Abstract
The paper studies the problem of fluid flow and fluid shear stress in canaliculi when the osteon is subject to external mechanical loading and blood pressure oscillation. The single osteon is modeled as a saturated poroelastic cylinder. Solid skeleton is regarded as a poroelastic transversely isotropic material. To get near-realistic results, both the interstitial fluid and the solid matrix are regarded as compressible. Blood pressure oscillation in the Haverian canal is considered. Using the poroelasticity theory, an analytical solution of the pore fluid pressure is obtained. Assuming the fluid in canaliculi is incompressible, analytical solutions of fluid flow velocity and fluid shear stress with the Navier-Stokes equations of incompressible fluid are obtained. The effect of various parameters on the fluid flow velocity and fluid shear stress is studied.
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Abbreviations
- r i :
-
inner radius of osteon
- r o :
-
outer radius of osteon
- \(\hat \sigma \) :
-
total stress vector for entire medium
- \(\hat C\) :
-
drained elasticity tensor
- \(\hat A\) :
-
Biot effective stress coefficient vector
- \(\hat \varepsilon \) :
-
apparent strain tensor of solid matrix
- Λ:
-
reciprocal of Biot modulus
- p :
-
interstitial fluid pressure
- ζ :
-
fluid content variation
- k :
-
intrinsic Darcy’s law permeability
- µ:
-
fluid viscosity
- ω :
-
axial loading frequency
- Ω:
-
blood pressure frequency
- I n :
-
first kind modified Bessel function of order n
- K n :
-
second kind modified Bessel function of order n
- λ :
-
dimensionless spatial variable
- α :
-
specific value of λ
- ϕ :
-
porosity
- ɛ 0 :
-
strain amplitude
- K rr :
-
permeability
- K f :
-
compressibility
- \(p^{B_0 } \) :
-
blood pressure amplitude
- a :
-
ratio of osteon inner radius and outer radius
- l :
-
length of canaliculi
- u :
-
fluid velocity vector
- ρ :
-
interstitial fluid density
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Project supported by the National Natural Science Foundation of China (No. 11032005)
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Liu, S., Wang, F. & Liu, R. Fluid flow and fluid shear stress in canaliculi induced by external mechanical loading and blood pressure oscillation. Appl. Math. Mech.-Engl. Ed. 36, 681–692 (2015). https://doi.org/10.1007/s10483-015-1932-7
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DOI: https://doi.org/10.1007/s10483-015-1932-7