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Abbreviations
- ρ:
-
distance from axis of a tube
- r :
-
radius of tube
- g(r) :
-
tension as a function of tube radius
- x :
-
axial distance on a tube
- x 0 :
-
a reference value ofx
- t :
-
time
- t 0,t 1 :
-
reference times
- P :
-
fluid pressure
- f(t) :
-
spatial pressure gradientP x for rigid tube flow
- R :
-
radius of convergence forf(t) when it is analytic
- F(x, t) :
-
spatial pressure gradientP x for elastic tube flow
- v :
-
fluid velocity
- V :
-
defined by equations (6.4) (6.5) or by (6.6), a particular solution forv of the equation of motion for rigid tube flow
- V(ρ,t;F(x, t)) :
-
the functionV above withf(t) replaced byF(x, t)
- ∝ A V dA :
-
Integral over a cross section areaA at positionx
- L 2 V :
-
the result of applying a certain non-linear integral operatorL 2 to the functionV
- v(0) :
-
initial iterate for elastic tube flow, same asV above
- v (1) :
-
equalsV +L 2 V, the iterate for elastic tube flow to be used in clinical work
- d :
-
volume density of blood, approximately one in CGS units
- d w :
-
surface density of blood, approximately one in CGS units
- v l :
-
velocity of lateral pumping, a gross parameter
- K :
-
correction for mass loss
- K (i) :
-
iterates forK
- P s (x, t) :
-
a reference pressure in the tube
- a :
-
value ofP Sx when it is constant
- μ:
-
viscosity coefficient
- ν:
-
dynamic viscosity, μ/d
- τ:
-
period of a periodic pressure
- \(\mathfrak{F}\) :
-
frequency, i/τ
- α2 :
-
second dimensionless parameter,\(\frac{{r^2 }}{v}\mathfrak{F}\), necessary in pulsatile rigid tube flow
- E(t) :
-
energy entering a cross section of tube atx 0
- E w :
-
work done by the fluid in moving the walls
- E n :
-
error bound in truncating series for∝ A V dA aftern terms
References
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This work is sponsored by NIH Grant HE 08500-01.
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Lieberstein, H.M. Determination of the tension-stretch relation for a point in the aorta from measurement in vivo of pressure at three equally spaced points. Acta Biotheor 17, 49–94 (1965). https://doi.org/10.1007/BF01602210
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DOI: https://doi.org/10.1007/BF01602210