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
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
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1
Background and model
2
Mathematically related models
3
Riemann problem and mathematical structure
4
L–M and R–M curves L–M curve in Case IIl Examples L–M curve in Case IVl Examples
5
Conclusions and outlook
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Background and model
Figure: http://www.hopkinsmedicine.org
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Background and model
Blood in compliant vessels of medium and large diameter
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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
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
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 ∂tA + ∂x(Au) = 0, ∂t(uA) + ∂x(Au2 + AΨ
ρ ) − ΨAx ρ
= −Ru, ∂tK = 0, where A(x, t) - cross–sectional area of vessel u(x, t) - average velocity in cross sectional area
5/28
Background and model
Governing equations ∂tA + ∂x(Au) = 0, ∂t(uA) + ∂x(Au2 + AΨ
ρ ) − ΨAx ρ
= −Ru, ∂tK = 0, where A(x, t) - cross–sectional area of vessel u(x, t) - average velocity in cross sectional area ρ - constant density of blood
5/28
Background and model
Governing equations ∂tA + ∂x(Au) = 0, ∂t(uA) + ∂x(Au2 + AΨ
ρ ) − ΨAx ρ
= −Ru, ∂tK = 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 pressure1 Ψ(x, t) = K(x) A A0 m − 1
5/28
Background and model
Material property variable K(x) = √π (1 − ν2)R0 E(x)h(x) √A0
A0, R0 - 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 K(x) = √π (1 − ν2)R0 E(x)h(x) √A0
A0, R0 - equilibrium cross sectional area and radius h - thickness of vessel walls E - Young’s modulus of elasticity ν - Poisson ratio
R = 2πνR
δ
due to viscous resistance of blood
6/28
Background and model
Material property variable K(x) = √π (1 − ν2)R0 E(x)h(x) √A0
A0, R0 - equilibrium cross sectional area and radius h - thickness of vessel walls E - Young’s modulus of elasticity ν - Poisson ratio
R = 2πνR
δ
due to viscous resistance of blood Homogeneous simplified blood flow model ∂tA + ∂x(Au) = 0, ∂t(uA) + ∂x(Au2 + AΨ
ρ ) − ΨAx ρ
= 0, ∂tK = 0, Wave speed c(A, K) :=
ρ ΨA =
ρ
A0
m . (1)
6/28
Mathematically related models
The Baer-Nuntiato model for two-phase mixtures2 a - solid, b - gas
∂ca ∂t + ua ∂ca ∂x = Πa
c
∂caρa ∂t + ∂caρaua ∂x = Πa
ρ
∂caρaua ∂t + ∂(caρau2
a + capa)
∂x = pb ∂ca ∂x + Πa
M
∂caρaEa ∂t + ∂caua(ρaEa + pa) ∂x = pbua ∂ca ∂x + Πa
E
∂cbρb ∂t + ∂cbρbub ∂x = −Πa
ρ
∂cbρbub ∂t + ∂(cbρbu2
b + cbpb)
∂x = −pb ∂ca ∂x + −Πa
M
∂cbρbEb ∂t + ∂cbub(ρbEb + pb) ∂x = −pbua ∂ca ∂x + −Πa
E
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Mathematically related models
Duct flow, discontinuous duct area, extended model3
∂a ∂t = 0, ∂aρ ∂t + ∂aρv ∂x
= 0,
∂aρv ∂t
+ ∂a(ρv2+p)
∂x
= p ∂a
∂x, ∂aρE ∂t
+ ∂av(ρE+p)
∂x
= 0, a(x) duct area, ρ(t, x), v(t, x), p(t, x) density, velocity and pressure respectively
(al, ρl, vl, pl) (ar, ρr, vr, pr) 3Ee Han, M. Hantke, W., J. Hyperbolic Diff. Eqns. 9(3) (2012) 403-449
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Mathematically related models
Shallow water system discontinuous bottom topography extended model4
∂z ∂t = 0, ∂h ∂t + ∂hv ∂x = 0, ∂hv ∂t + ∂(hv2+g h2
2 )
∂x
= gh ∂z
∂x,
z(x) bottom topography, h(t, x), v(t, x), g water depth, velocity and gravitational constant respectively
4Ee Han, W., Quart. Appl. Math., to appear
9/28
Riemann problem and mathematical structure
Initial data (A, u, K) = (AL, uL, KL), x < 0, (AR, uR, KR), x > 0.
10/28
Riemann problem and mathematical structure
Initial data (A, u, K) = (AL, uL, KL), x < 0, (AR, uR, KR), x > 0. Serves as building block in theory and numerical methods
10/28
Riemann problem and mathematical structure
Initial data (A, u, K) = (AL, uL, KL), x < 0, (AR, uR, KR), x > 0. Serves as building block in theory and numerical methods Applications Figure: Abdominal aortic aneurysms (AAA)
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Riemann problem and mathematical structure
Initial data (A, u, K) = (AL, uL, KL), x < 0, (AR, uR, KR), x > 0. Serves as building block in theory and numerical methods Applications Figure: Abdominal aortic aneurysms (AAA) Figure: Surgical treatment
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Riemann problem and mathematical structure
Eigenvalues: λ0 = 0, λ1 = u − c, λ2 = u + c Eigenvectors: R0 = 1
(u2−c2) c2
, R1 = 1 u − c , R2 = 1 u + c . 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
Eigenvalues: λ0 = 0, λ1 = u − c, λ2 = u + c Eigenvectors: R0 = 1
(u2−c2) c2
, R1 = 1 u − c , R2 = 1 u + c . 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 (u2 = c2 ) R0 → Rk as λk → 0 when k = 1, 2
separated and coincide with each other
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Riemann problem and mathematical structure
The 1–wave curve T1(wL) and 2–wave curve T2(wR) are classical given by Tk(wq) = {w|u = uq ± fq(A; wq)} where fq(A; wq) :=
2 m
ρ
A0
m − cq
if A ≤ Aq
cq √m+1
Aq
m+1 − 1 1 − Aq
A
1
2
, if A > Aq and q = L when k = 1 and q = R when k = 2 T1(wL) is strictly decreasing while T2(wR) is strictly increasing in the (u, Ψ) phase plane
12/28
Riemann problem and mathematical structure
The additional stationary wave curve satisfies Aoutuout = Ainuin (2) 1 2 ρu2
Aout A0 m − 1
1 2 ρu2
in + Kin
Ain A0 m − 1
Denote wout = J(Kout; win, Kin)
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Riemann problem and mathematical structure
The additional stationary wave curve satisfies Aoutuout = Ainuin (2) 1 2 ρu2
Aout A0 m − 1
1 2 ρu2
in + Kin
Ain A0 m − 1
Denote wout = J(Kout; win, Kin) Define a velocity function φ(u; win, Kin, Kout) by inserting (2) into (3) φ(u; win, Kin, Kout) =
1 2 ρu2 + Kout
A0u
m − 1
2 ρu2 in
−Kin
A0
m − 1
Riemann problem and mathematical structure
The additional stationary wave curve satisfies Aoutuout = Ainuin (2) 1 2 ρu2
Aout A0 m − 1
1 2 ρu2
in + Kin
Ain A0 m − 1
Denote wout = J(Kout; win, Kin) Define a velocity function φ(u; win, Kin, Kout) by inserting (2) into (3) φ(u; win, Kin, Kout) =
1 2 ρu2 + Kout
A0u
m − 1
2 ρu2 in
−Kin
A0
m − 1
Consider u∗ = sgn(uin)
ρ
Ain|uin|
A0
m
1 m+2 then the properties of the velocity function
are the following:
1
φ(u; win, Kin, Kout) decreases if u < u∗;
2
φ(u; win, Kin, Kout) increases if u > u∗;
3
φ(u; win, Kin, Kout) has the minimum value at u = u∗ and |u∗| = c∗, where c∗ = c(Kout, A∗) is defined in (1) and A∗ = Ainuin
u∗
.
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L–M and R–M curves
Closely resemble those of the Euler equations in discontinuous ducts and the shallow water equations with discontinuous bottom topography 5
5Han et al. (2011,2012,2013)
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L–M and R–M curves
Closely resemble those of the Euler equations in discontinuous ducts and the shallow water equations with discontinuous bottom topography 5 The discontinuous material property variable can be viewed as the limiting case of piecewise linear material property variables
5Han et al. (2011,2012,2013)
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L–M and R–M curves
Closely resemble those of the Euler equations in discontinuous ducts and the shallow water equations with discontinuous bottom topography 5 The discontinuous material property variable can be viewed as the limiting case of piecewise linear material property variables Stationary wave to the Riemann problem is a 0 width transition layer located at x = 0
5Han et al. (2011,2012,2013)
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L–M and R–M curves
Closely resemble those of the Euler equations in discontinuous ducts and the shallow water equations with discontinuous bottom topography 5 The discontinuous material property variable can be viewed as the limiting case of piecewise linear material property variables Stationary wave to the Riemann problem is a 0 width transition layer located at x = 0 We merge the stationary wave curve into the 1– or 2–wave curves and name these L–M and R–M curves
5Han et al. (2011,2012,2013)
14/28
L–M and R–M curves
Closely resemble those of the Euler equations in discontinuous ducts and the shallow water equations with discontinuous bottom topography 5 The discontinuous material property variable can be viewed as the limiting case of piecewise linear material property variables Stationary wave to the Riemann problem is a 0 width transition layer located at x = 0 We merge the stationary wave curve into the 1– or 2–wave curves and name these L–M and R–M curves The L–M curve starting from (AL, uL) to (AM, uM), consists of a 1–shock curve S1, a 1–rarefaction curve R1, and/or the stationary wave curve (not necessarily present)
5Han et al. (2011,2012,2013)
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L–M and R–M curves
Closely resemble those of the Euler equations in discontinuous ducts and the shallow water equations with discontinuous bottom topography 5 The discontinuous material property variable can be viewed as the limiting case of piecewise linear material property variables Stationary wave to the Riemann problem is a 0 width transition layer located at x = 0 We merge the stationary wave curve into the 1– or 2–wave curves and name these L–M and R–M curves The L–M curve starting from (AL, uL) to (AM, uM), consists of a 1–shock curve S1, a 1–rarefaction curve R1, and/or the stationary wave curve (not necessarily present) The algorithm for finding the exact Riemann solutions: calculate (AM, uM) for given Riemann initial data construct the L–M and R–M curves
5Han et al. (2011,2012,2013)
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L–M and R–M curves
Case Il AP l
2
max ≥ A0
KL
1
m ; uL ≤ cL; ul
c ≤ ul sc
Case IIl AP l
2
max ≥ A0
KL
1
m ; uL ≤ cL; ul
c > ul sc
Case IIIl AP l
2
max ≥ A0
KL
1
m ; uL > cL; ϕ
KL ; wL
Case IVl AP l
2
max ≥ A0
KL
1
m ; uL > cL; ϕ
KL ; wL
KL ; wL
Case Vl AP l
2
max ≥ A0
KL
1
m ; uL > cL; τ
KL ; wL
Case V Il AP l
2
max < A0
KL
1
m
where AP l
2
max =
A0
mKL
1
m m
2 uL + cL
2
m ,
if uL ≤ 0 ALxu0
1,
if uL > 0 and xu0
1 is the solution
to xm+2 − xm+1 −
uL cL 2 x + 1 = 0 The critical velocity ul
c = m m+2 uL + 2 m+2 cL and ul sc =
ρ(m+2) (KR−KL)
KR
KL
m+2 −1 15/28
L–M and R–M curves
Case Ir uR + cR ≥ 0; ur
c < −ur sc
Case IIr uR + cR ≥ 0; ur
c ≥ −ur sc
Case IIIr uR + cR < 0; ϕ
KR ; wR
Case IVr uR + cR < 0; ϕ
KR ; wR
KR ; wR
Case Vr uR + cR < 0; τ
KR ; wR
where ϕ(κ; wq) =
m+2 2
uq
cq
2m
m+2 κ 2 m+2 −
Aq
m κ − m
2
uq
cq
2
ˆ A1q Aq
−2 −
A1q Aq
m + Aq
A0
−m (4) and τ(κ; wq) = m + 2 2 F
2m m+2
q
κ
2 m+2 −
A0 Aq m κ − m 2 F 2
q − 1 +
A0 Aq m . (5)
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L–M and R–M curves L–M curve in Case IIl
2
max ≥ A0
KL
m; uL ≤ cL; ul
c ≤ ul sc Possible wave configurations with uM > 0
1–wave 2–wave x t wL wR wM w− 0–wave
Figure: A
1–wave 1–wave 2–wave x t wL wR wM wc ˜ wc 0–wave
Figure: B
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L–M and R–M curves L–M curve in Case IIl
Case IIl: A
P l
2
max ≥ A0
KL
1
m ; uL ≤ cL; ul
c ≤ ul sc
The L–M curve CL(wL) =
3
P l
k(wL):
P l
1(wL) = {w|w ∈ T1(wL) with u ≤ 0}
P l
2(wL) = {w|w = J(KR; w−, KL) w− ∈ T1(wL) and 0 < u < uc}
P l
3(wL) = {w|w ∈ T1(wc) with u > uc}
where wc = J(aR; ˜ wc, aL) and ˜ wc ∈ T1(wL)
18/28
L–M and R–M curves L–M curve in Case IIl
Case IIl: A
P l
2
max ≥ A0
KL
1
m ; uL ≤ cL; ul
c ≤ ul sc
The L–M curve CL(wL) =
3
P l
k(wL):
P l
1(wL) = {w|w ∈ T1(wL) with u ≤ 0}
P l
2(wL) = {w|w = J(KR; w−, KL) w− ∈ T1(wL) and 0 < u < uc}
P l
3(wL) = {w|w ∈ T1(wc) with u > uc}
where wc = J(aR; ˜ wc, aL) and ˜ wc ∈ T1(wL)
−2 2 4 6 8 10 −1 1 2 3 4 5 x 10
5
P1(QL) P2(QL) P3(QL) u Ψ
Theorem of monotonicity. L–M curve is continuous and decreasing in the (u, ψ) phase plane
18/28
L–M and R–M curves L–M curve in Case IIl
Case IIl: A
P l
2
max ≥ A0
KL
1
m ; uL ≤ cL; ul
c ≤ ul sc
The L–M curve CL(wL) =
3
P l
k(wL):
P l
1(wL) = {w|w ∈ T1(wL) with u ≤ 0}
P l
2(wL) = {w|w = J(KR; w−, KL) w− ∈ T1(wL) and 0 < u < uc}
P l
3(wL) = {w|w ∈ T1(wc) with u > uc}
where wc = J(aR; ˜ wc, aL) and ˜ wc ∈ T1(wL)
−2 2 4 6 8 10 −1 1 2 3 4 5 x 10
5
P1(QL) P2(QL) P3(QL) u Ψ
Theorem of monotonicity. L–M curve is continuous and decreasing in the (u, ψ) phase plane The Riemann solution uniquely exists
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L–M and R–M curves Examples
Example: the solution with wave configuration A in Case IIl by Toro and Siviglia Table: Initial and intermediate states
K A (m2) u (m/s) F wL 2000003.266554 3.0 × 10−4 −2.6575 × 10−5 −7.88826 × 10−7 w− 2000003.26654 2.01185 × 10−4 12.81037 0.420195 wM 40000.06533 6.34034 × 10−4 4.0648486 0.107114 wR 40000.06533 3 × 10−4 6.123 × 10−6 1.28516 × 10−6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.3 1.82 2.34 2.86 3.38 3.9 4.42 4.94 5.46 5.98 6.5 x 10
−4
vesessl area (m2) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.02 1.28 2.58 3.88 5.18 6.48 7.78 9.08 10.38 11.68 12.98 velocity (m/s)
Figure: The exact vessel area and velocity at t = 0.012 s.
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L–M and R–M curves Examples
Example: the solution with wave configuration B in Case IIl Table: Initial and intermediate states
K A (m2) u (m/s) F wL 2000003.266554 3.0 × 10−4 −2.6575 × 10−5 −7.88826 × 10−7 ˜ wc 2000003.26654 2.0034 × 10−4 12.938557 0.424847 wc 40000.06533 4.82994 × 10−4 5.36675 1 wM 40000.06533 4.13503 × 10−4 6.18445 1.197995 wR 40000.06533 1.17 × 10−4 1.5 × 10−7 3.984 × 10−8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.65 1.3 1.95 2.6 3.25 3.9 4.55 5.2 x 10
−4
vesessl area (m2) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.05 1.25 2.55 3.85 5.15 6.45 7.75 9.05 10.35 11.65 12.95 velocity (m/s)
Figure: The exact vessel area and velocity at t = 0.012 s.
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L–M and R–M curves Examples
Case IVl: A
P l
2
max ≥ A0
KL
1
m ; uL > cL; ϕ
KL ; wL
Possible wave configurations with uM > 0
1–wave 2–wave x t wL wR wM w− 0–wave
(a) A
S0
1(w−)
2–wave x t wL wR wM w− w+ KL KR K 0–wave
x=0
(b) E
1–wave 2–wave x t wL wR wM ¯ wL 0–wave
(c) F
21/28
L–M and R–M curves L–M curve in Case IVl
Case IVl: A
P l
2
max ≥ A0
KL
1
m ; uL > cL; ϕ
KL ; wL
The L–M curve CL(wL) =
4
P l
k(wL):
P l
1(wL)
= {w|w ∈ T1(wL) with u < 0} , P l
2(wL)
= {w|w = J(KR; w−, KL) and w− ∈ T1(wL) 0 < u < ¯ ˆ u1L
3(wL)
= {w|w = J(KR; w+, K); w+ = S0
1(w−); w− = J(K; wL, KL), K ∈]KR, KL[} ,
P l
4(wL)
=
w1L) u > ˆ ¯ uL
22/28
L–M and R–M curves L–M curve in Case IVl
Case IVl: A
P l
2
max ≥ A0
KL
1
m ; uL > cL; ϕ
KL ; wL
The L–M curve CL(wL) =
4
P l
k(wL):
P l
1(wL)
= {w|w ∈ T1(wL) with u < 0} , P l
2(wL)
= {w|w = J(KR; w−, KL) and w− ∈ T1(wL) 0 < u < ¯ ˆ u1L
3(wL)
= {w|w = J(KR; w+, K); w+ = S0
1(w−); w− = J(K; wL, KL), K ∈]KR, KL[} ,
P l
4(wL)
=
w1L) u > ˆ ¯ uL
1 2 3 4 5 4 5 6 7 8 x 10
4
P l
1(wL)
P l
2(wL)
P l
3(wL)
P l
4(wL)
u ψ
Theorem of Uniqueness. If
ρ¯ u2
L
mKR − ˆ ¯ AL ¯ AL > 0, the L–M curve on
(u, ψ) phase plane is monotonically decreasing The Riemann solution uniquely exists
22/28
L–M and R–M curves L–M curve in Case IVl
Case IVl: A
P l
2
max ≥ A0
KL
1
m ; uL > cL; ϕ
KL ; wL
The L–M curve CL(wL) =
4
P l
k(wL):
P l
1(wL)
= {w|w ∈ T1(wL) with u < 0} , P l
2(wL)
= {w|w = J(KR; w−, KL) and w− ∈ T1(wL) 0 < u < ¯ ˆ u1L
3(wL)
= {w|w = J(KR; w+, K); w+ = S0
1(w−); w− = J(K; wL, KL), K ∈]KR, KL[} ,
P l
4(wL)
=
w1L) u > ˆ ¯ uL
0.5 1 1.5 −4.35 −4.3 −4.25 −4.2 −4.15 −4.1 −4.05 x 10
4
P l
2(wL)
P l
3(wL)
¯ ˆ wL ˆ ¯ wL
P l
4(wL)
u Ψ
Theorem of Nonuniqueness. If
ρ¯ u2
L
mKR − ˆ ¯ AL ¯ AL < 0, the L–M curve on
(u, ψ) phase plane contains a bifurcation. For the given initial data we have three possible solutions, with wave configurations A, E, and F respectively
23/28
L–M and R–M curves Examples
Example: the unique solution with wave configuration E in Case IVl
K A (m2) u (m/s) F wL 40000.065331 4.2248 × 10−4 10.380259 2.0 w− 29017.111011 4.03365 × 10−4 10.872157 2.488099 w+ 29017.111011 1.607526 × 10−3 2.728076 0.441869 wM 28000.045732 1.700599 × 10−3 2.578769 0.419263 wR 28000.045732 1.098391 × 10−3
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10
−3
vesessl area (m2) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 11 12 velocity (m/s)
Figure: The exact vessel area and velocity at t = 0.02 s, where ϕ
KR
KL ; wL
τ KR
KL ; wL
ρ¯ u2 L mKR − ˆ ¯ AL ¯ AL = 4.814028 24/28
L–M and R–M curves Examples
Example: the nonunique solutions with wave configurations A, E and F
K A (m2) u (m/s) F wL 58136.483963 1.0 × 10−6 6.655409 4 wR 56973.754284 5.038 × 10−6
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.8 1.6 2.4 3.2 4 4.8 5.6 6.4 7.2 8 x 10
−6
vesessl area (m2) A E F 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7 velocity (m/s) A E F
Figure: The Exact vessel areas and velocities at t = 0.15 s, where ϕ
KR
KL ; wL
τ KR
KL ; wL
ρ¯ u2 L mKR − ˆ ¯ AL ¯ AL = −5.156529 25/28
L–M and R–M curves Examples
The L–M curve is monotone decreasing in the (u, Ψ) phase plane in Cases IIl, IIIl, and V Il
26/28
L–M and R–M curves Examples
The L–M curve is monotone decreasing in the (u, Ψ) phase plane in Cases IIl, IIIl, and V Il In Case IVl it is also monotone decreasing if
ρ¯ u2
L
mKR − ˆ ¯ AL ¯ AL > 0
26/28
L–M and R–M curves Examples
The L–M curve is monotone decreasing in the (u, Ψ) phase plane in Cases IIl, IIIl, and V Il In Case IVl it is also monotone decreasing if
ρ¯ u2
L
mKR − ˆ ¯ AL ¯ AL > 0
The Riemann solution uniquely exists
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L–M and R–M curves Examples
The L–M curve is monotone decreasing in the (u, Ψ) phase plane in Cases IIl, IIIl, and V Il In Case IVl it is also monotone decreasing if
ρ¯ u2
L
mKR − ˆ ¯ AL ¯ AL > 0
The Riemann solution uniquely exists The bifurcation occurs in Case IVl if
ρ¯ u2
L
mKR − ˆ ¯ AL ¯ AL < 0
and always in Case Vl
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L–M and R–M curves Examples
The L–M curve is monotone decreasing in the (u, Ψ) phase plane in Cases IIl, IIIl, and V Il In Case IVl it is also monotone decreasing if
ρ¯ u2
L
mKR − ˆ ¯ AL ¯ AL > 0
The Riemann solution uniquely exists The bifurcation occurs in Case IVl if
ρ¯ u2
L
mKR − ˆ ¯ AL ¯ AL < 0
and always in Case Vl The nonunique Riemann solutions occur due to the bifurcation
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L–M and R–M curves Examples
The L–M curve is monotone decreasing in the (u, Ψ) phase plane in Cases IIl, IIIl, and V Il In Case IVl it is also monotone decreasing if
ρ¯ u2
L
mKR − ˆ ¯ AL ¯ AL > 0
The Riemann solution uniquely exists The bifurcation occurs in Case IVl if
ρ¯ u2
L
mKR − ˆ ¯ AL ¯ AL < 0
and always in Case Vl The nonunique Riemann solutions occur due to the bifurcation The cases for the R–M curves are analogous to the cases of the L–M curves
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Conclusions and outlook
For any given Riemann initial data we obtained all possible exact solutions to the simplified blood flow model with discontinuous material properties The behaviors of the L–M and R–M curves for all basic cases were completely analyzed May help to assess numerical schemes for much more complicated and realistic models Possibly a tool to evaluate the effect of the stent graft to the body Planned: Numerical schemes based on these exact Riemann solutions
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Conclusions and outlook
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