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    • GLPG0634 Thus the calculated Navier Stokes equations for an

    2018-11-09

    Thus, the calculated Navier–Stokes equations for an incompressible conducting fluid which include the density of the EM field force as the driving force can be expressed in the following way in differential form:
    Here μ is the effective viscosity, defined as the sum of the laminar and the turbulent viscosity, i.e., μ=μ+μ is the strain rate tensor. Simulating the melt flow in the induction furnace with an alternating EM field using the k-ε turbulence model allows to calculate the average velocities and the flow structure that are in substantial agreement with the experimental data [3,15,16].
    Verification of the mathematical model To check the validity of the model we numerically studied the motion of the molten metal for a fixed current value IRMS=200 A and for a variety of values (Fig. 3). The graph demonstrates that the changes in the maximum axial velocities on the axis of the cylindrical volume depend on the AC frequency in the inductor. This functional relationship reflects the effectiveness of applying the EM field to the flow and is usually expressed in terms of the dimensionless ratio of the cylinder diameter to the thickness of the GLPG0634 depth (D/δ). The shape of the curve shown, containing a peak and then monotonic behavior of the function after it, is typical for the interaction between the alternating magnetic field and the conducting material in it. [17] The numerical experiment also yielded a characteristic flow pattern and the distribution of the axial velocity on the cylinder axis (r = 0) time-averaged over the duration of the experiment exposed to the constant Lorentz force (Figs. 2 and 4). The profile corresponds to the theoretical model of flow formation under the action of an alternating magnetic field, as two maximum values of the measured velocity , corresponding to the divergent flows of two toroidal eddies along the axis, are present in the graph. In addition, the curve can be seen passing through the x-axis at 35mm, which corresponds to the region of minimal axial velocities (eddies rotation) in the central region of the melt volume [3–5]. Also notice the agreement between the maximum velocities obtained numerically and experimentally. For the physical experiment, the following parameters were chosen in accordance with the technical characteristics of the equipment to create the EM field in steady mode: the AC frequency in the inductor =150Hz and the effective current value IRMS=200 A.
    Melt flow under pulse conditions To estimate the effect of the changing frequency on the behavior of the fluid flow, we compared the intensities of the flow velocity fluctuations for pulse frequencies in the range from 0.05 to 10Hz. For this purpose, we calculated the standard deviations of the measured axial velocity from its mean value over the whole duration of the experiment. The results of these calculations allowed to estimate how actively the fluid responded to a pulse or a constant external action (Fig. 5). The solid horizontal line on the graph marks the value of the measured velocity fluctuation for the flow developing under the influence of the constant Lorentz force. Analysis of the experimental data has shown that a pulsed EM field force with a frequency in the range from 0.50 to 10Hz has little effect on the flow: the values of the velocity fluctuations are close to the one obtained for the steady case. This result can be explained by the inertia of the fluid, which is consistent with the experimental data of other studies [10,15]; such conditions can be regarded as quasi-steady. The most significant influence on melt flow was from the pulsed modes of the Lorentz force in the range of = 0.05–0.20Hz. At the same time, the maximum of axial velocity fluctuations is observed for the frequency = 0.10Hz. It is well known, that in the melt flow driven by an alternating magnetic field, there exist the velocity pulsations with a relatively low period. As shown by the authors of [3], this period depends on the EM force strength and the geometry of the vessel containing the fluid, and as mentioned in Ref. [4], it is associated with the revolution period of toroidal eddy structures in the flow (), which are generated by an alternating electromagnetic field. The characteristic frequency (the notation adopted by the authors of Ref. [18]) corresponding to this period was determined as follows: