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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
September 14-18, 200914
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
September 14-18, 200931
Results: PM ́ ´rms
A>0 A<0
IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling
September 14-18, 200932
Results: PM ́ ´rms
A > 0
A < 0
IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling
September 14-18, 200933
Results: PM rms and flow enhancement
IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling
September 14-18, 200934
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
September 14-18, 200936
Results: N0De at 4π rads /s
IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling
September 14-18, 200937
Results: N0De at 4π rads /s
IMA Workshop: Flowing Complex Fluids: Rheological Measurements and Constitutive Modelling
September 14-18, 200938
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