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 r A D x C y C z (A.29) @x @y @z @A @A @A @A @A @A r A 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 r A D .rA / C C z (A.34) r @r r r @ @z 1 @A @A @A @A 1 @.rA / @A r A D i z C i r z C i r r r @ @z @z @r z r @r @ (A.35) 156 Appendixes 1Ð8