Z. Angew. Math. Phys. 61 (2010), 929–947 c  2010 Birkh¨auser Verlag Basel/Switzerland 0044-2275/10/050929-19 published online January 12, 2010 DOI 10.1007/s00033-009-0054-7 Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP Numerical study of forced and free convective boundary layer flow of a magnetic fluid over a flat plate under the action of a localized magnetic field E. E. Tzirtzilakis, N. G. Kafoussias and A. Raptis Abstract. The two-dimensional, steady, laminar, forced and free convective boundary layer flow of a magnetic fluid over a semi-infinite vertical plate, under the action of a localized magnetic field, is numerically studied. The magnetic fluid is considered to be water-based with temperature dependent viscosity and thermal conductivity. The study of the boundary layer is separated into two cases. In case I the boundary layer is studied near the leading edge, where it is dominated by the large viscous forces, whereas in case II the boundary layer is studied far from the leading edge of the plate where the effects of buoyancy forces increase. The numerical solution, for these two different cases, is obtained by an efficient numerical technique based on the common finite difference method. Numerical calculations are carried out for the value of Prandl number Pr = 49.832 (water-based magnetic fluid) and for different values of the dimensionless parameters entering into the problem and especially for the magnetic parameter Mn, the viscosity/temperature parameter Θ r and the thermal/con- ductivity parameter S ∗ . The analysis of the obtained results show that the flow field is influenced by the application of the magnetic field as well as by the variation of the viscosity and the thermal conductivity of the fluid with temperature. It is hoped that they could be interesting for engineering applications. Mathematics Subject Classification (2000). 76W05, 76R05, 76R10, 76D99. Keywords. FHD · Magnetic fluids · Free-forced convective flow · FDM. 1. Introduction Ferro hydrodynamics (FHD) deals with the mechanics of magnetic fluid motion influenced by strong forces of magnetic polarization. A magnetic Ferro fluid consists of a stable colloidal dispersion of sub-domain magnetic particles in a liquid carrier. Magnetic fluids have been used commercially for a num- ber of years in numerous devices, such as rotating shaft seals and exclusion seals, loud speakers, dampers, inclinometers etc, whereas magnetorheological fluids have achieved commercial use in hydraulical devices, loudspeakers and in grinding applications. According to Berkovski and Bashtovoy [1], numerous patents and a great number of scientific papers have been published, related to the preparation, properties and application of magnetic fluids [2–5]. In order to examine the flow of a magnetic fluid, under the action of an applied magnetic field, mathematical models have been developed by many investigators [6–9]. The two classical problems in fluid mechanics, namely the Blasius boundary layer flow along a flat plate and the stagnation point flow, were extended for a saturated Ferro fluid, under the combined influ- ence of thermal and magnetic field gradients in [10]. The flow of a viscous Newtonian fluid past a linearly stretching surface in otherwise quiescent surroundings was first considered in [11]. Similar problems for a micro polar fluid or for inelastic power law fluids were studied by many authors [12–16]. The problem studied in [16] was extended in [17] by assuming that the magneto-thermo-mechanical coupling is not described by a linear function of temperature difference as in [16], but by a non linear one, the expression of which was used in [18]. Another classical problem in fluid mechanics is the free or the forced convective