Appendixes 1–8

Appendix 1: The Equation of Continuity for Incompressible Fluid

Rectangular (Cartesian) coordinates (x, y, z)

@ @ @ .v / C .v / C .v / D 0 (A.1) @x x @y y @z z

Circular cylindrical coordinates (r, ,z)

1 @ 1 @ @ .rv / C .v / C .v / D 0 (A.2) r @r r r @ @z z

Spherical coordinates (r,, ®)

1 @ 1 @ 1 @ r2v C .v sin/ C .v / D 0 (A.3) r2 @r r r sin @ r sin @' '

Appendix 2: The Conservation of Mass for Species

Rectangular (Cartesian) coordinates (x, y, z) @c @c @c @c @2c @2c @2c C v C v C v D D C C (A.4) @t x @x y @y z @z @x2 @y2 @z2

Circular cylindrical coordinates (r, ,z) @c @c 1 @c @c 1 @ @c 1 @2c @2c C v C v C v D D r C C (A.5) @t r @r r @ z @z r @r @r r2 @ 2 @z2

S. Asai, Electromagnetic Processing of Materials, Fluid Mechanics 151 and Its Applications 99, DOI 10.1007/978-94-007-2645-1, © Springer ScienceCBusiness Media B.V. 2012 152 Appendixes 1Ð8

Spherical coordinates (r, , ®) @c @c 1 @c 1 @c 1 @ @c C v C v C v D D r2 @t r @r r @ ' r sin @' r2 @r @r 1 @ @c 1 @2c C sin C r2 sin @ @ r2sin2 @'2 (A.6)

Appendix 3: The Equation of Energy for Incompressible Fluid

Rectangular (Cartesian) coordinates (x, y, z) @T @T @T @T @2T @2T @2T c C v C v C v D C C (A.7) p @t x @x y @y z @z @x2 @y2 @z2

Circular cylindrical coordinates (r, ,z) @T @T 1 @T @T 1 @ @T 1 @2T @2T c C v C v C v D r C C p @t r @r r @ z @z r @r @r r2 @ 2 @z2 (A.8)

Spherical coordinates (r,, ®) @T @T 1 @T 1 @T c C v C v C v p @t r @r r @ ' r sin @' 1 @ @T 1 @ @T 1 @2T D r2 C sin C (A.9) r2 @r @r r2sin @ @ r2sin2 @'2

Appendix 4: The Components of the Energy Flux q

Rectangular (Cartesian) coordinates (x, y, z)

@T q D (A.10) x @x @T q D (A.11) y @y @T q D (A.12) z @z Appendix 5: The Equation of Motion for a Newtonian Fluid with Constant and 153

Circular cylindrical coordinates (r, ,z)

@T q D (A.13) r @r 1 @T q D (A.14) r @ @T q D (A.15) z @z

Spherical coordinates (r, , ®)

@T q D (A.16) r @r 1 @T q D (A.17) r @ 1 @T q D (A.18) ' r sin @'

Appendix 5: The Equation of Motion for a Newtonian Fluid with Constant and

Rectangular (Cartesian) coordinates (x, y, z) (x-component) @v @v @v @v @p @2v @2v @2v x C v x C v x C v x D C x C x C x C f @t x @x y @y z @z @x @x2 @y2 @z2 x (A.19)

(y-component) @v @v @v @v @p @2v @2v @2v y C v y C v y C v y D C y C y C y C f @t x @x y @y z @z @y @x2 @y2 @z2 y (A.20)

(z-component) @v @v @v @v @p @2v @2v @2v z C v z C v z C v z D C z C z C z C f @t x @x y @y z @z @z @x2 @y2 @z2 z (A.21) 154 Appendixes 1Ð8

Circular cylindrical coordinates (r, ,z) (r-component) @v @v v @v v2 @v @p r C v r C r C v r D @t r @r r @ r z @z @r @ 1 @.rv / 1 @2v 2 @v @2v C r C r C r C f (A.22) @r r @r r2 @ 2 r2 @ @z2 r

(™-component) @v @v v @v v v @v 1 @p C v C C r C v D @t r @r r @ r z @z r @ @ 1 @.rv / 1 @2v 2 @v @2v C C C r C C f (A.23) @r r @r r2 @ 2 r2 @ @z2

(z-component) @v @v v @v @v @p z C v z C z C v z D @t r @r r @ z @z @z 1 @ @v 1 @2v @2v C r z C z C z C f (A.24) r @r @r r2 @ 2 @z2 z

Spherical coordinates (r, , ®) (r-component) ! @v @v v @v v @v v2 C v2 @p r C v r C r C ' r ' D @t r @r r @ r sin @' r @r 1 @2 1 @ @v 1 @2v C .r2v / C sin r C r C f (A.25) r2 @r 2 r r2sin @ @ r2sin2 @'2 r

(™-component) ! @v @v v @v v @v v v v2 cot 1 @p C v C C ' C r ' D @t r @r r @ r sin @' r r r @ 1 @ @v 1 @ 1 @.v sin / C r2 C r2 @r @r r2 @ sin @ 1 @2v 2 @v 2cot @v C C r ' C f (A.26) r2 sin2 @'2 r2 @ r2 sin @' Appendix 6: Differential Operation for Scalars and Vectors in Rectangular . . . 155

(®-component) @v @v v @v v @v v v v v 1 @p ' C v ' C ' C ' ' C r ' C ' cot D @t r @r r @ r sin @' r r r sin @' 1 @ @v 1 @ 1 @.v sin / C r2 ' C ' r2 @r @r r2 @ sin @ 2 1 @ v' 2 @vr 2cot @v C C C C f' r2 sin2 @'2 r2sin @' r2 sin @' (A.27)

Appendix 6: Differential Operation for Scalars and Vectors in Rectangular (Cartesian), Cylindrical and Spherical Coordinates

Rectangular (Cartesian) coordinates (x, y, z)

@f @f @f rf D i C i C i (A.28) @x x @y y @z z @A @A @A rA D x C y C z (A.29) @x @y @z @A @A @A @A @A @A rA D i z y C i x z C i y x (A.30) x @y @z y @z @x z @x @y

@2f @2f @2f r2f D C C (A.31) @x2 @y2 @z2

2 2 2 2 r A D i xr Ax C i y r Ay C i zr Az (A.32)

Circular cylindrical coordinates (r,™,z)

@f 1 @f @f rf D i C i C i (A.33) @r r r @ @z z 1 @ 1 @A @A rA D .rA / C C z (A.34) r @r r r @ @z 1 @A @A @A @A 1 @.rA / @A rA D i z C i r z C i r r r @ @z @z @r z r @r @ (A.35) 156 Appendixes 1Ð8

1 @ @f 1 @2f @2f r2f D r C C (A.36) r @r @r r2 @ 2 @z2 @ 1 @.rA / 1 @A @2A 2 @A Œr2A D r C r C r (A.37) r @r r @r r2 @ 2 @z2 r2 @ @ 1 @.rA / 1 @2A @2A 2 @A Œr2A D C C C r (A.38) @r r @r r2 @ 2 @z2 r2 @ 1 @ @A 1 @2A @2A Œr2A D r z C z C z (A.39) z r @r @r r2 @ 2 @z2

Spherical coordinates (r, , ®)

@f 1 @f 1 @f rf D i C i C i (A.40) @r r r @ r sin @' ' 1 @ 1 @.sin A / 1 @A rA D .r2A / C C ' (A.41) r2 @r r r sin @ r sin @' ( ) 1 @ sinA' @A 1 1 @A @.rA / rA D i C i r ' r r sin @ @' r sin @' @r 1 @.rA / @A C i r ' r @r @ (A.42) 1 @ @f 1 @ @f 1 @2f r2f D r2 C sin C (A.43) r2 @r @r r2sin @ @ r2sin2 @'2 @ 1 @ 1 @ @A 1 @2Ar Œr2A D .r2A / C sin r C r @r r2 @r r r2 sin @ @ r2sin2 @'2 2 @.A sin / 2 @A ' r2 sin @ r2 sin @' (A.44) 1 @ @A 1 @ 1 @ 1 @2A Œr2A D r2 C .A sin / C r2 @r @r r2 @ sin @ r2sin2 @'2 2 @A 2cot @A C r ' r2 @ r2 sin @' (A.45) 1 @ @A 1 @ 1 @ 1 @2A Œr2A D r2 ' C A sin C ' ' r2 @r @r r2 @ sin @ ' r2sin2 @'2 2 @A 2cot @A C r C r2sin @' r2 sin @' (A.46) Appendix 8: Integral Theorems 157

Appendix 7: Vector Identities

.A B/ C D A .B C/ D .C A/ B (A.47)

A .B C/ D B.A C/ C.A B/ (A.48)

r.rA/ D 0 (A.49)

r.rf/D 0 (A.50)

r.fg/ D f rg C grf (A.51)

r.A B/ D .A r/B C .B r/A C A .rB/ C B .rA/ (A.52)

r.f A/ D f rA C .A r/f (A.53)

r.A B/ D B .rA/ A .rB/ (A.54)

r.A B/ D A.rB/ B.rA/ C .B r/A .A r/B (A.55)

r.f A/ Drf A C f rA (A.56)

.rA/ A D .A r/A .1=2/r.A A/ (A.57)

r.rA/ Dr.rA/ r2A (A.58)

Appendix 8: Integral Theorems

Line Integral of a Gradient Z b rf dl D f.b/ f.a/ (A.59) a 158 Appendixes 1Ð8

Divergence Theorem Z I rA dV D A dS (A.60) V S Corollaries Z I rfdV D f dS (A.61) V S Z I rA dV D A dS (A.62) V S

Stokes’ Theorem I Z A dl D .rA/ dS (A.63) L S

Corollary I Z f dl D rf dS (A.64) L S

Tables A to F

Table A Conversion factors for quantities having following dimensions The dimensions of MLtÐ2 (Force) Given a quantity in Multiply by table these units value to convert to these units gcmsÐ2 (dynes) N D kg m sÐ2 (Newtons) gcmsÐ2 110Ð5 N D kg m sÐ2 105 1

The dimensions of MLÐ1 tÐ2 (Pressure, Momentum flux) Multiply by Given a table value to quantity in convert to these gcmÐ1 sÐ2 kg mÐ1 sÐ2 these units units (dynes cmÐ2) (N mÐ2)(Pa) atm mmHg gcmÐ1 sÐ2 110Ð1 9.869 10Ð7 7.501 10Ð4 kg mÐ1 sÐ2 10 1 9.869 10Ð6 7.501 10Ð3 atm 1.013 106 1.013 105 1 760 mm Hg 1.333 103 1.333 102 1.316 10Ð3 1 (continued) Table A 159

Table A (continued) The dimensions of ML2tÐ2 (Work, Energy, Torque) Multiply by table Given a value to quantity in convert to kg m2 sÐ2 these units these units gcm2 sÐ2 (absolute (ergs) joules) cal hp-hr kw-hr gcm2 sÐ2 110Ð7 2.390 10Ð8 3.725 10Ð14 2.778 10Ð14 kg m2 sÐ2 107 1 2.390 10Ð1 3.725 10Ð7 2.778 10Ð7 Thermochemical 4.184 107 4.184 1 1.559 10Ð6 1.162 10Ð6 calories Horsepower 2.685 1013 2.685 106 6.416 105 1 7.457 10Ð1 hours Absolute 3.600 1013 3.600 106 8.604 105 1.341 1 kilowatt- hours

The dimensions of MLÐ1 tÐ1 (Viscosity, Density times diffusivity, Concentration times diffusivity) Given a quantity in Multiply by table value to these units convert to these units gcmÐ1 sÐ1 (poises) kg mÐ1 sÐ1 Centipoises gcmÐ1 sÐ1 D (poises) 1 10Ð1 102 Pa s D kg mÐ1 sÐ1 10 1 103 Centipoises 10Ð2 10Ð3 1

The dimensions of MLtÐ3 TÐ1 (Thermal conductivity) Multiply by table Given a quantity in value to convert to these units these units gcmsÐ3 KÐ1 kg m sÐ3 KÐ1 (ergs sÐ1 cmÐ1 KÐ1) (W mÐ1 KÐ1) cal sÐ1 cmÐ1 KÐ1 gcmsÐ3 KÐ1 110Ð5 2.390 10Ð8 kg m sÐ3 KÐ1 105 1 2.390 10Ð3 cal sÐ1 cmÐ1 KÐ1 4.184 107 4.184 102 1

The dimensions of L2tÐ1 (Momentum, Thermal, Molecular and Magnetic diffusivities) Given a quantity in these Multiply by table value to units convert to these units cm2 secÐ1 m2 secÐ1 Centistokes cm2 secÐ1 110Ð4 102 m2 secÐ1 104 1106 Centistokes 10Ð2 10Ð6 1 (continued) 160 Appendixes 1Ð8

Table A (continued) The dimensions of MtÐ3 TÐ1 (Heat transfer coefficients) Given a Multiply by table quantity in value to convert these units to these units kg sÐ3 KÐ1 gsÐ3 KÐ1 (W mÐ2 KÐ1) cal cmÐ2 sÐ1 KÐ1 WcmÐ2 KÐ1 gsÐ3 KÐ1 110Ð3 2.390 10Ð8 10Ð7 kg sÐ3 KÐ1 103 1 2.390 10Ð5 10Ð4 cal cmÐ2 sÐ1 KÐ1 4.184 107 4.184 104 1 4.184 WcmÐ2 KÐ1 107 104 2.390 10Ð1 1

The dimensions of MLÐ2 tÐ1 (Mass transfer coefficients) Given a quantity in these Multiply by table value to units convert to these units gcmÐ2 sÐ1 kg mÐ2 sÐ1 gcmÐ2 sÐ1 110 kg mÐ2 sÐ1 10Ð1 1

Table B Constants 1. Mathematical constants e D 2.71828 :::.. ln 10 D 2.30259 :::.. D 3.14159 :::.. 2. Physical constants Gas law constant R D 1.987 cal g-molÐ1 KÐ1 D 82.06 cm3 atm g-molÐ1 KÐ1 D 8.315 107 gcm2 sÐ2 g-molÐ1 KÐ1 D 8.315 103 kg m2 sÐ2 kg-molÐ1 KÐ1 Standard acceleration of gD 980.665 cm sÐ2 D 9.80665 m sÐ2 gravity 7 Ð1 Ð1 Joule’s constant (mechanical Jc D 4.1840 10 erg cal D 4.1840 J cal equivalent of heat) Avogadro’s number NQ D 6.02 1023 molecules g-molÐ1 Boltzmann’s constant k D R/NQ D 1.380 10Ð16 erg KÐ1 D 1.380 10Ð23 JKÐ1 Stefan-Boltzmann constant D 1.355 10Ð12 cal sÐ1 cmÐ2 KÐ4 D5.671 10Ð8 WmÐ2 KÐ4 Electron charge e D 1.602 10Ð19 C (coulomb) Speed of light in a vacuum c D 2.99792 108 msecÐ1 Ð12 Ð1 2 4 Ð1 Ð3 Permittivity in vacuum "0 D 8.85 10 farad m (A s kg m ) Ð7 Ð1 Ð2 Ð2 Permeability in vacuum m0 D 4 10 henry m (kgmA s ) Ð31 Mass of electron me D 9.108 10 kg Table C 161 Ð1 scm CGSesu CGSesu statvolt CGSesu Ð11 Ð2 Ð11 Ð11 CGSesu cm Ð2 10 10 10 CGSesu CGSesu statcoulomb statampere 10 11 11 10 9 9 9 9 / 10 10 / 2 / 10 10 2 10 10 3/ 3 2 b b 3 3/ b 3 b 2 b 2 3 3 3 .1= 3 3 3 .1= b b b b b b .1= .1= .1= D D D D D D D D D statvolt D D Ð4 Ð1 10 statamp 5 CGSemu CGSemu CGSemu centimeter (cm) gram (g) dyne erg erg sec CGSemu CGSemu CGSemu 10 3/ b 2 3 5 7 7 Ð1 Ð1 8 Ð9 9 9 3 10 .1= 10 10 10 10 10 1 second (s) 10 10 10 10 10 b D D D D D D D D D D D D D D A Ð1 1V Ð1 D Ð1 Ð1 Ð1 1WA 1Cs Ð1 D 1Js 1CV 1VA D Ð1 D D D ) 1JC 1VsA Ð2 Ð1 D D 0.0000010 ˙ 2.9979250 D 3 b where Capacitance 1 farad (F) CurrentCurrent density 1 1 A ampere m (A) Potential 1 V Electric field 1 V m Resistance 1 ohm ( MassTimeForceWork (Energy)Power 1 1 kilogram joule (kg) (J) 1 second (s) 1 Newton (N) 1 watt (W) Table C ConversionThe table following for table Gaussian gives MKS the units. relations betweenQuantity the units of common quantities in theLength rationalized MKS system and in the Gaussian (CGS) system Charge Rationalized MKS 1 meter (m) 1 coulomb (C) Inductance Gaussian 1 H 162 Appendixes 1Ð8 2 cm = 2 s 5 20 2 9 7 Ð11 Ð11 Ð2 Ð6 10 10 10 10 10 10 10 10 10 Ð1 11 3 9 9 3 3 3 b / / / b b b 10 = 2 2 2 dyne erg 4 10 10 erg s 3/ 3/ 3/ 3 3 3 b b b b b b 5 7 7 2 10 .1= 10 3 3 4 .1= .1= 3 1 4 .1= .1= .1= 4 b b b 2 cm = 2 s 20 Oe 5 10 Ð1 Ð3 Gb / 10 10 2 dyneerg 10 10 Mx G erg s 3 b 5 7 7 8 6 Ð1 8 4 Ð1 9 9 Ð9 10 10 10 4 10 10 10 4 10 10 10 .1= 10 2 5 Ð11 Ð2 10 10 10 10 Ð1 Ð3 11 3 9 9 3 b / 10 b 10 = 2 dyneerg 10 10 4 10 10 erg s 3/ 3/ 3 b b b 5 7 7 4 9 2

10 3 4 3 4 3 1 11 .1= 4=10 4=10 .1= .1= b b b Ð1 Fm Ð9 10 Ð1 1T(tesla) 10 Hm D 4/ Ð1 Ð7 Ð2 Ð1 Ð2 10 2 (ohm) 3 b 1A(amp) 1 J (joule)1 W (watt) 10 10 1AT(amp-turn) 1Wb(weber) 1=. 4 1 N (newton)1Vm 10 1 C (coul.) 1 H (henry)1 F (farad) 10 1 V (volt) 1ATm 1Wbm 1Cm 1 m m0 m 0 I W P F, f E ˚ Q L C V V H B R " 0.0000010 ˙ 2.9979250 D 3 b Current Permeability of free space Work (energy) Power Table D Comparison amongQuantity MKS, Gaussian, CGS emu and CGSForce esu systems Electric potential Electric field Magnetic potential SignMagnetic field intensity Magnetic flux MKSA systemMagnetic flux density Charge Charge density Electric resistance Inductance Gaussian systemCapacitance Permittivity of free space CGS emuwhere CGS esu Table E 163 (JIM, Maruzen, (W/m K) K 0.6 293 Kaitei 3 edition kin-zoku deta bukku mho/m) Ð6 6 10 (10 4 ¢ /s) 5 2 (Wiley, New York, 1966) and in 10 (m m ) 3 kg/m 3 (10 ¡ 1 Ð Ð 0.0238 293 ) 2 Ð2 s/m The Electromagnetodynamics of Fluids 10 N Ð3 (10 Viscosity Density Magnetic diffusivity Electric conductivity Thermal conductivity Temp. 0.14 at 1,700 C 4.4 at 1,650 C 6.86 at 1,650 C 1.12 at 1,600 C 0.714 at 1,600 C 20% 0.25 at 1,600 C 3 O 2 30%) 2 Slag (Al Sea water 1 1.025 2 Liquid metals CaO 50% SiO Table E Properties of conducting fluid Liquid steelLiquid mercuryLiquid sodiumNaK 5.5at Alloy1 1,500 C 1.55(Na 22% K 0.705 78%) NaK Alloy2 7.08 at 1,600 0.5 C(Na 56% K 44%) Wood’s 1.09 metal at 1,600 13.55 C 0.6(Pb 50%Bi 25% 0.927Sn 12.5% Cd 12.5%) 2 0.85Air (sea 0.730 level) at 1,560 C 0.744 0.0765 0.89 1.825 33.4 at 1,600 C 0.30 1.05 1.07 0.266 10.4 8.85 2.66 3 8.8 85 9 24.3 25.8 293 373 13.4 293 293 373 From W.F.Hughes, F.J.Young, in Tokyo, 1993), p. 16 164 Appendixes 1Ð8 (continued) (N/m) f Surface tension ) 2 s/m N Ð3 Viscosity (10 0.343 at 210.9 C ¢ mho/m) 6 Electrical conductivity (10 ) 3 kg/m 3 (10 ¡ 2.26 at 1,100 C9.20 at 962 C 4.317.72 at at 870 600 C C 0.651 at 750 C 2.80 at 600 C 1.4 at 800 C 0.996 at 600 C 1.54 0.363 at at 600 500 C C 0.600 at 500 C 5.445 at 1,100 C9.81 at 1,000 3.66 C at 46.1 C0.441 at1,000 C 0.81612.88 at1,000 at C 300 C 0.6520.676 at at 806 700 C C 0.737 at 1.185 350 at C 844 C 3.18 at 350 C 0.431 at 500 C 1.01 at 200 C 0.455 at 285.5 C 0.136 at 700 0.394 C at 354 C Table F Properties of liquidLiquid metals metals (MKS units) (at.no.)Al (13)Bi (83) Melting point (C)Cd (48) 660.2 Boiling Point (C)Cs (55) Density 271.0 321 2,950 28.5 1,477 2.38 765 at 660 C 10.0 705 at 300 C 5.1 at 657 8.01 C at 330 C 0.775 at 1.84 300 at C 28 C 2.97 at 330 C 2.9 at 700 C 1.66 at 2.73 304 at C 30 C 0.520 at 750 C 0.376 at 300 2.37 C at 350 C 0.630 0.564 at at 43.4 330 C C Ð Fe (26)Ga (31) 1,535 29.92 2,750 1,983 7.08 at 1,600 C 6.093 at 32.4 C 0.73 at 1,560 C 3.86 at 29.8 C 4.4 at 1,650 C 1.894 at 52.9 C 1.85 at 1,550 C 0.735 at 30 C In (49)Pb (82)Li (3) 156.4Hg (80) 327.4K (19) 179 Ð38.37 2,087 1,737 63.7 2,403 7.026 357 at 164 C 10.51 at 400 C 760 0.507 at 3.43 200 at C 154 C 13.55 1.084 at at 20 327 C C 0.819 2.21 at at 100 230 C C 1.016 at Ð 50 2.116 C at 441 C 7.6 at 0.442 64 at C 350 C 0.592 at 183.4 C 1.55 at 20 C Ð 0.340 at 170 C 0.465 at 0.515 20 at C 69.6 C 0.086 at 100 C Table F 165 (JIM, Maruzen, 0.110 at Ð11 C 0.120 at 250 C 0.468 at 103.7 C 0.146 at 700 C Kaitei 3 edition kin-zoku deta bukku 3.64 at 100 C 0.323 at 220.1 C 2.66 at 50 C 2.13 at 200 C 0.277 at 30 C 112 at 60 C (Wiley, New York, 1966) and in 9.0 at 1,300 C0.780 at 700 C6.64 at 704 C 4.76 at 1,340 C 6.57 at 800 C 5.41 at 350 C0.742 at 700 C 1.45 at 1,000 C0.703 at 700 2.80 C at 800 C 0.182 at 700 C 2.44 at 200 C6.94 1.05 at at 687 600 C C 0.200 at 250 1.88 C at 700 C 0.161 0.510 at at 700 500 C C 0.936 at 327 C 0.765 at 640 C 0.110 at 250 C 177 at 200 C The Electromagnetodynamics of Fluids Ð11 784 0.847 at 100 C (continued) Table F Rb (37)Ag (47)Na (11)Sn (50) 39.0Zn (30) 960.5NaK Alloy 1(Na 688 22% 97.8K 2,212 78%) NaK Alloy 2 1.475 at 39.0(Na 231.9 C 56% 9.3 1,621 at 960.5K C 44%) 419.5Wood’s metal 2,270 19 0.928 at 100 CNaCl standard 4.32 1,663 solution at 50 6.834 C at 409 C 6.04 1,518 at 1,000 C 6.92 at 419.5 C Ð 70 10.35 at 0.886 100 at C 100 C 2.1 at 0.673 231.9 at C 38 2.98 C at 1,200 C Ð Ð 2.83 at 419.5 C 0.686 at 3.03 103.7 at C 50 C 0.923 Ð at 995 C 1.201 1.91 9.8 at at at 18 240 73 C C C 0.206 3.17 at at 100 450 C C 0.546 0.526 at at 103.7 300 C C 0.226 at 0.785 20 at C 510 0.993 C at 85.6 C 0.100 at 19 C 108 at 281 18 at C 100 C 0.326 0.08 at at 100 20 C C From W.F.Hughes, F.J.Young, in Tokyo, 1993), p. 16 Notations

(Number 1 to 5 show the chapters where the notations are used)

A surface area (m2)2 A magnetic gravity ratio number (-) 5 A0 relative magnetic gravity ratio number (-) 5 A,A complex number (-) 3 A vector potential of a magnetic flux (T m) 4 a radius of curvature of a metal drop (m) 4 a1,a2 real number (-) 3 B,Be magnetic flux density (T) 1, 2, 3, 4, 5 B,B complex number (-) 3 B magnetic flux density vector (T) 3, 4, 5 Bi Bingham number (-) 2 Bmax,Bmin maximum and minimum amplitude of magnetic flux density (T) 4 Bx;By ;Bz magnetic flux density components (T) 3, 4 B0 magnetic flux density (T) 4 Bo (-) 2 b1,b2 real number (-) 3 _ _ bz,bx,bz,bx,b0 magnetic flux density (T) 3, 4 bx ;bz complex conjugate numbers of bx and bz, respectively 3 berž,beiž Kelvin function of order ž(-) 3 C circulation (m2/s) 3 c concentration (kg/m3)2 2 2 c1 constant fkg=.m s /g 3 3 cb bulk concentration (kg/m )2 3 cs surface concentration (kg/m )2 cp heat capacity at constant pressure fJ/(kg K)g 1, 2, 3 D diffusion coefficient (m2/s) 2 D displacement field (C/m2)1,2,3

S. Asai, Electromagnetic Processing of Materials, Fluid Mechanics 167 and Its Applications 99, DOI 10.1007/978-94-007-2645-1, © Springer ScienceCBusiness Media B.V. 2012 168 Notations

DI Damkohler-I¬ number (-) 2 DII Damkohler-II¬ number (-) 2 DIII Damkohler-III¬ number (-) 2 E energy density (J/m3)5 E electric field (V/m) 1, 2, 3 Ex;Ey ;Ez electric field components (V/m) 3 F body force vector (N/m3)4,5 F body force (N/m3)2 F function showing the shape of boundary (-) 3 Fo Fourier number (-) 2 Fr (-) 2 Frm magnetic Froude number (-) 5 f, f external body force (N/m3)2,4 fint intermittent frequency (1/ s) 4 3 fNz; fNr time average body force (N/m )3 3 fx,fy ;fz body force components (N/m )3,4 f friction coefficient (-) 2 Gr (-) 2, 3 Grm magnetic Grashof number (-) 5 g, g gravitational acceleration coefficient (m/s2)2,4,5 g0 apparent gravitational acceleration coefficient (m/s2)4 H magnetic field intensity (A/m) 1, 2, 5 H magnetic field intensity vector (A/m) 3, 5 Ha Hartmann number (-) 2, 3 Hd demagnetizing magnetic field intensity (A/m) 5 Heff magnetic field intensity induced in a substance (A/m) 5 Hex magnetic field intensity imposed from the outside of substance (A/m) 5 Hs magnetic field intensity at which a magnetic moment is satu- rated (A/m) 5 h heat transfer coefficient fW/(m2 K)g 2 h surface wave height (m) 2 h surface height (m) 4, 5 h0 liquid depth (m) 2 Iz, Isc current (A) 3 i imaginary number (-) 3, 4 ix, iy, iz unit vector in x-, y-andz-directions, respectively (-) 3 J electric current density (A/m2)3 J mass flux fkg=.m2s/g 2 J electric current density (A/m2)3,4 Js surface current (A/m) 3 2 Jx;Jy ;Jz;J electric current density (A/m )3,4 2 Jx massfluxinthex-direction fkg=.m s/g 2 Jp Bessel function of the first kind of order p (-) 3 Notations 169

2 jy electric current density (A/m )3 * jy conjugate of jy 3 k average mass transfer coefficient (m/s) 2 k constant (1/ m) 3 k Boltzmann constant (1.38 10Ð23 J/K) 5 k0 instantaneous mass transfer coefficient (m/s) 2 L, L0 characteristic length (m) 2, 3 L falling length of particle (m) 5 Le (-) 2 Lm characteristic length relating with magnetic gradient (m) 5 l diffusion length of mass (m) 2 M magnetic moment (A/m) 5 Mm magnetic moment of ferromagnetic materials (A/m) 5 Mn magnetic moment in non-magnetic materials (A/m) 5 Mam magnetic or Alfven« number (-) 3 Ms saturated magnetic moment (A/m) 5 mmol molecule mass (g/mol) 5 N Stuart number or interaction parameter (-) 2, 3 N demagnetizing factor (-) 5 Nx;Ny ;Nz components of demagnetizing factor in x-, y-andz-directions, respectively (-) 5 Nu (-) 2 n constant (1/ m) 3 n unit normal vector (1/ m) 3 P pressure (Pa) 2, 3 Pe heat Peclet number (-) 2 Pe mass Peclet number (-) 2 Pm magnetic pressure number (-) 3 Pr (-) 2 Prm (-) 3 pc pressure due to surface tension (Pa) 4 pdyn dynamic pressure due to molten metal motion (Pa) 4 ps static pressure of molten metal (Pa) 4 pm magnetic pressure (Pa) 3, 4 P pressure (Pa) 3 Q heat generation per unit volume (W/m3)2 Q volumetric flow rate (m3/s) 2 Q Ohmic dissipation per unit area (W/m2)3 q,q heat flux f J/(m2 s)g 2, 4 2 qx heat flux in the x-direction f J/(m s)g 2 qN Ohmic dissipation (J/m3)3,4 R Reaction term fkg=.m3 s/g 2 R half diameter of a cylindrical tube and spherical ball (m) 2 Ra (-) 2 170 Notations

Ram magnetic Rayleigh number (-) 5 Re (-) 2 Rem (-) 3 Ri electric inner resistance ()3 r axis in circular cylindrical and spherical coordinates (-) 2 r average curvature radius (m) 3 r particle size (m) 5 Sc (-) 2 Sh (-) 2 T temperature (K) 2 T period of a periodic function (s) 2, 4 T magnetization torque (N m) 5 T stress tensor (N/m2)3 Tb temperature in bulk flow (K) 2 Ts temperature at interface (K) 2 t time (s) 2, 3, 5 te contacting time of two fluids (s) 2 trel mechanical relaxation time (s) 4 U magnetization energy (J/m3)5 U electromotive force (V) 4 Ua;b;Uc magnetization energy in a- or b-axis and c-axis, respectively (J/m3)5 V volume of a vessel (m3)2 VT , Voc, Vg voltage (V) 3 v velocity (m/s) 2 v velocity vector (m/s) 3, 4, 5 vr ; v ; vx; vy ; vz velocity components (m/s) 3, 4 vx average flow velocity (m/s) 3 W energy density (J/m3)1,2 We (-) 2 Wm shielding parameter or magnetic (-) 3 w mass transfer rates (kg/s) 2 x axis in rectangular (Cartesian) coordinates (-) 2, 3, 4 x variable (-) 3 Y Yakob number (-) 2 y axis in rectangular (Cartesian) coordinates (-) 2, 3, 4 z axis in rectangular (Cartesian) coordinates (-) 2, 3, 4 ˇ thermal coefficient of volumetric expansion (1/ K) 2 2 2 “ constant (ˇ C i!=vm) (1/m) 3, 4 ˇr ;ˇi real and imaginary parts of ˇ (1/m) 3, 4 propagation constant (1/m) 3, 4 H enthalpy change (J/kg) 2 ı skin depth (m) 3, 4 ıij Kronecker delta (-) 3 © dielectric constant (F/m) 3 Notations 171

angle in circular cylindrical and spherical coordinates (-) 2 angle between the easy magnetization axis and the imposed magnetic field (rad) 5 thermal diffusivity (m2/s) 2 viscosity coefficient fkg=.ms/g 2, 3, 5 m magnetic permeability (H/m) 1, 2, 3, 4 7 m0 magnetic permeability in vacuum, 4 10 (H/m) in MKSA unit system, 1(-) in cgs Gauss unit system 5 r relative magnetic permeability (r 1 C )(-) 5 kinematic viscosity (momentum diffusivity) (m2/s) 2, 3, 5 2 m magnetic diffusivity (m /s) 3, 4 wave number (1/m) 2 thermal conductivity f J/(m s K)g 2, 3 wave length (m) 2 angular velocity (rad/s) 3 n n-th root of J0.nR/ D 0 (1/m) 2 density (kg/m3)1,2,3,4,5 00 density (g/cm3)5 3 f net charge density (C/ m )3 3 l ;p densities of liquid and particle, respectively (kg/m )5 e, es electric charge (C) 3 electric conductivity (S/m) 2, 3, 4 f surface tension (N/m) 2, 3, 4 2 ij stress tensor fkg=.ms/g 2, 3 momentum flux fkgm=sg=.m2 s/ 2 2 0 stress (N/m )2 R crystal alignment time in synchronized region (s) 5 s crystal alignment time (s) 5 yx flux of the x-momentum in the negative y-direction fkg=.ms2/g 2 ˆ velocity potential (m2/s) 2 magnetic scalar potential (T m) 3 ' angle in spherical coordinates (-) 2 magnetic susceptibility (-) 1, 2, 5 magnetic susceptibility tensor (-) 5 a:b;c magnetic susceptibilities in a- or b-axis and c-axis, respectively (-) 5 r relative susceptibility (-) 5 0 magnetic susceptibility in SI for E-H unit system (H/m) 5 00 magnetic susceptibility in cgs Gauss unit system (-) 5 ? magnetic susceptibility in the difficult magnetization axis (-) 5 3 g mass magnetic susceptibility (m /kg) 5 00 3 g mass magnetic susceptibility (c m /g) 5 3 m molar magnetic susceptibility (m /k mol) 5 172 Notations

00 3 m molar magnetic susceptibility (c m /mol) 5 med magnetic susceptibility of medium (-) 5 p magnetic susceptibility of precipitated substance (-) 5 ¨ angular frequency (rad/s) 2, 3, 4 ¨ rotation speed of magnetic field (rad/s) 5 ! vorticity vector (1/s) 3, 5 !r ;! ;!z vorticity components (1/s) 3 Index

A D Adhesive force, 94 Damkohler-I¬ number, 22 Agglomerating function, 91Ð92, 103Ð104 Damkohler-II¬ number, 22 Alfven« number, 78, 79 Damkohler-III¬ number, 22 Alfven« wave, 79 Demagnetizing factor, 122Ð124, 129 Alternating magnetic field, 1, 4, 64, 66, 67, 70, Demagnetizing magnetic field, 122 88, 90, 93Ð95, 103, 104 Diamagnetic substance, 116, 129, 144 Ampere’s law, 52, 69, 76 Dielectric constant, 53 Angular frequency, 36, 90 Diffusion coefficient, 10, 16, 24 Angular velocity, 50 Diffusion length, 24 Arc welding, 110 Displacement field, 53 Astronomy, 2 Driving function, 104

B E Bingham number, 22 Electric and magnetic fields, 1, 49, 76Ð77, Biot number, 22 87Ð96, 109 Body force, 18, 51, 57, 61, 64Ð67, 88, 90, 92, Electric and magnetic science, 1 93, 94 Electric conductivity, 49, 52, 92 Buoyancy force, 79 Electric current, 1, 4, 52, 53, 59, 60, 69, 87, 88, 90Ð94, 99, 100, 104, 106, 107 Electric energy, 1, 4, 5 Electric field, 52, 59 C Electrically conductive fluid, 1, 5, 55, 56, 87, Cartesian coordinates, 15, 21, 151Ð153, 155 95, 96, 103, 108 Characteristic length, 63, 78, 89, 146 Electrically conductive materials, 2, 107 Circular cylindrical coordinates, 12, 151Ð155 Electromagnetic Archimedes force, 92, 144 Circulation, 50Ð51 Electromagnetic brake, 101 Clad slab, 101Ð102 Electromagnetic casting, 96, 98 Coercive force, 122 Electromagnetic force, 1, 18, 49, 51, 54Ð55, Cold crucible, 1, 94, 100 87, 92, 96Ð109, 147 Conservation law, 11 Electromagnetic levitation, 1 Contacting time, 25, 26 Electromagnetic mixing, 93, 104 Crystal alignment time, 137, 139, 140, 141 Electromagnetic pump, 4, 74, 104 Czochralski method, 101, 107 Electromagnetic stirring, 1, 4

S. Asai, Electromagnetic Processing of Materials, Fluid Mechanics 173 and Its Applications 99, DOI 10.1007/978-94-007-2645-1, © Springer ScienceCBusiness Media B.V. 2012 174 Index

Electromagnetic ultrasonic wave, 94, 106 Kirchhoff’s voltage law, 60 Electromotive force, 95, 108 Kronecker delta, 19, 55 Enhanced Moses effect, 117Ð119, 146 Enthalpy change, 22 ESR, 110 L Laplace equation, 54 Levitating function, 94 F Levitation melting, 97 Faraday’s law, 52, 77 Lewis number, 22 Ferromagnetic materials, 114, 115 Lorentz force, 2, 54, 55, 70, 72, 73, 79, 90, 91, Ferromagnetic substance, 122Ð125 92, 94, 106, 113, 144 Fick’s first law, 10 Fleming’s right-hand rule, 95, 108 Flow suppressing function, 90, 91, 101Ð102 M Flow velocity detecting function, 95, 108Ð109 Magnetic anisotropy, 121Ð122 , 1 Magnetic Archimedes force, 145 Fourier number, 22 Magnetic , 143, 145 Fourier’s law of heat conduction, 10 Magnetic crystal alignment, 123, 127Ð143 Froude number, 22, 145 Magnetic diffusivity, 53 Magnetic energy, 2, 121 Magnetic field, 2, 121 G Magnetic flux density, 53, 91, 95, 99, 115, 116, Gauss’ law, 52, 71 120, 129 Geophysics, 2, 49 Magnetic Froude number, 145 Grashof number, 22, 79 Magnetic Grashof number, 146 Gravity changing function, 94Ð95 Magnetic gravity ratio number, 146 Gravity force, 18, 36, 54, 94, 113, 131, 132, Magnetic Mach number, 78, 79 134, 135, 143Ð146 Magnetic moment, 115, 119, 123 Magnetic permeability, 53, 115, 120, 121 Magnetic Prandtl number, 53, 78, 79 H Magnetic pressure, 64Ð67, 78, 79, 88, 91, 96, Harmonic function, 54 100, 103, 105 Hartmann number, 22, 58, 78, 79 Magnetic pressure number, 78, 79 Hartmann problem, 56, 59, 60 Magnetic Rayleigh number, 146 Heat flux, 9, 15 Magnetic Reynolds number, 62, 78, 79 Heat Peclet number, 22 Magnetic scalar potential, 53 Heat transfer coefficient, 21, 160 Magnetic Science, 1, 6 High frequency induction skull melting Magnetic slip casting, 132Ð142 method, 107 Magnetic susceptibility, 115Ð121, 124, 127, High magnetic field, 2, 3, 106, 113, 115, 127, 128, 129, 132, 133, 134, 144 132, 142, 143, 147, 148 Magnetic Womersley number, 78 Magnetism, 4, 119Ð122 Magnetization energy, 2, 122Ð128, 130, 137 I Magnetization force, 2, 113Ð119, 132, 143Ð147 Interaction parameter, 78, 79 Magnetohydrodynamics (MHD), 1, 2, 5, 6, Intermittent frequency, 99 49Ð87 Intermittent high frequency magnetic field, 99 Mass flux, 9, 25 Inverse Moses effect, 116Ð117 Mass magnetic susceptibility, 120, 121, 144 Mass Peclet number, 22 K Mass transfer coefficient, 25, 26, 28, 30, 31 Kinematic viscosity, 22, 53, 78 Maxwell stress, 54Ð55 Kinetic energy, 53, 78 Maxwell’s equation, 49 Index 175

Mechanical relaxation time, 93, 94 S Molar magnetic susceptibility, 120, 121 Schmidt number, 22 Molten metal, 1, 4, 5, 23, 49, 67, 87, 88, 90Ð96, Self-exciting dynamo, 2 99, 103, 104, 106, 107, 126 Separating function, 91Ð92, 103Ð104 Molten steel, 4, 96, 97, 99, 101, 108 Shape controlling function, 87Ð90, 93, 96Ð101 Momentum, 9Ð11, 17Ð21, 77, 119 Shape magnetic anisotropy, 122, 124Ð125 Momentum flux, 9, 20 Sherwood number, 22 Moses effect, 113, 114, 116Ð119, 146 Shielding parameter, 78, 79 Skin depth, 64Ð66, 90, 96 Soft contacting solidification, 96Ð100 N Spherical coordinates, 12, 20, 21, 30, 31, 151, Navier-Stokes equation, 20, 36, 51, 57, 62, 145 152, 156 Newton’s law of viscosity, 10 Splashing function, 94, 106Ð107 Newtonian fluid, 19, 51, 153Ð155 Steel-making process, 2, 5 Non-rotational force, 89, 90 Stress tensor, 17, 54Ð55 Nuclear fusion, 2 Stuart number, 22, 78, 79 Nuclear power, 2 Superconducting magnet, 4, 113, 115 Nusselt number, 22 Surface current, 77 Surface tension, 18, 21, 75, 94, 96, 106

O Ohm’s law, 49, 53, 57, 79 T Ohmic dissipation, 67 Temperature raising function, 95, 107Ð108 Oscillating function, 94, 104Ð106 Tensor, 9, 17, 75, 127, 128 Thermal coefficent, 22 Thermal coefficent of volumetric expansion, 22 P Thermal conductivity, 10, 21, 52 Paramagnetic substance, 116, 123, 144 Thermal diffusivity, 16, 22 Permanent magnet, 4, 108, 113 Thermal energy, 2, 4, 5 Pinch effect, 91 Transport phenomena, 9Ð47 Plasma, 5 Plasma physics, 2, 49 Prandtl number, 22 V Process Metallurgy, 6 VAR, 110 Propagation constant, 71 Vector potential of magnetic flux, 95 Velocity potential, 35 Viscosity coefficient, 10, 53, 132 R Vorticity, 50Ð52, 54, 80, 89, 93 Radius of curvature, 94 Rayleigh number, 22 Rectangular coordinates, 12, 13 W Relative magnetic gravity ratio number, 146 Wave length, 36 Relative magnetic permeability, 121 Wave number, 36 Relative susceptibility, 120 Wave suppressing function, 91, 103 Reynolds number, 22, 62 Weber number, 22 Right hand-Fleming’s law, 95, 108 Rotating magnetic field, 106, 133, 135, 136, 139, 141 Y Rotational force, 89 Yakob number, 22