**3.1 Pressure along flow axis**

Numerical calculations have been performed for a circular pipe with insulating wall under the conditions of a Reynolds number (*Re*) of 1000, a Hartmann number (*Ha*) of 100 (for the fully-developed MHD region) and a magnetic Reynolds number (*Rm*) of 0.001. The Hartmann number (relating to the applied magnetic field) is 100 from *z*=0 to *z1*, decreases linearly from *z*=*z1* to *z2*, and is zero from *z*=*z2* to *z0* (See Fig. 1(a)). The values of *z1*/*z2* are changed from 10/20 to 10/10.05. (Note that both *z1* and *z0*-*z2* are fixed to 10 in all the cases.) These values for the nondimensional numbers and parameters are selected in order to simulate those typical to laboratory scales and conditions.

Figure 2 shows calculated pressures along the flow axis, i.e. the *z*-axis, for the cases of *z1*/*z2* from 10/20 to 10/10.05. Figure 3 presents a calculated result only for the case of *z1*/*z2*=10/12 as a standard case, indicated by a solid line, together with a corresponding result for the magnetic-field inlet-region, indicated by a dotted line, which will be explained in Sec. 3.4. From *z*=0 to *z* ≈ *z1*, the pressure decreases steeply following the pressure drop of a fully-developed MHD flow. From *z* ≈ *z1* to *z* ≈ *z2*, the pressure decreases more sharply than in the region of *z*<*z1*, since a large Lorentz force is produced in the negative *z* direction as was mentioned in Chap. 1 and again will be explained in Sec. 3.2. In *z*>*z2*, the pressure decreases slowly, representing the frictional pressure drop as a non-MHD laminar flow.

The steeper the gradient of the applied magnetic field becomes, the more sharply the pressure decreases from *z* ≈ *z1* to *z* ≈ *z2*. However, the pressure drop through the magneticfield outlet-region becomes saturated for the steeper gradient of the magnetic field (See the cases of *z1*/*z2*=10/10.1 and 10/10.05). For the slower gradient of the magnetic field, the effect of the length along the flow axis (i.e. *z*-axis) contributes more to the pressure drop through the outlet region than the effect of the outlet region (Compare the cases of *z1*/*z2*=10/20 and 10/15).

212 Trends in Electromagnetism – From Fundamentals to Applications

The discretization of the equations is carried out by the finite difference method. The calculations are performed using a non-uniform expanding 15 x 15 x 30 grid with grid elements closely spaced near the channel wall of *r*=1 and the region between or around *z*=*z1* and *z*=*z2*. The first-order accurate upwind differencing is adopted for the fluid convection terms in Eq. (7). The solution procedure follows the MAC method that is widely used in

Even for the fully-developed region, it is difficult to obtain a stable numerical solution for large Hartmann numbers (Kumamaru & Fujiwara, 1999). In the present three-dimensional calculations, stable numerical solutions have been obtained for Hartmann numbers up to 100 and Reynolds numbers up to 1000 by applying the following means or procedures. (1) The grids are arranged closely near the wall of *r*=1, i.e. at *r*=0.0, … , 0.95, 0.97, 0.99, 0.995, 1.0, on referring to a velocity profile of the classical Hartmann flow, i.e. fully-developed MHD flow in infinite parallel plates (Kumamaru and Fujiwara, 1999). (2) Simultaneous linear equations on the pressure, i.e. Poisson equation, are solved not by the iterative method but by the elimination method. (3) First, a solution is obtained for *Re* (Reynolds number) of 0.01,

Numerical calculations have been performed for a circular pipe with insulating wall under the conditions of a Reynolds number (*Re*) of 1000, a Hartmann number (*Ha*) of 100 (for the fully-developed MHD region) and a magnetic Reynolds number (*Rm*) of 0.001. The Hartmann number (relating to the applied magnetic field) is 100 from *z*=0 to *z1*, decreases linearly from *z*=*z1* to *z2*, and is zero from *z*=*z2* to *z0* (See Fig. 1(a)). The values of *z1*/*z2* are changed from 10/20 to 10/10.05. (Note that both *z1* and *z0*-*z2* are fixed to 10 in all the cases.) These values for the nondimensional numbers and parameters are selected in order to

Figure 2 shows calculated pressures along the flow axis, i.e. the *z*-axis, for the cases of *z1*/*z2* from 10/20 to 10/10.05. Figure 3 presents a calculated result only for the case of *z1*/*z2*=10/12 as a standard case, indicated by a solid line, together with a corresponding result for the magnetic-field inlet-region, indicated by a dotted line, which will be explained in Sec. 3.4. From *z*=0 to *z* ≈ *z1*, the pressure decreases steeply following the pressure drop of a fully-developed MHD flow. From *z* ≈ *z1* to *z* ≈ *z2*, the pressure decreases more sharply than in the region of *z*<*z1*, since a large Lorentz force is produced in the negative *z* direction as was mentioned in Chap. 1 and again will be explained in Sec. 3.2. In *z*>*z2*, the pressure decreases slowly, representing the frictional pressure drop as a non-MHD laminar flow.

The steeper the gradient of the applied magnetic field becomes, the more sharply the pressure decreases from *z* ≈ *z1* to *z* ≈ *z2*. However, the pressure drop through the magneticfield outlet-region becomes saturated for the steeper gradient of the magnetic field (See the cases of *z1*/*z2*=10/10.1 and 10/10.05). For the slower gradient of the magnetic field, the effect of the length along the flow axis (i.e. *z*-axis) contributes more to the pressure drop through the outlet region than the effect of the outlet region (Compare the cases of

where *A*, *B* and *C* are the *x*, *y* and *z* components of *B*, respectively.

and thereafter *Re* is increased gradually to a final value, i.e. 1000.

simulate those typical to laboratory scales and conditions.

numerical calculations.

**3. Analysis results** 

*z1*/*z2*=10/20 and 10/15).

**3.1 Pressure along flow axis** 

Fig. 2. Pressures along *z*-axis for *z1*/*z2*=10/20 to 10/10.05.

Fig. 3. Pressures along *z*-axis for *z1*/*z2*=10/12.

The small pressure recovery, which was also pointed out in Chap. 1 and again will be expained in Sec. 3.2, is observed in the region near *z* ≈ *z2* for the cases of *z1*/*z2*=10/12~10/10.05. The pressure drop appears again outside the magnetic-field region. This may be due to rapid change in velocity distibution in this region, which will be explained in Sec. 3.4.

The pressure drops in the fully-developed region of *z*<*z1*, -Δ*p*/Δ*z*, are almost the same for all the cases. The pressure drops agree with a value calculated numerically by the authors for the fully-developed MHD flow, -Δ*p*/Δ*z* ≈ 0.123 (Kumamaru and Fujiwara, 1999), and also agree nearly with a value predicted by Schercliff's theoretical approximate equation, - Δ*p*/Δ*z* ≈ 0.118 (Schercliff, 1956; Lielausis, 1975), for the case of *Ha*=100 and *Re*=1000. As

Three-Dimensional Numerical Analyses on Liquid-Metal

0 0.2 0.4 0.6 0.8 1

x

0 0.2 0.4 0.6 0.8 1

x

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

0

0.2

0.4

0.6

y

Fig. 5(b).

0

(b) At *z*=10

**3.3 Velocity distribution** 

0.2

0.4

0.6

y

0.8

1

0.8

1

Magnetohydrodynamic Flow Through Circular Pipe in Magnetic-Field Outlet-Region 215

5

j

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

y

5

jx

5

j

y

5

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

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

jx

0

0 0.2 0.4 0.6 0.8 1

5

j

y

5

jx

x

0.2

0.4

0.6

y

0.8

1

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

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