e i e j = δ ij. (1.1)

Peieqìmena 1 DIANUSMATIKOS KAI TANUSTIKOS LOGISMOS 1 1.1 Sust mata suntetagmènwn.................................. 1 1.4 DiafoikoÐ telestèc..................................... 9 1.4.1 H ulik pa gwgoc.................................. 9 1.5 Tanustikìc logismìc..................................... 16 1.5.1 'Algeba tanust n.................................. 18 1.5.2 Tanustèc sth eustodunamik............................ 20 1.5.3 Apìklish tanustikoô pedðou............................. 25 1.5.4 PwteÔsousec dieujônseic kai analloðwtec tanust............... 27 1.5.5 Sumbolismìc deikt n kai h sômbash joishc................... 31 1.6 Oloklhwtik jew mata.................................. 34 1.6.1 To je hma tou Stokes............................... 34 1.6.2 To je hma thc apìklishc............................. 35 1.7 To je hma metafo c tou Renolds........................... 42 1.8 Pobl mata.......................................... 52 1.9 BibliogafÐa......................................... 60 i

ii PERIEQ OMENA

Kef laio 1 DIANUSMATIKOS KAI TANUSTIKOS LOGISMOS 1.1 Sust mata suntetagmènwn Me thn epilog ti n gammik ane thtwn dianusm twn, e 1, e 2 kai e 3,ston tisdi stato q o oðzetai èna sôsthma suntetagmènwn coodinate sstem. Ta dianôsmata e 1, e 2 kai e 3 antiposwpeôoun teic diafoetikèc metaô touc dieujônseic ston q o. To sônolo B={e 1, e 2, e 3 } eðnai mia b sh basis tou tisdi statou q ou. Sun jwc ta dianôsmata e 1, e 2 kai e 3 epilègontai na eðnai monadiaða. Sta tða sust mata suntetagmènwn pou mac endiafèoun, dhl. stic katesianèc Catesian, tic kulindikèc clindical kai tic sfaiikèc spheical suntetagmènec, ta tða dianôsmata thc b shc eðnai epðshc ojog nia metaô touc. 'Aa sta tða aut sust mata h b sh B={e 1, e 2, e 3 } eðnai ojokanonik : e i e j = δ ij. 1.1 K je di nusma u tou q ou autoô g fetai monos manta san gammikìc sunduasmìc twn e 1, e 2 kai e 3 : u = u 1 e 1 + u 2 e 2 + u 3 e 3. 1.2 Oi bajmwtèc posìthtec u 1, u 2 kai u 3 eðnai oi sunist sec tou u kai antiposwpeôoun ta megèjh twn pobol n tou u se k je mia apì tic basikèc dieujônseic. To di nusma u g fetai suqn san uu 1,u 2,u 3 apl sanu 1,u 2,u 3. To katesianì sôsthma suntetagmènwn,, z, me <<, << kai <z<, mac eðnai dh gnwstì. H b sh tou sumbolðzetai suqn me {i, j, k} {e, e, e z }. H an lush enìc dianôsmatoc v stic teic tou sunist sec faðnetai gafik sto Sq ma 1.1. Shmei noume ìti se ìla ta kef laia twn shmei sewn qhsimopoioôme deiìstofa ight-handed sust mata suntetagmènwn. Oi kulindikèc kai oi sfaiikèc polikèc suntetagmènec eðnai ta pio shmantik ojog nia kampulìgamma sust mata suntetagmènwn cuvilinea coodinate sstems. Oi kulindikèc polikèc suntetagmènec, θ, z, me 0, 0 θ<2π kai <z<, faðnontai sto Sq ma1.2mazðmetic katesianèc suntetagmènec. Hb shtou kulindikoô sust matoc suntetagmènwn apoteleðtai apì tða ojokanonik dianôsmata: to aktinikì di nusma e, to azimoujiakì gwniakì di nusma e θ, kai to aonikì di nusma e z. PaathoÔme ìti h azimoujiak gwnða θ peistèfetai gôw apì ton ona twn z. K je di nusma v analôetai kai oðzetai monos manta apì tic teic tou sunist sec vv,v θ,v z. Me th q sh apl n tigwnometik n tautot twn k je di nusma mpoeð na metasqhmatisteð apì to èna sôsthma sto llo. Ston PÐnaka 1.1 eðnai sugkentwmènoi oi 1

2 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc z z v z k,, z v k i j z v i v j Sq ma 1.1: Katesianèc suntetagmènec,, z me <<, << kai < z<. z z e z, θ, z z e e θ θ θ Sq ma 1.2: Kulindikèc polikèc suntetagmènec, θ, z me 0, 0 θ<2π kai <z<, kai to di nusma jèshc.

1.1. Sust mata suntetagmènwn 3, θ, z,, z,, z, θ, z Suntetagmènec = cos θ = 2 + 2 actan, > 0, 0 = sin θ θ = π +actan, < 0 2π +actan, > 0, < 0 z = z z = z DianÔsmata b shc i =cosθe sin θ e θ e =cosθi +sinθ j j =sinθe +cosθ e θ e θ = sin θ i +cosθ j k = e z e z = k PÐnakac 1.1: Sqèseic metaô katesian n kai kulindik n polik n suntetagmènwn. e θ e j i θ. Sq ma 1.3: Polikèc suntetagmènec, θ sto epðpedo.

4 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc z z e θ, θ, φ z θ e θ e φ φ φ Sq ma 1.4: Sfaiikèc polikèc suntetagmènec, θ, φ me 0, 0 θ π kai 0 φ 2π, kai to di nusma jèshc. basikoð tôpoi gia thn met bash apì tic katesianèc stic kulindikèc suntetagmènec kai antðstofa. An agno soume th suntetagmènh z kai peioistoôme sto epðpedo, oi kulindikèc polikèc suntetagmènec an gontai stic gnwstèc mac polikèc suntetagmènec, θ sto epðpedo pou faðnontai sto Sq ma 1.3. Oi sfaiikèc polikèc suntetagmènec, θ, φ, me 0, 0 θ π kai 0 φ<2π, faðnontai sto Sq ma 1.4 mazð me tic katesianèc suntetagmènec. EpishmaÐnoume ìti ta kai θ stic kulindikèc kai tic sfaiikèc suntetagmènec den eðnai ta Ðdia. H b sh tou sust matoc sfaiik n suntetagmènwn apoteleðtai apì tða ojokanonik dianôsmata: to aktinikì di nusma e,to meshmbinì di nusma e θ, kai to azimoujiakì di nusma e φ. K je di nusma v analôetai monos manta se teic sunist sec, vv,v θ,v φ, oi opoðec eðnai ta mèta twn pobol n tou v sta tða dianôsmata b shc. O metasqhmatismìc enìc dianôsmatoc apì tic sfaiikèc stic katesianèc suntetagmènec kai antðstofa gðnetai me th q sh twn sqèsewn tou PÐnaka 1.2. Pa deigma 1.1.1. B sh tou kulindikoô sust matoc suntetagmènwn Ja deðoume ìti h b sh B={e, e θ, e z } tou kulindikoô sust matoc suntetagmènwn eðnai ojokanonik. Epeid i i = j j = k k=1 kai i j = j k = k i=0, èqoume: e e = cosθ i +sinθ j cos θ i +sinθ j = cos 2 θ +sin 2 θ =1 e θ e θ = sin θ i +cosθ j sin θ i +cosθ j = sin 2 θ +cos 2 θ =1 e z e z = k k =1 e e θ = cosθ i +sinθ j sin θ i +cosθ j = 0 e e z = cosθ i +sinθ j k =0 e θ e z = sin θ i +cosθ j k =0

1.1. Sust mata suntetagmènwn 5, θ, φ,, z,, z, θ, φ Suntetagmènec = sin θ cos φ = 2 + 2 + z 2 actan 2 + 2 z, z > 0 π = sin θ sin φ θ = 2, z =0 π +actan 2 + 2 z, z < 0 actan, > 0, 0 z = cos θ φ = π +actan, < 0 2π +actan, > 0, < 0 DianÔsmata b shc i =sinθcos φ e +cosθcos φ e θ sin φ e φ e =sinθcos φ i +sinθsin φ j +cosθ k j =sinθsin φ e +cosθsin φ e θ +cosφe φ e θ =cosθcos φ i +cosθsin φ j sin θ k k =cosθe sin θ e θ e φ = sin φ i +cosφ j PÐnakac 1.2: Sqèseic metaô katesian n kai sfaiik n polik n suntetagmènwn. Pa deigma 1.1.2. To di nusma jèshc To di nusma jèshc position vecto oðzei th jèsh enìc shmeðou ston q o se sqèsh me èna sôsthma suntetagmènwn. Stic katesianèc suntetagmènec, = i + j + z k, 1.3 kai ètsi = 1 2 = 2 + 2 + z 2. 1.4 H an lush tou stic teic tou sunist sec faðnetai sto Sq ma 1.5. Stic kulindikèc suntetagmènec, to di nusma jèshc dðnetai apì thn = e + z e z me = 2 + z 2. 1.5 AÐzei na shmei soume ìti to mèto tou dianôsmatoc jèshc den eðnai Ðso me thn aktinik kulindik suntetagmènh. Tèloc, stic sfaiikèc suntetagmènec, = e me =, 1.6 dhl. to mèto eðnai h aktinik sfaiik suntetagmènh. An kai oi ekf seic 1.5 kai 1.6 gia to di nusma jèshc eðnai pofaneðc bl. Sq mata 1.2 kai 1.4 ja tic apodeðoume me th q sh metasqhmatism n suntetagmènwn ekin ntac apì thn 1.3. Stic kulindikèc suntetagmènec, = i + j + z k = cos θ cos θ e sin θ e θ +sin θ sin θ e +cosθ e θ +ze z = cos 2 θ +sin 2 θ e + sin θ cos θ +sinθcos θ e θ + z e z = e + z e z.

6 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc z = i + j + z k k i j z Sq ma 1.5: To di nusma jèshc,, se katesianèc suntetagmènec. Stic sfaiikèc suntetagmènec, = i + j + z k = sin θ cos φ sin θ cos φ e +cosθcos φ e θ sin φ e φ + sin θ sin φ sin θ sin φ e +cosθsin φ e θ +cosφe φ + cos θ cos θ e sin θ e θ = [sin 2 θ cos 2 φ +sin 2 φcos 2 θ] e + sin θ cos θ [cos 2 φ +sin 2 φ 1] e θ + sin θ sin φ cos φ +sinφcos φ e φ = e. Pa deigma 1.1.3. Pa gwgoi twndianusm twnb shc Ta dianôsmata b shc i, j kai k twn katesian n suntetagmènwn eðnai staje afoô den eat ntai apì th jèsh enìc shmeðou. Autì den alhjeôei gia ta dianôsmata b shc sta kampulìgamma sust mata suntetagmènwn. Ston PÐnaka 1.1 paathoôme ìti stic kulindikèc suntetagmènec e =cosθ i +sinθ j kai e θ = sin θ i +cosθ j. EÐnai faneì ìti ta e kai e θ eat ntai apì to θ. PaagwgÐzontac bðskoume: e θ = sin θ i +cosθ j = e θ kai e θ θ = cos θ i sin θ j = e.

1.1. Sust mata suntetagmènwn 7 Oi upìloipec qwikèc pa gwgoi twn e, e θ kai e z eðnai mhdenikèc. 'Etsi èqoume: e = 0 e θ = 0 e z = 0 e θ = e θ e θ θ = e e z θ = 0 1.7 e = 0 e θ = 0 e z = 0 Apì ton PÐnaka 1.2 blèpoume ìti e =e θ, φ, e θ =e θ θ, φ kai e φ =e φ φ. Gia tic qwikèc paag gouc twn dianusm twn b shc twn sfaiik n suntetagmènwn èqoume: e = 0 e θ = 0 e φ = 0 e θ = e θ e θ θ = e e φ θ = 0 1.8 e φ = sinθ e φ e θ φ =cosθ e φ e φ φ = sin θ e cos θ e θ Oi eis seic 1.7 kai 1.8 eðnai idiaðtea q simec sth metatop diafoik n telest n apì tic katesianèc se ojog niec kampulìgammec suntetagmènec. UpenjumÐzoume ìti to di nusma thc taqôthtac oðzetai wc e c: Se katesianèc suntetagmènec bðskoume u d. 1.9 u d = d d i + j + zk = i + d j + dz k. 1.10 EÐnai faneì ìti gia tic teic sunist sec thc taqôthtac isqôei u d, u d, u z dz. 1.11 Pa deigma 1.1.4. Sunist sec tou dianôsmatoc thc taqôthtac Ja boôme ti antipospwpeôoun oi sunist sec thc taqôthtac sta sust mata kulindik n kai katesian n suntetagmènwn. Se kulindikèc suntetagmènec, θ, z paagwgðzontac thn 1.5 bðskoume: u d = d e + z e z = d e + de + dz e z = d e + de dθ dθ + dz e z = u = d e + dθ e θ + dz e z 1.12

8 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc 'Aa gia tic sunist sec thc taqôthtac èqoume: u d, u θ dθ, u z dz. 1.13 Se sfaiikèc suntetagmènec, θ, φ, pa'noume apì thn 1.6: u d = d e = d e + de = d e e + θ dθ + e φ dφ = 'Aa oi sunist sec thc taqôthtac dðnontai apì tic u = d e + dθ e θ + sin θ dφ e φ 1.14 u d, u θ dθ, u φ sin θ dφ. 1.15

1.4. DIAFORIKO I TELEST EC 9 1.4 DiafoikoÐ telestèc Sthn pa gafo aut oðzoume ton telest klðshc, thn klðsh gadient bajmwtoô pedðou kaj c kai thn apìklish divegence kai ton stobilismì voticit dianusmatikoô pedðou. Pa deigma 1.4.1. Apìklish kai stobilismìc tou dianôsmatoc jèshc JewoÔme to di nusma jèshc se katesianèc suntetagmènec, Gia thn apìklish kai ton stobilismì tou èqoume = i + j + z k. 1.16 = + + = =3, 1.17 kai = i j k z = = 0 1.18 Oi eis seic 1.17 kai 1.18 isqôoun se k je sôsthma suntetagmènwn. 1.4.1 H ulik pa gwgoc O anagn sthc eðnai dh eoikeiwmènoc me tic ènnoiec thc meik c paag gou patial deivative kai thc olik c paag gou total deivative. Sthn pa gafo aut ja eis gagoume mia llh pa gwgo, polô shmantik sth eustodunamik pou eðnai gnwst san ulik pa gwgoc mateial deivative. Gia to skopì autì ja qhsimopoi soume èna pa deigma apì to fusikì kìsmo gia na katadeðoume th fusik shmasða kai tic diafoèc twn ti n poanafejeis n paag gwn. Ja upojèsoume ìti jèloume na met soume th jemokasða f tou neoô sflèna pot mi. Aut eðnai pofan c sun thsh tou dianôsmatoc jèshc kai tou qìnou, dhlad thc mof c f,t f,, z, t. Sto pa deigm mac upojètoume ìti to bajmwtì pedðo f eðnai paagwgðsimo. Meik qonik pa gwgoc Wc gnwstì gia na boôme th meik qonik pa gwgo thc f,, z, t paagwgðzoume wc poc to qìno t jew ntac ta, kai z staje. SumbolÐzoume th meik qonik pa gwgo me f. t,,z Sto pa deigm mac stekìmaste se mia gèfua sthn ìqjh kai metoôme th metabol thc jemokasðac sto Ðdio shmeðo akib c apì k tw mac, dhlad sflèna stajeì shmeðo tou q ou. Olik qonik pa gwgoc 'Otan to shmeðo mèthshc thc qonik c metabol c thc f den eðnai stajeì, tìte t = t i + t j + zt k. SÔmfwna me ton kanìna thc alusðdac, h olik qonik pa gwgoc thc ft,t,zt,t dðnetai apì thn df = f t + f d + f d + f dz

10 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc = i + j + k 2 = 2 2 + 2 2 + 2 2 u = u + u + u z D Dt = t + u + u + u z p = p i + p j + p k u = u u = + u + u z uz u i + u u z j + u u k u = u ii + u ij + u z ik + u ji u u = τ = + u jj + u z jk + u ki + u kj + u z kk + u u + u u + u u z i + u u z + u u z + u u z z k τ + τ + τ z i + u u + u u + u u z j τ + τ + τ z j + τz + τ z + τ zz k PÐnakac 1.3: BasikoÐ diafoikoð telestèc se katesianèc suntetagmènec,, z; ta p, u kai τ eðnai bajmwtì, dianusmatikì kai tanustikì pedðo, antðstoiqa.

1.4. DiafoikoÐ telestèc 11 u v = u v +v u + u v +v u fu = f u + u f u v = v u u v u = 0 fu =f u + f u u v = u v v u +v u u v u = u 2 u f =0 u u = 2u u +2u u 2 fg = f 2 g + g 2 f +2 f g f g = 0 f g g f =f 2 g g 2 f PÐnakac 1.4: Q simec tautìthtec me ton telest. Ta f kai g eðnai bajmwt en ta u kai v eðnai dianusmatik pedða. NoeÐtai ìti oi meikèc pa gwgoi eðnai suneqeðc.

12 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc = e + e θ 1 θ + e z 2 = 1 + 1 2 2 θ 2 + 2 2 u = u + u θ θ + u z D Dt = t + u + u θ θ + u z p = p e + 1 p θ e θ + p e z u = 1 u + 1 u θ θ u = 1 u z θ u θ e + + u z u u z e θ + u = u e e + u θ e e θ + u z e e z + 1 u θ u θ e θ e u u = τ = [ 1 u θ 1 u ] e θ z + 1 u θ θ + u e θ e θ + 1 u z θ e θe z + u e ze + u θ e ze θ + u z e ze z + + + + [ u u + u 1 u θ θ u θ [ u u θ + u 1 u θ θ θ + u [ u u z + u 1 θ u z θ + u u ] z z e z [ 1 τ + 1 τ θ θ + τ z ] + u u z e ] + u u θ z e θ τ ] θθ e [ 1 2 2 τ θ + 1 τ θθ θ + τ zθ τ ] θ τ θ e θ [ 1 τ z + 1 τ θz θ + τ ] zz e z PÐnakac 1.5: BasikoÐ diafoikoð telestèc se kulindikèc polikèc suntetagmènec, θ, z; ta p, u kai τ eðnai bajmwtì, dianusmatikì kai tanustikì pedðo, antðstoiqa.

1.4. DiafoikoÐ telestèc 13 = e + e θ 1 θ + e φ 1 sin θ φ 2 = 1 2 2 u = u + u θ θ + D Dt = t + u + u θ θ + + 1 2 sin θ sin θ θ θ u φ sin θ φ u φ sin θ φ + 1 2 2 sin 2 θ φ 2 p = p e + 1 p θ e θ + 1 p sin θ φ e φ u = 1 2 2 u + 1 sin θ θ u θ sin θ + 1 u φ sin θ φ u =[ 1 sin θ θ u φ sin θ sin 1 θ u θ φ ]e +[ 1 sin θ u φ 1 u φ]e θ +[ 1 u θ 1 u θ ]e φ u = u e e + u θ e e θ + u φ e e φ + 1 u θ u θ e θ e + 1 u θ θ + u e θ e θ + 1 u φ θ e θe φ + 1 sin θ u φ u φ e φ e + 1 sin θ u θ φ u φ cot θ e φ e θ + 1 sin θ u φ φ + u + u θ cot θ u u =[u u + u 1 u θ θ u θ + u 1 φ sin θ u φ u φ ] e +[u u θ + u 1 u θ θ θ + u + u 1 φ sin θ u θ φ u φ cot θ ] e θ u +[u φ + u 1 θ u φ θ + u 1 u φ φ sin θ φ + u + u θ cot θ ] e φ τ =[ 1 2 2 τ + 1 sin θ θ τ θ sin θ+ 1 +[ 1 3 3 τ θ + 1 sin θ θ τ θθ sin θ+ 1 sin θ +[ 1 3 3 τ φ + 1 sin θ θ τ θφ sin θ+ 1 sin θ τ φ e φ e φ sin θ φ τ θθ + τ φφ ]e τ φθ φ + τ θ τ θ τ φφ cot θ ]e θ τ φφ φ + τ φ τ φ τ φθ cot θ ]e φ PÐnakac 1.6: BasikoÐ diafoikoð telestèc se sfaiikèc polikèc suntetagmènec, θ, φ; ta p, u kai τ eðnai bajmwtì, dianusmatikì kai tanustikì pedðo, antðstoiqa.

14 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc,, z D Dt = t + u + u + u z, θ, z D Dt = t + u + u θ θ + u z, θ, φ D Dt = t + u + u θ θ + u φ sin θ φ PÐnakac 1.7: O telest c thc ulik c paag gou se di foa sust mata suntetagmènwn. ìpou u df = f t + u f, 1.19 = d = d i + d j + dz k h taqôthta me thn opoða kineðtai to shmeðo mèthshc. Sto paadeigm mac, antð na stekìmaste sth gèfua, mpaðnoume se mia mhqanokðnhth b ka kai k noume bìltec poc di foec kateujônseic, llote antðjeta poc to eôma kai llote mazð me to eôma, kai metoôme th jemokasða pleuik thc b kac. H qonik metabol thc jemokasðac pou paathoôme antanakl thn kðnhsh thc b kac. H taqôthta u sthn 1.19 eðnaihtaqôthta thc b kac. Ulik pa gwgoc Upojètoume t a ìti sb noume th mhqan kai af noume to eôma na mac paasôei, en suneqðzoume na metoôme th metabol thc jemokasðac. H qonik metabol thc jemokasðac pou metoôme eat tai apì thn taqôthta u tou eômatoc. Jètontac u =u sthn 1.19 èqoume thn pa gwgo Df Dt = f t + u f, 1.20 h opoða kaleðtai ulik pa gwgoc mateial deivative ousiastik ousi dhc pa gwgoc substantial deivative. O p toc ìoc thc ulik c paag gou apoteleð to metabatikì topikì mèoc kai o deôteoc to metafeìmeno sunagìmeno mèoc thc paag gou. Oi mofèc tou telest thc ulik c paag gou, D Dt = t + u 1.21 sta di foa sust mata suntetagmènwn faðnontai ston PÐnaka??. Gia thn ulik pa gwgo tou dianusmatikoô pedðou thc taqôthtac isqôei Du Dt = u t + u u. 1.22

1.4. DiafoikoÐ telestèc 15 Pa deigma 1.4.2. H eðswsh sunèqeiac H eðswsh sunèqeiac dhl. h eðswsh diat hshc thc m zac eðnai h ρ t + ρ u = 0. 1.23 Qhsimopoi ntac th deôteh tautìthta tou PÐnaka 1.4 h eðswsh paðnei th mof : ρ t + u ρ + ρ u = 0. 1.24 Qhsimopoi ntac ton oismì thc ulik c paag gou èqoume epðshc thn enallaktik mof : Dρ Dt + ρ u =0. 1.25 Pa deigma 1.4.3. Eis seic Eule HeÐswsh diat hshc thc om c gia ani dh o eðnai gnwst san eðswsh Eule: ρ Du Dt = p, 1.26 ìpou ρ h puknìthta kai p h pðesh. Se katesianèc suntetagmènec Du Dt kai = u t + u u u u = + u t + u uz u z + + u t u + u + u u z i + + u u z z k u z u t p = p i + p j + p k. + u u + u u + u z u j Apì tic pio p nw eis seic bðskoume eôkola tic teic sunist sec thc eðswshc Eule se katesianèc suntetagmènec: ρ ρ u t u t + u u + u u + u u + u u + u u z + u z u = p = p 1.27 ρ uz t + u u z + u u z + u u z z = p

16 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc 1.5 Tanustikìc logismìc 'Estw {e 1, e 2, e 3 } mia ojokanonik b sh tou R 3. Stic pohgoômenec paag fouc oðsame to eswteikì ginìmeno, e i e j = δ ij kaj c kai to ewteikì ginìmeno e i e j. To anoiktì ginìmeno e i e j ja to kaloôme monadiaðo ginìmeno du dac 1 apl c monadiaða du da unit dad. AÐzei na paath soume ìti en èna monadiaðo di nusma antiposwpeôei mia mìno dieôjunsh suntetagmènwn, mia monadiaða du da antiposwpeôei èna diatetagmèno zeôgoc dieujônsewn. 'Etsi ij ji. Oi monadiaðec du dec tou katesianoô sust matoc suntetagmènwn ston R 3 eðnai oi e c: ii, ij, ik ji, jj, jk ki, kj, kk 'Estw t a ta dianôsmata a, b R n. To anoiktì ginìmeno ab kaleðtai ginìmeno du dac dad poduct apl c duadikì dadic du da. An tìte a = a 1 i + a 2 j + a 3 k kai b = b 1 i + b 2 j + b 3 k ab = a 1 b 1 ii + a 1 b 2 ij + a 1 b 3 ik + a 2 b 1 ji + a 2 b 2 jj + a 2 b 3 jk + a 3 b 1 ki + a 3 b 2 kj + a 3 b 3 kk. 1.28 PaathoÔme ìti to ginìmeno du dac eðnai ènac gammikìc sunduasmìc twn monadiaðwn du dwn: ab = i=1 j=1 Ja oðsoume t a k poiec shmantikèc p eic metaô monadiaðwn du dwn: iginìmeno monadiaðwn du dwn ginìmeno teleðac: a i b j e i e j. 1.29 ij kl =i j k l = δ jk il 1.30 To apotèlesma thc p hc eðnai duadikì. PaathoÔme epðshc ìti den isqôei h antimetajetik idiìthta. ii Ginìmeno dipl c teleðac bajmwtì ginìmeno: ij :kl =i lj k =δ il δ jk 1.31 To apotèlesma thc p hc aut c eðnai bajmwtì. O anagn sthc mpoeð eôkola na deð ìti kl :ij =ij :kl iii Ginìmeno monadiaðac du dac me monadiaðo di nusma: ij k = i j k =δ jk i 1.32 To ginìmeno du dac me di nusma mac dðnei di nusma. Pofan c den isqôei h antimetajetik idiìthta: k ij =k i j = δ ki j 1.33 iv Ewteikì ginìmeno monadiaðac du dac me monadiaðo di nusma: ij k = i j k 1.34 k ij =k i j 1.35 1 Se aket biblða gia th monadiaða du da qhsimopoieðtai epðshc o sumbolismìc ei e j.

1.5. Tanustikìc logismìc 17 To apotèlesma thc p hc aut c eðnai duadikì. Den isqôei h antimetajetik idiìthta. Stic 1.30-1.35 qhsimopoioôme th legìmenh sunj kh enjèshc twn Chapman kai Milne h opoða jespðzei zeug wma apì mèsa poc ta èw. Oi pio p nw p eic genikeôontai eôkola gia ginìmena du dac. An ta a, b, c kai d eðnai dianôsmata tou R 3,tìte ab cd =a b c d =b c ad 1.36 ab :cd =a db c 1.37 ab c = a b c =b c a 1.38 c ab =c a b 1.39 ab c = a b c 1.40 UpenjumÐzoume ed ìti sth genik peðptwsh den isqôei h antimetajetik idiìthta: ab ba. 'Opwc ja doôme pio k tw, h isìthta isqôei mìno ìtan o duadikìc ab eðnai summetikìc. OÐzoume t a wc duadikì tanust deôtehc t hc second-ode tenso k je gammikì sunduasmì twn monadiaðwn du dwn: τ = τ ij e i e j 1.41 i=1 j=1 EÐnai faneì ìti ta ginìmena du dac eðnai tanustèc deôtehc t hc. K je tanust c τ mpoeð na gafeð sth mof τ =e 1, e 2, e 3 τ 11 τ 12 τ 13 τ 21 τ 22 τ 23 e 1 e 2 τ 31 τ 32 τ 33 e 3 akìma,ìtan oi dieujônseic e i e j ennooôntai, se mof pðnaka τ = τ 11 τ 12 τ 13 τ 21 τ 22 τ 23 1.42 τ 31 τ 32 τ 33 GenikeÔontac, odhgoômaste stouc poluadikoôc tanustèc an tehc t hc. 'Etsi ènac tiadikìc tanust c tðthc t hc eðnai gammikìc sunduasmìc monadiaðwn ti dwn, p.q. B = i=1 β ijk e i e j e k 1.43 j=1 k=1 'Ena bajmwtì pedðo eðnai tanust c mhdenik c t hc en èna dianusmatikì pedðo eðnai tanust c p thc t hc. Oi monadiaðec du dec eðnai fusik tanustèc deôtehc t hc. Autèc g fontai upì mof pðnaka wc e c: ii = 1 0 0 0 0 0, ij = 0 1 0 0 0 0, ik = 0 0 1 0 0 0, 0 0 0 0 0 0 0 0 0 Oi oismoð pou akoloujoôn eðnai oikeðoi apì th jewða pin kwn. 'Estw ab = a i b j e i e j i j

18 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc èna ginìmeno du dac. To ginìmeno du dac pou pokôptei an enall oume touc deðktec twn a kai b sto pio p nw joisma kaleðtai an stofo ginìmeno du dac kai sumbolðzetai me ab T : ab T = a j b i e i e j 1.44 i j EÐnai faneì ìti ab T = ba. OmoÐwc o an stofoc tou tanust deôtehc t hc τ = τ ij e i e j i j eðnai o τ T = i τ ji e i e j. 1.45 An τ = τ T,tìte o τ kaleðtai summetikìc, en ìtan τ = τ T o τ kaleðtai antisummetikìc. EÐnai faneì ìti ènac tanust c thc mof c aa eðnai summetikìc skhsh. j O monadiaðoc tanust c unit tenso deôtehc t hc sumbolðzetai me I kai oðzetai wc e c: I = δ ij e i e j 1.46 i=1 j=1 se mof pðnaka, I = 1 0 0 0 1 0. 0 0 1 1.5.1 'Algeba tanust n 'Opwc kai stouc pðnakec, to joisma dôo tanust n τ = i τ ij e i e j kai σ = j i σ ij e i e j j oðzetai wc e c: τ + σ = τ ij + σ ij e i e j. 1.47 i j EpÐshc, o bajmwtìc pollaplasiasmìc λτ, ìpou λ R, oðzetai wc e c: λ τ = i λτ ij e i e j 1.48 j Ac doôme ek nèou tic basikèc p eic metaô tanust n. itanustikì ginìmeno ginìmeno teleðac To tanustikì ginìmeno ginìmeno teleðac tenso o dot poduct dôo tanust n τ kai σ mac dðnei kat ta gnwst τ σ = τ ij e i e j σ kl e k e l = τ ij σ kl e i e j e k e l i j k l i j k l = τ ij σ kl δ jk e i e l = τ ij σ jl e i e l = i j i j k l l

1.5. Tanustikìc logismìc 19 τ σ = i j l τ ij σ jl e i e l. 1.49 PaÐnoume loipìn epðshc èna tanust tou opoðou h i, l sunist sa eðnai h τ ij σ jl j 'Opwc kai stouc tetagwnikoôc pðnakec, g foume σ σ = σ 2, σ σ 2 = σ 3 k.o.k. H antimetajetik idiìthta den isqôei sto tanustikì ginìmeno. An I eðnai o monadiaðoc tanust c tou mpooôme polô eôkola na deðoume ìti σ I = I σ = σ. 1.50 ii Ginìmeno dipl c teleðac bajmwtì ginìmeno Gia to legìmeno ginìmeno dipl c teleðac bajmwtì ginìmeno double dot o scala poduct dôo tanust n èqoume: τ : σ = τ ij e i e j : σ kl e k e l = τ ij σ kl e i e j : e k e l i j k l i j k l = τ ij σ kl δ il δ jk = τ ij σ ji δ ii δ jj = i j i j k l τ : σ = i τ ij σ ji. 1.51 j To apotèlesma thc p hc eðnai bajmwtì. Me paìmoio tìpo mpooôme na deðoume ìti τ : ab = τ ij a j b i 1.52 i j kai ab : cd = i a i b j c j d i 1.53 j iii Ginìmeno tanust - dianôsmatoc 'Estw τ tanust c kai a di nusma. Gia to ginìmeno τ a èqoume: τ a = τ ij e i e j a k e k = τ ij a k e i e j e k i j k i j k = τ ij a k δ jk e i = τ ij a j δ jj e i = i j i j k τ a = i j τ ij a j e i. 1.54

20 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc To apotèlesma thc p hc eðnai èna di nusma me i sunist sa thn τ ij a j. OmoÐwc, gia to ginìmeno a τ bðskoume ìti j a τ = i j a j τ ji e i. 1.55 PaathoÔme ìti genik τ a a τ. Hisìthta isqôei p nta ìtan o τ eðnai summetikìc. 1.5.2 Tanustèc sth eustodunamik O tanust c t sewn Sth eustodunamik, o tanust c iwd n t sewn viscousstesstenso pou ja ton sumbolðzoume me τ, τ = τ ij e i e j, 1.56 i j antiposwpeôei tic i deic t seic sflèna eustì. Stic katesianèc suntetagmènec èqoume thn pinakomof τ = τ 11 τ 12 τ 13 τ 21 τ 22 τ 23 τ = τ τ τ z τ τ τ z 1.57 τ 31 τ 32 τ 33 τ z τ z τ zz Otanust c t sewn eðnai summetikìc, dhlad τ = τ, τ z = τ z kai τ z = τ z. Oi diag niec sunist sec, τ,τ kai τ zz, kaloôntai k jetec t seic nomal stesses, en oi ewdiag niec sunist sec tou tanust t sewn kaloôntai diatmhtikèc t seic shea stesses. O olikìc tanust c t sewn σ total stess tenso oðzetai wc e c: σ = p I + τ 1.58 ìpou p h pðesh kai I omonadiaðoc tanust c. EÐnai faneì ìti o σ eðnai epðshc summetikìc. Otanust c p I kaleðtai tanust c t sewn pðeshc pessue stess tenso. EÐnai isìtopoc isotopic giatð ìtan d p nw se mia epif neia mac dðnei èlh taction epif neiac. P gmati, an n eðnai to monadiaðo k jeto se mia epif neia di nusma tìte n pi = pn I = pn = p Autì den sumbaðnei me ton tanust t sewn τ pou eðnai anisìtopoc. Oolikìc tanust c t sewn mpoeð na gafteð epðshc sth mof : σ = p 0 0 0 p 0 + τ 0 0 0 τ 0 + 0 τ τ z τ 0 τ z 1.59 0 0 p 0 0 τ zz τ z τ z 0 ìpou o p toc pðnakac dðnei tic t seic pðeshc, o deôteoc tic i deic k jetec t seic kai o tðtoc tic i deic diatmhtikèc t seic.

1.5. Tanustikìc logismìc 21 O tanust c klðsewn thc taqôthtac O tanust c klðsewn thc taqôthtac velocit-gadient tenso sumbolðzetai me u kai eðnai èna ginìmeno du dac. Stic katesianèc suntetagmènec bðskoume apl : u = i + j + k u i + u j + u z k = u = u ii + u ij + u z ik + u ji + u jj + u z jk + u ki + u kj + u z kk. 1.60 Pio sôntoma mpooôme na g oume u = i=1 j=1 u j i e i e j 1.61 se mof pðnaka u u u z u = u u u z. 1.62 u u u z Egazìmenoi an loga mpooôme na boôme ton tanust klðsewn thc taqôthtac se kulindikèc kai sfaiikèc suntetagmènec. Posoq : o tôpoc 1.61 den isqôei sflaut ta sust mata. Oi tanustèc ujm n paamìfwshc kai stobilismoô 'Opwc kai k je lloc tanust c deôtehc t hc, o tanust c klðsewn thc taqôthtac u mpoeð na gafteð wc joisma enìc summetikoô kai enìc antisummetikoô tanust wc e c: u = 1 2 [ u+ ut ]+ 1 2 [ u ut ] 1.63 ìpou to p to hmi joisma eðnai summetikìc tanust c kai to deôteo antisummetikìc tanust c. O summetikìc tanust c D = 1 2 [ u + ut ] 1.64 kaleðtai tanust c ujm n paamìfwshc ate of stain o ate of defomation tenso en o antisummetikìc tanust c Ω = 1 2 [ u ut ] 1.65 kaleðtai tanust c stobilismoô voticit tenso. SÔmfwna me touc pio p nw oismoôc mpooôme na g oume u = D + Ω. 1.66 O tanust c ujm n paamìfwshc D antiposwpeôei tic paamof seic tou eustoô kai eðnai mhdenikìc ìtan èqoume met jesh peistof steeoô s matoc solid bod tanslation o otation, afoô tìte den èqoume paamof seic. O tanust c stobilismoô Ω antiposwpeôei to stobil dec thc o c kai ètsi eðnai mhdenikìc se astìbilec oèc. Gia ton tanust ujm n paamìfwshc se katesianèc suntetagmènec èqoume D = 1 u j e i e j + u i e i e j = 2 i j i i j j

22 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc se mof pðnaka: D = 1 2 1 2 u D = i u + u u + u z j 1 uj + u i e i e j 1.67 2 i j 1 u 2 + u u 1 u 2 + u z 1 uz 2 + u 1 u 2 + u z u z. 1.68 EÐnai faneì ìti oi sunist sec tou D eðnai: D = u D = u D zz = u z D = D = 1 u 2 + u D z = D z = 1 uz 2 + u D z = D z = 1 uz 2 + u 1.69 OmoÐwc gia ton tanust stobilismoô bðskoume ìti se mof pðnaka Ω = 1 u 2 1 2 Ω = i j 1 2 uj i u i j 0 1 u 2 u 1 uz 2 u u 0 1 uz 2 u u u z 1 u 2 u z 0 e i e j 1.70. 1.71 Pa deigma 1.5.1 Gia na boôme touc u kai D se kulindikèc suntetagmènec, jewoôme to u wc du da sflautì to sôsthma: u = e + e θ 1 θ + e z u e + u θ e θ + u z e z. Lamb nontac upìh ìti ta monadiaða dianôsmata den eðnai staje sflautì to sôsthma suntetagmènwn bðskoume ìti u = e e u + e θ e 1 + e z e u + e u θ e θ + e u z e z u θ + e 1 θ u e θ + e 1 θe θ + e ze θ u θ + e ze z u z e θ θ = + e 1 θ u e θ θ θ + e 1 θe z u z θ

1.5. Tanustikìc logismìc 23 u = u e e + u θ e e θ + u z e e z + 1 u θ u θ e θ e + 1 u + u θ e θ e θ + 1 θ Gia ton an stofo u T èqoume: u z θ e θe z + u e ze + u θ e ze θ + u z e ze z. 1.72 u T = u e e + 1 + u θ e θe + 1 + u z e ze + 1 u θ u θ e e θ + u e e z u + u θ e θ e θ + u θ θ e θe z u z θ e ze θ + u z e ze z. 1.73 MpooÔme t a me b sh ton oismì na boôme tic sunist sec tou tanust ujm n paamìfwshc se kulindikèc suntetagmènec: D = u D θθ = 1 u + u θ θ D zz = u z D z = D z = 1 uz 2 + u D θ = D θ = 1 [ uθ + 1 ] u 2 θ D θz = D zθ = 1 uθ 2 + 1 u z θ 1.74 Pa deigma 1.5.2. H katastatik eðswsh Neut neiou eustoô H ulik katastatik eðswsh constitutive equation enìc eustoô eðnai h sun thsh pou apeikonðzei ton tanust ujm n paamìfwshc D ston tanust iwd n t sewn τ : τ = fd. 1.75 Sta Neut neia eust Newtonian fluids, h ulik sqèsh eðnai thc mof c τ =2ηD + k 23 η ui 1.76 ìpou η kai k stajeèc kai u to di nusma thc taqôthtac. Hstaje η kaleðtai diatmhtikì i dec shea viscosit apl c i dec en h k pou èqei pofan c tic Ðdiec mon dec me to i dec kaleðtai mazikì i dec bulk viscosit. To mìno pou gnwðzoume gia to mazikì i dec k eðnai ìti h tim tou eðnai akib c mhdèn gia monoatomik aèia qamhl c puknìthtac. Se ìlec tic llec efamogèc, lìgw èlleihc epilog n, to mazikì i dec tðjetai aujaðeta! Ðso me mhdèn R.B. Bid, R.C. Amstong and O. Hassage, Dnamics of Polmeic Liquids, John Wile & Sons, New Yok, 1987. 'Etsi h 1.76 gðnetai: τ =2ηD 2 η ui 1.77 3

24 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc H pio p nw eðswsh aplopoieðtai akìma peissìteo ìtan h o eðnai asumpðesth, dhlad ìtan h puknìthta ρ tou eustoô eðnai staje. Sthn peðptwsh aut, h eðswsh sunèqeiac paðnei th mof u=0, opìte: τ =2η D. 1.78 H 1.78 apoteleð thn ulik sqèsh Neut neiou eustoô se asumpðesth o. Me b sh ta apotelèsmata aut c thc paag fou, mpooôme polô eôkola na boôme tic sunist sec thc 1.78 se katesianèc kai kulindikèc suntetagmènec: Katesianèc suntetagmènec: τ = 2η u τ = 2η u τ zz = 2η u z u τ = τ = η + u uz τ z = τ z = η + u uz τ z = τ z = η + u 1.79 Kulindikèc suntetagmènec: τ = 2η u τ θθ = 2η u + u θ θ τ zz = 2η u z uz τ z = τ z = η + u [ τ θ = τ θ = η uθ + 1 ] u θ uθ τ θz = τ zθ = η + 1 u z θ 1.80 O tanust c metafeìmenhc om c O tanust c metafeìmenhc om c ρ uu = ρ u i u j e i e j 1.81 i=1 j=1

1.5. Tanustikìc logismìc 25 eðnai èna ginìmeno du dac. O summetikìc autìc tanust c g fetai se mof pðnaka wc e c ρ uu = ρ u 2 u u u u z 2 u u u u u z. 1.82 2 u z u u z u u z 1.5.3 Apìklish tanustikoô pedðou Ja bìume p ta thn apìklish tou duadikoô ab = j a j b k e j e k 1.83 k ìpou a kai b dianôsmata tou R 3, bðskontac to ginìmeno ab ìpou = i e i i 1.84 o telest c klðshc se katesianèc suntetagmènec: ab = = = i i j e i i j k k j a j b k e j e k = k a j b k i e i e j e k == a j b k j e k = k j i i a j b k j j j e k e i a j b k e j e k i k k a j b k i δ ij e k ab = i j a j b i j e i. 1.85 Me paìmoio tìpo mpooôme na boôme thn apìklish enìc tanust τ se katesianèc suntetagmènec: τ = i j τ ji j e i. 1.86 H i sunist sa tou τ eðnai j τ ji j. Pa deigma 1.5.3. H eðswsh diat hshc thc om c H eðs sh diat hshc thc om c gia k je eustì g fetai se dianusmatik mof wc e c: ρ Du Dt = p + τ + ρ g 1.87 ìpou g to di nusma thc epit qunshc thc baôthtac, g = g i + g j + g z k 1.88

26 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc EÐdame pohgoumènwc ìti stic katesianèc suntetagmènec Du Dt = u t + u u + u u + u u u z i+ t + u Apì thn E. 1.86 bðskoume ìti τ τ = + τ + τ z i + + u + u u + u z uz t + u u z + u u z + u z u j u z k 1.89 τ + τ + τ z τz j + + τ z + τ zz k. 1.90 MpooÔme loipìn na boôme tic teic sunist sec thc E. 1.87 antikajist ntac tic 1.88, 1.89 kai 1.90: -sunist sa: u ρ t + u u + u u + u u z = p + τ + τ + τ z + ρg -sunist sa: u ρ t + u u + u u + u u z = p + τ + τ + τ z + ρg z-sunist sa: uz ρ t + u u z + u u z + u u z z = p + τ z + τ z + τ zz + ρg z 1.91 GnwÐzoume ìti sthn peðptwsh Neut neiou eustoô se asumpðesth o u =0 o tanust c t sewn dðnetai apì thn τ =2η D = η [ u + u T ] 1.92 O anagn sthc mpoeð na deðei ìti isqôei tìte skhsh τ = η 2 u. 1.93 Antikajist ntac sthn 1.87 paðnoume thn eðswsh twn Navie-Stokes: Oi sunist sec thc 1.94 se katesianèc suntetagmènec eðnai: ρ Du Dt = p + η 2 u + ρ g. 1.94 -sunist sa: u ρ t + u u + u u + u u z = p 2 + η u 2 + 2 u 2 + 2 u 2 + ρg -sunist sa: u ρ t + u u + u u + u u z = p 2 + η u 2 + 2 u 2 + 2 u 2 + ρg z-sunist sa: uz ρ t + u u z + u u z + u u z z = p 2 + η u z 2 + 2 u z 2 + 2 u z 2 + ρg z 1.95

1.5. Tanustikìc logismìc 27 Oi pio p nw eis seic sumplh nontai me thn eðswsh sunèqeiac gia asumpðesth o : u + u + u z =0 1.96 'Etsi èqoume èna sôsthma tess wn meik n diafoik n eis sewn pou antistoiqoôn sta tèssea bajmwt pedða: u, u, u z kai p. Pa deigma 1.5.4. H apoklish tou telest metafeìmenhc om c Ja boôme thn apìklish tou tanust metafeìmenhc om c ρuu gia asumpðesth o. ρuu = ρ uu = ρ u i u j e i i j j = ρ u j u i u i + u j e i i j j j = ρ u j u i e i + ρ i j j i j = = ρ u i u j e i + ρ i j j i = ρ i u i ue i + ρ u u = u j u i j e i j u j u i j e i ρ uu = ρ u u+ρ u u. 1.97 Gia asumpðesth o, u=0 kai ètsi o p toc ìoc sto deiì mèloc thc 1.97 mhdenðzetai. 'Aa isqôei ρ uu = ρ u u. 1.98 'Etsi, gia asumpðesth o oi eis seic diat hshc thc om c paðnoun epðshc th mof ρ u t en gia asumpðesth Neut neia o isqôei + ρuu = p + τ + ρ g 1.99 ρ u t + ρuu = p + η 2 u + ρ g. 1.100 1.5.4 PwteÔsousec dieujônseic kai analloðwtec tanust Let {e 1, e 2, e 3 } be an othonomal basis of the thee dimensional space and τ be a second-ode tenso, τ = i=1 τ ij e i e j, 1.101 j=1 o, in mati notation, τ = τ 11 τ 12 τ 13 τ 21 τ 22 τ 23. 1.102 τ 31 τ 32 τ 33

28 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc If cetain conditions ae satisfied, it is possible to identif an othonomal basis {n 1, n 2, n 3 } such that τ = λ 1 n 1 n 1 + λ 2 n 2 n 2 + λ 3 n 3 n 3, 1.103 which means that the mati fom of τ in the coodinate sstem defined b the new basis is diagonal: τ = λ 1 0 0 0 λ 2 0. 1.104 0 0 λ 3 The othogonal vectos n 1, n 2 and n 3 that diagonalize τ ae called the pincipal diections, andλ 1, λ 2 and λ 3 ae called the pincipal values of τ. Fom Eq. 1.105, one obseves that the vecto flues though the suface of unit nomal n i, i=1,2,3, satisf the elation f i = n i τ = τ n i = λ i n i, i =1, 2, 3. 1.105 What the above equation sas is that the vecto flu though the suface with unit nomal n i is collinea with n i, i.e., n i τ is nomal to that suface and its tangential component is zeo. Fom Eq. 1.107 one gets: τ λ i I n i = 0, 1.106 whee I is the unit tenso. In mathematical teminolog, Eq. 1.108 defines an eigenvalue poblem. The pincipal diections and values of τ ae thus also called the eigenvectos and eigenvalues of τ, espectivel. The eigenvalues ae detemined b solving the chaacteistic equation, o detτ λi = 0 1.107 τ 11 λ τ 12 τ 13 τ 21 τ 22 λ τ 23 τ 31 τ 32 τ 33 λ =0, 1.108 which guaantees nonzeo solutions to the homogeneous sstem 1.108. The chaacteistic equation is a cubic equation and, theefoe, it has thee oots, λ i, i=1,2,3. Afte detemining an eigenvalue λ i, one can detemine the eigenvectos, n i, associated with λ i b solving the chaacteistic sstem 1.108. When the tenso o mati τ is smmetic, all eigenvalues and the associated eigenvectos ae eal. This is the case with most tensos aising in fluid mechanics. Eample 1.5.5. Pincipal values and diections a Find the pincipal values of the tenso τ = 0 z 0 2 0. z 0 b Detemine the pincipal diections n 1, n 2, n 3 at the point 0,1,1. c Veif that the vecto flu though a suface nomal to a pincipal diection n i is collinea with n i. d What is the mati fom of the tenso τ in the coodinate sstem defined b {n 1, n 2, n 3 }? Solution: a The chaacteistic equation of τ is λ 0 z 0=detτ λi = 0 2 λ 0 z 0 λ =2 λ λ z z λ =

1.5. Tanustikìc logismìc 29 2 λ λ z λ + z =0. The eigenvalues of τ ae λ 1 =2, λ 2 = z and λ 3 = + z. batthepoint0, 1, 1, 0 0 1 τ = 0 2 0 = ik +2jj + ki, 1 0 0 and λ 1 =2, λ 2 = 1andλ 3 =1. The associated eigenvectos ae detemined b solving the coesponding chaacteistic sstem: τ λ i I n i = 0, i =1, 2, 3. Fo λ 1 =2, one gets 0 2 0 1 0 2 2 0 1 0 0 2 n 1 n 1 = n z1 0 0 0 = 2n 1 + n z1 =0 0=0 n 1 2n z1 =0 = n 1 = n z1 =0. Theefoe, the eigenvectos associated with λ 1 ae of the fom 0,a,0, whee a is an abita nonzeo constant. Fo a=1, the eigenvecto is nomalized, i.e. it is of unit magnitude. We set n 1 =0, 1, 0 = j. Similal, solving the chaacteistic sstems 0+1 0 1 0 2+1 0 1 0 0+1 n 2 n 2 = n z2 0 0 0 of λ 2 = 1, and of λ 3 =1, we find the nomalized eigenvectos 0 1 0 1 0 2 1 0 n 3 n 3 = 0 0 1 0 0 1 n z3 0 n 2 = 1 2 1, 0, 1 = 1 2 i k and n 3 = 1 2 1, 0, 1 = 1 2 i + k. We obseve that the thee eigenvectos, n 1 n 2 and n 3 ae othogonal: 2 n 1 n 2 = n 2 n 3 = n 3 n 1 =0. c The vecto flues though the thee sufaces nomal to n 1 n 2 and n 3 ae: n 1 τ = j ik +2jj + ki =2j =2n 1, n 2 τ = 1 i k ik +2jj + ki = 2 1 k i = n 2, 2 n 3 τ = 1 i + k ik +2jj + ki = 2 1 k + i =n 3. 2 2 A well known esult of linea algeba is that the eigenvectos associated with distinct eigenvalues of a smmetic mati ae othogonal. If two eigenvalues ae the same, then the two lineal independent eigenvectos detemined b solving the coesponding chaacteistic sstem ma not be othogonal. Fom these two eigenvectos, howeve, a pai of othogonal eigenvectos can be obtained using the Gam-Schmi othogonalization pocess; see, fo eample, [3].

30 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc d The mati fom of τ in the coodinate sstem defined b {n 1, n 2, n 3 } is τ =2n 1 n 1 n 2 n 2 + n 3 n 3 = 2 0 0 0 1 0. 0 0 1 The tace, tτ,ofatensoτ is defined b tτ τ ii = τ 11 + τ 22 + τ 33. 1.109 i=1 An inteesting obsevation fo the tenso τ of Eample 1.5.5 is that its tace is the same equal to 2 in both coodinate sstems defined b {i, j, k} and {n 1, n 2, n 3 }. Actuall, it can be shown that the tace of a tenso is independent of the coodinate sstem to which its components ae efeed. Such quantities ae called invaiants of a tenso. 3 Thee ae thee independent invaiants of a second-ode tenso τ : I tτ = II tτ 2 = III tτ 3 = τ ii, 1.110 i=1 i=1 j=1 τ ij τ ji, 1.111 i=1 j=1 k=1 τ ij τ jk τ ki, 1.112 whee τ 2 =τ τ and τ 3 =τ τ 2. Othe invaiants can be fomed b simpl taking combinations of I, II and III. Anothe common set of independent invaiants is the following: I 1 = I = tτ, 1.113 I 2 = 1 2 I2 II = 1 2 [tτ 2 tτ 2 ], 1.114 I 3 = 1 6 I3 3I II +2III = detτ. 1.115 I 1, I 2 and I 3 ae called basic invaiants of τ. The chaacteistic equation of τ canbewittenas 4 λ 3 I 1 λ 2 + I 2 λ I 3 =0. 1.116 If λ 1, λ 2 and λ 3 ae the eigenvalues of τ, the following identities hold: I 1 = λ 1 + λ 2 + λ 3 = tτ, 1.117 I 2 = λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 = 1 2 [tτ 2 tτ 2 ], 1.118 I 3 = λ 1 λ 2 λ 3 = detτ. 1.119 The theoem of Cale-Hamilton states that a squae mati o a tenso is a oot of its chaacteistic equation, i.e., τ 3 I 1 τ 2 + I 2 τ I 3 I = O. 1.120 v. 3 Fom a vecto v, onl one independent invaiant can be constucted. This is the magnitude v= v v of 4 The component matices of a tenso in two diffeent coodinate sstems ae simila. An impotant popet of simila matices is that the have the same chaacteistic polnomial; hence, the coefficients I 1, I 2 and I 3 and the eigenvalues λ 1, λ 2 and λ 3 ae invaiant unde a change of coodinate sstem.

1.5. Tanustikìc logismìc 31 Note that in the last equation, the boldface quantities I and O ae the unit and zeo tensos, espectivel. As implied b its name, the zeo tenso is the tenso whose all components ae zeo. Eample 1.5.6. The fist invaiant Conside the tenso τ = 0 0 1 0 2 0 = ik +2jj + ki, 1 0 0 encounteed in Eample 1.5.5. Its fist invaiant is I tτ = 0+2+0 = 2. Veif that the value of I is the same in clindical coodinates. Solution: Using the elations of Table 1.1, we have τ = ik +2jj + ki = cosθ e sin θ e θ e z +2sinθe +cosθ e θ sinθ e +cosθ e θ + e z cos θ e sin θ e θ = 2 sin 2 θ e e +2sinθcos θ e e θ +cosθe e z + 2sinθ cos θ e θ e +2cos 2 θ e θ e θ sin θ e θ e z + cos θ e z e sin θ e z e θ +0e z e z. Theefoe, the component mati of τ in clindical coodinates {e, e θ, e z } is 2sin 2 θ 2sinθ cos θ cos θ τ = 2sinθ cos θ 2cos 2 θ sin θ. cos θ sin θ 0 Notice that τ emains smmetic. Its fist invaiant is I = tτ =2 sin 2 θ +cos 2 θ +0=2, as it should be. 1.5.5 Sumbolismìc deikt n kai h sômbash joishc So fa, we have used thee diffeent was fo epesenting tensos and vectos: a the compact smbolic notation, e.g., u fo a vecto and τ fo a tenso; b the so-called Gibbs notation, e.g., u i e i i=1 and τ ij e i e j i=1 j=1 fo u and τ, espectivel; and c the mati notation, e.g., fo τ. τ = τ 11 τ 12 τ 13 τ 21 τ 22 τ 23 τ 31 τ 32 τ 33

32 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc Ve fequentl, in the liteatue, use is made of the inde notation and the so-called Einstein s summation convention, in ode to simplif epessions involving vecto and tenso opeations b omitting the summation smbols. In inde notation, a vecto v is epesented as v i v i e i = v. 1.121 i=1 Atensoτ is epesented as τ ij τ ij e i e j = τ. 1.122 i=1 j=1 The nabla opeato, fo eample, is epesented as i i=1 i e i = i + j + k =, 1.123 whee i is the geneal Catesian coodinate taking on the values of, and z. The unit tenso I is epesented b Konecke s delta: δ ij δ ij e i e j = I. 1.124 i=1 j=1 It is evident that an eplicit statement must be made when the tenso τ ij is to be distinguished fom its i, j element. With Einstein s summation convention, if an inde appeas twice in an epession, then summation is implied with espect to the epeated inde, ove the ange of that inde. The numbe of the fee indices, i.e., the indices that appea onl once, is the numbe of diections associated with an epession; it thus detemines whethe an epession is a scala, a vecto o a tenso. In the following epessions, thee ae no fee indices, and thus these ae scalas: u i v i τ ii u i i 2 f i i u i v i = u v, 1.125 i=1 τ ii = tτ, 1.126 i=1 i=1 o u i i 2 f 2 i = u i=1 + u 2 f 2 i + u z = 2 f 2 = u, 1.127 + 2 f 2 + 2 f 2 = 2 f, 1.128 whee 2 is the Laplacian opeato tobediscussedinmoedetailinsection1.4. Inthefollowing epession, thee ae two sets of double indices, and summation must be pefomed ove both sets: σ ij τ ji σ ij τ ji = σ : τ. 1.129 i=1 j=1

1.5. Tanustikìc logismìc 33 The following epessions, with one fee inde, ae vectos: ɛ ijk u i v j ɛ ijk u i v j e k = u v, 1.130 f i τ ij v j u i t τ ji j u j u i j k=1 f e i i i=1 i=1 j=1 i=1 j=1 = f i + f j + f k = f, 1.131 τ ij v j e i = τ v 1.132 = u i e i = u 1.133 t t i = τ ji e i = τ 1.134 i j j = u i u j e i = u u. 1.135 i j j Finall, the following quantities, having two fee indices, ae tensos: u i v j σ ik τ kj u j i i=1 j=1 u i v j e i e j = uv, 1.136 i=1 j=1 i=1 j=1 σ ik τ kj e i e j = σ τ, 1.137 k=1 Note that u in the last equation is a dadic tenso. 5 It is eas to show that the continuit and momentum equations, u j i e i e j = u. 1.138 and in inde notation become and ρ ρ t u t + u u ui ρ t + ρ u = 0 1.139 = p + τ + ρ g, 1.140 ρ t + ρu i i = 0 1.141 u i + u j j = p i + τ ji j + ρg i. 1.142 5 Some authos use even simple epessions fo the nabla opeato. Fo eample, u is also epesented as iu i o u i,i, with a comma to indicate the deivative, and the dadic u is epesented as iu j o u i,j.

34 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc 1.6 Oloklhwtik jew mata Sthn pa gafo aut, ja suzht soume dôo jemeli dh jew mata thc dianusmatik c an lushc: to je hma tou Stokes kai to je hma thc apìklishc divegence theoem je hma tou Gauss. 1.6.1 To je hma tou Stokes JewoÔme ìti o anagn sthc eðnai eoikeiwmènoc me thn ènnoia tou posanatolismoô miac fagmènhc epif neiac S kaj c kai tou kleistoô sunìou thc S. Mia tètoia epif neia faðnetaisto Sq ma 1.6. 'Estw F èna dianusmatikì pedðo. UpenjumÐzoume ìti to epifaneiakì olokl wma F ds S kaleðtai o tou F diamèsou thc epif - neiac S, en to epikampôlio olokl wma F d S n S S Sq ma 1.6: Posanatolismènh fagmènh e- pif neia me posanatolismèno sônoo. F d S kaleðtai kuklofoða tou F gôw apì thn kleist kampôlh S. To je hma tou Stokes mac lèei ìti ho stobilismoô enìc dianusmatikoô pedðou F di mèsou miac epif neiac S isoôtai me thn kuklofoða tou F gôw apì to sônoo S thc S. Je hma 1.6.1 Je hma Stokes Stokes theoem 'Estw S mia posanatolismènh epif neia kai S to epðshc posanatolismèno thc sônoo. An to F eðnai èna leðo dianusmatikì pedðo, tìte F ds = F d 1.143 S S Paat hsh Epeid ds=nds, ìpou n to monadiaðo k jeto di nusma sthn epif neia S bl. Sq ma 1.6 kai d=tds, ìpou t to monadiaðo efaptomenikì di nusma sthn kampôlh S kai ds to stoiqei dec m koc tìou, h 1.143 g fetai epðshc wc e c: S F n ds = S F t ds 1.144 H sqèsh 1.144 mac lèei ìti to olokl wma thc k jethc sunist sac tou stobilismoô enìc dianusmatikoô pedðou F p nw se mia epif neia S isoôtai me to olokl wma thc efaptomenik c sunist sac tou F p nw sto posanatolismènosônoo S thc S. Pa deigma 1.6.1 Ja deðoume me dôo tìpouc ìti to dianusmatikì pedðo F = e z i + e z j + e z k eðnai sunthhtikì consevative. 1oc tìpoc

1.6. Oloklhwtik jew mata 35 AkeÐ na deðoume ìti h kuklofoða tou F gôw apì opoiad pote kleist kampôlh C eðnai mhdèn: F d =0. C 'Estw loipìn mia tuqoôsa kleist kampôlh C kai S mia epif neia pou èqei san sônoo th C, dhl. S=C. SÔmfwna me to je hma tou Stokes me thn poôpìjesh ìti oi C kai S eðnai posanatolismènec èqoume: F d = F ds C S 'Omwc F = i j k e z e z e z =e z e z i e z e z j +e z e z k = 0. Sunep c: 'Aa to pedðo F eðnai sunthhtikì. S F d =0. 2oc tìpoc Ja deðoume ìti to F eðnai pedðo klðsewn, dhl. ìti up qei bajmwtì pedðo φ,, z tètoio stef= φ. PaathoÔme ìti èna tètoio pedðoeðnai to φ = e z. 'Aa to F eðnai sunthhtikì. 1.6.2 To je hma thc apìklishc 'Estw Ω èna tisdi stato fagmèno qwðo kai Ω h kleist epif neia pou to f ssei. W- c gnwstì, epif neiec aut c thc mof c posanatolðzontai ètsi ste to monadiaðo k jeto di nusma na deðqnei poc ta èw, ìpwc faðnetai sto diplanì sq ma. n Ω Posanatolismènh kleist epif neia. To je hma thc apìklishc mac lèei ìti to qwikì olokl wma thc apìklishc enìc dianusmatikoô pedðou F sflèna tisdi stato fagmèno qwðo Ω eðnai Ðso metho tou F di mèsou thc epif neiac Ω tou Ω. Je hma 1.6.2 Je hma thc apìklishc je hma tou Gauss 'Estw Ω èna tisdi stato fagmèno qwðo kai Ω h posanatolismènh kleist epif neia pou f ssei to Ω. An F eðnai èna leðo dianusmatikì pedðo oismèno sto Ω, tìte FdV = F ds 1.145 Ω Ω Epeid ds = nds, h 1.145 paðnei epðshc th mof FdV = F n ds 1.146 Ω Ω

36 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc To je hma thc apìklishc epekteðnetai kai se tanustik pedða. Je hma 1.6.3 Je hma thc apìklishc gia tanustèc 'Estw Ω èna tisdi stato fagmèno qwðo kai Ω h posanatolismènh kleist epif neia pou f ssei to Ω. An τ eðnai èna leðo tanustikì pedðo oismèno sto Ω, tìte τ dv = τ n ds 1.147 Ω Ω Pa deigma 1.6.2 Ja upologðsoume to olokl wma S F nds ìpou F =2i + 2 j + z 2 k kai S h epif neia thc monadiaðac sfaðac pou oðzetai apì thn 2 + 2 + z 2 =1. Apì to je hma tou Gauss èqoume FdV = F n ds Ω ìpou Ω hmonadiaða sfaða. O apeujeðac upologismìc tou epifaneiakoô oloklh matoc eðnai boloc. PotimoÔme loipìn naupologðsoume to qwikì olokl wma. F n ds = FdV = 21 + + z dv =2 dv +2 dv +2 zdv. S Ω Ω Ω Ω Ω Lìgw summetðac, ta dôo teleutaða oloklh mata mhdenðzontai. Sunep c F n ds =2 dv =2V = 8π 3, S Ω S afoô h monadiaða sfaða èqei ìgko 4π/3. Pa deigma 1.6.3 'Estw F kai G dianusmatik pedða kl shc C 1 sto tisdi stato qwðo Ω pou f ssetai apì thn apl kleist epif neia Ω. JewoÔme ìti se k je shmeðo thc Ω to dianusmatikì pedðo F G eðnai efaptìmeno thc Ω. Ja deðoume ìti F GdV = G FdV 1.148 Apìdeih Apì th dianusmatik tautìthta èqoume: Apì to je hma tou Gauss paðnoume: F G dv = Ω Ω Ω Ω F G =G F F G 1.149 F G dv = [G F F G] dv Ω Ω F G ds = To pedðo F G ef ptetai thc Ω, a autì eðnai k jeto sto n: F G n =0. Ω F G n ds

1.6. Oloklhwtik jew mata 37 PaathoÔme ìti to epifaneiakì olokl wma mhdenðzetai kai ètsi 0= F G dv = [G F F G] dv = Ω Ω F GdV = Ω Ω G F dv. Pa deigma 1.6.4. H eðswsh sunèqeiac JewoÔme èna tisdi stato stajeì kai fagmèno ìgko eustoô Ω sflèna pedðo o c. O ujmìc o c m zac eustoô diamèsou thc Ω eðnai dm = ρ u n ds Ω ìpou m h m za, ρ h puknìthta kai u to di nusma thc taqôthtac. Apì to je hma thc apìklishc èqoume: dm = ρ u n ds = ρu dv. 1.150 Ω Ω To apotèlesma autì eðnai idiaðtea q simo sthn apìdeih thc eðswshc sunèqeiac. MpooÔme p ntwc na paath soume ìti an to eustì eðnai asumpðesto kai to qwðo Ω stajeì dhlad mh kinoômeno, h m za pou peikleðetai sto Ω eðnai staje. 'Aa dm =0 = Ω ρu dv =0 = Ω u dv =0 afoô h ρ eðnai staje. Epeid to qwðo Ω eðnai aujaðeto, sumpeaðnoume ìti u =0. 1.151 ApodeÐame ètsi thn eðswsh sunèqeiac gia asumpðesth o. OtÔpoctou Ostogadsk O tôpoc tou Ostogadsk eðnai sthn ousða to je hma thc apìklishc diatupwmèno se katesianèc suntetagmènec. Ton mnhmoneôoume ed diìti eðnai q simoc se aketèc efamogèc. Gia thn apìdeih tou tôpou ja qhsimopoi soume tic e c sqèseic gia th stoiqei dh epif neia ds kai ton stoiqei dh ìgko dv : n i ds = ddz, n j ds = ddz, n k ds = dd kai An F=F 1 i + F 2 j + F 3 k,tìte dv = dddz. F = F 1 + F 2 + F 3. Apomènei na metasqhmatðsoume ton ìo F nds pou emfanðzetai sto epifaneiakì olokl wma tou jew matoc thc apìklishc: F n ds = F 1 i n ds + F 2 j n ds + F 3 k n ds = F n ds = F 1 ddz + F 2 ddz + F 3 dd. Antikajist ntac tic pio p nw sqèseic sthn 1.146, paðnoume ton tôpo tou Ostogadsk: Ω F1 + F 2 + F 3 dddz = F 1 ddz + F 2 ddz + F 3 dd 1.152 Ω

38 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc Pa deigma 1.6.5 Ja upologðsoume to olokl wma I = ddz+ ddz+2zdd S ìpou S h kleist epif neia pou oðzetai apì to paaboloeidèc z α= 1 z z=1 2 2 2 + 2 + z =1, 0 <z<1 kai to epðpedo z=0 bl. diplanì sq ma. Me ton tôpotou Ostogadsk, èqoume gia to olokl wma I: I = V 1+1+2dddz =4 dddz. V 1 1 H epif neia S. PaathoÔme ìti I =4 1 0 dz dd =4 1 0 πα 2 dz =4π 1 0 1 z dz =4π ] 1 [z z2 =2π 2 0 Pa deigma 1.6.6 Ja upologðsoume to epifaneiakì olokl wma I = S 2 ddz + 2 ddz + z 2 dd p nw sthn ewteik ìh thc sfaðac 2 + 2 + z 2 = α 2. Apì ton tôpo tou Ostogadsk, èqoume: I = V 2 +2 +2z dddz ìpou V = {,, z R 3 2 + 2 + z 2 α 2 }. An kai mpooôme na doôme amèswc ìti to I eðnai mhdèn lìgw summetðac, ja to upologðsoume san skhsh sthn allag suntetagmènwn. Ja egastoôme loipìn se sfaiikèc suntetagmènec, θ, φ. GnwÐzoume ìti = sin θ cos φ, = sin θ sin φ, kai z = cos θ ìpou 0 α, 0 φ 2π kai 0 θ π Gia to stoiqei dh ìgko dv èqoume dv = dddz =,, z, θ, φ ddθdφ = 2 sin θddθdφ.

1.6. Oloklhwtik jew mata 39 Sunep c to tiplì olokl wma g fetai: I = = 2 = α4 2 = α4 2 α π 2π 0 0 2π 0 2π 0 2π 0 0 π 2sin θ cos φ +sinθ sin φ +cosθ 2 sin θddθdφ dφ sin 2 θ cos φ +sin 2 θ sin φ +sinθcos θ dθ 0 [ 1 dφ 2 θ 1 sin 2θ sin φ +cosφ+ sin2 θ 4 2 [ π ] 2 sin φ +cosφ+0 α 0 ] π 0 3 d dφ = πα4 [ cos φ +sinφ]2π 0 = πα4 4 4 1+1+0=0 Oi tautìthtec tou Geen 'Eqoume dh dei ìti to je hma thc apìklishc eðnai polô q simo sth metatop tipl n oloklhwm twn se epifaneiak ìtan ta deôtea apodeiknôontai eukolìtea kai antðstofa. 'Omwc h qhsimìthta tou den peioðzetai mìno sflautì to gegonìc, afoô me autì mpooôme na melet soume th fusik shmasða thc laplasian c kai aketèc meikèc diafoikèc eis seic. Sthn pa gafo aut ja apodeðoume tic tautìthtec tou Geen. UpenjumÐzoume ìti èna C 2 bajmwtì pedðo φ,, z kaleðtai amonikì hamonic an ikanopoieð thn eðswsh Laplace, dhl. an 2 φ=0. JewoÔme t a th sun thsh: ìpou φ kai ψ bajmwt pedða. Apì th dianusmatik tautìthta F = φ ψ 1.153 ff =f F + F f jètontac f = φ kai F = ψ bðskoume gia thn apìklish tou F: F = φ ψ = φ ψ + φ 2 ψ 1.154 Oloklh nontac thn 1.154 p nw sto fagmèno tisdi stato qwðo V èqoume: Apì to je hma thc apìklishc isqôei: V φ ψ dv = V V φ ψ dv = [ φ ψ + φ 2 ψ] dv. 1.155 S φ ψ n ds 1.156 ìpou S h epif neia pou f ssei to V. tautìthta tou Geen: Sundu zontac tic 1.155 kai 1.156 paðnoume thn p th φ ψ + φ 2 ψ dv = φ ψ n ds. 1.157 V S Enall ssontac ta φ kai ψ sthn pio p nw sqèsh, paðnoume: ψ φ + ψ 2 φ dv = ψ φ n ds. 1.158 V S

40 Kef laio 1:Dianusmatikìc kai tanustikìc logismìc Afai ntac thn 1.158 apì thn 1.157, paðnoume th deôteh tautìthta tou Geen h opoða eðnai epðshc gnwst wc summetikì je hma smmetical theoem: V φ 2 ψ ψ 2 φ dv = φ ψ ψ φ n ds. 1.159 S Sthn pio p nw apìdeih upojèsame ìti ta φ kai ψ èqoun suneqeðc deôteec paag gouc. Ta V kai S ikanopoioôn tic sunj kec tou jew matoc thc apìklishc. Shmei noume epðshc ìti h 1.159 mpoeð nflapodeiqjeð an jèsoume F = φ ψ ψ φ 1.160 kai egastoôme ìpwc sto p to mèoc skhsh. Pa deigma 1.6.7 An h sun thsh ψ eðnai lôsh thc eðswshc Laplace se k poio qwðo V pou f ssetai apì thn epif neia S, tìte: ψ ds =0 1.161 n S Apìdeih Jètontac φ =1sthn p th tautìthta Geen paðnoume 0 ψ +1 2 ψ dv = 'Omwc 2 ψ =0kai ψ n = ψ/n, opìte V V 2 ψdv = S S S ψ ds =0. n ψ n ds 1 ψ n ds Pa deigma 1.6.8 An oi φ kai ψ eðnai amonikèc, to qwikì olokl wma thc deôtehc tautìthtac tou Geen eðnai mhdèn, opìte φ ψ ψ φ n ds =0 Epeid ψ n = ψ/n kai φ n = φ/n, èqoume telik : φ ψ n ψ φ ds =0. n Paat hsh To je hma thc apìklishc, S S V FdV = S F n ds 1.162 metasqhmatðzei èna qwikì olokl wma miac diafoismènhc posìthtac sflèna epifaneiakì olokl wma sto opoðo o diafoikìc telest c tou èqei apaleifjeð. To je hma tou Stokes, F n ds = F t ds 1.163 S C