Amn Halloc Math Ecss E-mal : amn@sthths bpa : sthths/amn MATH EXERCISES GRADIENT DIVERGENCE CURL DEL NABLA OERATOR LALACIAN OERATOR CONTINUITY AND NAVIER-STOKES EQUATIONS VECTOR RODUCTS I and thn scala o dot podct cto o s podct In som boos s also consdd ot podct dnd b GRADIENT DIVERGENCE CURL DEL NABLA OERATOR LALACIAN OERATOR GRADIENT Lt ϕ b a scala ld Th adnt s th cto ld dnd b ad ϕ ϕ ϕ ϕ DIVERGENCE Lt R Q b a cto ld contnosl dntabl th spct to and Thn th dnc o s th scala ld dnd b R Q d CURL Th cl o s th cto ld dnd b Q R Q R R Q cl o Q R Q R cl DEL NABLA OERATOR Th cto dntal opato s calld dl o nabla / 7
Amn Halloc Math Ecss U can dnot ad d and cl as blo: ad ϕ ϕ d cl Not that s not th sam as Q R LALACIAN OERATOR Th Laplacan opato s dnd o a scala ld U b U U U U U and o a cto ld Q R b Q R Som omlas o pola and clndcal coodnats ola coodnats dm ϑ ϑ ϑ ϑ tansomaton: aa lmnt: da d d standad bass: [Rma : Not that a dpnd on hn mo om pont to pont ths s th ason h ths bass n som boos s calld local bass ] I n Catsan cood and ϑ th sam cto n pola coodnats thn ϑ [Rma : V can d ths omlas b calclatn th componnts o n th dctons o and Ths / 7
Amn Halloc Math Ecss smlal ϑ ϑ ] Clndcal coodnats : tansomaton: olm lmnt: d d d dv ϑ ϑ standad bass: I n Catsan cood and ϑ th sam cto n clndcal coodnats thn ha ollon cto componnts latonshp: ϑ [Rma : o ampl can t n th ollon a: ] scala ld: adnt: ad laplacan: cto ld: dnc: d cl: cl / 7
Amn Halloc Math Ecss EXERCISES nd a d b ad d and c cl nd ad d cl Whch on o th ollon nctons a b ln c p satss th Laplac qaton 0? 4 nd 5 Wt th nal tanspot qaton ϕ ϕu Γ ϕ S t φ thot opatos d cl o ad H U nctons ϕ Γ S a al nctons o t and 6 Whch on an o th ollon nctons 4 a ϕ b ϕ ϕ c satss th qaton ϕu Γ ϕ S? H Γ 5 U and S 4 7 nd hch on an o th ollon nctons a ϕ b ϕ 5 5 ϕ c satss th qaton ϕ d ϕu d Γadϕ S t h Γ U 4 and S 8 5 4 / 7
Amn Halloc Math Ecss 8 am 008 A Wt th nal tanspot qaton ϕ ϕu Γ ϕ S q t thot opatos d cl o ad H U nctons ϕ Γ S a al nctons o t and B Lt Γ U 4 nd S n th qaton q no that th ncton ϕ satss th qaton 9 Q6 am 008 Consd th ollon qaton ϕ ϕu Γ ϕ U 6 4 q t Lt Γ constant U nd th constant Γ n th qaton q no that th ncton ϕ t satss th qaton 0 I possbl nd o th n patal dats and a and b and c and d and 5 Hnt: Ncssa condton: I has contns dats thn th md dats o shold b qal Ths * s th ncssa condton o th stnc o a ncton that has th n dats Dtmn th al o a o hch th sstm o patal dntal qatons a and has soltons Thn nd pondn to ths al o a 5 / 7
Amn Halloc Math Ecss I possbl nd o th n patal dats and a and b and c and d and Hnt: Ncssa condton: I has contnos dats thn th md dats o shold b qal Ths Con : Con : Con : a th ncssa condton o th stnc o a ncton that has th n dats Dtmn th als o a and b o hch th sstm o patal dntal qatons a and b has soltons Thn nd pondn to ths als o a and b 4 W consd an ncompssbl dnst const stad stat aabls do not dpnd on tm sothmal Ntonan lo th a n loct ld V Us th ollon qatons contnt and Na Stos qatons to nd n psson o pss as a ncton o and h constant µ constant 00 0 and h 98m / s Incompssbl contnt qaton: 0 q Na Stos qatons: componnt: 6 / 7
Amn Halloc Math Ecss µ q t componnt: µ q t componnt: µ q4 t a V 4 0 b V 4 c V 4 5 am 009 A Consd th ollon qaton ϕ ϕu Γ ϕ U 6 6 4 8 4 t Lt Γ constant U 4 4 nd th constant Γ n th qaton q no that th ncton ϕ t satss th qaton q B W consd an ncompssbl dnst const stad stat aabls do not dpnd on tm sothmal Ntonan lo th a n loct ld V Us th ollon qatons contnt and Na Stos qatons to nd n psson o pss as a ncton o and h constant µ constant 00 0 and h 98m / s and V 6 4 4 6 am 009 W consd an ncompssbl dnst const stad stat aabls do not dpnd on tm sothmal Ntonan lo th a n loct ld V Us th ollon qatons contnt and Na Stos qatons to nd st paamt a and thn n psson o pss as a ncton o and h constant µ constant 00 0 and h 98m / s and V 5 a 7 / 7
Amn Halloc Math Ecss 7 Consd stad ncompssbl sothmal lamna statona Ntonan lo n a lon ond pp n th -dcton th constant ccla s-scton o ads R m Us th contnt and th Na-Stos qatons n clndcal coodnats to nd th loct ld V and th pss ld th ld lo satss th ollon condtons: c0 All patal dats th spct to tm t a 0 Stad lo c μ000 /m s and 000 /m c A Constant pss adnt / /50 a/m s appld n th hoontal as -as n o notaton: / /50 c Th lo s paalll to th as that s 0 and 0 c4 W assm that th lo s asmmtc Th loct dos not dpnd on that s 0 c5 Bonda cond No-slp bonda condton V ld V all : I thn 0 c6 Bonda condton : has mamm at 0 that s 0 0 --------------------------------------------------------------------------------------------- Th contnt and th Na-Stos qatons o an ncompssbl sothmal Ntonan lo dnst const st µ const th a loct ld V n Clndcal coodnats : Incompssbl contnt qaton 0 q a Na-Stos qatons n Clndcal coodnats: -componnt: t µ q b -componnt: t µ q c 8 / 7
Amn Halloc Math Ecss -componnt: t µ q d 8 Eam Mach 0 qston A 4ponts W consd an ncompssbl dnst const stad stat aabls do not dpnd on tm sothmal Ntonan lo th a n loct ld V c 4 b a Us th ollon qatons contnt and Na Stos qatons h constant µ constant 00 0 and 98m / s to nd: paamts a b and c n psson o pss as a ncton o and Th GRADIENT VECTOR th chan o aabls and bass Th adnt cto o th ncton s dnd as ad * I chan aabls to and plac bass ctos th n lnal ndpndnt ctos thn can pss th sam adnt cto ad n tms o aabls and ctos W smpl calclat th dats and n n aabls and pss as a lna combnatons o Thn sbsttt thos als nto * S th ollon ampl 7 W consd a scala ld n n clndcal coodnats h and bass ctos a nd th psson o th adnt ad n clndcal coodnats that s n tms o and a p th sam bass 9 / 7
Amn Halloc Math Ecss b c am 0; Q5 B ponts d ths s otn sd as a local bass o clndcal coodnats 8 am 06; Q6 A ponts D th Cach momntm qaton DV σ Dt Solton: S http://nomhannths/amn/ar_000/hl008/deriv_navier_stokespd ANSWERS AND SOLUTIONS: Solton: Q R a Snc d ha d 0 0 Ans a d ϕ ϕ ϕ b Snc ad ϕ ha o ϕ d ad d 00 Ans b ad d 00 d c cl Q R Ans c cl Solton: 0 / 7
Amn Halloc Math Ecss cl 0 Ths d cl 0 and tho ad d cl 000 0 Ans: ad d ot 000 0 0 0 Ans: Th ncton ln satss th Laplac qaton 4 Ans: d cl ad 6 5 Solton: ϕ d ϕu d Γadϕ Sφ t ϕ ϕ ϕ ϕ d ϕ ϕ ϕ d Γ Γ Γ Sφ t ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ t ϕ Γ S φ 6 Whch on an o th ollon nctons 4 a ϕ b ϕ ϕ c satss th qaton ϕu Γ ϕ S? H Γ 5 U and S 4 Solton : Th qaton ϕu Γ ϕ S can b ttn as / 7
Amn Halloc d ϕu d Γadϕ S 5ϕ 5ϕ 5ϕ d ϕ ϕ ϕ d 4 ϕ ϕ ϕ 5 ϕ 5 ϕ 5 ϕ 4 Math Ecss q 4 a Lt ϕ ϕ V calclat th dats oϕ and sbsttt n th lt hand sd LHS and ht hand sd o th qaton q ϕ ϕ ϕ 4 4 ϕ 5 ϕ 5 ϕ 4 LHS: 5 RHS 60 4 Whnc LHS RHS Ths th ncton ϕ 4 s not a solton to th qaton b ϕ ϕ LHS 4 RHS 4 Whnc LHS RHS and th ncton ϕ s not a solton to th qaton ϕ ϕ c Lt Thn LHS 4 6 RHS 7 4 Ths LHS RHS and th ncton ϕ s not a solton to th qaton Ans: Non o th nctons satss th qaton 7 Ans: ncton ϕ 5 satss th qaton 8 am 98 A Wt th nal tanspot qaton ϕ ϕu Γ ϕ S q t thot opatos d cl o ad H U nctons ϕ Γ S a al nctons o t and / 7
Amn Halloc Math Ecss B Lt Γ U 4 nd S n th qaton q no that th ncton ϕ satss th qaton Solton: A ϕ ϕu Γ ϕ S t ϕ d ϕu d Γadϕ S t ϕ ϕ ϕ ϕ d ϕ ϕ ϕ d Γ Γ Γ S t ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ S q t B W sbsttt Γ U 4 and ϕ n th qaton q and t ϕ 4ϕ 8φ ϕ ϕ ϕ 0 S 0 8 4 0 6 8 S Consqntl S 4 8 8 4 9 Q6 am 008 Consd th ollon qaton ϕ ϕu Γ ϕ U 6 4 q t Lt Γ constant U nd th constant Γ n th qaton q no that th ncton ϕ t satss th qaton Solton: ϕ ϕu Γ ϕ U 6 4 t ϕ d ϕu d Γadϕ d cl U 6 4 t c cl U 0 ha d cl U 0 / 7
Amn Halloc Math Ecss ϕ ϕ ϕ ϕ d ϕ ϕ ϕ d Γ Γ Γ 0 6 4 t ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ 0 6 4 q t W sbsttt U and ϕ t n th qaton q and t ϕ ϕ ϕ φ ϕ ϕ ϕ Γ Γ Γ 6 4 t Not that Γ s a constant 6 0 Γ Γ 6 4 8 4Γ Γ Ans: Γ 0 I possbl nd o th n patal dats and a and b and c and d and 5 Hnt: Ncssa condton: I has contnos dats thn th md dats o shold b qal * s th ncssa condton o th stnc o a ncton that has th n dats Ans: a C b C c C d No solton c th condton * s not llld 5 Solton a Snc and th dats a contnos th condton * s llld and can nd o th n dats In od to nd ntat th spct to th st o th qatons q 4 / 7
Amn Halloc Math Ecss and t q d C Ths C W ha ntatd th spct to tho th constant stll dpnd on No to nd C dntat and sbsttt n q and t: C C C C C nall sbstttn C C n ha C h C s a constant Ans: om a a Thn o a ha C Ans: a C b C c C d No solton c th condton Con s not llld Solton a a and Snc th condtons Con a llld and can nd o th n dats In od to nd ntat th spct to th st o th qatons q q q and t d C Ths 5 / 7
Amn Halloc Math Ecss C W ha ntatd th spct to tho th constant stll dpnd on and No to nd C dntat and sbsttt n q and t: C C C C C W ha ntatd th spct to tho th constant stll dpnd on and Ths C No sbstttn n q ha C C C C C nall sbstttn C C n ha C h C s a constant Ans: om a a b b b b Ths all th condtons a llld a and b o ths als o a and b t C Calclaton o th pss ld o a non loct ld o an ncompssbl stad stat sothmal Ntonan lo 4 Ans: a 8 8 C 6 / 7
Amn Halloc Math Ecss 7 7 b C c 4 4 4 C Solton a W sbsttt 4 0 n q4 and t not that al dats th spct to t a 0: Contnt qaton: 0 0 q dntcall llld Na Stos qatons: componnt: 6 q componnt: 6 q componnt: 0 q4 No q s 8 C * Sbsttton n q mpls C 6 C 8 C Hnc om * ha 8 8 C ** No sbsttt ** n q4 and t 0 0 C C C h C s a constant nall sbstttn C C n ** ha 8 8 C h C s a constant 5 Solton A: ϕ ϕ U Γ ϕ U 6 6 4 8 4 t ϕ d ϕu d Γadϕ d cl U 6 6 4 8 4 t c cl U 0 ha d cl U 0 ϕ ϕ ϕ ϕ d ϕ ϕ ϕ d Γ Γ Γ 6 6 4 8 4 t 7 / 7
Amn Halloc Math Ecss ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ 6 6 4 8 4 t q W sbsttt U 4 4 and ϕ t n th qaton q and t ϕ 8ϕ 8ϕ φ ϕ ϕ Γ Γ t Not that Γ s a constant 6 6 6 4 8 6Γ 6 6 4 8 4 0 6Γ Γ 5 Ans A: Γ 5 Solton B: W sbsttt 6 4 4 n q4 and t not that al dats th spct to t a 0: Contnt qaton: 0 0 q dntcall llld Na Stos qatons: componnt: 6 4 q componnt: 4 8 q componnt: 4 4 q4 No q s 8 4 C * Sbsttton n q mpls C 4 8 C 8 C Hnc om * ha 8 4 8 C ** No sbsttt ** n q4 and t 4 4 4 4 C C 4 C h C s a constant nall sbstttn C C n ** ha 8 4 8 4 C Ans B: Γ ϕ 6 6 4 8 4 8 / 7
Amn Halloc Math Ecss C 4 8 4 8 h C s a constant 6 Solton 5 a V st sbsttt a 5 n q and t not that al dats th spct to t a 0: Contnt qaton: 0 a a No ha 5 V U th Na Stos qatons t: componnt: 6 9 q componnt: 5 q componnt: 4 q4 No q s 6 9 C * Sbsttton n q mpls 5 5 C C C Hnc om * ha 5 6 9 C ** W sbsttt ** n q4 and t C 4 4 C C h C s a constant nall sbstttn C n ** ha C 5 6 9 Ans : C 5 6 9 h C s a constant Q7 Consd stad ncompssbl sothmal lamna statona Ntonan lo n a lon ond pp n th -dcton th constant ccla s-scton o ads R m Us 9 / 7
Amn Halloc Math Ecss th contnt and th Na-Stos qatons n clndcal coodnats to nd th loct ld V and th pss ld th ld lo satss th ollon condtons: c0 All patal dats th spct to tm t a 0 Stad lo c μ000 /m s and 000 /m c A Constant pss adnt / /50 a/m s appld n th hoontal as -as n o notaton: / /50 c Th lo s paalll to th as that s 0 and 0 c4 W assm that th lo s asmmtc Th loct dos not dpnd on that s 0 c5 Bonda cond No-slp bonda condton V ld V all : I thn 0 c6 Bonda condton : has mamm at 0 that s 0 0 Th contnt and th Na-Stos qatons o an ncompssbl sothmal Ntonan lo dnst const st µ const th a loct ld V n Clndcal coodnats : ---------------------------------------------------------------- SOLUTION Incompssbl contnt qaton 0 q a Na-Stos qatons n Clndcal coodnats: -componnt: t µ q b -componnt: t µ q c -componnt: 0 / 7
Amn Halloc Math Ecss t µ q d W choos as a tcal as an a n a hoontal plan and th lo s paalll th th -as W dnot loct cto V h and a -componnt - componnt and -componnt n clndcal coodnats Accodn to th assmptons ha 0 0 and dos not dpnd on Snc s th tcal as ha that cto - 00 h 98 m/s hch n clndcal coodnats s and 0 No sbsttt / /50 a/m μ000 /ms n th contnt and Na- Stos qatons: Snc 0 and 0 accodn to c contnt qaton n clndcal coodnats 0 s 0 / 7
Amn Halloc Math Ecss Ths tlls s that s not a ncton o thmo c loct dos not dpnd on assmpton c4 concld that dpnds onl on To smpl notaton dnot * No sbsttt and 0 / /50 a/m μ000 /ms n th Na-Stos qatons: Th -componnt o th Na-Stos qaton s: 0 q -c Th -componnt o th Na-Stos qaton: 0 q -c Th Z-componnt o th Na-Stos qaton h and 0 q -c 50 000 50 s: Stp W nd th pss In od to nd th pss sol q -c q -c and th qaton s 50 that 50 om ths qatons t C 50 / 7
Amn Halloc Math Ecss Stp W nd th loct componnt W sol q -c th bondas c5 and c6: 0 q -c 50 000 0 c5 0 0 c6 d Rma: Tchncall can t nstad d aabl om q -c ha 0 50 000 4 c s no a ncton o onl on C sbsttton 0 and c6 C 0 C sbsttton and c5 C 4 4 Ths 4 and V 0 0 4 Ans : C 50 V 0 0 4 / 7
Amn Halloc Math Ecss 8 W consd an ncompssbl dnst const stad stat aabls do not dpnd on tm sothmal Ntonan lo th a n loct ld 4 a b c V Us th ollon qatons contnt and Na Stos qatons h constant µ constant 00 0 and / 98 s m to nd: paamts a b and c n psson o pss as a ncton o and Incompssbl contnt qaton: 0 q Na Stos qatons: componnt: t µ q componnt: t µ q componnt: t µ q4 ------------------------------------------------------------- W sbsttt 4 a b c n q4 and t not that al dats th spct to t a 0: Contnt qaton: 0 a a q Ths 4 b c Na Stos qatons: componnt: q componnt: q componnt: 4 / 7
Amn Halloc Math Ecss q4 Th sstm q q q4 s solabl onl md dats a qal: 4 : c c Con 0 0 : b bc Con 0 0 : b b Con Ths c and b0 W sol smpld qatons and t C 6 4 8 4 5 Ans C 6 4 8 4 5 9 W consd a scala ld n n clndcal coodnats h and bass ctos a nd th psson o th adnt ad n clndcal coodnats that s n tms o and a p th sam bass b c am 0; Q5 B ponts d ths s otn sd as a local bass o clndcal coodnats 5 / 7
Amn Halloc Math Ecss Solton: In aabls ha ad q o clndcal coodnats ha st t th dats and n coodnats th aabl s n both cood sstms Soln th ollon sstm o and t ** W sbsttt th dats ** n q and t *** ad To sol poblms a b c and d mst pss as a lna combnatons o and sbsttt thm nto *** a om *** c ha mmdatl ad b om nd q b Thn pt om q b nto *** and t ad 6 / 7
Amn Halloc Math Ecss and at collctn componnts o ad c om ha q c ttn om q c nto *** s ad d W can sol d n th sam mann as n abc bt ths tm can st collct tms and t th slt: ad Ans: a ad b ad c ad d ad 7 / 7