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1 Cuvlna Coodnats Outln:. Otogonal cuvlna coodnat systms. Dffntal opatos n otogonal cuvlna coodnat systms. Dvatvs of t unt vctos n otogonal cuvlna coodnat systms 4. Incompssbl N-S quatons n otogonal cuvlna coodnat systms 5. Exampl: Incompssbl N-S quatons n cylndcal pola systms T govnng quatons w dvd usng t most basc coodnat systm,., Catsan coodnats: x x+ yj+ zk f gad f f f f + j+ k x y z F F F dv F F + + x y z j k culf F x y z F F F Laplacan f f f x y z f + + Exampl: ncompssbl flow quatons V DV ρ ( p+ γ z) + µ V Dt V ρ + V V ( p+ γ z) + µ V t V ρ + ( V V) V ω ( p+ γ z) + µ ( ) t V ω ω V ( V ) n t abov quaton, but tand to kp t complt vcto dntty fo V n quaton. Howv, onc t quatons a xpssd n vcto nvaant fom (as abov) ty can b tansfomd nto any convnnt coodnat systm toug t us of appopat dfntons fo t,,, and. Fquntly, altnatv coodnat systms a dsabl wc t

2 xplot ctan fatus of t flow at and o facltat numcal pocdus. T most gnal coodnat systm fo flud flow poblms a nonotogonal cuvlna coodnats. A spcal cas of ts a otogonal cuvlna coodnats. H w sall dv t appopat latons fo t latt usng vcto tcnqu. It sould b cognzd tat t dvaton can also b accomplsd usng tnso analyss. Otogonal cuvlna coodnat systms Suppos tat t Catsan coodnats ( xyz,, ) a xpssd n tms of t nw coodnats ( x, x, x ) by t quatons x x( x, x, x) y y( x, x, x) z z( x, x, x) w t s assumd tat t cospondnc s unqu and tat t nvs mappng xsts. Fo xampl, ccula cylndcal coodnats x cosθ y snθ z z.., at any pont P, x cuv s a stagt ln, x cuv s a ccl, and t x cuv s a stagt ln. T poston vcto of a pont P n spac s R x+ yj+ zk cosθ + snθ + z R ( ) ( ) j ( ) k fo cylndcal coodnats By dfnton a vcto tangnt to t x cuv s gvn by: R x + y j+ z k (Subscpt dnots patal dffntaton) x x x x So tat t unt vctos tangnt to t x cuv a R x,, x R x R W R x a calld t mtc coffcnts o scal factos, fo cylndcal coodnats θ, z T ac lngt along a cuv n any dcton s gvn by ds dr dr dx + dx + dx

3 Snc dr R dx dx and R and snc t x x a otogonal: x, j, j j An lmnt of volum s gvn by t tpl poduct d ( dx dx ) dx dxdxdx W snc t x a otogonal Fnally, on t sufac x constant, t vcto lmnt of sufac aa s gvn by ds dx dx dxdx Wt smla sults fo x and x constant ds dxdx ds dx dx. Dffntal opatos n otogonal cuvlna coodnat systms Wt t abov n and, w now pocd to obtan t dsd vcto opatos f f f x x x. Gadnt f + + By dfnton: df f dr f dx If w tmpoaly wt f λ + λ+ λ Tn by compason df f dx λ dx Not x λ x f λ x + + x x x + θ + z θ z x fo cylndcal coodnats f x cul gad f ) So tat by dfnton ( ( )

4 x Also x x So tat by dfnton ( ( f g) ) x x x. Dvgnc F ( F) + ( F ) + ( F ) ( F ) ( F ) ( F ) F + + ( F ) F usng ( ϕu) ϕ u+ u ϕ ( F ) usng ( F ) x Tatng t ot tms n a smla mann sults n F F + F + F x x x F ( F) ( F) ( F) + + θ z ( F) + ( F) + ( F) θ z ( ) ( ) ( ) fo cylndcal coodnats. Cul F x x x F F F ( F ) ( F ) ( F ) F + + ( F ) ( F)

5 ( F) usng ( ) ϕu ϕ u+ ϕ u and ( F) ( F) ( F) + + x x x ( F ) + ( F ) x x F x x ( ) F x x x F F F θ z F F F θ z F fo cylndcal coodnats x x x x x x + + x x x x x x + + θ θ z z + + θ θ z z.4 Laplacan actng on a scala f Laplacan actng on a vcto F ( F) ( F ) f f f Usng f + + x x x F F + F + F x x x and ( ) ( ) ( ) ( F )

6 Usng ( F ) + ( F ) + ( F) x x x x + ( F) + ( F ) + ( F ) x x x x + ( F) + ( F ) + ( F ) x x x x F x x x ( F ) F F F ( F ) ( F) ( F) ( F ) x x x x x x + ( F) ( F) ( F) ( F) x x x x x x + ( F) ( F) ( F) ( F) x x x x x x Combnng tos two tms gvs F F F ( ) ( ) ( F ) + ( F ) + ( F) x x x x ( F ) ( F) ( F) ( F ) x x x x x x + ( F ) + ( F ) + ( F) x x x x ( F) ( F) ( F) ( F) + ( F ) + ( F ) + ( F) x x x x ( F ) ( F ) ( F ) ( F ) x x x x x x x x x x x x

7 Fo cylndcal coodnats(,, z) of Laplacan opato actng on a scala f θ, θ,,, and us t dfnton f + + θ θ z z + + θ z θ z F F F F a ( ) + b θ + c z F F + F + θ + F z θ θ a ( F ) + ( F ) + ( F) x x x x ( F ) ( F ) ( F ) ( F ) x x x x x x ( F) + ( F) + ( F) θ z ( F) ( F) ( F) ( F) θ θ z z F F F F θ z F F F F F θ + θ z z F F F F θ z F F F F F + + θ θ z z F F F F F F θ θ z F F F F F + + θ θ θ z z F F F F F F θ θ z z

8 F F + θ θ F F F + + θ z z F F F F F F θ θ z F F F F F F θ z θ F F F F + + F θ z θ F F F θ b ( F ) + ( F ) + ( F) x x x x ( F ) ( F ) ( F ) ( F ) x x x x x x ( F) + ( F) + ( F) θ θ z ( F) ( F) ( F) ( F) z θ z θ F F F F θ θ z F F F F F + z θ z θ F F F F θ θ z F F F F F + z θ z θ F F F F θ θ θ θz F F F F F F F + + z θ z θ θ F F + θ θ F F + + θ θz

9 F + F F F F F F z θ z θ θ F F F F F F θ z θ F F F F F θ z θ F F F + θ c ( F ) + ( F ) + ( F) x x x x ( F ) ( F ) ( F ) ( F ) x x x x x x ( F) + ( F) + ( F) z θ z ( F) ( F) ( F) ( F) z θ θ z F F F F z θ z F F F F z θ θ z F F F F z θ z F F F F F F + z z θ θ z F z F + F F + + z z θ z F + F F F F F z z θ θz F F F F θ z F F F F θ z F

10 . Dvatvs of t unt vctos n otogonal cuvlna coodnat systms T last topc to b dscussd concnng cuvlna coodnats s t pocdu to obtan t dvatvs of t unt vctos,.. j x j Suc quantts a qud, fo xampl, n obtanng t at-of-stan and otaton tnso V j ε j j + j + T ω j j j V V To smplfy t notaton w dfn: R, R x T ( ) ( V V ) ( ) ( ) x Rx j and j x j x j Not tat j s symmtc,.. j j a + b + c Dvaton of ( ) x +

11 ( ) x + +. Dvaton of ( ) x + ( ) x + + +

12 . Dvaton of ( ) x + ( ) x + + +,.4 Dvaton of + ( ) ( ) x + ( ) ( ) x + ( ) ( ) x

13 ( ) ( ) a + b + c + a + b + c + + a + b + c +,.5 Dvaton of ( ) ( ) ,.6 Dvaton of ( ) ( )

14 Incompssbl N-S quatons n otogonal cuvlna coodnat systms 4. Contnuty quaton V x x x and V v + v+ v V ( v ) + ( v ) + ( v ) x x x Snc F ( F) + ( F ) + ( F ) V V V p V, (w p pzomtc pssu) t ρ V v + v + v, w can xpand t momntum quaton tm by tm 4. Momntum quaton + ( ) + ν Snc V v v v Local dvatv + + t t t t Convctv dvatv ( V ) V Snc v + v + v v v v V and V + + x x x V V V v + V v + V v V v ( ) ( )( ) ( )( ) ( )( ) ( )( )

15 ( v ) v ( v ) v ( v ) v + + x x x v v vv v v vv v v vv x x x x x x vv vv vv vv vv vv x x x v v v v v v vv x x x vv vv + + v v v v v v vv vv vv vv x x x ( V )( v ) v ( v ) v ( v ) v ( v ) + + x x x v v vv v v v v v v vv x x x x x x vv vv vv vv vv vv x x x v v v v v v vv x x x vv vv + + vv vv v v v v v v vv vv x x x ( V )( v ) v ( v ) v ( v ) v ( v ) + + x x x v v vv v v v v v v v v x x x x x x v v v v v v vv vv vv x x x

16 v v v v v v vv vv x x x vv + vv vv vv vv v v v v v v x x x p p p Pssu gadnt p + + x x x Vscous tm V V ( v ) + ( v ) + ( v ) x x x x ( v ) ( v ) ( v ) ( v ) x x x x x x + ( v ) + ( v ) + ( v ) x x x x ( v ) ( v ) ( v ) ( v) x x x x x x + ( v ) + ( v ) + ( v ) x x x x ( v) ( v ) ( v ) ( v ) x x x x x x 4.. Combn tms n ê dcton to gt momntum quaton n ê dcton v v v v v v v vv vv vv vv t x x x p ρ x + ν ( v ) + ( v ) + ( v ) x x x x ν ( v ) ( v ) ( v ) ( v ) x x x x x x

17 4.. Combn tms n ê dcton to gt momntum quaton n ê dcton v vv vv v v v v v v vv vv t x x x p ρ x + ν ( v ) + ( v ) + ( v ) x x x x ν ( v ) ( v ) ( v ) ( v ) x x x x x x 4.. Combn tms n ê dcton to gt momntum quaton n ê dcton v vv vv vv vv v v v v v v t x x x p ρ x + ν ( v ) + ( v ) + ( v ) x x x x ν ( v ) ( v ) ( v ) ( v ) x x x x x x 5. Exampl: Incompssbl N-S quatons n cylndcal pola systms 5. Contnuty quaton V n cylndcal coodnats (, θ, z) Fo cylndcal coodnats(,, z) θ, θ,, z V ( v) ( vθ ) ( vz) + + θ z ( v) + ( vθ ) + ( vz) θ z 5. Momntum quaton V + ( V ) V p + ν V n cylndcal coodnats(, θ, z ) t Fo cylndcal coodnats(,, z) ots a zo. ρ, θ, z and only θ, all θ,

18 5.. T -momntum quaton: vθ vθ vθ + v + + vz + ν v v v v v v p t θ z ρ θ 5.. T θ -momntum quaton: v v v v v p vv v F F v θ θ θ θ θ θ z ν F t θ z ρ θ θ 5.. T z-momntum quaton: v v v v v p + v + + v + ν v t θ z ρ z z z θ z z z z

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