@Article{fdmp.2005.001.247,
AUTHOR = {C.  Giessler, C.  Sievert, U.  Krieger, B.  Halbedel, D.  Huelsenberg, U.  Luedke, A.  Thess},
TITLE = {A Model for Electromagnetic Control of Buoyancy Driven Convection in Glass Melts},
JOURNAL = {Fluid Dynamics \& Materials Processing},
VOLUME = {1},
YEAR = {2005},
NUMBER = {3},
PAGES = {247--266},
URL = {http://www.techscience.com/fdmp/v1n3/24210},
ISSN = {1555-2578},
ABSTRACT = {Buoyancy driven motion of a highly viscous electrically conducting fluid under the influence of Lorentz forces is investigated theoretically and experimentally. This problem is relevant to the processing of glass, where it is of considerable interest to know whether electromagnetic forces can effectively improve mixing and help to avoid undesired flow patterns in glass melting furnaces. Two highly simplified models are proposed in which the fluid is assumed to be confined in a circular loop containing several localized resistive heating, convective cooling, and electromagnetic forcing elements. The first model is used to derive the scaling laws of the mean velocity and maximum temperature depending on the electric current density and magnetic field. The predictions of this model are found to be in agreement with numerical simulations and with experiments in a small-scale glass melting furnace. Using the second model which contains two cooling elements we demonstrate that the steady-state velocity in the general case is determined by a single nonlinear algebraic equation whose bifurcation structure reveals an unexpectedly subtle influence of the Lorentz force on buoyancy driven convection. It is shown that the system undergoes a transition from a nearly stably stratified "slow" mode for weak Lorentz forces, in which the velocity <i>v</i> and temperature θ<sub><i>H</i></sub> scale with the electric current density <i>J<sub>0</sub></i> as <i>v ~ const</i>. and θ<sub><i>H</i></sub> ~ <i>J<sub>0</sub><sup style="margin-left:-5px">2</sup></i> to a "fast" mode for strong Lorentz forces in which the scaling on the magnetic field <i>B<sub>0</sub></i> is <i>v ~ J<sub>0</sub>B<sub>0</sub></i> and θ<sub><i>H</i></sub> ~ <i>J<sub>0</sub><sup style="margin-left:-5px">4/3</sup>B<sub>0</sub><sup style="margin-left:-5px">-2/3</sup></i>. In a wide range of parameters this transition depends discontinuously on the magnetic field and has the character of a subcritical bifurcation involving hysteresis. The scaling laws imply that already a comparatively weak electromagnetic force can strongly modify buoyancy driven convection. The consequences of this finding for electromagnetic control of glass melting processes are briefly discussed.},
DOI = {10.3970/fdmp.2005.001.247}
}