Mathematical analysis of pulsatile flow and vortex breakdown. Tsuyoshi Yoneda (U of Tokyo) Euler eq. 1 / 17
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1 Mathematical analysis of pulsatile flow and vortex breakdown فلم ع ىو ش ش ن ى م ى ص ( ه ىفس م ى س) Tsuyoshi Yoneda (U of Tokyo) Euler eq. 1 / 17
2 Pulsatile flow problem ) 属 مخ فم ) 属 مخ م س- مى فخ كىل ى م -م ى ف ن ى ىقف ى ل ف ى ىقف س مق م م ط : كى و ى م موش : 3 R := {x R م ف R م م ى ء x x 2 2 < R, ٠ < x 3 < l} فل ق-ملى ى و ى属 l}. < x 3 < ٠ R, R = {x R 3 : x1 2 + x 2 2 = م : 属 ن ف لمقى ك مل م ف ى ف م م س- مى فخ م قى م ك ى موش t u + (u )u νىu = p, u م ى = ٠ R, u = ٠ م R ل ف t)) u = u(x, t) = (u 1 (x 1, x 2, x 3, t), u 2 (x 1, x 2, x 3, t), u 3 (x 1, x 2, x 3, و ى 属 p = p(x, t). Tsuyoshi Yoneda (U of Tokyo) Euler eq. 2 / 17
3 Recent research of the pulsatile flow (٢٠١٢) ف م ذ- مم 属 م مط- ى ث- ى ش م م مو R م م ى مو ن ل م مو ف م م مو م ف p 2 ل ف p 1 نة )/l. (p 1 p 2 ف لم م م مق فك مىلف ه مو s (p 1 p 2 )/l =: p مل م مل ى-م ى ى مىلف ه م م مو نة 属 ف ى ف مو ل مح فك م مخ م س- مى فخ 属 مخ م ى م ى ذ) 属 :( 属 م مو r = u s = (u 1, u 2, u 3 ) = (٠, ٠, p s ٤νl (R2 r 2 )), x1 2 + x 2 2. م فك مىلف ه م م ه ى ف ىك موش p 1 (t) p 2 (t) l = p o e int مق فك u o ى ه ىل م ك ى موش.م ى ى كىل ى م ى وكىو 属属 ى ك ن م م ف ه ى ق ىكى م م ى 属 و ى u 1 = u 2 = ٠. Tsuyoshi Yoneda (U of Tokyo) Euler eq. 3 / 17
4 Womersley number α N α = R ν م محمل مو ء مق ل مز ف ىك (م ف ف) مق ل مز فم مو.( s u o + u ى 属 مخ ىف مو م فك ىو ى) م ى كم م u s ل ف u o ه ى ق م ى م م ىمو ه ىل ككء ى ى ف ف ى ى ل مى مق م م ط م مffىل ف م م فم ى فومق م ى م فو فم ف ف مق ل مز ف ىك مو ن ى فى ف ى فومق ى ى ف مو كمffم مو ق لم ى م مل مق مم ى ى ف مو فو مل ك ك مو وش. مق ل مز فم ى 属 مو ن كمffم مو م فهى م ى لىل مو مم ى م م 属 ب ( م ك فو ى»ف) م ك Tsuyoshi Yoneda (U of Tokyo) Euler eq. 4 / 17
5 Vortex breakdown م قى مو م ى م مل فكى ف م (١٩٩٩) م م س ل ف هى ض- ف ى 属 مخ م س- مى فخ كى م ى ف لفم مو ن مىه 属 مخ. م ك-ل م ه ى ف و ى R ) 属 م ف وك ) م ىف ك فكى ل ى ك (٢٠١٦ حءت ع- خ- ب.ن.ك) ١٩٩٩ م م س- هى ض- :م هىء Tsuyoshi Yoneda (U of Tokyo) Euler eq. 5 / 17
6 ككف ى م ف فل ق مو م ف ل م 属 مك ىس 属 مخ م مو 属 مخ مق ل مز وهىو مو Euler equations in a pipe Ω R Inflow conditions t u + (u )u = p, u م ى = ٠ R, u(x, t) x3 =0 = (٠, ٠, U in (r, t)) 属 و ى U in > ٠, (r)g(t) U in = U s (r) + U o :م فك 属 مخ م ى ف ذ + (r) r j U s g. ه ى فم ك ى لى ف و ى属 ١ (r) r j U o 0 j 2 0 j 2 : in 属 U مخ ى ه ى فم ك ى لى ف خ :م فك 属 ل فم ق م ض ١. t) r j t k U in (r, 0 j,k 2 Tsuyoshi Yoneda (U of Tokyo) Euler eq. 6 / 17
7 Restrict to the axisymmetric flow ى كم ىل فى ف : z e ى كم ىل ى ف : θ e ى كم ىل فىلف : r e فو وك مق t) v r = v r (r, z, t) v θ = v θ (r, z, t) v z = v z (r, z, u = v r e r + v θ e θ + v z e z :ه ى属 ن مو ن ى ف v r, v z, v θ موش Axisymmetric Euler equations (no pressure on e θ -direction) t v r + v r r v r + v z z v r v θ 2 + r p = ٠, r t v θ + v r r v θ + v z z v θ + v r v θ = ٠, r t v z + v r r v z + v z z v z + z p = ٠, r (rv r ) + z v z r = ٠. Tsuyoshi Yoneda (U of Tokyo) Euler eq. 7 / 17
8 The key definitions.٠ > z u e ى فو ف م ف ى ى u ل ممح كم مو م ف ف属 ف مط Axis-length streamline Φ (definition) و ى属 zφ(z) = (u/u ez ) ( Φ(z), t) ٠ > t لم مح ء Φ(z) = ( R(z) ك Θ(z), R(z) ى Θ(z), z), R(z) = R( r 0, z, t) R( r 0, ٠, t) = r 0 Θ(z) = Θ(z, t). Rate of disturbing laminar profile (key definition) L 0 ( r 0, z, t) = r0 R + r R 1 L x ( r 0, z, t) := z j k r 0R + 1 j+k 3, (j,k) (0,1) 1 j+k 3 (j,k) (0,1) j z k r R 1 L t ( r 0, z, t) = t R t R 1 + t r0 R + t z0 R. Tsuyoshi Yoneda (U of Tokyo) Euler eq. 8 / 17
9 Remarks r مك ى ٢ ى L 0 ن م ف ىح R 1 = ١/ r0 R. ة. 属 مخ ف ى ف فكى مو ى (g,٠,٠) = u ى م فكى موش g. ف ن ٠ t L ل ف ٠ x ٢, L 0 L م فك ىو Pulsatile flow case (First theorem) ى ك ن و ف ى م مو مو ٠ θ u 0 (x) e ه ى ن ى ف R م x ف ء g فو وك g ١ g (t) g (t) (t t b ) L t ( r 0, z, t).( ف م م قف مو و ى 属 م ف ك) t t b ن Vortex breakdown case (Second theorem) مق ل و ىو ) ١/δ > (٠) θ r v نى فو وك > ٠ δ ى م مو ٠ > ϵ ف ء ١ (٠) θ r z v ( مى فل ق ق ل ف ه ى ف ه ىل م ك ى ف مو ١ x L 0 + L ١ (٠) θ 2 r v. ى ف مو ١/ϵ > t L مو. م لم ى ه ىل م ك م ف م م مو كى ف مو ف م ة Tsuyoshi Yoneda (U of Tokyo) Euler eq. 9 / 17
10 Sketch of the proof Strategy. م ه ف ى كىلف ك ف م ل ف ϵ/١ t L م ء. in U ل ف R ه ى ق v z ل ف v r م كم ىء م مو مك مه م ىل لا فا مو : r v ل ف v z ن ف ن ه ى 属 ن مو م فو مط v z (r, z, t) = ρ( R 1 (r, z, t), z, t)u z ( R 1 (r, z, t), ٠, t) = ρ( R 1, z, t)u in v r (r, z, t) = ( z R)( R 1 (r, z, t), z, t)u z (r, z, t). 属 و ى ρ( r 0, z, t) = ٢ r 0 r0 R( r 0, z, t) 2. Tsuyoshi Yoneda (U of Tokyo) Euler eq. 10 / 17
11 مج.(م ف - r z ) م ف فىلى م مو 属 مخ فىه ف هفج مو م محمل مط d dt Z (t) = v z (R (t), Z (t), t), Z (٠) = z 0 ل ف d dt R (t) = v r (R (t), Z (t), t), R (٠) = r 0 属 و ى Z (t) = Z (r 0, z 0, t) ل ف R (t) = R (r 0, z 0, t). translate the time t to the axis-length z 1 Z t: t = ى Z ن م م ى مو م محمل فك م 属 مو ٠ > z v مك ىس t (z, r 0, z 0 ). We can estimate the Lagrangian deformation (for fixed t) z0 Z z0 R r0 Z r0 R z Z 1 z R 1 r Z 1 r R 1 Tsuyoshi Yoneda (U of Tokyo) Euler eq. 11 / 17
12 . R ل ف Z ق v θ م كم فك مط فو مم م v θ 属 ن ى ف م م مو t v θ (R (t), Z (t), t) = v r (R (t), Z (t), t)v θ (R (t), Z (t), t) R (t) مم م 属 ى ف م ف 属 ا مو ه ى ء Formula of v θ { t م ٠) v θ (r, z, t) = v θ (r 0, z 0, 0 属 و ى r 0 = R 1 (r, z, t) ل ف z 0 = Z 1 (r, z, t). v r (R (r 0, z 0, t ), Z (r 0, z 0, t ), t } ) R (r 0, z 0, t dt ) Tsuyoshi Yoneda (U of Tokyo) Euler eq. 12 / 17
13 Lagrangian flow Φ (t) (definition) Axis-length trajectory فو وك مق Φ مج d dt Φ (x, t) = u(φ (x, t), t), Φ (x, ٠) = x م R. Φ(z) := (R(z) ك Θ(z), R(z) ى Θ(z), z) 1 Z Φ(z) = Φ (x, ن ى ف مل ى Θ(z) ل ف R(z) م وك م 属 ل ف t (z)) 1 (Z R(z) = R : ف مز t (z)). Connection between trajectory and streamline R (z) = v r (R(z), z, Z 1 t (z)) v z (R(z), z, Z 1 t (z)) R(z)Θ (z) = v θ(r(z), z, Z t 1 (z)) v z (R(z), z, Z t 1 (z)), = ( z R)( R 1 (R(z), z, Z 1 t (z)), z, Z 1 t (z)) Tsuyoshi Yoneda (U of Tokyo) Euler eq. 13 / 17
14 Arc-length trajectory فو وك مق ϕ مج ϕ(s) := Φ (x, t(s)) ل ف ϕ(x, ٠) = Φ (x, ٠) = x و ى属 s t(s) = u 1. Remark s ϕ(s) = ١ كم مه ف ى τ(s): كم ف ى n(s): كم ف ىق ى b(s): م ف ك κ(s): ى (s): T Key estimates on curvature and torsion. Θ ل ف Θ ه ى ق κ s κ T م ف ى م فك مط. م م م مو و ى 属 ى ف م م مو وك م م م 属 م مو ص مو ن م ف ف م ف ) م م م مو م ف ى م م 属 م ء.( ى ف ك فك م قف Tsuyoshi Yoneda (U of Tokyo) Euler eq. 14 / 17
15 ع- فق» - فو ممس.فملى كى م مه فى م مffىل ف م م 属 属 ن فو 属 ة.(٢٠٠٤) ه فط-فح مم ف ىهى م (٢٠١٤) لم م م م ى ى ϕ كميف و ه م -ك ف مو فم x R 3 ى ف ء مو ن ه ى فم مو ) θ, r, z) R 3 ( و ى x = ϕ( θ) + rn( θ) + zb( θ) 属 ف.( كميف و ه م -ك ف مو ه ف م ف مو م ف θ ل ف s م م ف ف فو م فو م属 وش θ r z ١ κ r zκ rt τ = ٠ ١ ٠ n. ٠ ٠ ١ b n r = :م ف ن ه ى ف و ه ى 属 ن مو م فو م 属 م نم موش ل ف z = b (١ κ r) 1 θ zt (١ κ r) 1 r rt (١ κ r) 1 z = τ. Tsuyoshi Yoneda (U of Tokyo) Euler eq. 15 / 17
16 Rewrite the Euler equation along the particle trajectory p τ = τ p = D t u := t u(φ (x, t), t) مو كف م فك م 属 م مو م م ف م فك ى م م ف م مه ة. ى ل ف م ف ك ه ى ق كمffم م م فك Lemma (rewrite the pressure term using curvature and torsion) ٢٠١٦ ءحزء ف فس-ف ف مذ ل ف ىك.نك) م فو م 属 كميف مو ه ء ( ى ف ف م ى ع- خ-ف فقف 属 ىو فث ٣κ t u + s κ u 2 = r t u ل ف T κ u 2 = z t u.. z κ r) 1 (١ rt κ r) 1 r (١ zt κ r) 1 θ (١ = τ فكمز :م ف ى م م م مو ى م موش r ( p τ) = r τ p = κ θ p r θ p T zp b) θ ) = κ( p τ) θ ( p n) T ( p ل ف r م ك)!! T κ s κ ن م ف ى م ن ى كىلف ك ف مك ل ى فك مط Tsuyoshi Yoneda (U of Tokyo) Euler eq. 16 / 17
17 ! فوش! ي مء!فلى ف ث! ى ملم م مش!م فء!ىي مء! كفش! ف و! فىكف ا!مي مى»ء!مىظ مىظ! ك فمق ىك مح!مى»ف ا! ف وس! فك ى! ف! قى ف س Tsuyoshi Yoneda (U of Tokyo) Euler eq. 17 / 17
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