Abstract
Arterial blood flow may be described as the almost periodic, unsteady-state flow of a non-Newtonian fluid through a flexible-walled, tapered, branching conduit. A generalized approach to the development of a realistic mathematical model is described. All assumptions involved in the analysis are evaluated critically in terms of their quantitative effect on the validity of the model. As a result some of the traditional assumptions are avoided. A numerical integration routine for obtaining solutions to the resulting nonlinear partial differential equations comprising the model is outlined. The possible application of the model to compute localized arterial wall elasticity is discussed. Successful implementation of this procedure could lead to a significant clinical diagnostic tool for vascular disease.
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Abbreviations
- a n ,b n :
-
constants in Equation (9)
- A, B :
-
constants in Equation (1)
- f :
-
general function notation
- i, j :
-
general incremental subscripts
- m, n :
-
constants in rheological equations (See Equations (2), (2a))
- n :
-
summation index in Equation (9) with upper limitN
- p :
-
fluid pressure
- P R :
-
fluid pressure at arterial wall
- r :
-
radial position
- R :
-
radius of artery
- t :
-
time
- u :
-
fluid velocity component inz direction
- v :
-
fluid velocity component inr direction
- z :
-
axial position
- Z :
-
length of arterial segment
- 0, 1, 2:
-
specific incremental subscripts.Greek Symbols
- \(\dot \gamma \) :
-
shear rate
- Δ:
-
finite increment such as ΔP, Δr, Δz, etc
- ε:
-
elasticity
- η:
-
non-Newtonian viscosity
- μ:
-
Newtonian viscosity
- π:
-
angle in radians
- ρ:
-
fluid density
- τ:
-
shear stress
- τ rr , τ rz , τ zz :
-
shear stress components as defined in Table 3
- ω:
-
fundamental frequency=2π/T, withT=the period
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Huckabe, C.E., Hahn, A.W. A generalized approach to the modeling of arterial blood flow. Bulletin of Mathematical Biophysics 30, 645–662 (1968). https://doi.org/10.1007/BF02476681
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DOI: https://doi.org/10.1007/BF02476681