On the Riemann problem for a system modeling blood flow Gerald - PowerPoint PPT Presentation

On the Riemann problem for a system modeling blood flow Gerald Warnecke Institute of Analysis and Numerics Faculty of Mathematics Otto-von-Guericke-Universit¨ at May, 2014 Joint work with: Ee Han Eleuterio Toro Annunziato Siviglia University of Bremen, Germany University of Trento, Italy ETH Z¨ urich, Switzerland 1/28

Outline Background and model 1 Mathematically related models 2 Riemann problem and mathematical structure 3 L–M and R–M curves 4 L–M curve in Case II l Examples L–M curve in Case IV l Examples Conclusions and outlook 5 2/28

Background and model Figure: http://www.hopkinsmedicine.org 3/28

Background and model Blood in compliant vessels of medium and large diameter 4/28

Background and model Blood in compliant vessels of medium and large diameter continuum and incompressible liquid in thin collapsible tubes 4/28

Background and model Blood in compliant vessels of medium and large diameter continuum and incompressible liquid in thin collapsible tubes Fig. 1 Assumed axially symmetric vessel configuration in three space dimensions at time t. Cross Figure: Assume blood vessel in 3D is axially symmetric axisymmetric incompressible Navier–Stokes equations + boundary conditions 4/28

Background and model Blood in compliant vessels of medium and large diameter continuum and incompressible liquid in thin collapsible tubes Fig. 1 Assumed axially symmetric vessel configuration in three space dimensions at time t. Cross Figure: Assume blood vessel in 3D is axially symmetric axisymmetric incompressible Navier–Stokes equations + boundary conditions Apply averaging procedure The radius of blood vessel is much smaller than length of blood vessel 4/28

Background and model Governing equations ∂ t A + ∂ x ( Au ) = 0 , ∂ t ( uA ) + ∂ x ( Au 2 + A Ψ ρ ) − Ψ A x = − R u, ρ ∂ t K = 0 , where A ( x, t ) - cross–sectional area of vessel u ( x, t ) - average velocity in cross sectional area 1 E. Toro and A. Siviglia, Commun. Comp. Phys. 13(2)(2011) 361-385. 5/28

Background and model Governing equations ∂ t A + ∂ x ( Au ) = 0 , ∂ t ( uA ) + ∂ x ( Au 2 + A Ψ ρ ) − Ψ A x = − R u, ρ ∂ t K = 0 , where A ( x, t ) - cross–sectional area of vessel u ( x, t ) - average velocity in cross sectional area ρ - constant density of blood 1 E. Toro and A. Siviglia, Commun. Comp. Phys. 13(2)(2011) 361-385. 5/28

Background and model Governing equations ∂ t A + ∂ x ( Au ) = 0 , ∂ t ( uA ) + ∂ x ( Au 2 + A Ψ ρ ) − Ψ A x = − R u, ρ ∂ t K = 0 , where A ( x, t ) - cross–sectional area of vessel u ( x, t ) - average velocity in cross sectional area ρ - constant density of blood Tube law for transmural pressure 1 �� A � m � Ψ( x, t ) = K ( x ) − 1 A 0 0 < m < 1 constant 1 E. Toro and A. Siviglia, Commun. Comp. Phys. 13(2)(2011) 361-385. 5/28

Background and model Material property variable √ π E ( x ) h ( x ) K ( x ) = √ A 0 (1 − ν 2 ) R 0 A 0 , R 0 - equilibrium cross sectional area and radius h - thickness of vessel walls E - Young’s modulus of elasticity ν - Poisson ratio 6/28

Background and model Material property variable √ π E ( x ) h ( x ) K ( x ) = √ A 0 (1 − ν 2 ) R 0 A 0 , R 0 - equilibrium cross sectional area and radius h - thickness of vessel walls E - Young’s modulus of elasticity ν - Poisson ratio R = 2 πνR due to viscous resistance of blood δ 6/28

Background and model Material property variable √ π E ( x ) h ( x ) K ( x ) = √ A 0 (1 − ν 2 ) R 0 A 0 , R 0 - equilibrium cross sectional area and radius h - thickness of vessel walls E - Young’s modulus of elasticity ν - Poisson ratio R = 2 πνR due to viscous resistance of blood δ Homogeneous simplified blood flow model ∂ t A + ∂ x ( Au ) = 0 , ∂ t ( uA ) + ∂ x ( Au 2 + A Ψ ρ ) − Ψ A x = 0 , ρ ∂ t K = 0 , Wave speed � � � � m A mK A (1) c ( A, K ) := ρ Ψ A = . ρ A 0 6/28

Mathematically related models Mathematically related models The Baer-Nuntiato model for two-phase mixtures 2 a - solid, b - gas ∂c a ∂c a Π a ∂t + u a = c ∂x ∂c a ρ a + ∂c a ρ a u a Π a = ρ ∂t ∂x + ∂ ( c a ρ a u 2 ∂c a ρ a u a a + c a p a ) ∂c a ∂x + Π a = p b M ∂t ∂x ∂c a ρ a E a + ∂c a u a ( ρ a E a + p a ) ∂c a ∂x + Π a = p b u a E ∂t ∂x ∂c b ρ b + ∂c b ρ b u b − Π a = ρ ∂t ∂x + ∂ ( c b ρ b u 2 ∂c b ρ b u b b + c b p b ) ∂c a ∂x + − Π a = − p b M ∂t ∂x ∂c b ρ b E b + ∂c b u b ( ρ b E b + p b ) ∂c a ∂x + − Π a = − p b u a E ∂t ∂x 2 N. Andrianov, W., J. Comput. Phys. 195 (2004) 234-264 7/28

Mathematically related models Mathematically related models Duct flow, discontinuous duct area, extended model 3  ∂a ∂t = 0 ,     ∂aρ ∂t + ∂aρv  = 0 ,   ∂x + ∂a ( ρv 2 + p ) ∂aρv = p ∂a ∂x ,  ∂t ∂x     + ∂av ( ρE + p ) ∂aρE  = 0 ,  ∂t ∂x a ( x ) duct area, ρ ( t, x ) , v ( t, x ) , p ( t, x ) density, velocity and pressure respectively ( a l , ρ l , v l , p l ) ( a r , ρ r , v r , p r ) 3 Ee Han, M. Hantke, W., J. Hyperbolic Diff. Eqns. 9(3) (2012) 403-449 8/28

Mathematically related models Mathematically related models Shallow water system discontinuous bottom topography extended model 4  ∂z ∂t = 0 ,     ∂h ∂t + ∂hv ∂x = 0 ,  ∂t + ∂ ( hv 2 + g h 2  2 )  ∂hv = gh ∂z ∂x ,  ∂x z ( x ) bottom topography, h ( t, x ) , v ( t, x ) , g water depth, velocity and gravitational constant respectively 4 Ee Han, W., Quart. Appl. Math., to appear 9/28

Riemann problem and mathematical structure Riemann problem � ( A L , u L , K L ) , x < 0 , Initial data ( A, u, K ) = ( A R , u R , K R ) , x > 0 . 10/28

Riemann problem and mathematical structure Riemann problem � ( A L , u L , K L ) , x < 0 , Initial data ( A, u, K ) = ( A R , u R , K R ) , x > 0 . Serves as building block in theory and numerical methods 10/28

Riemann problem and mathematical structure Riemann problem � ( A L , u L , K L ) , x < 0 , Initial data ( A, u, K ) = ( A R , u R , K R ) , x > 0 . Serves as building block in theory and numerical methods Applications Figure: Abdominal aortic aneurysms (AAA) 10/28

Riemann problem and mathematical structure Riemann problem � ( A L , u L , K L ) , x < 0 , Initial data ( A, u, K ) = ( A R , u R , K R ) , x > 0 . Serves as building block in theory and numerical methods Applications Figure: Abdominal aortic aneurysms (AAA) Figure: Surgical treatment 10/28

Riemann problem and mathematical structure Mathematical structure Eigenvalues: λ 0 = 0 , λ 1 = u − c, λ 2 = u + c Eigenvectors:       1 1 1 0  ,  ,  . R 0 = R 1 = u − c R 2 = u + c    ( u 2 − c 2 ) 0 0 c 2 0–wave (stationary wave), 1–wave, 2–wave 1 – and 2 –waves consist of shock and rarefaction waves 11/28

Riemann problem and mathematical structure Mathematical structure Eigenvalues: λ 0 = 0 , λ 1 = u − c, λ 2 = u + c Eigenvectors:       1 1 1 0  ,  ,  . R 0 = R 1 = u − c R 2 = u + c    ( u 2 − c 2 ) 0 0 c 2 0–wave (stationary wave), 1–wave, 2–wave 1 – and 2 –waves consist of shock and rarefaction waves Features Nonstrictly hyperbolic: Mutual position of stationary wave with the remaining two waves cannot be determined a priori Resonance occurs at the critical state ( u 2 = c 2 ) R 0 → R k as λ k → 0 when k = 1 , 2 T. Liu (1982): In the resonant state waves of different families are not well separated and coincide with each other 11/28

Riemann problem and mathematical structure Elementary wave curves of Riemann problem The 1 –wave curve T 1 ( w L ) and 2 –wave curve T 2 ( w R ) are classical given by T k ( w q ) = { w | u = u q ± f q ( A ; w q ) } where �� �  � m � mK q 2 A − c q , if A ≤ A q   m ρ A 0  f q ( A ; w q ) := �� 1 ��� � � � m +1 2 c q 1 − A q A  − 1 , if A > A q √ m +1   A q A and q = L when k = 1 and q = R when k = 2 T 1 ( w L ) is strictly decreasing while T 2 ( w R ) is strictly increasing in the ( u, Ψ) phase plane 12/28

Riemann problem and mathematical structure Stationary wave curve The additional stationary wave curve satisfies A out u out = A in u in (2) �� A out �� A in � m � m 1 � 1 � 2 ρu 2 2 ρu 2 out + K out − 1 = in + K in − 1 (3) A 0 A 0 Denote w out = J ( K out ; w in , K in ) 13/28

Riemann problem and mathematical structure Stationary wave curve The additional stationary wave curve satisfies A out u out = A in u in (2) �� A out �� A in � m � m 1 � 1 � 2 ρu 2 2 ρu 2 out + K out − 1 = in + K in − 1 (3) A 0 A 0 Denote w out = J ( K out ; w in , K in ) Define a velocity function φ ( u ; w in , K in , K out ) by inserting (2) into (3) � m 2 ρu 2 + K out �� � 1 A in u in − 1 2 ρu 2 φ ( u ; w in , K in , K out ) = − 1 in A 0 u � m �� � A in − K in − 1 A 0 13/28