S P ΩR U w = b SP Ω Rw Ub

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1 SP =ΩRw/ Ub SP = ΩRw/ Ub

2 S P ΩR U w = b SP Ω Rw Ub

3 (,, ) S = f R R χ P b e Rb Re χ grb/ Ub ρru b b / σ ( µ,ζ, ω) ( x, yz, ) x = kcos θsinh η= kµζ 1/ 1/ y = k(1 µ ) (1+ζ ) cos ω 1/ 1/ z = k(1 µ ) (1+ζ ) sin ω Rw Ω χ

4 hµ hζ ω h = r/ µ = r/ ζ = r/ ω = r ( x, y, z) µ µ H = h h µ ζ H = hζ h ζ ω H = hω h ω eµ = hµ / hµ hµ = hµ / Hµ eζ = hζ/ hζ hζ = hζ/ Hζ eω = hω/ hω hω = hω/ Hω Φ=0 Φ

5 Φ= kµ(1+ζ )(1 ζcot ζ)( U cos χ+ωr sin χ) Z 1 1 b w 1/ 1/ kzyζ (1 +ζ ) ( Ubsin χ ΩRwcos χ) (1 µ ) cos ω 1/ 1/ +Ωk sin χ (1 +ζ ) ( X ζ)µ (1 µ ) sin ω X 1 1 3ζcot ζ 3+(1+ζ ) =ζ+ (6 ζ + 3)cot ζ 6ζ ζ(1+ζ ) { )cot } 1 (1 ){(1 ) cot } Y = +ζ ζ(1+ζ ζ Z = +ζ +ζ ζ ζ 0 V = Φ W Ω ( Rw+ r) Rw xy x χ ( Ub = Ubcos χ, Ubsin χ,0) Ω= ( Ω cos χω, sin χ,0) Rw = (0,0, Rw) r = ( x, y, z) e ζ 0 1 Φ Vµ = { W cos χ+ω ( z+ Rw )sin χ} ex e H µ µ W z Rw e e y x ez eζ { sin )cos } y ζ { cos sin } χ Ω( + χ Ω χ Ω χ 1 Φ Vω = { W cos χ+ω ( z+ Rw )sin χ} ex e H ω ω W z Rw e e y x ez eζ { sin )cos } y ζ { cos sin } χ Ω( + χ Ω χ Ω χ ζ ζ 1 Φ Vζ = { W cos χ+ω ( z+ Rw )sin χ} ex e H ζ ζ W z Rw e e y x ez eζ { sin )cos } y ζ { cos sin } χ Ω( + χ Ω χ Ω χ ζ ζ

6 0 { V = Z (1 +ζ )( W cos χ+ωr sin χ) (1 µ ) µ 1 1/ w 1/ Y W Rw + (1 +ζ ) ( sinχ Ω cosχ)µ cosω ( X ) } Ωk χ + ζ ζ + (1 µ ( ζ) ω µ + ζ ) 1/ 1/ sin (1 ) ) sin ( 1/ 1/ Vω = Y( Wsin χ ΩRw cosχ) sin ω Ωkcos χ (1 +ζ ) (1 µ ) +Ωksin χ Xµ cosω V ζ = 0 p Ω ϖ = ρ V gh const. ( µ s,ζ 0,ωs) σ 1 1 µ ω µ µ ω ω ρ,,,, V gh 0 + Ω ϖ + + = R1 R ϖ H (1/ R1+ 1/ R) H = xcos χ+ ysin χ { d z) ( xsin ycos ) } 1/ ϖ= ( + + χ χ

7 1/ 1 1 ζ(ζ + 1) ζ + = + R R k( ζ +µ ) k( ζ + 1) ( ζ +µ ) 1 3/ 1/ 1/ µ σζµ(4ζ + µ + 3) ρk( ζ +µ ) ( ζ + 1) 1/ 1/ gkζcosχ+ gk sin χ( ζ + 1) µ (1 µ ) cosω 5/ 1/ kd { 1/ 1/ Ω ( ζ + 1) µ (1 µ ) sinω+ω µζ sin χ µ ( ζ + 1)(sin ω+ cos χcos ω) + µ ζ ζ + χ χ µ ω = 1/ 1/ ( 1) ( 1) sin cos (1 ) cos 0 k } ω { 1/ 1/ k( ζ + 1) (1 µ ) gsin χsin ω+ω Rw cosω+ω kζµ sin χcosχsin ω 1/ 1/ +Ω k sin χ ( ζ + 1) (1 µ ) sin ωcosω = 0 } ζ ζ V ζ 0 χ (1 µ ) 1/ µ 0 UZ (1 +ζ ) (1 µ ) + Ycos ω( Uχ Ω R) Ωkχsin ω=0 1 1/ 1/ b b w V ω = 0 Ω k +ζ µ + Y ωu χ Ω R )+ΩkXχ ω=0 1/ 1/ (1 ) (1 ) sin ( b w cos 1/ 4 k 1 R Ω w σζ Ω µ Ω kζ +ζ+ ( ζ + 1) + sinω 1+ χcosω=0 ρgk ( ζ + 1) g 1+ζ g g Ω kζ Ω R 1 sin w + χ ω+ cosω=0 g g

8 Ω kζ/ g Ω k / Ub kgzχ/ωrwub ΩR ΩR Y U = kgx χ χ w w b Ub gkx χ 4σζ ( ζ + 1) = +ζ Z ΩR ρgk ζ + w ( 1) / µ 0 Ub gk Z 4σζ = +ζ ζ + 1 ρgk ( ζ + 1) Ub, Ω, Rw, ζ,χ Ω Rw = U b χ X Z 4σ ζ + 1 = 1 ρgk ( ζ + 1) Yζ( Z X ) Rb ζ χ K Ub gk KZ 4σζ = +ζ ζ + 1 ρgk ( ζ + 1)

9 Ω Rw = U b χ X KZ 4σ ζ + 1 = 1 ρgk ( ζ + 1) Yζ( Z X / K) σρ cm /sec g = 981 cm/ sec ζ R b ζ ζ R b ζ ζ w ζn 1mm

10 Bubble rise velocity U b (cm/sec) Solid line : Pure water Dot line : Impure water Equivalent bubble radius R b (mm) Reynolds numbmer R e =U b R b /ν (at T = 300K) Solid line : Pure water Dot line : Impure water Equivalent bubble radius R b (mm) SP 1mm 1mm.3mm.3mm

11 Angular velocity product spiral radius Ωd (cm/sec) ζ n ζ w Equivalent bubble radius R b (mm) Angular velocity product spiral radius Ωd (cm/sec) ζ n ζ w Bubble rise velocity U b (cm/sec) 1.mm 1mm.3mm Angular velocity product spiral radius Ωd (cm/sec) ζ n ζ w Equivalent bubble radius R b (mm) Angular velocity product spiral radius Ωd (cm/sec) ζ n ζ w Bubble rise velocity U b (cm/sec)

12 .3mm.3mm SP Dimensionless Number S p =Ωd/U b ζ n ζ w Equivalent bubble radius R b (mm) SP > < SP < 60 SP < 40

13 SP = ΩRw/ Ub H R V SP U b e h g r Φ Ω χ ρ σ ϖ 1 b n w x x

14 y z y z µ µ ω ω ζ ζ