On the high frequency oscillatory tube flow of healthy human blood

Abstract

In this paper pressure gradient vs. volume flow rate calculations over a wide range of oscillatory frequencies for oscillatory tube flow of healthy human blood are performed using the non-homogeneous hemorheological model of Moyers-Gonzalez et al. [M.A. Moyers-Gonzalez, R.G. Owens, 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.A. Moyers-Gonzalez, R.G. Owens, 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.A. Moyers-Gonzalez, R.G. Owens, J. Fang, A non-homogeneous constitutive model for human blood. Part III. Oscillatory flow, J. Non-Newtonian Fluid Mech. 155 (2008) 161–173]. Results at low (2 Hz) oscillatory frequencies are shown to be in close conformity to the experimental data of Thurston [G.B. Thurston, The effects of frequency of oscillatory flow on the impedance of rigid, blood-filled tubes, Biorheology 13 (1976) 191-199] and the behaviour may be interpreted using a linear viscoelastic model. As the oscillatory frequencies increase a resonant frequency at which flow rate amplitude enhancement occurs is encountered. For frequencies greater than the resonant frequency the pressure gradient amplitude required to maintain a constant volume flow rate amplitude increases with the oscillatory frequency. For very high frequency oscillations we use a multiple time scales technique in conjunction with our non-homogeneous hemorheological model to solve for the leading order flow variables. It is found that the leading order expressions for the cell number density, average aggregate size and r r -component of elastic stress (i.e. that due to the red blood cells) are functions only of the radial component r. The O ( 1 ) elastic shear stress is shown to be zero, so that, for sufficiently large values of the oscillatory frequency, the red cell contribution to the total shear stress tends to zero. Using our multiple time scales method it is also shown that the model behaves in the very high frequency regime like a generalized linear viscoelastic fluid, having a radially dependent complex viscosity. This allows us to explain the computed results using asymptotic expressions for the in phase and π / 2 out of phase components of the pressure gradient in a linear viscoelastic fluid. In particular, we may predict the apparent complex viscosity of human blood in very high frequency oscillatory tube flow.