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# Anthropology of Sexuality 113 112

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This 18 page Class Notes was uploaded by Mary Gutkowski on Friday October 23, 2015. The Class Notes belongs to 113 112 at University of Iowa taught by Staff in Fall. Since its upload, it has received 42 views. For similar materials see /class/228063/113-112-university-of-iowa in anthropology, evolution, sphr at University of Iowa.

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Date Created: 10/23/15

Curvilinear Coordinates Outline 1 Orthogonal curvilinear coordinate systems 2 Differential operators in orthogonal curvilinear coordinate systems 3 Derivatives of the unit vectors in orthogonal curvilinear coordinate systems 4 lncompressible N S equations in orthogonal curvilinear coordinate systems 5 Example lncompressible N S equations in cylindrical polar systems The governing equations were derived using the most basic coordinate system ie Cartesian coordinates x xi yj 2k 6f 6f 6f A rad V 1 k g f f 6x 6yJ 62 divFVFE ai 6x 6y 62 i j 12 curlfVgtltF 3 E 3 6x 6y 62 Fl Fz Fa Z Z Z LaplacianV2fa 6x 6y 62 Example incompressible ow equations V V 0 pg Vpy2uV2V p YVVV Vpy2uV2V p VVV Vgtltco Vpy2uVVV Vgtltco o V X V WV V 0 in the above equation but retained to keep the complete vector identity for VZV in equation However once the equations are expressed in vector invariant form as above they can be transformed into any convenient coordinate system through the use of appropriate definitions for the V VVx and V2 Frequently alternative coordinate systems are desirable which either exploit certain features of the flow at hand or facilitate numerical procedures The most general coordinate system for uid flow problems are nonorthogonal curvilinear coordinates A special case of these are orthogonal curvilinear coordinates Here we shall derive the appropriate relations for the latter using vector technique It should be recognized that the derivation can also be accomplished using tensor analysis 1 Orthogonal curvilinear coordinate systems Suppose that the Cartesian coordinates x yz are expressed in terms of the new coordinates x1x2x3 by the equations x xx1x2x3 y yx1x2x3 z zxlx2x3 where it is assumed that the correspondence is unique and that the inverse mapping exists For example circular 2311 1 coordinates x r cos 9 y r sin 9 Z Z ie at any pointP x1 curve is a straight line x2 curve is a circle and the x3 curve is a straight line The position vector of a point P in space is R yj 21 Rrcos6irsin 9jzl for 39 quot coordinates By definition a vector tangent to the x1 curve is given by x R 1 xiii ynj 2 11 Subscript denotes partial differentiation So that the unit vectors tangent to the q curve are quot R i quot R z quot RX 81 hi e2 hi e3 2 Where h R1 are called the metric coefficients or scale factors h 1 he r hz 1 for l coordinates The arc length along a curve in any direction is given by dsz dRdR hfdxf h dx h32dx32 Since dR Rx dxl hldxl l and RI he and since the x1 are orthogonal An element of volume is given by the triple product W me x adxz zmdxs s amdwxzwg Where since the x1 are orthogonal l gtlt z 53 Finally on the surface cl constant the vector element of surface area is given by dSi hzdxz z X hsdxs s ihzhsdxzdxs With similar results for x2 and x3 constant 6152 zhshidxsdxi 6153 shihzdxidxz 2 Differential operators in orthogonal curvilinear coordinate systems With the above in hand we now proceed to obtain the desired vector operators 21 Gradient Tjsr39 By de nition df Vf 61R 61x If we temporarily write Vf z z l3 3 Then by comparison df fx xx Mdgt9 A zi h 699 Vzii ig 1 6 A hi 6961 h 6 Vzi rliA coordinates 6r r66 Note qu e h A 41 h 699 So that by de nition curl grad f 0 wa Vxe 0 h x Alsoe 1e 2gtlte 3szgtltVx3 hi5 h So that by de nition V fo Vg 0 V3JV LJV LJO ma am 22 Divergence f vFvltae1gtvltzezgtvltae3gt VFl lVhzh3Fl l using VwnguuVgp ma l V hi F1 39 v l JV 2v 30 ma mmg ma aa aa 1 6 h F amaaa J Treating the other terms in a similar manner results in 1 6 6 6 h F F hF WM 2 1gt6x2ltm n6st 2 3 1 6 6 6 VF ltragtgltzgtgltrmgt 1301171 for r w in coordinates r 6r r 66 62 Fs s lx16thlA 16m I 16thl hi hiaaq hlaxz zhjaxs A6 e6 2F a 1 e3 M1 MaiM ow I A quot3 m Q L e hm 36x2 hl l hZ Z hj S 1 6 V X F i 1511th 6x2 6x3 V X F l coordinates I 1 83b 3 Q to O 24 Laplacian acting on a scalar V2VV 1 i iiriw iiri i hi 6x1 6x2 hi ax2 h 6x3 21aaa1aaa V r r r 6r 6r 66 r66 62 62 16 6 1 6 1 6 16 6 r r r6r 6r r66 r66 r62 62 25 Laplacian acting on a vector Using Vf ii z ii 3 hl 6x1 h2 6x2 hj 6x3 1 imm mm mm and VF Wile 39 VVF 1 a 1 a 6 0 A MWEV39ZWEWWWWHCI 1 a 1 a 6 0 A TED1WMFawm wzm ez 1 a 1 a 6 0 A 7673 111213WWWW39IEszm es 11a 11622 1123 UsingVgtltFLi i 1 111121169 6x2 613 111 1111 111 VXVgtltF 1Li 6 i 12 e 1a1eaxz111aaxl 61 61111J6 W1 6 Wm hllhz im m Combinin those two terms ives 1a 1 a 6 Ewhjfi gw 7 1111211 Himm q 1113112211121gt11111111111116ka i ltm 1e1112111111 2 itlt11gtlt11gtD 2lt11gtlt11gtDA ii ltmagtltma1111EH 3 hllhz 23 611111e11D h2 169 126 For quot coordinatesr6z h1 h 1 5 he r a hz 1 and use the definition of Laplacian operator acting on a scalar V2 f V2fl E 3 1 11 3 3 r 6r 6r 66 r66 62 62 16 6 1 62 62 r 6r r2 662 622 li iiii r6r 6r2 r2 662 622 A A A 1 2 6F A F 2 6F A A VZF ae be cez VZF1 r 2F1 1 26 62 V2FZ r r Za eg VZEeZ a 16 1 a 6 0 mimiawwmWWW ltmrimgtD gm gwm ltrmltmgt ltrmgt ltrEgt ltEgt rgltEgt gm iifitiiwaa fiiiif ifri i a 1 6Fl lan 6F3 5 1 r 6r r66 62 2 2 13136E1 6 r6 1 r 66 r 6r r66 62 6z6r 1F 16Fl 62Fl 16FZ162FZ ast r2 1 r6r 6r2 krz 66 r6r66 6r6z ll6FZ62FZ 162Fl r62Flr62E r 66 666r r662 622 6z6r r 1 16Fl 62Fl 16F 22 F 2 r2 1 r6r 6r2 r2 66 6r r 1 6F 2 2 1 62Fl 62Fl 2 3 r2 66 6 r2 662 622 2 1 IBFMZFI MEIIBZEWFI lr6rl6r2 r2 61939er 6192 622 r2 U7 619 36171 16211 62Fl 1 6FZ r2 1 r2 66 1 621171 1 62E 62E 1 2 6172 1 r 6r2 r2 662 622 r2 662 622 1E1515EAEH L wa iwgt1ampltwWJ ltmgtltmgt ltr gt l a a a a 1 a a F rF rF F tailed 3 62 JD 6rr6r 2 661D 11 l Er r r66 r 6r 66 62 3 l f i 3 1 62 r66 62 6r r 6r 66 11 1F 0F l r66r16r r66 62 16273 aze 61F6FZ 16171 r6266 622 arkr 2 6r r66 11 62E 1621 62 r 66 666r r 662 6662 1 62173 aze FZ I 16172 laze 16Fl 1621171 r r6266 622 r2 r 6r 6r2 r2 66 r6r66 1 6E 2 1 1 aze 2 3 r2 66 6 r2 662 6 2 3 62F1 16 62F1i 2 1 62 622 r2 r 6r 6r2 r2 66 6r r 6r 6r2 r2 662 I 622 r2 r2 66 136E162E BZE 3E r6r 1612396212391621723962172 le26F1 r2 662 622 r2 r2 66 6r i WWWWH 16wm1 gwmw 1 3 l Flr 1 2 62 r 6r 66 6 l i i l i r 6r 62 6r 66 r66 62 1 6Fl 62Fl 6F3 621173 1621173 6213 r r r 62 6r62 6r 6r2 r 662 6662 6 1 2 1 2 2 62F 2 62 622 a 1 2 1 laiaziiazF1 2 2 r r 6r 6r2 r2 662 6 1611 621173 1 621173 621173 r 6r 6r2 r2 662 622 1 6 6F3 62F3 1 62F3 62F3 r 2 2 2 2 6r 6r r 66 62 6Fl 6Fl 16172 61173 r6r 3 Derivatives of the unit vectors in orthogonal curvilinear coordinate systems The last topic to be discussed concerning curvilinear coordinates is the procedure to obtain the 6 A A der1vat1ves of the un1t vectors 1e a el e1 Such quantities are required for example in obtaining the rateof strain and rotation tensor 6 VV 1 1 EU 361 4 3vvvv my ey 17 2VV VV To simplify the notation we de ne R rs R h a n 9 9 Note that r1 1s symmetr1c 1e r1 r r1 hfl r2 hz z r3 h3 3 r11 a l b 2 c 3 hl l hl 11 H 2 hll l hl 12 3 hill l hl lll H 31 Derivation of n 2 3 h r1r1h2 Ii39lii hlhll Ii39liz lhhiz Ii39lis hihis rlr20 61 1r20 699 gtr1139r2r139r21 0 gtr1139r2 r139r12 gtr1139r3r139r31 0 gtr1139r3 2 1139113 gt 1 11 3913 hIhIS W112 1 11 hll l z 33 I lu l hl n W113 hi hi A h A h A 32 Derlvatlon 0f e22 el e3 2 rz39rzzhz rz39rzzzhzhzz rz39rnzhzhn rz39rzszhzhzs rr20 6 rr 1 Lo 6x2 gtr1239r2r139r220 gt1 221 1 1 2r21 gtr22 3911 h2h21 1 21 30 6 1 r gt 2 30 6x2 gtr2239 3r239r320 gtr22 3r239r23 gtr2239 3h2h23 A A 1111 A 1 hzhllehe 223ehehe r3 r3h32 LAquot LAquot LAquot LAquot LAquot dquot H II F S F a g s 1 11 30 a gtr1339r3r139r33 0 gtr33 T1 r3 1 31 gt r33 3911 h3h31 1 21 30 fir1 gt 2 30 6x3 gtr23 1 3r2 133 0 gt r33 1 2 r3 1 32 gt r33 1 2 hjhjz W6 A A A A h 2 e2 hnes hnes h3e33 2 r33 e1 W51 hl 34 Derivation of 32 Z a 1 1391 20 gtEIi39rz11339rzli39r230 1 3 6 rZr30 gtar2r3r21r3r2r310 2 r31 10 gtir3r1r32rr 6x2 gtr32r r gtr32r1 r2r ar321 1 I I gt1 321 10 1 32 ae1 be2 ce3 hjze3 hie32 1 32 ae1 be2 ce3 hme2 h e3 35 Derivation of 21 2 2 10 gtr2139r3r139r23 0 gtr21 1 r1 39rzs gtr21 1 13 T12 gtr21r30 1 21 hllel hlell 1 21 hlel h21e2 1 12 hll l hl 12 1 12 1 21 hl l hll l 36 Derivation of 13 ltsgt lt1gto gtr339r12r1339r2 0 gtr239r13 r339r12 gtr239r13 r239r31 gtr2r1 0 1 13 hIS l hl IS 1 13 hIS l hn s 1 31 hn s h3 31 1 31 1 13 hIS l hn s 4 Incompressible NS equations in orthogonal curvilinear coordinate systems 41 Continuity equation VV 0 1 6 6 6 ma a MMEEME and V vle1 vze2 23e3 Wen MV2M2V30 Since VF V 1 111112113 699 6V 1 2 42 Momentum equatlonEV VV Vp VV V where p p1ezometr1c pressure p Since V vl 1 vz 2 v3 3 we can expand the momentum equation term by term Local derivative 6 V 51 6L 2 3 6t 6t 6t 6t Convective derivative V VV Since V vl 1 vz 2 23 3 and VV V liV ZiV 3i hi 599 h 3962 39 vvv VVv1 lVVv2 2VVv3 3 VVv1 l v3 v1 Bel 53 vv B IIVZBVA lvzvl l IVSBVIAI 1I lla39xllhlaxl rllhlaxlllISBJCS III h h 1 I ll 3 M939 32 35E vv26v1 v36v1 v1 6 21 v2 6 21 v3 6 21 A vlv1 A Viv1 A V3V1 A e1 e11 e12 e13 vz z v2 6v2 2v36v2 2 VI hl 03 h 03 03 zl i vlv2 6amp2 v2 6 22 A vzv2 6amp2 v3 6 22 A v3v2 6amp2 hi x1 2 hi 6x1 hi 6x2 2 hi 6x2 h 6x3 2 h 6x3 12 v3 6 22 J z vlv2 Viv2 zz v3v2 zs hi hi VS S v26v3 3v36v3 3 hl 6x1 h2 6x2 h3 6x3 v L vlv3 6e3 v2 6 23 A L Viv3 6e3 v3 6 23 2 v v3 6e3 I Cl 3 6393 039 6393 0362 173x33 6363 v v v v v v A va va va 13 23 33e3 1 3e31 2 3 en 3 3 e33 hi 39 112 59 h 59 hi 112 h 1V2V30V3 3 ljmg zj hi hi h hi 59 112 5962 h 59 hi Pressure gradient Vp ia p l i6p 2 i6p 3 6x1 h2 6x2 Viscous term VZV VZV zi gmwgmwgm e a We eaewsrewwnD e22gt Mil newevawewn es aemem eWJJ beavn ewiia 421 Combine terms in l direction to get momentum equation in l direction 6V1 V1 6V1 V2 6V1 V3 6V1 Vlvlhll VZVZh ll VIVShIS V3V3h31 at 39hlaxl39hlaxz39wxs39 m m 39 We M Lila p h 6961 1 6 1 6 6 6 Vzamahzlgvla hjhlvzEl llhzv3 465themu 52ewrew i 422 Combine terms in z direction to get momentum equation in z direction 6 22 I vzvlhl1 vlvlh12 I v1 6 22 I v2 6 22 I v3 6 22 I vaShl3 v3x23h32 ar39 hth hth 39hl xl 39hzalelg yg39 hzhj hth ii6P p h 362 1 6 1 6 6 6 milklEWWWWWW waaeoawawiaaamnawl 423 Combine terms in 3 direction to get momentum equation in 3 direction aV3 V3V1h31 VlvlhlS stzhsz Vzvzhzs V1 6V3 V2 6V3 V3 6V3 ar39 M We 39 m m 39wx hzaxz39wx 116p New 6 6 0 3 vig a XIWWEWHV2EM2V3H waaaawamlaaawamm 5 Example Incompressible NS equations in cylindrical polar systems lwl coordinates r6z 51 Continuity equation VV 0 in 39 39 coordinatesr 9zhl h 1 h2 he r h3 hz 1 1 6 6 6 VVErvrv grvz 0 1 6 1 6 6 7EVVI7V EVZO coordinatesr 9z 52 Momentum equation 2 YVVV invVZV in p 521 The rmomentum equation 2 0LVr n vzaijiriipv Vzv L2Vr 6t 6r r 66 62 r p 6r 522 The 6momentum equation a lvvgvaiV 9a 6V9 1 6p VZF LJ Iquot V Br Br r66 Z62 pr6t9 523 The zmomentum equation BVZ 6vzvi6iv 6 22 16pVVZV V 6t r r r66 Z62 p62

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