**3.3 Velocity distribution**

Figures 6(a), (b), (c), (d) and (e) show calculated velocity *vz* distributions at z=4.5, 10, 11, 12 and 17.5, respectively, for the case of *z1*/*z2*=10/12, i.e. the standard case. There is no significant difference among velocity distributions from *z*=0 to *z*=8. The velocity profile is a

Three-Dimensional Numerical Analyses on Liquid-Metal

the same as that at *z*=11, shown in Fig. 6(c).

laminar flow with a peak value of ~2.

MHD flow in the region of *z*>*z2*.

**3.4 Comparison with magnetic-field inlet-region** 

Magnetohydrodynamic Flow Through Circular Pipe in Magnetic-Field Outlet-Region 217

flat one, particularly in the direction of applied magnetic field, having a peak value of ~1.1, as shown in Fig. 6(a). The velocity distribution in this region agrees nearly with a profile

The velocity distribution at *z*=10, shown in Fig. 6(b), is still nearly flat. The velocity profile changes sharply at *z* ≈ 11 and shows what is called an M-shape distribution having a peak near the wall, as shown in Fig. 6(c). This is because the Lorentz force acting in the negative *z*direction suppresses the flow in the *z*-direction in the fluid bulk region, though small Lorentz force acts in the negative z-direction in the region near the wall of *x*=1. The velocity distribution at z=12, at the outlet of applied magnetic field, shown in Fig. 6(d), is still nearly

The velocity profile changes sharply, from *z*=12 to *z* ≈ 13, from the M-shape distribution, shown in Fig. 6(d), to a parabolic distribution typical to a non-MHD flow, shown in Fig. 6(e). The pressure decrease from *z*=12 to *z* ≈ 13, shown in Fig. 3, is attributable to this sharp change in velosity distribution. It is considered that the pressure decreases largely since the velocity increases quickly in the fluid bulk region. No significant difference exists among velocity profiles from *z* ≈ 13 to *z*=22. The velocity profile is a parabolic one of a non-MHD

Figure 7 shows schematically the applied magnetic field in the *y*-direction, the induced currents in the *x*-*z* plane including the directions of Lorentz force and the pressure along the *z*-axis, in the inlet region and the outlet region of the magnetic field. The larger Lorentz force acts and thus the larger pressure drop occurs in the inlet and outlet regions than in the fullydeveloped MHD region for the reason mentioned in Chap. 1. On the other hand, a smaller Lorentz force may act in the flow direction and thus a small pressure recovery may occur in the first section of the inlet region and in the last section of the outlet region also for the reason mentioned in Chap. 1. The pressure drop behavior is not completely symmetric, since the fully- developed non-MHD flow enters the calculation domain in the inlet-region

Figure 8 presents pressures along *z*-axis for the magnetic-field inlet-region, calculated by the authors and presnted in a previous paper (Kumamaru et al, 2007). The calculation parameters for Fig. 8 are the same as for Fig. 2, except for the Hartmann number change along z-axis. The Hartmann number (relating to the applied magnetic field) is 0 from *z*=0 to *z1*, increases linearly from *z*=*z1* to *z2*, and is 100 from *z*=*z2* to *z0*. The pressure change for the case of *z1*/*z2*=10/12, i.e. a standard case, in the inlet region, indicated by a dotted line, is

The pressure decreases slowly following the drop in a non-MHD laminar flow from *z*=0 to *z* ≈ *z1*. The pressure recovery appears clearly in the region near *z* ≈ *z1*. The pressure decreases more rapidly in the region from *z* ≈ *z1* to *z* ≈ *z2* than in the fully-developed MHD region of *z*>*z2*. The pressure decreases rapidly following the drop of a fully-developed

Figure 9 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, in the magnetic-field inlet-region. The distribution in the

while the fully-developed MHD flow enters the domain in the outlet-region.

also compared with the corresponding case in the outlet region in Fig. 3.

calculated by the authors for the fully-developed MHD flow (Kumamaru, 1999).

Fig. 6. Velocity distribution for *z1*/*z2*=10/12.

216 Trends in Electromagnetism – From Fundamentals to Applications

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Fig. 6. Velocity distribution for *z1*/*z2*=10/12.

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(a) At *z*=4.5 (d) At *z*=12

flat one, particularly in the direction of applied magnetic field, having a peak value of ~1.1, as shown in Fig. 6(a). The velocity distribution in this region agrees nearly with a profile calculated by the authors for the fully-developed MHD flow (Kumamaru, 1999).

The velocity distribution at *z*=10, shown in Fig. 6(b), is still nearly flat. The velocity profile changes sharply at *z* ≈ 11 and shows what is called an M-shape distribution having a peak near the wall, as shown in Fig. 6(c). This is because the Lorentz force acting in the negative *z*direction suppresses the flow in the *z*-direction in the fluid bulk region, though small Lorentz force acts in the negative z-direction in the region near the wall of *x*=1. The velocity distribution at z=12, at the outlet of applied magnetic field, shown in Fig. 6(d), is still nearly the same as that at *z*=11, shown in Fig. 6(c).

The velocity profile changes sharply, from *z*=12 to *z* ≈ 13, from the M-shape distribution, shown in Fig. 6(d), to a parabolic distribution typical to a non-MHD flow, shown in Fig. 6(e). The pressure decrease from *z*=12 to *z* ≈ 13, shown in Fig. 3, is attributable to this sharp change in velosity distribution. It is considered that the pressure decreases largely since the velocity increases quickly in the fluid bulk region. No significant difference exists among velocity profiles from *z* ≈ 13 to *z*=22. The velocity profile is a parabolic one of a non-MHD laminar flow with a peak value of ~2.
