Fig. 4.1: Pressure and stress[Stationary fluid]

Pressure[Solid]

Stress© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.2: Telemetry of pressure by
the communicating tube
[Catheter]
Pressure in pipe
Measured pressure

Level

Fluid

[Communicating tube]© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.3: Continuity
v2
A1

v1

A2

DensityCompressibility
Incompressible fluid
Law of conservation of mass
© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.4: Expression of Bernoulli

P1P2v2A1v1

A2(1/2)ρv 2＋p ＝constant(4.2)© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.5: Head drop

Density:ρ

Drop: h⇒Pressure
difference: ρghGravitational acceleration: g© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.6: Shunt
Flow path and control fluid path
Valve

Flow path© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.7: Principle of siphon
AtmosphericpressurehGravity
1.0×103 kg m－3×9.8 m s－2×10 m＝98 kPa(4.4)

101.325 kPa－98 kPa＝3.325 kPa(4.5)© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.8: Collapse(a) Vein collapse(b) Blood removal cannula

Left atrium

Vein
Collapse

Blood

Vessel wall

Removal cannula© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.9: Hook elastic bodyStress: τ

Slope:
elastic modulus: E
Strain: ε© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.10: Shear ratev2

τyv2

Δv
v1v1
Flow

γ＝Δv／y(4.6)© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.11: Newtonian fluid
Shear stress: τSlope: coefficient ofviscosity: η

η＝τ／γ(4.7)

Shear rate: γ© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.12: Viscosity with temperature
Easy to be shear[Gas]
Molecular velocity[Low temperature][High temperature][Liquid]
Hard to be shear© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, JapanFig. 4.13: Viscosity with shear rate
Viscosity: η, Pa s

0.010

non-Newtonian fluid
Hematocrit: Ht
Blood: Ht＝40％, 310 K

0.005
Water: 293 K
050100150
Shear rate: γ, s-1© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.14: Rouleau formation

Erythrocyte

Rouleau© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.15: Viscosity tracings withvibrating electrode
Fixed electrodeVibratingelectrodeBlood© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.16: Measurement of local
viscosity with vibrating electrode
Fixed electrodeVibratingelectrode
AlbumenYolk© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.17: Maxwell model
Elastic element
Viscosity element
Stress: τStress relaxation

Strain: ε

Time: tTime: t© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.18: Kelvin-Voigt model
Elasticity
element

Viscosity
element

Stress: τStrain: ε

Time: tCreep deformation
Time: t© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, JapanFig. 4.19: Circulation resistance
Cardiac output: Q

Lung

Q
Pulmonary arterypressure: PpHeart

Right atrial
pressure: Pr[Pulmonary circulationresistance: Rp](Bronchial circulation)
Left atrial
pressure: Pl
Aortic pressure: PaQ

Systems[Systemic circulationresistance: Rs]© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.20: Velocity distribution in pipe(a) Plug flow(b) Hagen-Poiseuilleflow© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.21: Force balance in cylinder inflowv
lr

Δprv

Fr+drF+dF© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.22: Cylinders of fluid in flow
through pipev6v1v
2v
3vv5 4arv© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.23: Distribution of velocity(a2)
2r(a1)

rvv
(b2)
2r

(b1)

r(a) Plug flowv’(b) Hagen-Poiseuilleflowv’’© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.24: Inlet region

WallDistribution of velocity

Inlet region

Wall© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.25: Boundary layer

MainstreamBoundary layer
Wall© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.26: Couette flowv

dWallFluidWall

γ＝v／d(4.37)© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.27: Flow between rotatingcone and stationary platev

ωrRdθ

γ＝v／d＝r ω／(r θ)＝ω／θ(4.40)© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.28: Cone-plate viscometer

Rotation
TorqueViscosity
Rotating cone
Couette flow

FluidStationary plate© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.29: Clotting between rotatingcone and stationary plate(a) Blood between rotatingcone and stationary plate(b) Torque tracings during
clot formation

Torque T
T1
T0
t0
Clot formation

Rc＝(T1－T0)／T1

Clotformation
Time t(4.41)© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, JapanFig. 4.29: Clot formation between rotatingcone and stationary plate(c) Clot; cone (left), plate (right)
γ＝430 s－1
Rc＝0.45

Shear rate
Clotting ratio

43 s－1
Rc＝0.70
4.3 s－1
Rc＝0.90© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.29: Clot formation between
rotating cone and stationary plate(d) Shear rate γ and Clotting ratio Rc
Rc
1.00.5500γ, s－11000© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.30(a): Counter rotating rheoscopeVelocity[Rotating disc]
Velocity[Fluid]Stationary plane(observation)
Erythrocyte[Rotating disc]

Microscope© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.30(b): Counter rotating rheoscope
Motor
Rotating disk
FluidMicroscope© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.31(a): Velocity distribution inflow between parallel walls

ydl
γw＝6 Q ／(b d 2)b(4.62)© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.31(b): Force balance in flow
between parallel walls
lyy

F+dF
y+dyFv
Δpv

dF＝b dy ΔP(4.46)© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.32: Deformation and exfoliationof cell in flow
(1)(2) Deformation(3) Exfoliation
Flow

0.2 mm© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.33: Flow channel between
parallel walls
Microscope

Flowchannel© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.34: Flow channel system withparallel wall for microscopic observation

Flow
channelMicroscope

Syringe pump

© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.35: Extension of cellflow

0.1 mm© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.36: Movement, deformation,
proliferation, orientation, anddifferentiation of cell
Movement
Deformation

Proliferation

Orientation
Differentiation© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.37: Falling sphere
Viscosity

ρ2dρ1v
GravityDη∝d 2(ρ1－ρ2 )／v(4.63)© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.38: Magnus effectForceFlowRotation© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.39: Axis concentration?
WallVelocity
Axis

Erythrocytes© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.40: Secondary flow inbend tube

Main flow

Secondary flow component
in the cross section© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.41: Secondary flow betweencylinder (Taylor vortex)
Rotating inner cylinder
Main flow

Secondary flow

Stationary outer cylinder© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.42: Secondary flow between
rotating cone and stationary plate
Rotating cone

Stationary plate
Main flow

Secondary flow
© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.43: Flow between rotating outercylinder and stationary inner cylinder
Stationary inner cylinder

Rotating outer cylinder© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.44: Flow between stationary convex
cone and rotating concave cone(a)(b) Concave and convex cones

Stationaryconvex coneRotating
concave cone

50 mm

© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.45: Pressure in pulsatile flow
Pressure
Aortic pressure
Left ventricularpressureTime
Acceleration

Deceleration

SystoleDiastole© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.46: Compliance of tube wall
Low complianceHigh compliancePulsatileflow

Deformation with compliance of wall

Low complianceHigh compliance© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.47: Clot formation andhemolysis with shear rate
Erythrocyte

Fibrin[Clot]
Platelet

WallLow[Shear rate]

High

Erythrocyte[Deformation][Hemolysis]© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.48: Tracing(a) Laminarflow
Re＝ρv x／η(b) Turbulent
flow(4.64)© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.49: Streamline
Streamline

Flow vector© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.50: Vortex
WallWallExfoliation
Vortex(a) Low Reynolds
number(b) High Reynolds
number© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.51: Artificial ventricle(a)(c)(b)© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan

Fig. 4.52: Clot in artificial ventricle
Valve

Valve

Valve

Valve

Clot© Shigehiro Hashimoto 2013, Published by Corona Publishing Co., Ltd. Tokyo, Japan