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Appendixes 1–8 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)  à @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)  à @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 1–8 Spherical coordinates (r, , ®)  à @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  à 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)  à  à @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)  à  à @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,, ®)  à @T @T 1 @T 1 @T c C v C v C v p @t r @r r @ ' r sin @'  à  à 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)  à  à @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)  à  à @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)  à  à @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 1–8 Circular cylindrical coordinates (r, ,z) (r-component)  à @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 Ä @ 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)  à @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 @ Ä @ 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)  à @v @v v @v @v @p z C v z C z C v z D @t r @r r @ z @z @z  à 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  à 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 @ Ä Â Ã 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)  à @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 @' Ä Â Ã 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  à  à  à @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  à  à 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 1–8  à 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 @  à 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)  à  à 1 @ @f 1 @ @f 1 @2f r2f D r2 C sin C (A.43) r2 @r @r r2sin @ @ r2sin2 @'2  à @ 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)  à 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)  à 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 1–8 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 MLt–2 (Force) Given a quantity in Multiply by table these units value to convert to these units gcms–2 (dynes) N D kg m s–2 (Newtons) gcms–2 110–5 N D kg m s–2 105 1 The dimensions of ML–1 t–2 (Pressure, Momentum flux) Multiply by Given a table value to quantity in convert to these gcm–1 s–2 kg m–1 s–2 these units units (dynes cm–2) (N m–2)(Pa) atm mmHg gcm–1 s–2 110–1 9.869 10–7 7.501 10–4 kg m–1 s–2 10 1 9.869 10–6 7.501 10–3 atm 1.013 106 1.013 105 1 760 mm Hg 1.333 103 1.333 102 1.316 10–3 1 (continued) Table A 159 Table A (continued) The dimensions of ML2t–2 (Work, Energy, Torque) Multiply by table Given a value to quantity in convert to kg m2 s–2 these units these units gcm2 s–2 (absolute (ergs) joules) cal hp-hr kw-hr gcm2 s–2 110–7 2.390 10–8 3.725 10–14 2.778 10–14 kg m2 s–2 107 1 2.390 10–1 3.725 10–7 2.778 10–7 Thermochemical 4.184 107 4.184 1 1.559 10–6 1.162 10–6 calories Horsepower 2.685 1013 2.685 106 6.416 105 1 7.457 10–1 hours Absolute 3.600 1013 3.600 106 8.604 105 1.341 1 kilowatt- hours The dimensions of ML–1 t–1 (Viscosity, Density times diffusivity, Concentration times diffusivity) Given a quantity in Multiply by table value to these units convert to these units gcm–1 s–1 (poises) kg m–1 s–1 Centipoises gcm–1 s–1 D (poises) 1 10–1 102 Pa s D kg m–1 s–1 10 1 103 Centipoises 10–2 10–3 1 The dimensions of MLt–3 T–1 (Thermal conductivity) Multiply by table Given a quantity in value to convert to these units these units gcms–3 K–1 kg m s–3 K–1 (ergs s–1 cm–1 K–1) (W m–1 K–1) cal s–1 cm–1 K–1 gcms–3 K–1 110–5 2.390 10–8 kg m s–3 K–1 105 1 2.390 10–3 cal s–1 cm–1 K–1 4.184 107 4.184 102 1 The dimensions of L2t–1 (Momentum, Thermal, Molecular and Magnetic diffusivities) Given a quantity in these Multiply by table value to units convert to these units cm2 sec–1 m2 sec–1 Centistokes cm2 sec–1 110–4 102 m2 sec–1 104 1106 Centistokes 10–2 10–6 1 (continued) 160 Appendixes 1–8 Table A (continued) The dimensions of Mt–3 T–1 (Heat transfer coefficients) Given a Multiply by table quantity in value to convert these units to these units kg s–3 K–1 gs–3 K–1 (W m–2 K–1) cal cm–2 s–1 K–1 Wcm–2 K–1 gs–3 K–1 110–3 2.390 10–8 10–7 kg s–3 K–1 103 1 2.390 10–5 10–4 cal cm–2 s–1 K–1 4.184 107 4.184 104 1 4.184 Wcm–2 K–1 107 104 2.390 10–1 1 The dimensions of ML–2 t–1 (Mass transfer coefficients) Given a quantity in these Multiply by table value to units convert to these units gcm–2 s–1 kg m–2 s–1 gcm–2 s–1 110 kg m–2 s–1 10–1 1 Table B Constants 1.
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