Blood flow through a curved artery Giuseppe Pontrelli Istituto per - PowerPoint PPT Presentation

1 Blood flow through a curved artery Giuseppe Pontrelli Istituto per le Applicazioni del Calcolo (IAC) CNR, Roma, Italy Amabile Tatone DISAT, Facolt` a di Ingegneria University of L’Aquila, Italy

2 Circulatory system

3 K.B. Chandran, W.M. Swanson, D.N. Ghista, H.W.Vayo, Oscillatory Flow in Thin-Walled Curved Elastic Tubes, Annals of Biomedical Engineering 2, 392–412 (1974)

4 Project - Mechanical model - Problem formulation - Solution procedure - Results (velocity, pressure, tangential stress) starting from scratch and self contained

5 Laser Doppler ultrasound measurement (C. Guiot, Univ. Torino)

6 Motivations - To provide velocity patterns for flux measurement - To get some ansatz for one dimensional modelling

7 Basic references axis rigid wall elastic wall flow cur Dean (1927) steady Morgan, Kiely (1953) str Womersley (1955) Womersley (1955, 1957) unsteady Atabek (1968) Lyne (1970) cur Smith (1975) Chandran et al. (1974) unsteady Mullin, Greated (1980) Review paper: Berger, Talbot, Yao (1983).

8 Outline - 3D Navier-Stokes equations (blood) - 2D membrane model (wall) - Linearization - Wave propagating over a Poiseuille flow - Curvature as a small perturbation - Computer algebra for equation generation - Numerical results and visualization

9 a R = 0 . 1

10 Model features and assumptions - 3D unsteady fully developed flow - Planar curved axis (toroidal shape) - Elastic wall

11 Wall model (membrane) div S + b = 0 b = − T n − ρ w ¨ u ρ w wall mass density S membrane stress tensor T fluid stress tensor

12 Linear elastic isotropic wall hE   1 − σ 2 ( ǫ θθ + σǫ ψψ ) 2 hG ǫ θψ   ˆ S ( E ) =     hE   2 hG ǫ θψ 1 − σ 2 ( ǫ ψψ + σǫ θθ ) E Young modulus G shear modulus σ Poisson ratio h wall thickness

13 Blood model (newtonian fluid) div T − ρ a = 0 ∇ v + ( ∇ v ) T � � T = − p I + µ div v = 0 ρ mass density µ viscosity

14 Blood model (Navier-Stokes) � ∂ v � ∂t + ( ∇ v ) v = −∇ p + µ ∆ v ρ div v = 0 v = ˙ u (no slip condition at the wall)

15 Toroidal coordinate system ψ R θ a

16 Wave propagation

16 Wave propagation steady flow in a curved tube + small oscillatory motion

16 Wave propagation steady flow in a curved tube + small oscillatory motion χ ( r, ψ ) e i ( ωt − kz ) χ ( r, ψ ) + ˜ ¯

16 Wave propagation steady flow in a curved tube + small oscillatory motion χ ( r, ψ ) e i ( ωt − kz ) χ ( r, ψ ) + ˜ ¯ Linearization of the flow equations over ¯ χ .

17 Perturbation method χ = χ 0 + εχ 1 + ε 2 χ 2 + ε 3 χ 3 +... ε := a (Curvature parameter) R χ 1 ) e i ( ωt − kz ) (¯ χ 0 + ε ¯ χ 1 ) + (˜ χ 0 + ε ˜

18 Scaling λ = ω a λ ≪ 1 k, ∂w u � a v � a � a � � � ∂z w = O w = O = O , , λ λ w λ v �→ ( u, v, w )

19 Different stages - Linearization (wave amplitude) - Perturbation (curvature) - Scaling (wave length)

20 Linearized flow equations v �→ ( u, v, w ) � ∂u ¯ � ∂t − 2 ¯ w w w w sin ψ ¯ = ρ R + r sin ψ � ∂ 2 u ∂ 2 u ∂r 2 + 1 ∂r + 1 sin ψ − ∂p ∂u ∂u ∂r + µ ∂ψ 2 + r 2 R + r sin ψ r ∂r cos ψ r 2 − 2 vR cos ψ ∂ψ − u ∂u ∂v + ∂ψ − r ( R + r sin ψ ) 2 r 2 r ( R + r sin ψ ) u sin 2 ψ � 2 sin ψ ( R + r sin ψ ) 2 − 2 v sin ψ cos ψ ∂w (radial) − ∂θ − ( R + r sin ψ ) 2 ( R + r sin ψ ) 2

21 Linearized flow equations � ∂ 2 v ∂ 2 v � ∂v ¯ � ∂t − 2 ¯ w w w cos ψ ¯ = − 1 ∂r 2 + 1 ∂r + 1 ∂ψ 2 + 2 w ∂p ∂v ∂u ρ ∂ψ + µ r 2 r 2 R + r sin ψ r r ∂ψ uR cos ψ sin ψ cos ψ ∂v ∂ψ − v ∂v + r ( R + r sin ψ ) 2 + ∂r + r 2 R + r sin ψ r ( R + r sin ψ ) � v cos 2 ψ 2 cos ψ ∂w (circumferential) − ∂θ − ( R + r sin ψ ) 2 ( R + r sin ψ ) 2

22 Linearized flow equations � ∂ 2 w ∂ 2 w ∂r 2 + 1 ∂r + 1 sin ψ ρ∂w R ∂p ∂w ∂w ∂t = − ∂z + µ ∂ψ 2 + r 2 R + r sin ψ r R + r sin ψ ∂r � cos ψ ∂w w + (axial) ∂ψ − ( R + r sin ψ ) 2 r ( R + r sin ψ )

23 Wall equations u �→ ( η, ξ, ζ ) 2 3 „ « η + ∂ξ η sin ψ + ξ cos ψ + ∂ζ sin ψ ρ w h∂ 2 η » p − 2 µ∂u – hE ∂ψ ∂θ 6 7 ∂t 2 = − + 6 7 1 − σ 2 a 2 ( R + a sin ψ ) 2 6 7 ∂r r = a 4 5 2 3 „ 2 η + ∂ξ « + ξ cos ψ + ∂ζ sin ψ − σhE ∂ψ ∂θ 6 7 6 7 1 − σ 2 6 a ( R + a sin ψ ) 7 4 5 (radial)

24 Wall equations ∂ψ + ∂ 2 ξ 2 ∂η η + ∂ξ 0 ρ w h∂ 2 ξ » 1 ∂ψ − v ∂u r + ∂v – hE ∂ψ 2 ∂ψ 6 B ∂t 2 = − µ + + cos ψ 6 B 1 − σ 2 6 a 2 r ∂r a ( R + a sin ψ ) @ r = a 4 ∂ 2 ζ 2 3 − ξ sin ψ + sin ψ ∂η 1 3 η sin ψ + ξ cos ψ + ∂ζ ∂ψ + 5 + σhE ∂ψ∂θ 6 7 ∂θ C 7 − 6 7 C 7 ( R + a sin ψ ) 2 1 − σ 2 6 7 a ( R + a sin ψ ) A 4 5 ∂ 2 ζ 2 3 ∂ 2 ξ ∂θ 2 − cos ψ∂ζ ∂θ∂ψ 6 7 ∂θ + hG a ( R + a sin ψ ) + 6 7 ( R + a sin ψ ) 2 6 7 4 5 (circumferential)

25 Wall equations ρ w h∂ 2 ζ » 1 w sin ψ – ∂u R + a sin ψ + ∂w ∂t 2 = − µ ∂θ − R + a sin ψ ∂r r = a ∂ 2 ξ 2 3 ∂θ + ∂ 2 ζ ∂η 2 3 sin ψ∂η ∂θ + cos ψ∂ξ ∂θ + hE 5 + σhE ∂ψ∂θ ∂θ 2 6 7 6 7 + 6 7 6 7 1 − σ 2 ( R + a sin ψ ) 2 1 − σ 2 6 7 a ( R + a sin ψ ) 4 4 5 ∂ 2 ξ 2 3 ∂ψ∂θ + ζ sin ψ + cos ψ ∂ζ cos ψ∂ξ ∂θ − ζ cos 2 ψ ∂ 2 ζ 1 ∂ψ 6 7 + hG ∂ψ 2 + + 6 7 6 a 2 ( R + a sin ψ ) 2 7 a ( R + a sin ψ ) 4 5 (axial)

26 Perturbation method χ = χ 0 + εχ 1 + ε 2 χ 2 + ε 3 χ 3 +... ε := a (Curvature parameter) R χ 1 ) e i ( ωt − kz ) (¯ χ 0 + ε ¯ χ 1 ) + (˜ χ 0 + ε ˜

27 0-th order and 1-st order solutions (steady flow) Dean’s solution

28

29 0-th order solution (unsteady flow) Womersley’s solution - Straight axis - Homogeneous equations - Axisymmetric solution - Frequency equation

30 Womersley’s solution Axisymmetric flow in a straight elastic tube −5 x 10 4 2 u 0 0 Real part −2 Imaginary part −4 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −3 x 10 20 15 10 w 0 5 0 −5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y

31 1-th order solution (unsteady flow) du 1 dy + u 1 y − v 1 y − ikaw 1 = − ( ikayw 0 + u 0 ) „ 2 d 2 u 1 dy 2 + 1 « u 1 + 2 v 1 „ du 0 dy + 2 ika w 0 + 2 a ¯ « du 1 y 2 − a dp 1 w 0 y 2 + iα 2 dy − dy = − w 0 y µ ν „ 2 d 2 v 1 dy 2 + 1 v 1 + 2 u 1 y + 2 ikaw 0 + 2 a ¯ dv 1 « y 2 − ap 1 „ u 0 w 0 « y 2 + iα 2 dy − µy = − w 0 y ν „ 1 ! d 2 w 1 w 1 + ika 2 dy − ika 2 y dy 2 + 1 « dw 1 dw 0 y 2 + iα 2 dy − µ p 1 = − p 0 y µ r ρ ω ( α := a Womersley number) µ

32 Wave solution Assembling the 0-th and 1-th order solutions ( χ 0 + εχ 1 ) e i ( ωt − kz )

33 Full solution Superposing the steady solution χ 1 ) + ( χ 0 + εχ 1 ) e i ( ωt − kz ) (¯ χ 0 + ε ¯ Steady flow Unsteady flow Harmonic form e Im ( k ) z � � � � �� χ = ¯ χ + Re (˜ χ ) cos ωt − Re ( k ) z − Im (˜ χ ) sin ωt − Re ( k ) z

34 Numerical results E = 10 7 dynes/cm 2 h = 0 . 05 cm σ = 0 . 5 ω = 2 π s − 1 a = 0 . 5 cm ρ = ρ w = 1 g/cm 3 µ = 0 . 04 g/ cm s d ¯ p 0 A = 26000 dyne/cm 2 dz = 7 dyne/cm 3 ∆ y = 0 . 02

2 3 35 2 1.5 u 0 + ε u 1 (cm s −1 ) v 0 + ε v 1 (cm s −1 ) 1 1 0 −1 0.5 −2 0 −3 −0.5 −4 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 4 x 10 50 2.605 p 0 + ε p 1 (dyne cm −2 ) 40 w 0 + ε w 1 (cm s −1 ) 30 2.6 20 10 0 2.595 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 y y Unsteady solution: ˜ χ 0 + ε ˜ χ 1 ε = 0 (continuous line), ε = 0 . 05 (dashed line), ε = 0 . 1 (dotted line).

36 Secondary flow at t = 0 , z = 0 : ˜ χ 0 + ε ˜ χ 1 Σ = 0.007548 Σ = 0.009315 ε =0 ε =0.0001 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 y y Σ = 0.03902 Σ = 0.3902 ε =0.001 ε =0.01 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 y y

37 u ) 2 + ( Re ˜ � v ) 2 . Σ = max ( Re ˜ r,ψ

38 χe i ( ωt − kz ) , ε = 0 . 1 Secondary flow at z = 0 : ¯ χ + ˜ Σ = 7.705 Σ = 9.255 t = 0.105 t = 0.23 −1 0 1 −1 0 1 Σ = 6.894 Σ = 2.227 t = 0.355 t = 0.48 −1 0 1 −1 0 1

39 χe i ( ωt − kz ) , ε = 0 . 1 Secondary flow at z = 0 : ¯ χ + ˜ Σ = 4.769 Σ = 5.154 t = 0.605 t = 0.73 −1 0 1 −1 0 1 Σ = 2.794 Σ = 4.559 t = 0.855 t = 0.98 −1 0 1 −1 0 1

40 Vorticity curves at z = 0 , ε = 0 . 1 1000 800 600 400 Ω * ( ⋅ , 0) (s −1 ) 200 0 −200 t =0.105 t =0.23 −400 t =0.355 t =0.48 −600 −800 −1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y