on the oscillatory tube flow of healthy human bloodrouleaux and the description of non-homogeneous...

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 20091

On the oscillatory tube flow ofhealthy human blood

Miguel A. Moyers-GonzalezDepartment of Mathematics and Statistics,

University of Canterbury, New Zealand

Robert G. OwensDépartement de mathématiques et de statistique,

Université de Montréal, Canada

Jiannong FangENAC-ICARE-GEOLEP

EPFL, Switzerland

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 20092

• Blood rheology

rouleaux, linear viscoelastic properties and modelling, validation

• Oscillatory tube flow

linear viscoelastic fluid W<<1 and W>>1

non-homogeneous model small volume flow rate amplitude

high frequency oscillations• Results

quadrature component of pressure gradient phase shift, flow enhancement and loss angle leading order behaviour of macroscopic variables

• Conclusions

Outline

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 20093

Blood rheology: composition of blood

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 20094

Blood rheology: yield stress (Picart et al. (1998))

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 20095

Blood rheology: aggregation and disaggregation

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 20096

Blood rheology: shear and complex viscosity

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 20097

(From Y. Liu and W. K. Liu, Rheology of red blood cell aggregation by computer simulationJ. Comput. Phys. 220 (2006) 139-154. )

Blood rheology: modelling of rouleaux

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 20098

Blood rheology: a non-homogeneous model

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 20099

Blood rheology: a non-homogeneous modelBlood rheology: a non-homogeneous model

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200910

(i,k-i) kk+j (k,j)

+

k (i,k-i) (k,j) k+j

−

Blood rheology: a non-homogeneous model

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200911

Blood rheology: a non-homogeneous model

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200912

Experimental validation. 1: triangular shear-rate

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200913

Experimental validation. 1: triangular shear-rate

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

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Experimental validation. 1: triangular shear-rate

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200915

Experimental validation. 1: triangular shear-rate

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200916

Experimental validation. 1: triangular shear-rate

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200917

Experimental validation. 2: peripheral cell-free layer

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200918

Oscillatory tube flow: parameters

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200919

Oscillatory tube flow: basic equations

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200920

Oscillatory tube flow: linear viscoelastic fluid

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200921

Oscillatory tube flow: linear viscoelastic fluid

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200922

Oscillatory tube flow: non-homogeneous model

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200923

Oscillatory tube flow: MTSE: leading order variables

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200924

Oscillatory tube flow: MTSE: leading order variables

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200925

Oscillatory tube flow: MTSE: leading order variables

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200926

Oscillatory tube flow: MTSE: leading order variables

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200927

Oscillatory tube flow: MTSE: leading order variables

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200928

Oscillatory tube flow: MTSE: leading order variables

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200929

Oscillatory tube flow: MTSE: complex viscosity η∗

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200930

Oscillatory tube flow: MTSE: complex viscosity η∗

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

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Results: PM ́ ´rms

A>0 A<0

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

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Results: PM ́ ´rms

A > 0

A < 0

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

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Results: PM rms and flow enhancement

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

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Results: PM rms and flow enhancement

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200935

Results: A and ϕ

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

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Results: N0De at 4π rads /s

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

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Results: N0De at 4π rads /s

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

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Results: N0De at 400π rads /s

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200939

Results: N0De at 400π rads /s

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200940

• An extensively tested “sticky dumbbell” model has been developed, allowing for aggregation and fragmentation of rouleaux and the description of non-homogeneous blood flow.

• The behaviour of healthy human blood in oscillating tube flow has been described via computations of the pressure gradient-volume flow rate relationship.

• A multiple time scales analysis has shown that red cell number density, average aggregate size and the rr- component of the elastic stress are functions of r alone, to leading order.

• Current work is directed towards correct prediction (with HI) of cell migration the thickness of the cell-free layer and the Fahraeus-Lindqvist effect.

Conclusions

IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling

September 14-18, 200941

• RGO, A new microstructure-based constitutive model for human blood, J. Non-Newtonian Fluid Mech., 140 (2006) 57-70.

• J. Fang and RGO, Numerical simulations of pulsatile blood flow using a new constitutive model, Biorheology, 43 (2006) 637-660.

• M. Moyers-Gonzalez, RGO and J. Fang, A non-homogeneous constitutive model for human blood. Part I: model derivation and steady flow, J. Fluid Mech., 617 (2008) 327-354.

• M. Moyers-Gonzalez and RGO, A non-homogeneous constitutive model for human blood. Part II: Asymptotic solution for large Péclet numbers, J. Non-Newtonian Fluid Mech., 155 (2008) 146-160.

• M. Moyers-Gonzalez, RGO and J. Fang, A non-homogeneous constitutive model for human blood. Part III: Oscillatory flow, J. Non-Newtonian Fluid Mech., 155 (2008) 161-173.

• M. Moyers-Gonzalez, RGO and J. Fang, On the high frequency oscillatory tube flow of healthy human blood, J. Non-Newtonian Fluid Mech. (2009), in press.

References