Keywords: magnetohydrodynamics, micropolar flow, ferrofluid.

Micropolar magnetohydrodynamic flow of magnetic fluids concerns colloidal suspensions of ferromagnetic particles in a carrier magnetic liquid. The particles are considered as rigid magnetic dipoles. The application of an external magnetic field prevents the rotation of each particle, thus increasing the effective viscosity of the fluid and creating an additional magnetic pressure. The system of equations that governs the magnetic fluids flow includes the equations of continuity, momentum, energy, angular momentum and magnetization as well as the equations of Maxwell, all coupled to each other [1,2,3,4].

The present paper deals with the incompressible flow of a Newtonian carrier magnetic liquid including a small concentration of magnetic particles under the effect of an arbitrarily orientated applied magnetic field. We present the development of a new three-dimensional model where the momentum equation incorporates explicitly a theoretical expression for the extra viscosity, which is generated by the effect of the magnetic field. In our generalized three-dimensional method, we uncouple the evolved partial differential equations and we obtain a closed analytical solution for the internal angular momentum, as a function of the magnetization and the magnetic field. We also derive a closed expression for the magnetization, which contains the vorticity field and the magnetic field. Substituting the internal angular momentum and the magnetization in the momentum equation we obtain a new expression for the Navier-Stokes momentum equation, where the effective viscosity, which appears explicitly as an additional term, and the gradient of the magnetic pressure, are expressed in terms of the applied magnetic field.

We apply our model to the laminar flow within a straight square duct under the effect of an externally applied magnetic field. The system of the involved equations is solved using the continuity, vorticity and pressure variational equations (CVP) computational method [5]. Numerical results are presented for the development of the flow under the effect of the concentration and the magnetization of the magnetic particles and the strength of the external magnetic field.

1

B. Berkovski, V. Bashtovoy, "Magnetic Fluids and Applications Handbook", Begell House, Inc., New York, 1996.

2

R.W. Fox, A.T. McDonald, "Introduction to Fluid Mechanics", John Wiley & Sons, Inc., Third Edition, New York, 1985.

3

V.G. Bashotovoy, B.M. Berkovski, A.N. Vislovich, "Introduction to Thermomechanics of Magnetic Fluids, Hemisphere Publishing Corporation", New York, 1988.

4

R.E. Rosensweig, "Ferrohydrodynamics", Dover Publications, Inc., New York, 1997.

5

V.D. Sakalis, P.M. Hatzikonstantinou, P.K. Papadopoulos, "A Numerical Procedure for the Laminar Developed Flow in a Helical Square Duct", ASME J. Fluids Eng., 127, 136-148, 2005. doi:10.1115/1.1852483

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