On the oscillatory tube flow of healthy human blood Miguel A. Moyers-Gonzalez Department of Mathematics and Statistics, University of Canterbury, New Zealand Robert G. Owens Département de mathématiques et de statistique, Université de Montréal, Canada Jiannong Fang ENAC-ICARE-GEOLEP EPFL, Switzerland 1
Outline Blood rheology rouleaux, linear viscoelastic properties and modelling, validation Oscillatory tube flow Results linear viscoelastic fluid W<<1 and W>>1 non-homogeneous model small volume flow rate amplitude high frequency oscillations quadrature component of pressure gradient phase shift, flow enhancement and loss angle leading order behaviour of macroscopic variables Conclusions 2
Blood rheology: composition of blood 3
Blood rheology: yield stress (Picart et al. (1998)) 4
Blood rheology: aggregation and disaggregation 5
Blood rheology: shear and complex viscosity 6
Blood rheology: modelling of rouleaux (From Y. Liu and W. K. Liu, Rheology of red blood cell aggregation by computer simulation J. Comput. Phys. 220 (2006) 139-154. ) 7
Blood rheology: a non-homogeneous model 8
Blood rheology: a non-homogeneous model 9
Blood rheology: a non-homogeneous model + (i,k-i) k k+j (k,j) k (i,k-i) (k,j) k+j 10
Blood rheology: a non-homogeneous model 11
Experimental validation. 1: triangular shear-rate 12
Experimental validation. 1: triangular shear-rate 13
Experimental validation. 1: triangular shear-rate 14
Experimental validation. 1: triangular shear-rate 15
Experimental validation. 1: triangular shear-rate 16
Experimental validation. 2: peripheral cell-free layer 17
Oscillatory tube flow: parameters 18
Oscillatory tube flow: basic equations 19
Oscillatory tube flow: linear viscoelastic fluid 20
Oscillatory tube flow: linear viscoelastic fluid 21
Oscillatory tube flow: non-homogeneous model 22
Oscillatory tube flow: MTSE: leading order variables 23
Oscillatory tube flow: MTSE: leading order variables 24
Oscillatory tube flow: MTSE: leading order variables 25
Oscillatory tube flow: MTSE: leading order variables 26
Oscillatory tube flow: MTSE: leading order variables 27
Oscillatory tube flow: MTSE: leading order variables 28
Oscillatory tube flow: MTSE: complex viscosity η 29
Oscillatory tube flow: MTSE: complex viscosity η 30
Results: PM rms A>0 A<0 31
Results: PM rms A<0 A>0 32
Results: PM rms and flow enhancement 33
Results: PM rms and flow enhancement 34
Results: A and ϕ 35
Results: N0De at 4π rads /s 36
Results: N0De at 4π rads /s 37
Results: N0De at 400π rads /s 38
Results: N0De at 400π rads /s 39
Conclusions 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 rrcomponent 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. 40
References 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. 41