylinder and the electromagnetic force to the left region of the cylinder. For Hartmann numbers lower than 10 the centrifugal force dominates and the axial velocity is shifted to the right region, while for Hartmann numbers greater than 10 the electromagnetic force dominates and the axial velocity is shifted to the left region of the cylinder. A secondary flow is generated as a result of the curvature of the duct. This secondary flow is strongly suppressed under the presence of the external magnetic field. The contribution of the curvature in the axial pressure gradient is significant only under the absence of the magnetic field. The magnetic field significantly increases the axial pressure gradient by orders of magnitude.

1

N.B. Morley, S. Smolentsev, R. Munipalli, M.-J. Ni, D. Gao. M. Abdou, "Progress on the Modelling of Liquid Metal, Free Surface, MHD Flows for Fusion Liquid Walls", Fusion Engineering and Design, 72, 3-34, 2004. doi:10.1016/j.fusengdes.2004.07.013

2

P.A. Bakalis, P.M. Hatzikonstantinou, "MHD and thermal flow between isothermal vertical concentric cylinders with rotation of the inner cylinder", Numerical Heat Transfer, 59(11), 836-856, 2011. doi:10.1080/10407782.2011.578013

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