HYDRODYNAMIC AND HYDROMAGNETIC STABILITYstaff.ustc.edu.cn/~zuojin/arts/ST2-5.pdf · 2012. 3....

HYDRODYNAMIC AND

HYDROMAGNETIC STABILITYBY

S. CHANDRASEKHAR

$ X Æ $ � A � & 9

' � � M~�3lÆ*l 3l6�{rAo

2000 � 12 � 20)

4�&I df?7 1

§1 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1§2 wQH� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1§3 RQCx�%f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2§4 YT0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

II ��|lO�"}�Oy�$71. Bénard �z 7§5 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7§6 aiPu2wx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7§7 (w`zkwQ�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

(a) �P�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8(b) �`�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9(c) *P-�M< . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10(d) '�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

§8 Boussinesq nG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12§9 0%�`�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

(a) `_}7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16§10 QCx�%\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

(a) Mk9�%2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18§11 M[P2Chui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19§12 `|(e2�u��5)\*mJbPu . . . . . . . . . . . . . . . . . . . . . 20§13 f%ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

(a) C�f%ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21(b) C�f%ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

§14 f%ui2�zk/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25§15 mJbPu2v�\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

(a) 6V`_2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27(b) NP`_2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28(c) �`6V`_5�`NP`_2\ . . . . . . . . . . . . . . . . . . . . . . . 35(d) =�tLw2Y�>Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

§16 u>�" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36(a) �` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36(b) �G5Q�Gu> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37(c) +`Gu> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39(d) =GGu> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41(e) lFi2u>�" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

§17 f%\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43I

II 3�)§18 (w}�d�M[P2w� . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

III ��|lO�"}�Oy�$72. >�S' 63§19 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63§20 Helmholz 5 Kelvin [i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63§21 �g$Wjn}2(w`zk�� . . . . . . . . . . . . . . . . . . . . . . . . . 67§22 Taylor-Proudman [i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68§23 g$(w}z2_x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71§24 �g$(w}�M[Pu�Fi2,6 . . . . . . . . . . . . . . . . . . . . . 72§25 �`�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73§26 Mev(M[P�ds2tL��`f%ui . . . . . . . . . . . . . . . . . 75

(a) �`f%ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76§27 �Mev(M[P�ds2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

(a) �d�`6V`_tL2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . 78(b) �d�`NP`_tL2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . 81(c) ��`�NP2!�`�6V2`_tLw2\ . . . . . . . . . . . . . . 85(d) T

23 [;2�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

§28 Mev(M[P�ds9�jÆ2�`5u>�" . . . . . . . . . . . . . . 88(a) �`u>��G5�G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90(b) +`G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92(c) � T → ∞ s(2��Px . . . . . . . . . . . . . . . . . . . . . . . . . . 94

§29 �d/MePv(2�d��``_�6VtLw2\ . . . . . . . . . . . . . 94(a) mJ�� (215) 2h2Px . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

§30 �d%`|(emJ2��Fi2f%ui . . . . . . . . . . . . . . . 101(a) f%ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

§31 /M[Pv(2�d��dLa`_}72\ . . . . . . . . . . . . . . . . . . . 104§32 P = 0 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106§33 f%ui2�zk/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

(a) �`|(e�M[2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108(b) �`|(e�D!2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

§34 � Ω 5 g TR��s2tL . . . . . . . . . . . . . . . . . . . . . . . . . 110§35 �g$(w}�M[P�d2w� . . . . . . . . . . . . . . . . . . . . . . . . 111

(a) 9w� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113(b) 9�w� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

IV ��|lO�"}�Oy�$73. ��,P 119§36 n(wzk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119§37 n(wzkwQ�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119§38 �un&2�`��L-/3 . . . . . . . . . . . . . . . . . . . . . . . . . 120

3�) III(a) Y℄(w�`sn&23.� Joule -� . . . . . . . . . . . . . . . . . . . 121(b) �℄�`N'

IV 3�)(b) �k��5u>LkWn&gk2�f\ . . . . . . . . . . . . . . . . 180

VI g"}HVH}Oy�$7�/V 183§55 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183§56 �`�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

(a) S3 L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185(b) QCx�%f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185(c) `_}7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186(d) Mk& . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

§57 � β = constant, γ = constant sM[PChui2℄*P . . . . . . . . . . . . 188§58 � β 5 γ �)02tLw2��f%ui . . . . . . . . . . . . . . . . . . . . . 191

(a) f%ui2�zk/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191§59 �d(wyÆ�M[P2�d . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

(a) u>�" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195§60 �dy4Æ�M[P2�d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

(a) b = c = 1 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205(b) b(r) = 1 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213(c) c(r) = 1 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

§61 g$v(wy}�M[P�d2L� . . . . . . . . . . . . . . . . . . . . . . . 215(a) �`v62Y�k|&2m� . . . . . . . . . . . . . . . . . . . . . . . 215(b) �`�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216(c) `_}7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218(d) f%ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219(e) f%ui2�zk/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

§62 g$v(wy}Mev(�d2L� . . . . . . . . . . . . . . . . . . . . . . . . 222§63 �d�yaiIR2�-C" . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

VII COUETTE "��$7 231§64 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231§65 aiPu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231§66 Rayleigh ,� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231§67 Y* Couette (`M[P2%flD . . . . . . . . . . . . . . . . . . . . . . . . 235

(a) R Lagrange A�m�2�� . . . . . . . . . . . . . . . . . . . . . . . . . . 236(b) m = 0 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237(c) m 6= 0 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

§68 �g$(w�2D!�H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240(a) Ω = constant 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240(b) Ω = A+B/r2,m = 0 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . 244

§69 �d*P Couette ( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247§70 �`�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

(a) � µ > η2 s(`2M[P . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

3�) V§71 �`|(e�M[2'%rtLw2\ . . . . . . . . . . . . . . . . . . . . . . 253

(a) � σ = 0 s�mJbPu2\ . . . . . . . . . . . . . . . . . . . . . . . . . 254(b) 0bY� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256(c) ��z�\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259(d) � µ→ 1 s2nG\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262(e) � (1 − µ) → ∞ s2:j\ . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

§72 �dM[PChui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266§73 �`|(e�Mes�dH%rtL2\ . . . . . . . . . . . . . . . . . . . . . 268

(a) mJ�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269(b) � η = 12 s20bY� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

§74 g$x�\%*P(`M[Pw� . . . . . . . . . . . . . . . . . . . . . . . . 277(a) Rz�w��[� µ = 0 s2�_ Taylor 0 . . . . . . . . . . . . . . . . . 278(b) �_ Taylor 0v Ω2/Ω1 2�P�6�58��Y� . . . . . . . . . 283

VIII �0`toF�p�"(�$7 293§75 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293§76 ��`%��-}*P(`2M[P . . . . . . . . . . . . . . . . . . . . . . . . 293

(a) �`�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294(b) � σ = 0 smJbPu2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . 295(c) 0bY� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

§77 ��A�szpksg$�-\%*P(`2M[P . . . . . . . . . . . . . 299(a) (R2 −R1) ≪ 12 (R1 +R2) s2�`�� . . . . . . . . . . . . . . . . . . . . 300(b) σ = 0, µ = 0 smJbPu2\ . . . . . . . . . . . . . . . . . . . . . . . . . 301(c) Y�2ai\� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303(d) w�Y�vU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

§78 ��szpks�;x�\%Y*(`2M[P . . . . . . . . . . . . . 307(a) l�(`tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309(b) g$��s2�.tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

§79 ��szpksg$�;x�\%Y*(`2M[P . . . . . . . . . . 317(a) �`�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318(b) '%rtLw2)\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319(c) � µ > 0 smJbPu2nG\ . . . . . . . . . . . . . . . . . . . . . . . . 320(d) jw�Y�2UK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

IX g��} Couette "(�y�$7 327§80 �x��Wjnw2n(wzk�� . . . . . . . . . . . . . . . . . . . . . . . . 327§81 �j��J2n&�s��-� Couette (`2M[P . . . . . . . . . . . 328

(a) m = 0 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331(b) m 6= 0 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

§82 ��`n&TR��sg$�w�2D!�H . . . . . . . . . . . . . . . . . 333(a) Ω = constant 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

VI 3�)(b) � m = 0, Ω = A+B/r2 s��'%rtLw��`n&2M[\*I . . 334

§83 ��Jd�V(`s�-�2 Couette (`M[P . . . . . . . . . . . . . . . 337(a) m = 0 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339(b) m 6= 0 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

§84 ��5A�n&s�-� Couette (`2M[P . . . . . . . . . . . . . 339(a) Ω = 0 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

§85 n(wzk}2-� Couette (`M[P��`�� . . . . . . . . . . . . . . . 341(a) `_}7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343(b) �M[P�d�T7Me�u(s�u`|(e2�� . . . . . . . . . 343(c) '%rtL2)\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

§86 � µ > 0 smJbPu2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345(a) ��f%ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346(b) mJbPu2{\�mJJ�� . . . . . . . . . . . . . . . . . . . . . . . . 346(c) �'N[jtL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350(d) 'N[jtL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351(e) 0bY� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354(f) :jJ7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

§87 �.tLwmJbPu2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355(a) µ = −1 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

§88 �`�n&�s%��-}-�(`2M[P . . . . . . . . . . . . . . . . 361(a) �M[PT7Me�u(�ds`|(e/*�u�� . . . . . . . . . 361(b) DG�`lsz(`2M[P . . . . . . . . . . . . . . . . . . . . . . . . . 362

§89 ��n&�s*P(`M[P2w� . . . . . . . . . . . . . . . . . . . . . 364X �""}�$7� Rayleigh-Taylor y�$7 365

§90 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365§91 %��"�(w�C2mJ��`�� . . . . . . . . . . . . . . . . . . . . . . . 365

(a) �(wC_j^mj3z . . . . . . . . . . . . . . . . . . . . . . . . . . . 367§92 Y*PtL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

(a) V�`9�`_%%2��)ek"�(wtL . . . . . . . . . . . . . . 369(b) eke0f\2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

§93 ��Fi2f%ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371(a) f%ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

§94 M�`9�`_%%2"�*P(w2tL . . . . . . . . . . . . . . . . . . . . 375(a) ν1 = ν2 2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377(b) � ν1 = ν2, ρ2 > ρ1, 5 S = 0 sJM[Px� . . . . . . . . . . . . . . 378(c) � ν1 = ν2, ρ2 < ρ1, 5 S = 0 s3.2x� . . . . . . . . . . . . . . . . . . 380(d) �zz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

§95 g$2L� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384(a) ��"�(wM�`9�`_%%2tL . . . . . . . . . . . . . . . . . . 387(b) eke0f\2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

3�) VII§96 f_n&2L� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

(a) ��"�(wM�`9�`_%%�M[tL . . . . . . . . . . . . . . 390(b) "�(wM�`9�`_%%�M[tL . . . . . . . . . . . . . . . . . . . 394

§97 9�n&2L� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395§98 *P(wy2D! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

(a) �`��5aa2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397(b) Kelvin x��Y*tLw2\ . . . . . . . . . . . . . . . . . . . . . . . . . 398(c) �d�.tLw2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399(d) `_}75mJ�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400(e) Kelvin x�23.G��0bY� . . . . . . . . . . . . . . . . . . . . . . . 401

§99 *P�;2D! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404(a) Y*tL2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405(b) �.tLw2\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

XI �""}�$7 −− Kelvin -Helmhotz y�$7 409§100 �`�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409§101 ��[d�9��`2(wM�`9�`_%%2tL . . . . . . . . . . . 411

(a) Ymj3z2tL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411(b) mj3z2M[PL� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

§102 �f\2 U v Kelvin-Helmholtz M[P2L� . . . . . . . . . . . . . . . 413§103 ρ 5 U 7�f2(w}2 Kelvin-Helmholtz M[P . . . . . . . . . . . 417

(a) � ρ = ρ0e−βz 5 U = U0z/d s2%fY� . . . . . . . . . . . . . . . . . . 418§104 �`�"�Y�2Y*(w,l�M[P2�`r3 . . . . . . . . . . . . 420§105 g$L� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

(a) M�`9�`_%%2��"�(w�9��℄v�`2tL . . . . 425(b) mJ��2lD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

§106 9�n&2L� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432(a) �`��2n&L� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434(b) f_d(`��2n&L� . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

XII S"V`�$7 439§107 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439§108 Y�+x�2�zM[P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

(a) �zM[P2u:5�[mJ=

VIII 3�)(b) =;T(2RzM[P . . . . . . . . . . . . . . . . . . . . . . . . . . . 460(c) *PL� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

§112 "��n&v�wT(RzM[P2L� . . . . . . . . . . . . . . . . . 464(a) ℄�'N

3�) IXXVI g V m}�'��z�m:H_ 521

§127 �u��C2�.�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521XVII a3VM3\�� 525

§128 Y�k|&2�.mJ��w . . . . . . . . . . . . . . . . . . . . . . . . . . 525§129 wQY�k5G&2QCPx . . . . . . . . . . . . . . . . . . . . . . . . . . 526

XVIII dYaoÆ:��m:62 529§130 0%n�26WP��`r3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 529§131 v$�n5f%ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

XIX -�kCj{�t�qwRg 535§132 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535§133 �;�℄�j`_Pu2(0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

(a) V� Cn(x) 5 Sm(x) 2{% . . . . . . . . . . . . . . . . . . . . . . . . . . 536§134 �;�℄�j5yj`_Pu2(0 . . . . . . . . . . . . . . . . . . . . . . . 537

(a) QC{% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538(b) �R2�(0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540(c) 7O9{%Dy(0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540

I eg�8§1 OD(w`zk��r`�3�aa��u:�d>-(`�"#dd�`ain�^�2%�`Q℄�):G2tLw��[d(`�℄2M[P�Q�Vd>-(`^2M[�"℄�p�'pÆ(w`zkM[PPu�"av'N(w�n&}�`2B��"�Æ(`M[PPu2bD�>�n(w`zk o}2Pu��bGd(w`zkM[P2n(w`zkM[PPu�Q%,6(w`zk5n(w`zkM[Py�}2�-LDPu��'D}�Ta=eQwQui2m��5℄�M[PPu2wQH��§2 fhA9Ta�W℄�`(w`zn��a2�u��[dMe�m+>`n�2fY℄�`�s%2(0�X1, X2, · · · , Xj �[3n�2�HÆ0�>-Æ0=�QbGjn�+kLk29Æ0�mJ�(Mk&2Æ0�n�TRz2x�0szpk�Ekpk�n&5g$�66�,6>�n�2M[Ps (�℄e[2�HÆ0 X1, · · · , Xj), Ta��p609�[n�v%�`2�I��mp"sW�n��*�`�>��`��:!*��)n��:g%P�(eo��l2��M��`��tLw�Ta>n�v>`�`�M[2��J��tLw�Ta>a�M[2�~��j�℄��m 2�`M��6)%7n��M[2�T��D�2^℄�`�-��n��M[2��>�M[P/�/Y℄��`�}n�mM�� X1, · · · , Xj }`|(e2_�=V)w��[3

∑

(X1, · · · , Xj) = 0. (1)>`[3�(w`zkM[Py�2��v��7)�S2��,6�`(wn�M[Ps�Ta�Wn�Æ0}℄�`��f\2�LaÆ0DG)0��>`6fÆ0��-m bs�n�=VM[(e!n*M[(e�Ta>��LaÆ0�[bs�n��6fÆ02>`b�dÆM[�̀ |M[(e℄��>��(e%pvId%�`2/k�"+2��*F�2�aa6��HP�"+ (t;�*�HP2F�); aa�6��!/k"+}2D!"+ (t�*F�). �`��tL�wM[P*M[P2!n��`|(em�Q��Me�`x��J��tLw��d2!nm�Q`|(e2D!�`��℄�[2mJ=

2 I �Fs6>�#R)n2��TQÆ0w\��℄R2M[P}��`%2A�BS2k��jPu�`|(evd��m 2z0 (t;> kc) =�}P2�vdLa^℄z0�M[2�:r��M[P�ds�'d>`m z02 (Me2t;D!2�T76�2tL) �`=Q��aa=I�eQ��u>�"�a29�Lk�Vz0 λc = 2π/kc �[��La9}7w�I2�`Pu�6)jJI2%f�0�Me�v62�Ta6)&�`-%�

A(̟, z, θ, t) =

+∞∑

m=−∞

∫ +∞

−∞Am,k(̟, t)e

i(kz+mθ)dk (4)

(L} z ��Lk� ̟ �g2�g� θ ��AG�) uy�vd^℄R k 5 m [%2x�2n�M[P�bG��0�P�(e�℄yv6P�Ta6)&�`-%9yS5(

§3 ��1Kpi 30� Y ml (ϑ, ϕ), G��A(r, ϑ, ϕ) =

∞∑

l=0

+l∑

m=−lY ml (ϑ, ϕ)A

ml (r, t) (5)

(L} r guI2�g� ϑ 5 ϕ �y�G), uy�vdR l 5 m %2^℄x�2n�M[P�)I$+2��I��^℄tLw�Y\&�`-%9��(L2QCx��R2/ k %vId�`m Me2�x��w�I��x�2%V�-Æ0�Æ0k M�[�$xm�Æx�%V�2^℄Æ0�℄Æ>2i\�Ta6)5Q (�R2/)

A(r, t) =

∫

Ak(r, t)dk (6)�u�. (Yw%) �`2��6)#RdQCx��:r�Ta�!p6)wG�2\Y!Ts%�P�Ak(r, t) = Ak(r)e

pkt (7)L} pk ��`�[2)0� p 2wj k >s>`)0b�6�d k %2x��h�>�s%�P%g��`��=V�*Æ0 pk. ��{\sY\PC`_}7(U0�YX�2NP`_�Y℄,lIz26V`_). �.��vd$/e[2 pk, ��=^℄��\ (*[e!m), :r�Pu)\7vId�x��[ pk. �.��d pk 2mJb�`m z02�`2M[(e}%%�>ms� (X1, · · · , Xj) }&M[PowM[Po%%2_�∑(X1, · · · , Xj) �_�∑k 2�K�o�0��-m I℄! ∑, n�M[s�Q�2�`x�vII�=��-aa2 ∑k j ∑ OZ2I�j��6)�� §2 }eQ2��`|(e (Me5D!) \%2�p��d� pk 2w� p(r)k !ms�a2Y� p(i)k �-�!m�vd^℄2 k, 0� p(r)k = 0 /�/p(i)k = 0, �M[PCh2u�℄*�-��-x�2M[P�ds�TanO℄/M[P�0��d/M[P��{ p(i)k =� ∑k IM�[�:r��/M[PtLw�iDY\>s�`BS2M[P�d2x��)�D!2mJ=

4 I �Fs6§4 � BiQ0Ta$x$+2��M[PPu2\�)�}�mJP�(`2Æ0=% X1, · · · , Xj�[`|(e2o��m�iDY�s�&�Æ0Y;9�-YT0��e2�>�YT0}J�v2� Reynolds 0 R, a2H;G��

R = Lv/ν (11)L} L �n�2mJ+k� v �Me(`}2��MkLk� ν �(w2�`*Pn0��vdm 2Pu�V�[3�-m 2YT0�vddg$2�;�4\%2*P(`M[PPu�Æ0��427� R1 5 R2, g$2GMk Ω1 5 Ω2, )��`*Pn0 ν. 7edlD>`Pu��R2YTH;�η =

R1R2

, µ =Ω1Ω2, T =

4Ω21R41

ν2(1 − µ)(1 − µ/η2)

(1 − η2)2 (12)Q�� T 2[3�}2GMk Ω, Lk L, 5�`*Pn0 ν H;9YT0 4Ω2L4/ν2,�[ig$n�s�6a7 Taylor 0�bG��y�w>��29�(w�2�M[PPus�R2YTÆ0��R =

gαβ

kνd4, P = ν/κ (13)L} g ��z�Mk� d �(w�^k� β ��Ekpk� α, k 5 ν %p�w{35n0�'En05�`*Pn0�>l[32 R 67 Rayleigh 0� P � Prandtl 0��[in& H TRw2'N(ws�x)R2YTÆ0�

Q =µ2H2σ

ρνd2 (14)L} µ �n'

§4 g&tO 53. H. Alfvén, Cosmical Electrodynamics ,Internatinal Series of Monographs on Physics,

Oxford, England, 1950.

4. Lyman Spitzer, Physics of Fully Ionized Gases, Interscience Tracts on Physics and

Astronomy, No. 3, Interscience Publishers, Inc. New York, 1956.

5. J. W. Dungey, Cosmic Electrodynamics, ,Cambridge Monographs on Mechanics and

Applied Mathematics, Cambridge, England, 1958.)w=+PH0vQ%2#u℄L��6. S. Lundquist,� Study in megneto-hydrodynamics �� Arkiv för Fysik, 5, 297-347,

(1952).

7. T. G. Cowling,� Solar electrodynamics �� The Sun, chapter 8, edited by G.P.Kuiper, University of Chicago Press, Chicago, 1953

8. W. M. Elsasser,� Hydromagnetism I. A. Review � American J. Physics, 23,590-609 (1955) 5, 297-347, (1952); � Hydro,agneticsm. II. A Review, � ibid, 24,85-110 (1956)

9. G. H. A. Cole,� Some aspect of magnetohydrodynamics�� Advances in Physics,5, 452-497(1956)Q%m�Æ�)wwH�}�I2|Z�

10. S. Chandrasekhar,� Problems of stability in hydrodynamics and hydromagnetics�� Monthly Notices Roy. Astron. Soc. London, 113, 667-78 (1953)11. S. Chandrasekhar,� Thermal convection �� Deadalus, 86, 323-39 (1957)

Eddington 2BHY6�12. A. S. Eddington, The Internal Constitution of the Stars, , P.201, Cambridge,

England, 1926.

6 I �Fs6

II ��}mP�#~��Pz�%81. Bénard �{

§5 OD>��2(w�}2�M[P�dPu��;dms(w`zkM[P2^{�j�00k5ai�j2mJ�0�&aT7�%g$5n&*I2Pu�Ta�ms(w6�D�2L`�`2}��0kiD=�N�2�$xMw�^Tw2℄�(w�`J72^{℄�Px�7r�Taj�>��2(w�}2M[PT7�.#u�jJ�u2[i�C�5CD0#Rd,6>`Pu��Q0�=�,62�Y℄g$5n&L�2J)�2Pu�g$5n&*I=�)J20V},6�§6 ��|�Z|,6�`9�2(w��!>��DG�`�Ekpk�6>�Ekpk7�Ekpk��Vdw�352Gk'�>�(wUX�(wq�>�>�qX��2(w*��℄�M[P�Vd>��M[P��%(w=℄�:%�2}).9M[P��%(w2>�6�}

�*a6℄*P,u��>�TasH:G2�Ekpk��M[PQ�\`Y\/!�[b�(w�M[)��[G��d2J�w�� 1900 �2 Bénardw��v(��Q℄�V Count Rumford(1797) 5 James Thomson (1882) l��2�� §18 }�Ta=m+ Bénard w�5Law��>lTaHQ�w>-w�N�2��wC|Æ��wI��M[P�d\`��zY\℄/!�`�_b2�Ekpk�/!�_EkpksW\�d2�`�℄Me2u>YzmJ��M[PQ�sAQ�d2�(w�}#d�n�2u>�0�Rw�C|2u�>-u>f/��6TSN7Q+`G�"��1. (� Bénard �H892��(wM[��Q�\�)I�w2iDwX� Lord Rayleigh �`2DH}eQ2�Rayleigh2y�ms�w>��2(w�2M[P�[d�`YT2Æ0b�

R =gαβ

kνd4 (1)L} g ��z�Mk� d �(w�2Hk� β(=| dT/dz |) �:G2"�2�Ekpk�

α, k 5 ν %p�(w2w{35n0�'En05�`*Pn0� R [37 Rayleigh 0�Rayleigh j��ms�M[P�[�� R /!�`�_b Rc s�d�:r��2iDPu��"a09�[ Rc? >�Q0=�\�2��Pu�§7 %�*�Bfh8��.�[i §6 }m+2�M[PPu�V�ek5Ek�f\2*P(w(`2`zk�u��QVTa=eQ>-wQ��

7

8 II ZOD`([P`DNegy 1. Bénard fZ

� 1. tX�4 Bénard v?� Bénard w�9:4�4>w��W(w2ek�Ar xj(j = 1, 2, 3) 2(0� uj(j = 1, 2, 3) � xj ��2Mk%��5��s��)Ta=&{5�[IRd Cartesian 3�(a) B4/&VTa℄

∂ρ

∂t+

∂

∂xj(ρuj) = 0 (2)>`��m�x�C�:7��`$/w{IvajJ{%�6)/*

∂

∂t

∫

V

ρdτ = −∫

V

∂

∂xj(ρuj)dτ (3)L} dτ(= dx1dx2dx3) �w{�t�h� Gauss [i�`o2w{{%6d� V 2N`�-mj S I2{%�:r

∂

∂t

∫

V

ρdτ = −∫

S

ρujdSj (4)L} dSj m��`jmj{t dS f_2|�oa2x6d dS.[| dSj j�v63

dSlm �?d2�a2%6d dS �Wj�jÆ2�Lj{�] �� (4) m�Æ�`�w��%��`�[�uwÆx2f\MsMk&�Y�k2�

§7 ([h#}�FnU 9(b) s�&V�5Q�`��\`�℄Y�Æ\�wIz3 Pij , aTR� xj ��2�Aj{I�j{tf_d xi. >`Iz�[�j(w}If"�2M<�2�J;V

eij =1

2

(

∂ui∂xj

+∂uj∂xi

)

(7)(w`zk2�`wQ�[� Pij 5 eij P��:rPij = ̟ij + qij;klekl (8)L} ̟ij ��`v63��tL eij = 0 w� Pij }�d ̟ij , qij;kl ��`DR3�vd��P(w�vdWjn2$/g$5��v� (8) G��f2�>�{ ̟ij 5

qij;kl ��P3�:rTa℄̟ = −pδij (9))�

qij;kl = λδijδkl + µ(δikδjl + δilδjk) (10)L}f p, λ 5 µ � xi 2$/(0�&�� (9) 5 (10) �3v� (8), Ta℄Pij = −pδij + 2µeij + λδijekk (11)& p [37Y℄Ifs� xi [2��Psz�:r

Pii = −3p = −3p+ 2µeii + 3λekk (12)R p 2[3�℄λ = −2

3µ (13)

Pij = −pδij + 2µeij −2

3µδijekk (14)�>`��}2 µ �*Pn0�a�Ar2$/(0�� (14) }QUd µ 2�[37*PIz�R pij m��Ta℄

pij = µ

(

∂ui∂xj

+∂uj∂xi

)

− 23µ∂uk∂xk

δij (15)vd��6s(w�*PIz3℄l)�2G�pij = µ

(

∂ui∂xj

+∂uj∂xi

)

(16)IRIz Pij , Ta6)5Q�`2(w`zk��Ta℄ρ∂ui∂t

+ ρuj∂ui∂xj

= ρXi +∂Pij∂xj

(17)L} Xi �TR�(wI2$/"z2C i %��>`��}�3 Pij , Ta℄ρ∂ui∂t

+ ρuj∂ui∂xj

= ρXi −∂p

∂xi+

∂

∂xj

{

µ

(

∂ui∂xj

+∂uj∂xi

)

− 23µ∂uk∂xk

}

(18)

10 II ZOD`([P`DNegy 1. Bénard fZvd*Pn0 µ �)026s(w�� (18) )\7ρ∂ui∂t

+ ρuj∂ui∂xj

= ρXi −∂p

∂xi+ µ∇2ui (19)>� Navier − Stokes &V�2u�G�� (17) m�`�C�>6)w�$/2N`w{ V I2{%'Q�Ta��

∫

V

(

ρ∂ui∂t

+ ρuj∂ui∂xj

)

dτ =

∫

V

ρXidτ +

∫

V

∂Pij∂xj

dτ (20)�!vOo2C�`{%jJ%�{%��sIR Gauss [iv`o2C��jJfh�/∫

V

{

ρ∂ui∂t

− ui∂

∂xj(ρuj)

}

dτ +

∫

S

ρuiujdSj =

∫

V

ρXidτ +

∫

S

PijdSj (21)R�P��6)�� (21) Oo2C�`{%�∫

V

(

ρ∂ui∂t

+ ui∂ρ

∂t

)

dτ =∂

∂t

∫

V

ρuidτ (22):r�� (21) 6)59∂

∂t

∫

V

ρuidτ =

∫

V

ρXidτ +

∫

S

PijdSj −∫

S

ρuiujdSj (23)>��m�Æ�`�w��%(w2�`�[w{ V Æ`2f\) ui(��/�/v i {5), �w{ V IjJ{%�/

1

2

∫

V

ρ∂

∂tu2i dτ +

1

2

∫

V

ρuj∂

∂xju2idτ =

∫

V

ρuiXidτ +

∫

V

ui∂Pij∂xj

dτ (24)�!�n�bGjw�� (20) 2 (23) 2fh�6)��

1

2

∫

V

ρ∂

∂tu2i dτ

=

∫

V

ρuiXidτ +

∫

S

uiPijdSj −1

2

∫

S

ρu2iujdSj −∫

V

Pij∂ui∂xj

dτ

(25)>`��eQÆ�%��uw V Æ`�2f\

§7 ([h#}�FnU 11= −pejj + 2µe2ij −

2

3µ(ejj)

2 (27)`o2C��−pejj = −p

∂uj∂xj

=p

ρ

(

∂ρ

∂t+ uj

∂ρ

∂xj

)

=p

ρ

dρ

dt(28):r�>`2w{%m�Vd(ws[BS2Æ�"�� (27) }kg2��

Φ = 2µe2ij −2

3µ(ejj)

2 (29)�[�m��(w0t}*PBS26�2�-�Ms� Φ �w�Q[2�--Ts� Φ 6)m�96"G�Φ = 4µ(e212 + e

223 + e

231) +

2

3µ[(e11 − e22)2 + (e22 − e33)2 + (e33 − e11)2] (30)vd6s[(w ejj = 0, I2 Φ �

Φ = 2µe2ij (31)

(d) |�&VTa$x/QÆVx5`�C'Q2�P��5�`��kw2��=m��C�Ta='*�V��C6)'Qy�M[PsTaV�2'��TÆ�`B�2)0�(w�Aw{Æ2�6)57ǫ =

1

2u2i + cV T (32)L} cV �6-U�� T �Ek�h��w{ V Æ(w}a�2�/m�℄

∂

∂t

∫

V

ρǫdτ =� V 2`_ S I Pij 2Ts<+� V 2(w0tI"zTs−�! S 2'�M<−�!x�`℄! S 2�v(M<

=

∫

S

uiPijdSj +

∫

V

ρuiXidτ +

∫

S

k∂T

∂xjdSj −

∫

S

ρǫujdSj

(33)

L} k �(w2'�n0�IR�� (25),(27),(28) 5 (29), 6)&�� (33) `oC��:57∫

S

uiPijdSj =1

2

∂

∂t

∫

V

ρu2i dτ +1

2

∫

S

ρu2iujdSj −∫

V

ρuiXidτ−

−∫

V

p∂uj∂xj

dτ +

∫

V

Φdτ (34)o�C=�5CD�2!��G��∫

S

k∂T

∂xjdSj =

∫

V

∂

∂xj

(

k∂T

∂xj

)

dτ (35))�

−∫

S

ρǫujdSj = −∫

S

ρ(1

2u2i + cV T )ujdSj

= −12

∫

S

ρu2iujdSj −∫

V

∂

∂xj(ρujcV T )dτ

(36)

12 II ZOD`([P`DNegy 1. Bénard fZ~u)I��℄

∫

V

∂

∂t(ρcV T )dτ =

∫

V

∂

∂xj

(

k∂T

∂xj

)

dτ −∫

V

p∂uj∂xj

dτ+

+

∫

V

Φdτ −∫

V

∂

∂xj(ρcV Tuj)dτ

(37):7>`��vd$/�uwe�9u2�Ta�[℄∂

∂t(ρcV T ) +

∂

∂xj(ρcV Tuj) =

∂

∂xj

(

k∂T

∂xj

)

− p∂uj∂xj

+ Φ (38)IR�P��6)&)I��)\7ρ∂

∂t(cV T ) + ρuj

∂

∂xj(cV T ) =

∂

∂xj

(

k∂T

∂xj

)

− p∂uj∂xj

+ Φ (39)�� (2), (14), (17),(29) 5 (39) �(w`zkwQ��aaY\)(w2(e��T7~N�vd��V�2(w�Ta6)5Qρ = ρ0[1 − α(T − T0)] (40)L} α �w{35n0� T0 � ρ = ρ0 s2Ek�

§8 Boussinesq m�I�V�'(w`zk��s�Y℄v)0�t;>vB32�n0 (µ, cV , α 5k), TQ�W�:r��'Q2��Fi℄*2��0� Boussinesq�zeQ2��^{w�&;�wQ��6))\��Ekf\BS2ek5Lan02f\hh��2s�Q�>�tG��>-tLw�)\��)(w2w{35n0�`%>`�w7��2�vdx)V�*2Yw5�w�w{35n0 α ^[2�3� 10−3 *10−4. vd/! 10o 2Ekf\�Ta>�ek2f\X{� 1%. Lan02f\ (vId$xeQ2ekf\) �[��2��>�f\%��)6)O��℄�`��2r"��`��2xz� ρXi }2 ρ2f\�O��>�:7V δρXi = α∆TXi(L} ∆T �CQ�2Ekf\2Lk) BS2�Mk6)�?2�r0�aand�`��}2P�Mk��:r�TÆ�"z�}�Ta6)&Q��`��}2 ρ �T)0�� §9 }Ta=Ts�vdm 2�M[PPu�YT Boussinesq nG�:7�Ta=��>�tLw�R §7 }Fi2��6)'Q�2%�`��)Bousinesq nG7wX2��Q℄�℄�2�aaeQÆiD��P oj��/j-2wX�h�)Im+�Ta&�P�� (5) R

∂uj∂xj

= 0 (41)�z�:7�� (5) 2Ooj`oU� α R2�h� uj 2>`}7�*PIz32m��pij = µ

(

∂ui∂xj

+∂uj∂xi

)

(42)

§9 sChnU 13L}wd�iV�Ta6)& µ �T)0[i��>-nG2K$Æ��`�� (17) f9∂ui∂t

+ uj∂ui∂xj

=1

ρ0

∂p

∂xi+

(

1 +δρ

ρ0

)

Xi + ν∇2ui (43)L} ν(= µ/ρ0) m�(w2�`*Pn0� ρ0 ��j�2�"Ek T0 w2ek�uoδρ = −ρ0α(T − T0) (44)w`,6'�� (39), Ta& cV 5 k �T)0�&aa�*0%/2"`�Ta6)O��`o2� −pdivu, *P-�� Φ ��O��:7�h�� (43) 5 (44), �(Mk2�� [α∆T | X | d] 12 , L} d �Cn�2�`Lk�:r Φ �vd'��2�7

µα | X | d/k (45)>`Ub�vd�)2�w�� d ∼ 1cm 5 | X |∼ g(:�zBS2�Mk) � 10−7 t10−8. �>-tLw�'�� (39) )\7

∂T

∂t+ uj

∂T

∂xj= κ∇2T (46)L} κ(= k/ρ0cV ) �(w2'En0�� (41),(43),(44) 5 (46) �� Boussinesq nGw2wQ��

§9 �+N*8�,6�`Y�29��w��aDGÆ�`�EkpkoY℄(w�`�:rP�(e�uj ≡ 0 T ≡ T (λjxj) (47)L} λ = (0, 0, 1) �f_��2�A|��Y�`s�(w`zk��j�{sz%�V�� (2�� (18))∂p

∂xi= ρXi = −gρλi (48)�u�L}

ρ = ρ0[1 + α(T0 − T )] (49)L} ρ0 5 T0 ��>�`_I2(wek5Ek�Ek2�u��∇2T = 0 (50)�5Q�� (39) s��Me�� (50) 2}7w�$x�[Æ(w2'�n0 k �jEk TY�2)0��>�tLw>`�[�;i2�j�`2Pu�I2�� (50) 2\�

T = T0 − βλjxj (51)

14 II ZOD`([P`DNegy 1. Bénard fZβ �DG2�Ekpk�j\vI2ek%��

ρ = ρ0(1 + αβλjxj) (52)h�>` ρ 2m�� (48) 6){%�/p = p0 − gρ0(λixi +

1

2αβλiλjxixj) (53)�� (47),(51),(52) 5 (53) L0℄�`�� uj m��`(ew2Mk�f\Æ2Ek%�7

T ′ = T0 − βλjxj + θ (54)JJ�� δp m�sz%�2f\��Ta=/Q�u>��`(e2�`��2P\G��z,6) Boussinesq nG7wX2Pu��!O��0%�`}2�R5lWR�� (43) 5�� (46) eQ (2�� (62))∂ui∂t

= − ∂∂xi

(

δp

ρ0

)

+ gαθλi + ν∇2ui (55))�∂θ

∂t= βλjuj + κ∇2θ (56)Mk&��DGY�k�∂ui∂xi

= 0 (57)℄�2��TR Boussinesq �[�wFi2��Q��Ta�6)/*j�� (55)-(57)�2��Fi2��∂ρ

∂t+ uj

∂ρ

∂xj= −ρ∂uj

∂xj, (58)

∂T

∂t+ uj

∂T

∂xj= κ∇2T − p

ρcV

∂uj∂xj

+1

ρcVΦ (59))�

ρ∂ui∂t

+ ρuj∂ui∂xj

= − ∂p∂xi

+∂

∂xj

{

µ

(

∂ui∂xj

+∂uj∂xi

)

− 23µ∂uj∂xj

}

− gρλi (60)�Y℄�`2(e∂ρ

∂xj= λjρ0αβ (61)V�`Ek θ BS2ekf\V0w��eQ

δρ = −αρθ = −αρ0(1 + αβλjxj)θ (62):r�� (58) 2OoQ�2�R��℄:3 α, `oI2�Y℄>`:3�uo�Vd α ∼ 10−3 − 10−4, Ta6)O��Oo2� λjujαβ, �'Q u 2Y�kPx�IR>`�w��$9tLw� u }-�� Φ ��R2�Ta/* θ 2��∂θ

∂t= βλjuj + κ∇2θ (63)

§9 sChnU 15aj�� (56) ��,6w�`�� (60), �zTa��6)& µ �T)0[i�:7� µ 2f\2R0�[� αδρ, �aa>) uj Q�s�6)O�>`R02f\�Ta/*\��∂ui∂t

= −1ρ

∂

∂xiδp+

µ

ρ∇2ui + gαθλi (64))��>`��}�~�6)�Q� ρ 2��5I ρ0. :r�6)/Q`j) Boussinesq�[7wX2��H�l*�� (55)-(57), �� (55) }�!IRS3

curlk = ǫijk∂

∂xj(65)* k %2��!� δp/ρ0. �

ωi = ǫijk∂uk∂xj

(66)m�R�℄∂ωi∂t

= gαǫijk∂θ

∂xjλk + ν∇2ωi (67)v>`��u�gk�℄

∂

∂tǫijk

∂ωk∂xj

= gαǫijkǫklm∂2θ

∂xl∂xjλm + ν∇2ǫijk

∂ωk∂xj

(68)IRPxǫijkǫklm = δilδjm − δimδjl (69)��

ǫijk∂ωk∂xj

= ǫijkǫklm∂2um∂xj∂xl

=∂

∂xi

(

∂uj∂xj

)

−∇2ui = −∇2ui (70)bG��ǫijkǫklm

∂2θ

∂xj∂xlλm = λj

∂2θ

∂xj∂xi− λi∇2θ (71):r�� (68) f9

∂

∂t∇2ui = gα

(

λi∇2θ − λi∂2θ

∂xi∂xj

)

+ ν∇4ui (72)��R λi >)�� (67) 5 (72), /∂ζ

∂t= ν∇2ζ (73))�

∂

∂t∇2w = gα

(

∂2θ

∂x2+∂2θ

∂y2

)

+ ν∇4w (74)L}ζ = λjωj , w = λjuj (75)

16 II ZOD`([P`DNegy 1. Bénard fZ�R5Mk2 z− %�Ta℄�� (2�� (56))∂θ

∂t= βw + κ∇2θ (76)�� (73),(74) 5 (76) �V�20%�`��H�>`��H2\V�PC-`_}7��=Rv�m+>-}7�

(a) -�m�(w�u��j z = 0 5 z = d \%��>�`�jI�-`_}7Y\/*PC��-mj2Px09�Y\�{θ = 0, w = 0, � z = 0, d (77):7�mj z = 0 5 z = d DG)E��Yf\2�~��>-mjI�Mk2�%Y\!m��La%�-�!m��[d z = 0, d 2mjPx�Ta=%�b�-mj��aI`YX�2NPmj5Yl�IzTR26Vmj��z,6NPmj��>`mjI(w�`YX��/�/Mk2�% w, 5Mk29�% u 5 v e7��:r

u = 0, v = 0, w = 0, �NPmjI (78):7vdmjI2^℄ x 5 y, >`}7Y\PC�w�P��∂u

∂x+∂v

∂y+∂w

∂z= 0 (79)/Q

∂w

∂z= 0 �NPmjI (80)6VmjI2`_}7�Pxz = Pyz = 0 (81):7��P� −pδij Y℄A�%�}7 (81) �d*PIz2% pxz 5 pyz 7�� 1

pxz = µ

(

∂u

∂z+∂w

∂x

)

, pyz = µ

(

∂v

∂z+∂w

∂y

)

(82):7vd�-mjI^℄2 x 5 y, w !m�:rw (82) /*∂u

∂z=∂v

∂z= 0 �6VmjI (83)w�P�� (79) v z 20%�/

∂2w

∂z2= 0 �6VmjI (84)�dR2f_% ζ, `_}76)wI`2Y��'/*�:7ζ =

∂v

∂x− ∂u∂y

(85)

1 Ub>(+'~8 (81) JSeZB 3nkÆ?ÆSPA�Æs1Æ�{{3��+J (3D 10 1Æ §94e). 1�#b�A&?�+JÆ�~8 (81) Z℄S��+1&�a�eB 3nk�

§10 ��1Kp� 17w�� (78) 5 (83) /ζ = 0 �NPmjI (86))�∂ζ

∂z= 0 �6VmjI (87)

§10 sy1a

18 II ZOD`([P`DNegy 1. Bénard fZRYT2flD�� (92)-(94) 2\UK�e�j��A[L] = d, [T ] = d2/ν (97)�a = kd, σ = pd2/ν (98)m�z05s%)0��Ta� x, y 5 z m�R+k d 2:�Am�2Wj�� (92)5 (93) f9

(D2 − a2)(D2 − a2 − σ)W =(gα

νd2)

a2Θ (99))�(D2 − a2 − Pσ)Θ = −

(

β

κd2)

W (100)L} D = d/dz )� P(= ν/κ) �(w2 Prandtl 0� [W 5 Θ �℄aa�)2T�aaY℄h�� (97) 2�AjJYT\�] jr℄�2`_}7�Θ = 0, W = 0, � z = 0, 1 (101))�

DW = 0, � z = 0, 1, 0��`�-mj�NP2 (102)t;DW = 0, � z = 0, D2W = 0, � z = 1 (103)0�>��-mj�NP2�X�mj�6V2��!�� (99) 5 (100) \%! Θ, /

(D2 − a2)(D2 − a2 − σ)(D2 − a2 − Pσ)W = −Ra2W (104)L}R =

gαβ

κνd4 (105)� Rayleigh 0�>��`�u Θ 26"2��

(a) X�U{+E��0�$x/*�� (99) 5 (100) 2m\�Ta6)�!0w��[Mk29�%�v>`PujJ{\�h��`(0 φ 5 ψ, & u 5 v m�9G�u =

∂φ

∂x− ∂ψ∂y

, v =∂φ

∂y+∂ψ

∂x(106):r

−∂w∂z

=∂u

∂x+∂v

∂y=∂2φ

∂x2+∂2φ

∂y2= −a2φ (107))�

dζ =∂v

∂x− ∂u∂y

=∂2ψ

∂x2+∂2ψ

∂y2= −a2ψ (108)

§11 Negy`�� 19:rφ =

1

a2∂w

∂z, ψ = − d

a2ζ (109)Ta℄

u =1

a2

(

∂2w

∂x∂z+ d

∂ζ

∂y

)

=i

a2(axDW + aydZ) exp[i(axx+ ayy) + σt] (110))�

v =1

a2

(

∂2w

∂y∂z− d∂ζ

∂x

)

=i

a2(ayDW + axdZ) exp[i(axx+ ayy) + σt] (111))I~n u 5 v jMk2�%5R\%2��℄Fi/3�aa�>`Pum P2�u�

§11 {�':�y_�Ta=��vd�lD2Pu�M[P2Chui�℄*2��>�8 σ �w0��M[P2`|(eV σ = 0 mJ��G = (D2 − a2)W (112))�

F = (D2 − a2)(D2 − a2 − σ)W = (D2 − a2 − σ)G (113)h�� (99), ��-mjI}7 Θ = 0, �dF = 0 � z = 0, 1 (114)IR F , W PC2��

(D2 − a2 − Pσ)F = −Ra2W (115)��TaTsvd^℄Q2 R, σ �w0�� (115) >) F ∗ (F 2x[

20 II ZOD`([P`DNegy 1. Bénard fZ�!32%�{%�/∫ 1

0

WD2G∗dz = −∫ 1

0

DWDG∗dz =

∫ 1

0

G∗D2Wdz (120)\u,6*`_}7 W = 0 5�� DW = 0 � G∗ = (D2 − a2)W ∗ = 0(��dm 2�-mj�NP2�6V2) {%QY2�%!m�:r

∫ 1

0

WF ∗dz =

∫ 1

0

G∗{(D2 − a2)W − σ∗W}dz

=

∫ 1

0

| G |2 dz − σ∗∫ 1

0

W (D2 − a2)W ∗dz(121)�u}R%�{%�℄

∫ 1

0

WD2W ∗dz = −∫ 1

0

| DW |2 dz (122)~u�� (118), (121), 5 (122), /∫ 1

0

{| DF |2 +a2 | F |2 +Pσ | F |2}dz−Ra2∫ 1

0

{| G |2 +σ∗[| DW |2 +a2 |W |2]}dz = 0 (123)>`��2w�5Y��[%p7��VY0�%7�/im(σ)

{

P∫ 1

0

| F |2 +Ra2∫ 1

0

[| DW |2 +a2 |W |2]dz}

= 0 (124)��vd℄�2Q2 R > 0, WQ/Æ2�Q2�:rim(σ) = 0 (125)>msvd R > 0, σ �w0�vd>`Pu�M[P2Chui℄*�

§12 l�v��{8�Yr`�xpx�|:7vd^℄Q2 Rayleigh 0 (�>�vd^℄2�Ekpk), σ �w0�:r�wM[*M[2!n�[�d�M[2(ew�:r��u`|(e2��!# σ = 0/*2�Ta℄(D2 − a2)2W =

(gα

νa2Θ

)

(126)5(D2 − a2)Θ = −

(

β

κd2)

W (127)�>-��\%! Θ, /(D2 − a2)3W = −Ra2W (128)p62>`��2\��PC`_}7

{

W = 0, (D2 − a2)2W = 0 � z = 0, 1DW, tD2W = 0 � z = 0, 1, h��-mj�[� (129)

§13 Ip 21!��j�� (126) 5�� (127) \%! W , /(D2 − a2)3Θ = −Ra2Θ (130)`_}7�

Θ = 0, (D2 − a2)Θ = 0 � z = 0, 1D(D2 − a2)Θ tD2(D2 − a2)Θ = 0 � z = 0, 1h��-mj2Px�[ (131)�I`2v� (�� (128) 5 (129) t;�� (130) 5 (131)) }�Ta℄+R20%��Y\PC+``_}7�=`� z = 0, =`� z = 1; �.tLw�Ta>WR�:7j��m 2 R b�Pu^℄��\�:r�Ta℄�d R 2mJbPu�M[PPu{\=��R2��s~2�vd�`e[2 a2, TaY\�[ R 2J9b�h� a2, >/Q2�%b��Q�M[P2�_ Rayleigh 0�

§13 o

22 II ZOD`([P`DNegy 1. Bénard fZ~�w�� (136) 5�� (137), 6)/Q∫ 1

0

GiGjdz = 0 0� i 6= j (138)Oh�J`?p��bP Gj �� i = j, �� (136) eQRja

2

∫ 1

0

G2jdz =

∫ 1

0

[(DFj)2 + a2F 2j ]dz (139)a& Rj m�9�`Q2[{%2U��w�Ta=ms�`o��RAw2mJ(0m�s�eQ2v�

R =

∫ 1

0[(DF )2 + a2F 2]dz

a2∫ 1

0 G2dz

=I1a2I2

(140)�℄MePx�7ÆTs>�MeP�� R �mh�$/(0 W 2�� (140) /Q2mJb� W (TÆPC℄_5�P) V�PC (134) eQ2`_}7�W δR �� W ℄�`%2f\ δWs R 2f\� δW j W 2`_}7�0S2��>�δW = 0, δF = 0 � z = 0, 1DδW = 0, tD2δW = 0 (141)�� z = 0, 1 [�-mj2Px[�h�� (140),

δR =1

a2I2

(

δI1 −I1I2δI2

)

=1

a2I2(δI1 −Ra2δI2) (142)L}

δI1 = 2

∫ 1

0

(DFDδF + a2FδF )dz (143))�δI2 = 2

∫ 1

0

[(D2 − a2)W ][(D2 − a2)δW ]dz (144)�vId W f% δW 2 I1 5 I2 2f%�%�{%nb\J�--��δI1 = −2

∫ 1

0

δF (D2 − a2)Fdz (145))�δI2 = 2

∫ 1

0

W (D2 − a2)2δFdz = 2∫ 1

0

WδFdz (146):rδR = − 2

a2I2

∫ 1

0

δF{(D2 − a2)F +Ra2W}dz (147)

§13 Ip 23w�� (147), 6)/Q δR = 0, 0�(D2 − a2)F = −Ra2W (148):��$a8��`d�h�-�m�D��K0+ δF [= (D2 − a2)2δW ], �^ δR = 0 v&V (148) DÆU?�WbP F �Y�[: R �DÆ��h�DA��)ITsN�ÆR�� (140) eQ2mJb2MePx��Ta=ms�J92 Rb�w��`�%b�$x'*(0 Gj G9�`QC��Ta=�[aa�Q�\2�}/∫ 1

0

GiGjdz = δij (149)�G = (D2 − a2)W (150)L} W �$/℄_2�(0�oPCMe2`_}7��[ G 6)R(0 Gj 2wQ�-%�:rG =

∞∑

j=1

AjGj (151)L}Aj =

∫ 1

0

GGjdz (152)�W G ��\2��1 =

∫ 1

0

G2dz =

∞∑

j=1

∞∑

k=1

AjAk

∫ 1

0

GjGkdz =

∞∑

j

A2j (153)~n* G 2-%� (151), ℄W =

∞∑

j=1

AjWj (154))�F =

∞∑

j=1

Aj(D2 − a2)2Wj =

∞∑

j=1

Aj(D2 − a2)Gj (155)w�� (155), /Q

(D2 − a2)F =∞∑

j=1

Aj(D2 − a2)3Wj = −a2

∞∑

j=1

AjRjWj (156)

24 II ZOD`([P`DNegy 1. Bénard fZJJ>`��>) F , �� z 2�3ÆjJ{%�/

∫ 1

0

F (D2 − a2)Fdz = −a2∞∑

j=1

AjRj

∫ 1

0

WjFdz

= −a2∞∑

j=1

AjRj

{ ∞∑

k=1

Ak

∫ 1

0

Wj(D2 − a2)Wkdz

}

= −a2∞∑

j=1

∞∑

k=1

AjAkRj

∫ 1

0

(D2 − a2)Wj(D2 − a2)Wkdz

= −a2∞∑

j=1

∞∑

k=1

AjAkRj

∫ 1

0

GjGkdz

= −a2∞∑

j=1

A2jRj

(157)

:r∫ 1

0

F (D2 − a2)Fdz = −∫ 1

0

[(DF )2 + a2F 2]dz = −a2∞∑

j=1

A2jRj (158)t;�IR�� (153), 6)5Q

∫ 1

0

[(DF )2 + a2F 2]dz − a2R1 = a2

∞∑

j=1

A2jRj −R1

= a2∞∑

j=2

A2j(Rj −R1)(159)~��JJ2{5m7��:r

R1a2 ≤

∫ 1

0

[(DF )2 + a2F 2]dz (160)L}6/�oj�� Aj = 0, j = 2, · · · s9u� �&�iG (160) ;�_��w F J` R1 2o�:"��(b) ��0+h;"a6)�!R Θ m�2mJbPu/*C�f%ui��

G = (D2 − a2)Θ (161)Ta& Θ 2�u��59(D2 − a2)2G = −Ra2Θ (162)a2`_}7�

Θ = 0, G = 0 � z=0,1DG = 0, tD2G = 0 (163)��d� z = 0, 1 [�-mj2Px�

§14 Ip `D#}�� 25�'dm mJb Rj 2\Rwj j p�R Gi('d Ri) >) Θj PC2��J� z 2�3Æ{%�/*∫ 1

0

Gi(D2 − a2)2Gjdz = −Rja2

∫ 1

0

Θj(D2 − a2)Θidz (164)�!%�{%�`of9

Rja2

∫ 1

0

[(DΘi)(DΘj) + a2ΘiΘj]dz (165),6*`_}7 Θ = 0,� z = 0, 1, {%QY2�%!m�bG��32%�{%\J�/

∫ 1

0

GiD2(D2 − a2)Gjdz = −

∫ 1

0

DGiD(D2 − a2)Gjdz

= +

∫ 1

0

D2Gi(D2 − a2)Gjdz

(166)�,62 G 2`_}7�{Q2�%!m�:r�Ta/*Rja

2

∫ 1

0

[(DΘi)(DΘj) + a2ΘiΘj ]dz =

∫ 1

0

[(D2 − a2)Gi][(D2 − a2)Gj ]dz (167)w�� (167) /Q∫ 1

0

[(D2 − a2)Gi][(D2 − a2)Gj ]dz = 0 � i 6= j (168))� (� i = j)Rj =

∫ 1

0[(D2 − a2)Gj ]2dz

a2∫ 1

0 [(DΘ)2 + a2Θ2]dz

(169)��6)msR)Iv�eQ2 R �℄MePx�uo`oAw2�%� R 2J%mJb�§14 o6)w�� (94) /Q�:7�� σ = 0 s�

(D2 − a2)Z = 0 (170)>`��^℄PC Z 2`_}7 (2�� (96)) 2��\� [�Me}7w� Z 2!m6)w�� (73) /Ql�FiP2Ts�:7�� ∂ζ/∂t = 0,∇2ζ = 0, 0��&[2� Laplace ��^℄��\�a2�'0�*X2`_I!m�] :r�>s�V�� (110) 5 (111) eQ2Mk9�%f9u =

1

a2∂2w

∂z∂x, v =

1

a2∂2w

∂z∂y(171)mp��

w = W (z) cos axx cos ayy (172)

26 II ZOD`([P`DNegy 1. Bénard fZL}a2 = a2x + a

2y (173)�

u = −DWa2

ax sin axx cos ayy, v = −DW

a2cos axx sin ayy (174)��,6�Au�Æ(w�*P-�2�"M

§15 X��fZ`�B� 27�� (177) 5�� (182) ��eQρν

4a2d2

∫ 1

0

G2dz =ρκν2

4gαβa4d6

∫ 1

0

[(DF )2 + a2F 2]dz (184)t;59R =

gαβ

κνd4 =

∫ 1

0 [(DF )2 + a2F 2]dz

a2∫ 1

0G2dz

(185)>`��j�� (140) ��a'QÆ� §13 ,62�`f%ui2C�`�:r�Ta6)eQ>`f%ui2aiÆ'0w�tz"�f��.�?�Æ-_��ty/d"��qf3�D(�pq��0I�Æ�|W{l�§15 xpx�|��Mz��Tal*� §12 }Rv�m+2mJbPuI�Ta=/*0w=�tL2\�(a) �``_mj�6V2��>�tLw�

W = (D2 − a2)2W = 0, D2W = 0 � z = 0, 1 (186)(b) �``_mj�NP2�>s�

W = (D2 − a2)2W = 0, DW = 0 � z = 0, 1 (187)(c) �``_mj\� (0 y = 0) �NP2�!�``_mj�6V2�>s

W = (D2 − a2)2W = 0, � z = 0, 1DW = 0 � z = 0, D2W = 0 � z = 1 (188)ww��Æ2v�[w�6/P2GkY'�tL (b) ��B�J2��ImjDG6V)e��s�tL (c) �℄�2�tL (a)(�zM Rayleigh ,6), 0��A�|w��j℄��)mp2"7}7w (r0��w�-K�2�wI6`); ��>�tLiD℄B��:7a^�`~�\�:�℄�-UI�

(a) 2Z-��>�tLw�`_}7 (186) �{W = D2W = D4W = 0 � z = 0, 1 (189)V>`��M W (67 (128)) PC�6)�Q D6W = 0, � z = 0, 1. V�� (128) v z 0%�u�TaO//Q D8W = 0� z = 0, 1. �!v�� (128) 2j��0%�6)3�/*��`_Iv W 2$0R'0!m�:r

D(2m)W = 0 � z = 0, 1 5 m = 1, 2, · · · (190)

28 II ZOD`([P`DNegy 1. Bénard fZwr6[�{2��[�W = A sinnπz (n = 1, 2, · · ·) (191)L} A ��`)0� n ��`N0�&>`\�3�� (128), /Q �&V

R = (n2π2 + a2)3/a2 (192)vde[2 a2, � n = 1 s R Q�J%b�:rR =

(π2 + a2)3

a2(193)JJ>`�n2%3��vd%da2^℄ Rayleigh 0�z07 a 2�`=�M[2��

Rayleigh 06d (193) �eQ2bs�>-�`=�`|M[2�� Rayleigh 0/! (193)eQ2bs��2�`=�M[2�vdM[P�d2�_ Rayleigh0��[d}7∂R

∂a2= 3

(π2 + a2)2

a2− (π

2 + a2)3

a4= 0 (194)t;

3a2 = π2 + a2, a2 = π2/2 (195)I2 R b�Rc =

(32π2)3

12π

2=

27

4π2 = 657.5 (196)�`|M[PQ�2�`=Vz+

λ =2π

k=

2π

ad (197)mJ�

(b) `Pu2�`�-�j2v6P�7�eS2�& z 2uI��d�`�j\%2}%�jI�:r�(w=M�u� z = ± 12 \%�TaY\p6)w��2\(D2 − a2)3W = −Ra2W (198)aPC2`_}7�

W = DW = (D2 − a2)2W = 0 � z = ± 12 (199)�z��*�h�S3 (D2 − a2)3 2$DPx�)�� z = ± 12 Y\PC2`_}7�� (198) 2m\%7V�2�b�$D\5OD\�w�.P,6�J92(e�Y℄%XI2$D\��z2�e��℄�`%XI z = 0 2OD\�~�� (198) �2�\6)m�7)wG�\2W�W = e±qz (200)

§15 X��fZ`�B� 29L} q2 ��(q2 − a2)3 = −Ra2 (201)2h��

Ra2 = τ3a6 (202)Ta�� (201) 2h�q2 = −a2(τ − 1), q2 = a2[1 + 1

2τ(1 ± i

√3)] (203)t;��h�Ta℄+`h 2

±iq0, ±q, ±q∗ (204)L} q0 = a(τ − 1) 12 ,

re(q) = q1 = a{1

2

√

(1 + τ + τ2) +1

2(1 +

1

2τ)} 12

im(q) = q2 = a{1

2

√

(1 + τ + τ2) − 12

(1 +1

2τ)} 12

(205)w�� (203), Ta--��=Y℄R2w��n{

(q20 + a2)2 = a4τ2

(q2 − a2)2 = 12a4τ2(−1 ± i

√3)

(206)

(i) u*��z,6$D\�~��Ta6)5QW = A0 cos q0z +A cosh qz +A

∗ cosh q∗z (207)L} A0 5 A �)0�J;�`\I�w (207) Ta/*DW = −A0q0 sin q0z +Aq sinh qz +A∗q∗ sinh q∗z (208))�

(D2 − a2)2W = A0(q20 + a2)2 cos q0z +A(q2 − a2)2 cosh qz+

A∗(q∗2 − a2)2 cosh q∗z (209)IR�n (206), Ta6)& (209) 59(D2 − a2)2W = 1

2a4τ2{2A0 cos q0z + (i

√3 − 1)A cosh qz}

−12a4τ2{(i

√3 + 1)A∗ cosh q∗z} (210)

2 A0n�=1y\��

30 II ZOD`([P`DNegy 1. Bénard fZ:r�`_}7 (199) �{∣

∣

∣

∣

∣

∣

∣

cos 12q0 cosh12q cosh

12q

∗

−q0 sin 12q0 q sinh 12q q∗ sinh 12q∗cos 12q0

12 (i

√3 − 1) cosh 12q − 12 (i

√3 + 1) cosh 12q

∗

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

A0

A

A∗

∣

∣

∣

∣

∣

∣

∣

= 0 (211)vd�`��\� (211) }�H2J��!m�>`}7�∥

∥

∥

∥

∥

∥

∥

1 1 1

−q0 tan 12q0 q tanh 12q q∗ tanh 12q∗1 12 (i

√3 − 1) − 12 (i

√3 + 1)

∥

∥

∥

∥

∥

∥

∥

= 0 (212)wC=J}.C�J�u&Y�T) −√3/2, /∥

∥

∥

∥

∥

∥

∥

1 1 1

−q0 tan 12q0 q tanh 12q q∗ tanh 12q∗0

√3 − i

√3 + i

∥

∥

∥

∥

∥

∥

∥

= 0 (213)-%>`J�2J��℄im{(

√3 + i)q tanh

1

2q} + q0 tanh

1

2q0 = 0 (214)u I

(i)SaE��0v+g�+l`��59!��G��−q0 tan

1

2q0 = im

{

(√

3 + i)(q1 + iq2)sinh q1 + i sin q2cosh q1 + cos q2

}

(215))\�� (215) 2`o�℄−q0 tan

1

2q0 =

(q1 + q2√

3) sinh q1 + (q1√

3 − q2) sin q2cosh q1 + cos q2

(216)

§15 X��fZ`�B� 31

a

R

0 1 2 3 4 5 6 7 80

5000

10000

15000

20000

25000

30000

35000

40000

45000

1

2

� 2. xf�T&F (�� 1) 8QF (�� 2), �f�|2 a 4�b�f�O℄4 Rayleigh 2�L}�6)R�� (205) ��2�d q0, q1 5 q2 2[3�� (216) �~n a 5 τ = (Ra/a4) 13 2�`/��Y\�!�uTnjJ0b{\��vd�`e[2 a,�[ τ , �JR (202){QI2mJb�R>��{\>`Pu�zV Pellow 5 Southwell (9�r Low $xK��R�2 (��2) ��/QÆv�\��>`Pu2�[2[i��V Reid 5 Harris sQ2�`aeQ2Y�0m I 5� 2 ^��6)'Q�(0�a = 3.117, R = 1707.762 (217)�*a2J%b�� A0 = 1 s��d W 5 F 2I2\�

W = cos q0z − 0.06151664 coshq1z cos q2z + 0.10388700 sinhq1z sinh q2z

(218)

(a2R)−23F = cos q0z + 0.12072710 coshq1z cos q2z+

+0.001331473 sinhq1z sin q2zL}q0 = 3.973629; q1 = 5.195214; q2 = 2.126096 (219)

W 5 F 2\0m II 5� 3 ^��


z0 0.1 0.2 0.3 0.4 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1

2

� 3. �bbaol9PR4vNy�b~O℄*g4o^ W (�� 1) 8(a2R)−

23 F (�� 2).

z0 0.1 0.2 0.3 0.4 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2

� 4. ��bbaol9PR4�"�bbaol98X4vNy�b~O℄*g4o^ W (�� 1) 8 (a2R)− 23 F (�� 2).

§15 X��fZ`�B� 33(ii) �*�

)w,6OD\��Ta℄W = A0 sin q0z +A sinh qz +A

∗ sinh q∗z (220)

w>`\/Q2mJJ�� (2�� (213))∥

∥

∥

∥

∥

∥

∥

1 1 1

q0 cot12q0 q coth

12q q

∗ coth 12q∗

0√

3 − i√

3 + i

∥

∥

∥

∥

∥

∥

∥

= 0 (221)

34 II ZOD`([P`DNegy 1. Bénard fZt II(i) R`D�?�Æ/u*f�*k;�� W h (a2R)− 23Fz We (a

2R)−23 Fe Wo (a

2R)−23 Fo

0.000 0.9384834 1.1207271 0.0000000 0.0000000

0.010 0.9377396 1.1200748 0.0698930 0.0728662

0.020 0.9355102 1.1181191 0.1394197 0.1453850

0.030 0.9318004 1.1148639 0.2082157 0.2172109

0.040 0.9266188 1.1103153 0.2759205 0.2880024

0.050 0.9199780 1.1044832 0.3421794 0.3574235

0.060 0.9118936 1.0973760 0.4066453 0.4251459

0.070 0.9023850 1.0890103 0.4689809 0.4908506

0.080 0.8914750 1.0794012 0.5288600 0.5542300

0.090 0.8791900 1.0685672 0.5859696 0.6149890

0.10 0.8655597 1.0565291 0.6400120 0.6728475

0.11 0.8506176 1.0433099 0.6907054 0.7275412

0.120 0.8344005 1.0289350 0.7377866 0.7788233

0.130 0.8169488 1.0134318 0.7810122 0.8264663

0.140 0.7883063 0.9968295 0.8201596 0.8702628

0.15 0.7785204 0.9791598 0.8550291 0.9100266

0.16 0.7576421 0.9604558 0.8854448 0.9455945

0.17 0.7357256 0.9407526 0.9112557 0.9768262

0.18 0.7128290 0.9200870 0.9323374 1.0036059

0.19 0.6890136 0.8984973 0.9485924 1.0258427

0.20 0.6643445 0.8760233 0.9599518 1.0434710

0.21 0.6388900 0.8527060 0.9663757 1.0564510

0.22 0.6127224 0.8285876 0.9678542 1.0647689

0.23 0.5859170 0.8037116 0.9644079 1.0684367

0.24 0.5585532 0.7781220 0.9560889 1.0674925

0.25 0.5307138 0.7518638 0.9429810 1.0619998

0.26 0.5024749 0.7249824 0.9252006 1.0520474

0.27 0.4739567 0.6975237 0.9028964 1.0377482

0.28 0.4452229 0.6695339 0.8762509 1.0192390

0.29 0.4163808 0.6410589 0.8454794 0.9966785

0.30 0.3875314 0.6121447 0.8108314 0.9702473

0.31 0.3587797 0.5828367 0.7725905 0.9401465

0.32 0.3302342 0.5531798 0.7310741 0.9065908

0.33 0.3020074 0.5232179 0.6866346 0.8698180

0.34 0.2742159 0.4929939 0.6396586 0.8300749

0.35 0.2469799 0.4625492 0.5905684 0.7876218

0.36 0.2204238 0.4319239 0.5398210 0.7427274

0.37 0.1946762 0.4011558 0.4879092 0.6956670

0.38 0.1698697 0.3702806 0.4353619 0.6467188

0.39 0.1461412 0.3393317 0.3827446 0.5961605

0.40 0.1236321 0.3083394 0.3306597 0.5442657

0.41 0.1024879 0.2773310 0.2797474 0.4913001

0.42 0.0828590 0.2463303 0.2306868 0.4375168

0.43 0.0649003 0.2153572 0.1841964 0.3831516

0.44 0.0487714 0.1844274 0.1410353 0.3284183

0.45 0.0346369 0.1535520 0.1020048 0.2735031

0.46 0.0226663 0.1227369 0.0679496 0.2185585

0.47 0.0130345 0.0919828 0.0397596 0.1636974

0.48 0.0059215 0.0612842 0.0183718 0.1089859

0.49 0.0015130 0.0306295 0.0047726 0.0544365

0.50 0.0000000 0.0000000 0.0000000 0.0000000

§15 X��fZ`�B� 35>'p\q0 cot

1

2q0 = im

{

(√

3 + i)(q1 + iq2)sinh q1 − i sin q2cosh q1 − cos q2

}

(222))\>`JJ2��Ta/*q0 cot

1

2q0 =

(q1 + q2√

3) sinh q1 − (q1√

3 − q2) sin q2cosh q1 − cos q2

(223))�� (223) 7wX�V Reid 5 Harris �Q2Y��m I 5 II }eQ�vdJ92ODx�� W 5 F 2\�W = sin q0z − 0.01707389 sinhq1z cos q2z + 0.00345645 coshq1z sin q2z

(a2R)−23F = sin q0z + 0.01153032 sinhq1z cos q2z+

+0.01305820 coshq1z sin q2z (224)L}q0 = 7.137877; q1 = 9.110819; q2 = 3.789330 (224’)vd a(=3.117) b2C�`�x��>�x�w/*2 Rayleigh 0�℄B�2 (2CJ0� 71(d)). vd W 5 Θ, I2\�

W = sin q0z − 0.045302 sinhq1z cos q2z + 0.012173 coshq1z sin q2z

(a2R)−23F = sin q0z + 0.033193 sinhq1z cos q2z+

+0.033146 coshq1z sin q2z (225)L}q0 = 7.2569; q1 = 7.3711; q2 = 3.6643. (225’)

(c) DA2Z-�fDA�tLw2\�6)wJJ2%V}eQ2C�`ODx�2Y��Q�:7�s~��>�OD\PCj6VmjvI2� z = 0[2`_}7��>�W,D2W , 5 (D2 − a2)2W � z = 0 sÆ�!m�I��PC�`NP`_2�I2(w�Hk7 d s2OD\��6)eQ��>�tL2\��I2(w�HkI7 12d. �VHk[3 a 5 R s�vd>�tL�Ta6)BRmII 2Y�

a = 5.365/2 = 2.682, Rc = 17610.39/16 = 1100.65 (226)

(d) ! �6��^6�Ta$x,62=�tLw2Y�>Y��m III }�t III(i)6�! �6�-k-_�FP�.nk3nQ Rc a 2π/a�af7W3 657.511 2.2214 2.828�afOQ3 1707.762 3.117 2.016�a7W�aOQ 1100.65 2.682 2.342

36 II ZOD`([P`DNegy 1. Bénard fZ§16 !d�_Ta$x'*��M[%�sQ�2�`V�`m[2z0mJ��=2�"�(Æ�[2�>�Vd�`e[2z0 a 6)�!Yw{2�%\9�`QC%�o�vId�%\2z6)�!$/2G/5AW��`2iD�%>�Yw{26�P��0�Tawv6P,6jJ�[�t;�u>�"6)(J��iD6)m+/*2u>Yz2�utL�:7�>`9�jÆ�Y℄�`I5��/U2�:r℄iV%7�`|(e2N`��[�e9Q�2{`G�u>[�v62mj�>�(Æ2v6�{{`G6)�6`=GG�Q�Gt��2+`G� 3>�m tLw2u>Yz�b/y�2��zs�eQ��Ta>Q��[2�Hu>�"2%3℄Rs�Tae2��`�Au>��26TQ��Au>2[�f_2�Lmj�v62��u>[If_��Mk2�pk!m�u>�n�Q��u>[If_��Mk2�pk!m2}7�{

(n·∇⊥)w = 0, (227)L} n �u>[I2��A|�∇⊥ =

(

∂

∂x,∂

∂y, 0

)

(228)vd,62Pu�}7 (227) �d�{f_du>[29�Mk%!m�7'Q>�I�Tal��w�h�� (91) 5 (171),w = F (x, y)W (z) (∇⊥F = −a2F ), (229))�

u =1

a2∂2w

∂x∂z, v =

1

a2∂2w

∂y∂z(230):r

∇⊥w = W∇⊥F, u⊥ =DW

a2∇⊥F, (231))�

u⊥w =1

a2DW

W∇⊥w. (232)�/�`�[2�� ∇⊥w 2!m/�/��2��I u⊥ 2!m��!Y�9u�

(a) ℄��^℄2��d6�29�Wj�0 x s�Q��)�2�"��>�tLw�u>�Yw�|+2�l�;67 �2��w = W (z) cos

2π

Lx (233)

3 1� n ��|aH3a1Æ�YH (= π(1− 2/n)) U 2π Z℄O13K�;sUb�\^ 1− 2/n = 2/m,M m �aO1�?��piw n = 3, 4, 6, � m = 6, 4, 3 t&q:v�

§16 lD`A 37L} L(= 2π/a) �)0�vId>`�d w 2\�u = −DW

a22π

Lsin

2π

Lx, v = 0. (234):r��/>`z`2��Y℄9��`�℄

u = 0, x = ±(n+ 12

)L, 5 ± nL (235)L} n ��`N0�vd (235) �eQ2 x, w 2pk�S42��S42Hk� L;j�`z2z+��(b) ',f&,�!,6\

w = W (z) cos2π

Lxx cos

2π

Lyy, (236)L} Lx 5 Ly ��`� x 5 y ��2z+�aaj a 2�n�

a2 = 4π2(

1

L2x+

1

L2y

)

(237)h� (236) 2\�w = W (z), �I (±nLx, 0), [±(n+ 12 )Lx,± 12Ly], · · ·w = −W (z), �I [±(n+ 12 )Lx, 0], (±nLx,± 12Ly], · · · (238))�

w = 0, �_� x = ± 14Lx, 34Lx, · · ·.w = 0, �_� y = ± 14Ly, 34Ly, · · ·. (239)L} n ��`N0�Mk29�%�

u = −DWa2

2π

Lxsin

2π

Lxx cos

2π

Lyy,

v = −DWa2

2π

Lycos

2π

Lxx sin

2π

Lyy. (240):r�

u = 0, �/ x = 0,± 12Lx,±Lx, · · · 5 y = ± 14Ly, 34Ly, · · ·,v = 0, �/ y = 0,± 12Ly,±Ly, · · · 5 x = ± 14Lx, 34Lx, · · · (241)w u⊥ 5 nabla⊥w \%2Ur�n�6/

∂w/∂x = 0 �/ u = 0 2{,∂w/∂y = 0 �/ v = 0 2{, (242):r�u>�`7 Lx 5 Ly 2�G�u>}�`2�"�) x = 0, y = 0 7};�)

x = ±Lx, y = ±Ly 7`_�MLau>�


� 5. xf�Iv?�;Bl�4*��

� 6. xf�Iv?�;Bl�4*��w (240) eQ2 u 5 v 2m��Ta��Lxu± Lyv = −

2π

a2DW sin 2π

(

x

Lx± yLy

)

(243)w>`��6/Lxu+ Lyv = 0 vd x/Lx + y/Ly = 0 5 ± 12 (244)Lxu− Lyv = 0 vd x/Lx − y/Ly = 0 5 ± 12 (245)��>�(�VQC2��C9�`�vG��9�jÆ�`2(V��dx

dy=u

v=LyLx

[sin(2πx/Lx)][cos(2πy/Ly)]

[cos(2πx/Lx)][sin(2πy/Ly)](246)�[�{%>`��Ta/

sin2πx

Lx= Constant (sin 2πyLy)

L2y/L2x (247)w>`��Q2(0� 5 ^��L} Ly = √3Lx/2.�!Wr Lx = Ly, wI`2Y�6)/*Q�Gu>tL2\�I2�`(0�

6 ^��

§16 lD`A 39(c) U-,�!+`G�"2\� Christopherson ��2�`2\�

w =1

3W (z)

{

2 cos2π

L√

3x cos

2π

3Ly + cos

4π

3Ly

}

(248)>`\2z�G��w =

1

3W (z)

{

cos4π

3L

(√3

2x+

1

2y

)

+ cos4π

3L

(√3

2x− 1

2y

)

+ cos4π

3Ly

}

(249)5w =

1

3W (z)

{

4 cos4π

3L

(√3

2x+

1

2y

)

cos4π

3L

(√3

2x− 1

2y

)

cos4π

3Ly − 1

}

(250)��Ta=ms L kÆ+`G2`��zTa�T\ (248) ��w�m�`>z07 a 2�`�Ta℄∇2⊥w = −

1

3W (z)

{

2

(

2π

3L

)2

(3 + 1) cos2π

L√

3x cos

2π

3Ly +

(

4π

3L

)2

cos4π

3Ly

}

= −(

4π

3L

)2

w (251):r�a = 4π/3L (252)R (249) �m�2\�LwQ+`Gv6Pf/s~��Ta5Q

x = ̟ cos θ, y = ̟ sin θ (253)� √3

2x± 1

2y = ̟ sin(60o ± θ) (254)Ta6)&�� (249) �:59

w =1

3W (z)

{

cos

[

4π̟

3Lsin(θ + 60o)

]

+ cos

[

4π̟

3Lsin(θ + 120o)

]}

+1

3W (z)

{

cos

[

4π̟

3Lsin(θ)

]}

(255)w}~2w(̟, θ) = w(̟, θ + 60o) (256)TÆ�duIg$ 60o 2fP�>`\� X 5 y ��m�Q�HP�:r

w(x+ nL√

3, y + 3mL) ≡ w(x, y) (257)L} n 5 m �$/N0�� y ��Hz2z+62� x ��2z+2 sqrt3 J�h�� (250), Ta'*w(0) = W (z) (258)

40 II ZOD`([P`DNegy 1. Bénard fZ)��/=}_vy = ±3

4L, x

√3 + y = ±3

2L, x

√3 − y = ±3

2L (259)aam+Æ+`G GHIJKL(2� 7a),

w = −13W (z) (260)��+`I

(√3

2L,±1

2L

)

, (0,±L),(

−√

3

2L,±1

2L

)

(261)aa�+`G2XI ABCDEF ,w = −1

2W (z) (262)9�Mk%�

u =1

a2∂2w

∂x∂z= −DW

3a24π

L√

3sin

2π

L√

3x cos

2π

3Ly,

v =1

a2∂2w

∂y∂z= −DW

3a24π

3L

(

cos2π

L√

3s+ 2 cos

2π

3Ly

)

sin2π

3Ly (263)w>-��/*

u = 0 � x = 0, x = ±√32 L, y = ± 34Lv = 0 � y = 0, y = ± 32L (264):r��! OD(u = 0 �x = 0), BC(u = 0 �x = L√3/2), 5 OI(v = 0 �y = 0) Y℄(`�wg$ 60o \2fP�Ta/QYD�Y℄(`�!>+`6`=GG2`�aaG9+`G ABCDEF , �Y℄(`�!w+`G2};*+`G2`2f�w u⊥ 5

∇⊥w 2QU�n�6[Uj�+`6`=GG2d�2[j�v62mj�℄�h�(`�f_d KJ(u = 0, � y = 3L/4), Ta/Q�/ÆO+`G GHIJKL2`Y℄(`�>j�/>-�2` w �)0��p2��>`9�jÆ2(`�"0� 7b ^��f_�jÆ2(`2�;�UK

§16 lD`A 41

� 7a. -bIv?4�:[{vd (265) �eQ2 v 5 w, �� dy/v = dz/w, �u/ (y, z) �jÆ2(�6)--�{%/Q{

1 + 2 cos(2πy/3L)

1 + cos(2πy/3L)sin

2π

3Ly

}2

W (z) = constant (267)bG��w (266) 6/� (x, z) �jÆ2(V{(

1 − cos 2πL√

3x

)

sin2π

L√

3

}23

W (z) = constant (268)eQ�w (267) 5 (268) ��Q2(0� 8 ^��(d) !�,�!>�tLY\�glD�,67+`G2�"6)6��,67=GG�"��Au>�R6`=GG OMN(2� 7a) m�2�a2`+� L√3. �/>`=GG2`Y℄A��`��a2XI w 2b� (7��; C [2b2�J�2/�).(e) K}3��!t�

Chiristopherson\2�� Bisshopp��2�Bisshopp2\'YGNeQ~� x−5 y− ��[2�HPz2JFi2u>�"�V�`e[2>z0mJ�w��`m[2�� (U0>� y− ��) �℄$[�Hz2��%��Ta&\59G�w = W (z)

{

A cos

[

a

(

1 − 1m2

)12

x

]

cosa

my + cos a(y − y0)

}

(269)


� 7b. �;Bl�-bIv?48 w ��T 1,2,· · ·,12 l�4��xK4dÆw = 0.75, 0.5, 0.25, 0,−0.20,−0.25,−0.30,−0.325,−0.36,−0.40,−0.44,−0.48.��Q-bI (X[�_�) K4 w dÆ − 1

3; ��-bIZI4dÆ

− 12. *�l8 w �ÆSD4�

� 8. �fE�b&2y��O℄&�u��x8Bl�-bIv?�4*��*� (TL�K\4d��^) Æj��C (267)(�#�a�) 8�C (268)(�#�yl), *�T

_g4*2dl��

§17 Ip� 43L} m �d 1 2N0� A 5 y0 �)0��Ta��%M cos a(y − y0) ^E%2�HP�Lu�M�32�� x−��2�HPz06)}>2z07 a.) ~��\ (269) PC>-}7�:r

∇⊥ cos[

a

(

1 − 1m2

)12

x

]

cosa

my = −a2 cos

[

a

(

1 − 1m2

)12

x

]

cosa

my (270)5

∇⊥w = −a2w (271)0�^�{2�� x− 5 y− ��2�`z+�λx =

2π

a

m√m2 − 1

, λy =2π

am (272)9��Mk%�

u = −DWa

(

1 − 1m2

)12

A× sin[

a

(

1 − 1m2

)12

x

]

cosa

my (273)5

v = −DWa

{

A

mcos

[

a

(

1 − 1m2

)12

x

]

sina

my + sin a(y − y0)

}

(274)0� y0 = 0, �A�_

x = ± 12nλx, y = ± 12nλy (275)2Mk!m�:r�\

w = W (z)

{

A cos

[

a

(

1 − 1m2

)12

x

]

cosa

my + cos ay

}

(276)m+Æ��u>�"�L}�Au>��`��2`7 λx 5 λy 2�G��Au>Æ2�`�U� §(b) },62)��Gu>�


� 9. y�C w = 13W0{

cos[ 13k(x

√8 + y)] + cos[ 1

3k(x

√8 − y)] + cos ky

}�S4>�v?�#�5(D2 − a2)F = −Ra2W (278)6W2`_}7�

F = 0 W = DW = 0, � z = ± 12 (279)Q0� §15 }eQ2�PC�� (277) 5 (278) 2`_}72\�$(0�O(0�,6$D\�� F �$(0�o:7a�� z = ± 12 [!m�6)&a-%9g}�0G�F =

∑

m

Am cos[(2m+ 1)πz] (280)� (280)�}v m 2{5�6)�W7w 0 * ∞; ��>�Y℄Y�2�:7Ta=&Æ0Am 'T�f%Æ0 (2w`).

F V (280) �eQ�Ta�[ W , a��(D2 − a2)2W =

∑

m

Am cos[(2m+ 1)πz] (281)2�`\�PCDG2�d W 2`_}7�,6*�� (281) 2P�Ta6)& W m�952G�W =

∑

m

AmWm(z) (282)L} Wm ��(D2 − a2)2Wm = cos[(2m+ 1)πz] (283)2�`\�PC`_}7Wm = DWm = 0,� z = ± 12 (284):7�� (283) �DR2�>`\=M{Q��wI�6��[2��Ta=s/\2~�G��

§17 Ip� 45� §13 }m+2f%ui��d�%\∫ 1

2

− 12[(DF )2 + a2F 2]dz, (285)vd}

∫ 12

− 12[(D2 − a2)W ]2dz (286)DG)02 F 2>�f\�B3 Ra2 T7�`�[2 Lagrange >3�vd F 2$9f\�Ta6)6��%\

J =

∫ 12

− 12[(DF )2 + a2F 2]dz −Ra2

∫ 12

− 12[(D2 − a2)W ]2dz (287)�Ta& F j7℄�[0�2Æ02�`��wd>-Æ0�%\�:r�0� F 2 (280) �,69�`

46 II ZOD`([P`DNegy 1. Bénard fZv6P�s~2�l*�� (288), R�� (289) 5 (293), ℄J =

1

2

∑

m

A2mγ2m+1

−Ra2∑

m

∑

n

An(n | m)Am (295)uoTaY\h�Æ0 An �>`m��2�%b�:r��{∂J

∂An= 0 (n = 0, 1, · · ·) (296)Ta/*

∑

m

{

δmn2a2γ2m+1R

− (n | m)}

Am = 0 (n = 0, 1, · · ·) (297)>m�Æ�d Am 2RuP��H�vd>`��H2�`��\�n�2J��[!m�:r�6)/*}7‖ δmna2γ2m+1R

− 2(n | m)‖ = 0 (298)a��`�H2mJ��G��Ta�d,6 (280)�2` n `n0 A1, A2, · · · , An, #Lan06d��>bGd& F,69 n `Æ02

§17 Ip� 47\>` Pm 5 Qm 2��H�Ta��

Pm = (−1)m(2m+ 1)πγ22m+1

a+ sinh asinh

1

2a

Qm = (−1)m2(2m+ 1)πγ22m+1

a+ sinh acosh

1

2a

(303)�H (n | m) 2tL��V(n | m) =

∫ 12

− 12

{

Pm cosh az +Qmz sinh az + γ22m+1 cos[(2m+ 1)πz]

}

× cos[(2m+ 1)πz]dz (304)t;V(n | m) = 1

2γ22m+1δmn + Pm

∫ 12

− 12coshaz cos[(2m+ 1)πz]dz

+Qm

∫ 12

− 12z sinh az cos[(2m+ 1)πz]dz (305)eQ� (305) �``2{%--{Q�Ta��

∫ 12

− 12coshaz cos[(2m+ 1)πz]dz = 2(−1)n(2n+ 1)πγ2n+1 cosh

1

2a

∫ 12

− 12z sinh az cos[(2m+ 1)πz]dz = (−1)n(2n+ 1)πγ2n+1

× (sinh 12a− 4aγ2n+1 cosh

1

2a)

(306):rTa℄(n | m) = 1

2γ22m+1δmn + (2n+ 1)πγ2n+1(−1)n

×{2Pm cosh1

2a+Qm(sinh

1

2a− 4aγ2n+1 cosh

1

2a)} (307)&V (303) eQ2 Pm 5 Qm �3)I��u)\�Ta��

(n | m) = 12γ22m+1δmn − 8a(−1)m+n(2n+ 1)(2m+ 1)π2γ22n+1γ22m+1

cosh2 12a

sinh a+ a(308)0�^�{2�>�d n 5 m �v62�:r�mJJ�� (298) 2~�G��

‖(

1

a2Rγ2m+1− γ22m+1

)

δmn + (−1)m+n16aπ2cosh2 12a

sinha+ a

×(2n+ 1)(2m+ 1)γ22n+1γ22m+1‖ = 0 (309)v R 2�RnG�6)h�W[mJ�H2 (0, 0) tL6d��uO�La�eQ�>�dj� cosπz T7 F 2

48 II ZOD`([P`DNegy 1. Bénard fZt;�:7 γ1 = (π2 + a2)−1,R =

(π2 + a2)3

a2{1 − 16aπ2 cosh2 12a/[(π2 + a2)2(sinh a+ a)]}(311)>`v�eQ

R = 1715.08, � a = 3.117 (312)>Y\j a �sv��S2Y� (1707.76) jJUK�:r��}Y℄f%Æ0�>��s/2vk�U 12% ,�I��K�kI�>`vk�[d)w�w��IRf%�2!�}�Ta\Æ~u F 5 W 2DR0%��wITav>`Puv��\Æ”2-3 R ”.�!�% F -%�2B��Ta6�6)��' R bs�*lW2vk�>0m IV^��Ox� (� §15 }�Ta��>�x��d�`7�[�`76V2`_2tLw)�6)Rf%�/*�:r��Ta�W F 2-%G�

F =∑

m

Am sin 2mπz (313)vd Wm, I2\�Wm = Pm sinh az +Qmz coshaz + γ2m

2 sin 2mπz (314)L} Pm 5 Qm �{%)0�oγ2m =

1

4m2π2 + a2(315)`_}7�[Æ Pm 5 Qm, Ta��

Pm = (−1)m2mπγ22m

sinh a− a cosh1

2a

Qm = (−1)m+14mπγ22m

sinh a− a sinh1

2a

(316)�HtL (n | m) ��V(n | m) =

∫ 12

− 12sinh 2nπz{Pm sinhaz +Qmz coshaz + γ2m2 sin 2mπz}dz (317)eQ��!{Q{%��

(n | m) = 12γ22mδmn − 32(−1)m+namnπ2γ22mγ22n

sinh2 12a

sinh a− a (318)mJJ��℄G�‖(

1

a2Rγ2m− γ22m

)

δmn + (−1)m+n64aπ2sinh2 12a

sinh a− amnγ22mγ

22n‖ = 0 (319)�!#mJ�H (1, 1) tL6d��uO�La^℄2��eQ R 2�RnG�:r�Ta/*

R =1

a2γ2[1 − 64aπ2γ22 sinh2 12a/(sinh a− a)](320)

§18 ([ kGDNegy`I 49t;��3 γ2, ℄R =

(π2 + a2)3

a2{1 − 64aπ2 sinh2 12a/[(π2 + a2)2(sinh a− a)]}(321)>`v�eQ

R = 17803.24, � a = 5.365 (322)>Y\jv��S2Y� 17610.39 jJUK��!j��% F -%�2�/*2Y��m IV }eQ�Ta�u'*�!f%�6)/*KW2vk�u IVX1,'#u}� Rayleigh Q(i);2ba = 3.117�SoH R = 1715.080�SoH R = 1707.938 A2/A1 = +0.028973>SoH R = 1715.080 A2/A1 = +0.028963;

A3/A1 = −0.002694w�℄ R = 1707.76(ii)E2ba = 5.365�SoH R = 17803.24�SoH R = 17621.74 A2/A1 = +0.06304>SoH R = 17611.84 A2/A1 = +0.062945;

A3/A1 = −0.010088w�℄ R = 17610.39§18 %�1XQ{�':�ZE�>�V�Taa+XGiDj-2�d(w}�d�M[2�-w�qT�Q0Ta�%�eQ2�0Q02%ju^ms2�℄�dj℄>` o2w�� BénardR2�Q�Vd Bénard 2u℄k2w�/*Æ℄�2Y��Rayleigh jJÆ`2iDy��wI�>`BT� Rayleigh �`2DH}2v%C"�(a) Bénard 5<

Bénard ��`�)C�^k�� 1 +d�tld2(w�IjJw��Mrd�`DG)E29�e'1I (2� 10). I�mj�)�6Vmjuoj=YOZ�EkK9� Bénard Rai)0�2�(wjJw��mpLB�2�(w*k2TR�R+\2sW5XYT7W*k�w��^℄tLw� Bénard ��>�mj2Ek�:"��`�[2s:�(w�+(\u~�Q%\2u>�`�/*u>Æ�℄�`�};I6�j�nu>OZ2`_w4� Bénard %Æ�u>�"�-!�}2�`Rr��`q�2P�Rr�L}u>V��`}6�k2OI�f9�℄f_[2wD}`*J}`2�{`G�C�`vG�2Rr}�̂ ℄u>f9��+`G��*��


� 10. Bénard 4w�'t� 1, � Bénard u�892

§18 ([ kGDNegy`I 51�[Y℄~"Zm�te�(��EC = C2R = 4.1854ζ × area (324)L}N(2�A��/�NF2�A�#��N`z2�A�4m��A�$� (:3

4.1854 �s�). ~u�� (323) 5 (324), Ta℄'��EC = C2R = 4.1852k

∆T

d× area (325):r�0�Y℄�Zm��'��}�7Æ:G�`2Ek� ∆T , ��

}V��!2N(2��W ∆T P"��QUr)0��[2�/*2>�P�n�M[P�ds=M�D�:r�� (C2,∆T ) �}��Ek ∆T /!M[P�d2bs�Ta�[��*�n2Da�>�Da�[�6)��2��} C2R 2�Y℄Æ�te ζ: :7�0�Ta�[j℄ C2R 2�`�% q ste ζ, �M[P�ds� C2 2%�3Æ� q 6�2�f\2�>�Y��J2�Pm#d�� 11 eQÆ Schmidt 5 MilvertonT2 (C2,∆T ) ��>-�sW�ms�M[PQ�s=�dP�n2Da�w�I��

-�2N�mÆ�f9�( ζ: ��%�=�!'�5.T_*[dK9Ek2b}�5=Y}��2uW�2w�} (2w`2 (c)), >-Zm6)B9�u^℄C|2vk��6/2[��( ζ 2�6)/Q��32(w}2�_F��>��2�( ζ R�A k∆T/d(a�'��2b) m��e2�R k∆T/d m�2�(x)67 Nusselt 0�

Nu = ζ/(k∆T/d) (326)

Nusselt 0vd Rayleigh 02�6)YQY� Rayleigh 0�R =

gαβ

κνd4 (327)�>��}�Ta�[��*%�d`d 1 2�Wb�?F�*�- R b�Q�>�tL2>` R b��dM[P�d2�_ Rayleigh 0�

Schmidt 5 Milverton IR`asQ2ui��[Æ[d�`�[�j\%29�9�}�M[P�d2�_ Rayleigh 0�waa2w�Y��Q2b (~�� 11 }), �dm V }�6)'*aa/*2�_ Rayleigh 02bjiDb 1708 2U��2;2�u VSchmidt g Milverton 6=��_[F

1 2 3 4J2Fl 23.2o 22.75o 19.80o 17.63ox2Fl 24.9o 23.75o 20.85o 18.47oRc 1970 1580 1850 1670

Rc(mean) = 1770 ± 140

(c) Silveston �"�5


� 11. o�;Hw;�4EB3Gm!lX�4�!O*4B�4��4�ho��O℄R4f��S Schmidt-Milverton wk�yt��S4�a Rayleigh 2�o V �gS��w�}�7Æs/K2�35vk� Schmidt 5 Milverton 2w�$xM Schmidt5 Saunders, Malkus, Silveston 5L`"�-w�}�*5W�2wQui��2�Taj�L}2w�Æ0�3J2�� Silveston 2w��T7�ua+2v��Silveston 2w��*0� 12a, b ^��Rw�R2(w�� a; a�_�7 19.8cm 2�((w��^k6f�(w��u��`�1 b 5 c, 5�32�`℄xw}�4\%�w`2�1 b V�v6O2�1�aa\%��;2��

e. I1��V�v6O2�1H9�aa\%Q:/�}�jFg��Y6)EY�2e9w>��jJb(�"Y9�/�`Fg�j3��Jw!�`Fg��Q��9nbs�j�V�2�DG��)E*I�nb9=D�Ek2%f\��Vd�wIQ(53(��}�2Fg�}2w/*K2~-�nb923A5QA��R g m�2��`�1��6`=GG2GI[�R2u��rx7�q2^D^%2��!�RHWk�2^D�(w�2^k6)� 1.45 * 13mm \%f\��!�`DU� (i), }Y6>�2�Zm.%�>��aa\%Q:℄Fg� k 2�v�e'1H9��!>`Fg��s�*e[Ek2Y�jJnb��!}>��2EkOnd>� b 2Ek��Zm6).%�7Æ:Gw�}2Me}7�N`&rRÆ�`XJ s 5�4}+ r.�u(w2�1 b 5 c, 5�U1 i 2Ek�R��Ar2�N$��w�1�!2".�-7rW�6)��! f 2nbe9�3A5QA2Ek�}R��sR2&r0� 12b ^��>sI1 (� 12a }2 c) M_��2�Tw}1 a �z��N$��1Æ�2%=Æ�kw7Æ82�*�sW2�7Æ�r6�2Æ0�3ÆjJw�� Silveston TÆR9�RÆLa2�w�>-�w2ai)0��m VI }�*k�)W2�Y (AK350) BSÆmp2�/�

§18 ([ kGDNegy`I 53

� 12a. ��`!4 Silveston y�'t

� 12b. �m��4 Silveston y�'t

54 II ZOD`([P`DNegy 1. Bénard fZu VIE&Bj�-�Zmh��2$['�n0jJ~��M��t7!-2N��A�!m�VN( (�/) 5IRN`z (4m) 2>{eQ�w}.Tam+!2��[2�Zm� Silveston ��Q��( ζ 2 vb��jJw�s��/*Æ`uV�r0�DT�w�j3&r\`TiB2=Y�&�w��* 80o− 100oC uDG>`Ek��`%s6)w��&>[i2�wB3&r}/Q,jw�1mj cOZ�x!��t72N(5�!Fg� f 2�92(�x!SN}/ b 5 c \%2E��*s[2b�J:G�hh�}7M[nO�`%s)J%��7Æ^℄O`�v��t7 b, c 5 i 2EkjJÆ�1��1szQbJ�b��ÆEk�nb�92Ek�N(5Nse��Ek�2%��jJ}5Y-siQ�0�Rnb9B2� 4, w�1 c, w�[j5�U�162�Zm��S/st2>N2)~�/*(��2�pP2�� Silveston ��w��)Æ�>�(�>�6J2�Silveston w�2J�Y�0� 13 ^��wI�RÆ0�2�w��2}7wR2w�Y��RYT2 Nusselt 05 Rayleigh0T�s�e�p�I��2�"�I�>�w�26/P5vk2�`~�2Ts�ws/2�3� 103 − 104 2 Rayleigh 02Y�%f� Silveston �QM[P�d2�_ Rayleigh 02b�

Rc(w�) = 1700 ± 51 (328)4 W:3)�?*kRBFl fR�

§18 ([ kGDNegyÌ 55>jiDb 1708 2;/?,�(d) Y6&"UH�M[s�`2u>�"2Q��)��d\J2G�}/>��6)R�k�58�jJ�� Bénard 6�2��'d>�b�� 1 ��r�wM[Py�℄{)Y�>��$xR*Ælve2�k��Ta=)�m+m�2��)�aa�09/*2�� 15 � Schmidt 5 Milverton RJL�/*2892��}�}R:T��*�2�$:2�=#d!b�Cz2��0�15c, d ^�� 16� Schmidt 5 Saunders Rp�2WL/*2892�mj��Ekpks�89~��!}%?2�w�I�d8$��wIw�?TQ2��I8$Æ��N`892Hk�3Æ�8$f\��HP2�b/�/2��wI��?TQ2�f�8$J%2Ar�jww��?TQ2�f�8$J2Ar��p2�℄e�2�w�w�`2��w}%�?TQ2�Ys~2f�8$/�/��}%�jI�[2f�Ekpk6)O�� 16b, c, �^j3�j\%2N`%r (j� 15 �)G). mJ�� b, c}j�2-�� 16a}�℄ 11 ù>��16b, c }�℄ 22 ù>�>-u>20�Q�sI29�+k7 22.9cm. :�u>2Hk%p� 2.1cm 5 1.0cm, ��7�1\%2�g2�J��wI� Schmidt 5 Saunders��Æ��u>�`(Æ-%�u>2Hk��^k2�J�� 16b, ceQ2}7~��/QÆ`|M[P2�3��a�~�w`|}7%��wQ2u>�"Y℄�*$9~�f\�� 16d 2tL�?��>l Rayleigh0� 130,000;ams��2u>�"$x(ÆMWx2�(�`^z�� ((R� 16d j(RLa89℄I��892O`�j3��2��* 11 `f�?2�u��`7��Y℄Rf�?�~�}2f��Vd(w}2:T


� 13. �fTd��y`!4 Silveston y�[ (◦ ;� + p'� × (�k� • �[ AK3, ♠ �[ AK350, △ >Z��I2�Æ Mull 8 Reiher 4).Nusselt 2Æ�xf Rayleigh 2ZSZ4�

� 14. �fTd��y�O℄RDp4 Silveston y�[ (• �[ AK350,⋄�[ AK3, (�k� △ p'� ◦ ;). �/2��qou__�/�y�O℄Rf4 Rayleigh 29 1700±51(l^xW4kEdÆ 1708).

§18 ([ kGDNegy`I 57

� 15. Schmidt 8 Milverton TKM�gS � 16. Schmidt 8 Saunders T��m't4!x*f4�� Proc.Roy.Soc. gS4!x*f4�� Proc.Roy.Soc.(London) A, 152,586 (1935). (London) A, 152,586 (1938). ��42�gS2y�

(a) d=1.1cm, R=12,000, ∆T = 0.55oC

(b) d=0.5cm, R=3,500, ∆T = 1.7oC

(c) d=0.5cm, R=3,500, ∆T = 1.7oC

(d) d=1.1cm, R=130,000 ∆T = 4.0oC

� 17a. SilvestongS4!x*f4��Qb4��4 Rayleigh 2Æ 1500,ab4��4 Rayleigh 2Æ 1800. ��/y��4ImÆ 7mm.


� 17b. Silveston gS4!x*f4��Imu Rayleigh 2�;#��2. S.C. Brown,� Count Rumford discovers thermal convection��Daedalus, 86, 340-3,

(1957).�H� 2, Brown eQ�v(��q�V William Prout �zB32k2�3. W. Prout, � Bridgewater Treatises �� 8, 65, edited by W. Pickering, London,

1834.)w�wH� 2 } Prout DH2%1��`��Ta2n}Y℄�`��2qRYm�>��2_x��7rTa��sQ�`)n �R (Convectio,��1to;) ahm�'p2Y��oj!"�`)n (|�f/)) �p��wI�6W�`2u>�"29�(w�}2v(� James Thomson �z��*2�4. J. Thomson,� On a changing tesselated structure in certain liquids �� Proc. Phil.

Soc. Glasgow, 13, 464-8, (1882).)w�wH� 4 }%12Æ-��w�w^[2Ek/Q>`C_Cz2Eks�>��2Q��GN��jH.�=Y}2mjI�w2e�℄��(w�wEk/Qb}=YEk�'�II�C)G9f\/2>��eWJYz��Ta��m+5\�2>��`�0��!�95La�w~�>�WJYzs#d2��` (Thomson) :r>z9Æ��x)67v(b2m tG��`eQ�>�tL��mj�e�2Cz��%�w[�KW2Ek�eQÆjw-W>��BS2v(b2��p�>�mj��j�!N`�w2w��`��)�2��tG��iD5��jL�A,6[-��d�M[P2�u[w��u%z**k2TR2�� H. Bénard:

§18 ([ kGDNegy`I 595. H. Bénard,� Les Tourbillons cellulaires das une nappe liquide �� Revue générale

des Science pures et appliquées, 11, 1261-71 , and 1309-28 (1900).

6. ———,� Les Tourbillons cellulaires dans une nappe liquide transportant de la chaleurpar convection en régime permanent �� Annales de Chimie et de Physique, 23,61-114 (1901).�iD�j�wXDH� Lord Rayleigh 52�

7. Lord Rayleigh,� On convective currents in a horizontal layer of fluid when thehigher temperature is on the under side �� Phil. Mag. 32, 529-46, (1916). alsoScientific Papers, 6, 432-46, Cambridge, England, 192

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