**3.2 Induced current distribution**

Figure 4 illustrates induced electric current distribution in the *x-z* plane at *y*=0 for the case of *z1*/*z2*=10/12, i.e. the standard case. Figures 5(a), (b) and (c) give induced current distributions in the *x-y* planes at *z*=4.5, 10 and 12, respectively, for the same case. On the right side of each figure, is shown the magnitude of the (nondimensional) induced current vector in the each coordinate direction. In Figs. 5(a) through (c), the induced current vector is reduced by a factor of 4 for two vectors from the wall at each circumferential angle, in order to make the figures compact.

In the fully-developed region from *z*=0 to *z* ≈ 8, the induced current, flowing mainly in the negative *x*-direction, does not change in the *z*-direction, as shown in Fig. 4. This constant induced current produces constant Lorentz force (acting in the negative *z* direction) and results in constant pressure drop along the *z*-axis as shown in Fig. 3. The induced current returns by passing in an extremely thin region very near the wall, as shown in Fig. 5(a). Almost no induced current (less than 10-2) flows in the region from *z* ≈ 14 to *z*=22, as shown in Fig. 4, since no magnetic field is applied.

In the magnetic-field outlet-region from *z* ≈ 8 to *z* ≈ 14, the induced current forms a loop mainly in the *x-z* plane, as shown in Fig. 4. The induced current is larger in the outlet region than in the fully-developed region. This is because the electric resistance of the induced current loop in the outlet region is much smaller than the resistance of the loop in the fullydeveloped region. The induced current can return in the large downstream region in the outlet region, although the current needs to return only in the extremely thin region near the wall in the fully-developed region.

214 Trends in Electromagnetism – From Fundamentals to Applications

mentioned in Chap. 1, no experimental data on the pressure drop through the magneticfield outlet-region have been reported. However, pressure drops through the magnetic-field inlet-region calculated numerically by the authers agreed nearly with those estimated by an

0 5 10 15 20 25

5

j

x

25

j

z

z

Figure 4 illustrates induced electric current distribution in the *x-z* plane at *y*=0 for the case of *z1*/*z2*=10/12, i.e. the standard case. Figures 5(a), (b) and (c) give induced current distributions in the *x-y* planes at *z*=4.5, 10 and 12, respectively, for the same case. On the right side of each figure, is shown the magnitude of the (nondimensional) induced current vector in the each coordinate direction. In Figs. 5(a) through (c), the induced current vector is reduced by a factor of 4 for two vectors from the wall at each circumferential angle, in

In the fully-developed region from *z*=0 to *z* ≈ 8, the induced current, flowing mainly in the negative *x*-direction, does not change in the *z*-direction, as shown in Fig. 4. This constant induced current produces constant Lorentz force (acting in the negative *z* direction) and results in constant pressure drop along the *z*-axis as shown in Fig. 3. The induced current returns by passing in an extremely thin region very near the wall, as shown in Fig. 5(a). Almost no induced current (less than 10-2) flows in the region from *z* ≈ 14 to *z*=22, as shown

In the magnetic-field outlet-region from *z* ≈ 8 to *z* ≈ 14, the induced current forms a loop mainly in the *x-z* plane, as shown in Fig. 4. The induced current is larger in the outlet region than in the fully-developed region. This is because the electric resistance of the induced current loop in the outlet region is much smaller than the resistance of the loop in the fullydeveloped region. The induced current can return in the large downstream region in the outlet region, although the current needs to return only in the extremely thin region near the

existing equation based on experimental data (Kumamaru 2007; Lielausis, 1975).

0

Fig. 4. Induced currents in *x*-*z* plane at *y*=0 for *z1*/*z2*=10/12.

0.2

0.4

0.6

x

**3.2 Induced current distribution** 

order to make the figures compact.

in Fig. 4, since no magnetic field is applied.

wall in the fully-developed region.

0.8

1

Fig. 5. Induced currents in *x*-*y* plane for *z1*/*z2*=10/12.

The induced current flows mainly in the negative *x*-direction from *z* ≈ 8 to *z* ≈ 11. Hence, in this region, a larger Lorentz force than in the fully-developed region acts in the negative *z*direction, and a larger pressure drop is produced along the *z*-axis as shown in Fig. 3. On the other hand, the induced current flows mainly in the positive *x*-direction from *z* ≈ 11.5 to *z* ≈ 13. Thus, the Lorentz force is exerted in the positive *z*-direction, and a small pressure recovery along the *z*-axis happens from *z* ≈ 11.5 to *z* ≈ 12 as shown in Fig. 3. (No external magnetic field is applied from *z* ≈ 12 to *z* ≈ 13.) Also in the outlet region, there exists an induced current loop which returns in an extremely thin region near the wall, as shown in Fig. 5(b).
