﻿ 一种骨小管中液体流动产生的流量及切应力模型
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 力学学报  2016, Vol. 48 Issue (5): 1208-1216  DOI: 10.6052/0459-1879-16-046 0

### 引用本文 [复制中英文]

[复制中文]
Wu Xiaogang , Yu Weilun , Wang Zhaowei , Wang Ningning , Cen Haipeng , Wang Yanqin , Chen Weiyi . A CANALICULAR FLUID FLOWMODEL ASSOCIATED WITH ITS FLUID FLOWRATE AND FLUID SHEAR STRESS[J]. Chinese Journal of Ship Research, 2016, 48(5): 1208-1216. DOI: 10.6052/0459-1879-16-046.
[复制英文]

### 文章历史

2016-02-06 收稿
2016-05-11 录用
2016-05-18网络版发表

1 模型建立 1.1 骨单元模型

 $\sigma _{ij} = C_{ijkl} \varepsilon _{kl} - \alpha _{ij} p$ (1)
 $p = M\left( {\xi - \alpha _{ij} \varepsilon _{ij} } \right)$ (2)

 $\sigma _{ij,j} = \rho \ddot {u}_i ^{\rm s}$ (3)

 $\dot {\xi } = - q_{i,i}$ (4)

 $q_i = - \dfrac{k_{ij} }{\mu }(p_{,i} + \rho _{\rm f} \ddot {u}_i^{\rm s})$ (5)

 $\left. \begin{array}{*{35}{l}} \rho \ddot{u}_{i}^{\text{s}}={{C}_{ijkl}}{{\varepsilon }_{kl,j}}-{{\alpha }_{ij}}{{p}_{,j}} \\ \dot{p}-M\left[ \left( {{k}_{ij}}/\mu \right)\left( {{p}_{,ii}}+{{\rho }_{\text{f}}}\ddot{u}_{i,i}^{\text{s}} \right)-{{\alpha }_{ij}}{{{\dot{\varepsilon }}}_{ij}} \right]=0 \\ \end{array} \right\}$ (6)

 $\left. \begin{array}{*{35}{l}} {{C}_{ijkl}}{{\varepsilon }_{kl,j}}-{{\alpha }_{ij}}{{p}_{,j}}=0 \\ \dot{p}-M\left[ \left( {{k}_{ij}}/\mu \right){{p}_{,ii}}-{{\alpha }_{ij}}{{{\dot{\varepsilon }}}_{ij}} \right]=0 \\ \end{array} \right\}$ (7)

 \begin{align} & {{p}_{m}}=\frac{M{{C}_{11}}\left( \alpha {{c}_{m}}+{\alpha }'{{\varepsilon }_{z0}} \right)}{{{C}_{11}}+M{{\alpha }^{2}}}\cdot \\ & \left[ \frac{{{\text{I}}_{0}}\left( Cr \right){{\text{K}}_{1}}\left( Cb \right)+{{\text{K}}_{0}}\left( Cr \right){{\text{I}}_{1}}\left( Cb \right)}{{{\text{I}}_{0}}\left( Ca \right){{\text{K}}_{1}}\left( Cb \right)+{{\text{I}}_{1}}\left( Cb \right){{\text{K}}_{0}}\left( Ca \right)}-1 \right]{{\text{e}}^{\text{i}\omega t}} \\ \end{align} (8)
 \begin{align} & {{p}_{n}}=\left\{ \left( {{p}_{hn}}+\frac{M{{C}_{11}}\left( \alpha {{c}_{n}}+{\alpha }'{{\varepsilon }_{z0}} \right)}{{{C}_{11}}+M{{\alpha }^{2}}} \right) \right.\cdot \\ & \left[ \frac{{{\text{K}}_{1}}\left( Cb \right){{\text{I}}_{0}}\left( Cr \right)+{{\text{I}}_{1}}\left( Cb \right){{\text{K}}_{0}}\left( Cr \right)}{{{\text{I}}_{0}}\left( Ca \right){{\text{K}}_{1}}\left( Cb \right)+{{\text{I}}_{1}}\left( Cb \right){{\text{K}}_{0}}\left( Ca \right)} \right]- \\ & \left. \frac{M{{C}_{11}}\left( \alpha {{c}_{n}}+{\alpha }'{{\varepsilon }_{z0}} \right)}{{{C}_{11}}+M{{\alpha }^{2}}} \right\}{{\text{e}}^{\text{i}\omega t}} \\ \end{align} (9)
 图 1 骨单元系统的分层结构示意图 Figure 1 The hierarchical model for osteon system.
 图 2 两种骨单元模型 Figure 2 Two osteon models

 $C = \sqrt { {\rm i}\omega \mu \left( {C_{11} + M\alpha ^2} \right) / \left( {kMC_{11} } \right)}$ (10)

1.2 骨小管液体流动及切应力模型

 图 3 骨小管模型 Figure 3 Fluid flow model of the canalicular canal

 ${p}'({z}') = \left[{p_0 (a) + \dfrac{p_0 (b) - p_0 (a)}{l}\left( {{z}' - a} \right)} \right] {\rm e}^{{\rm i}\omega t}$ (11)
 $u({r}') = \dfrac{p_0 (a) - p_0 (b)}{{\rm i}\omega \rho _{\rm f} l} \left[{1 - \dfrac{{\rm I}_0 \left( {\beta {r}'} \right)}{{\rm I}_0 \left( {\beta R} \right)}} \right] {\rm e}^{{\rm i}\omega t}$ (12)

 $\beta = \sqrt {{\rm i}\omega / \mu }$ (13)

 $Q = \int_0^R {2\pi {r}'} u({r}') d {r}'$ (14)
 $\tau _{\rm w} = \mu \left. {\dfrac{\partial u({r}')}{\partial {r}'}} \right|_{{r}' = R}$ (15)

 $Q_s = \dfrac{2\pi }{{\rm i}\omega \rho _{\rm f} l}p_{0s} (b) \left[{\dfrac{R} {\beta }\dfrac{{\rm I}_1 \left( {\beta R} \right)}{ {\rm I}_0 \left( {\beta R} \right)} - \dfrac{R^2}{2}} \right] {\rm e}^{{\rm i}\omega t}$ (16)
 $\tau _{ws} = \dfrac{\mu \beta }{{\rm i}\omega \rho _{\rm f} l}p_{0s} (b)\dfrac{{\rm I}_1 \left( {\beta R} \right)}{ {\rm I}_0 \left( {\beta R} \right)} {\rm e}^{{\rm i}\omega t}$ (17)

2 数值参数

3 结果 3.1 载荷幅值和频率

 图 4 载荷大小(应变幅值)和载荷频率对流量($\left| Q \right|)$及切应力幅值($\left| {\tau _{\rm w} } \right|)$的影响 Figure 4 Effects of the loading frequency and strain amplitude on fluid flow rate amplitude($\left| Q \right|$) and fluid shear stress amplitude ($\left| {\tau _{\rm w} } \right|$)
3.2 应变率

 $\left| {Q_s } \right| = \left| {\dfrac{2\pi }{{\rm i}\omega ^2\rho _{\rm f} l}p_{0s}^\ast (b)\left( {\dfrac{R}{\beta }\dfrac{{\rm I}_1 \left( {\beta R} \right)}{{\rm I}_0 \left( {\beta R} \right)} - \dfrac{R^2}{2}} \right)} \right|\dot {\varepsilon }_z$ (18)
 $\left| {\tau _{{\rm w}s} } \right| = \left| {\dfrac{\mu \beta }{{\rm i}\omega ^2\rho _{\rm f}l}p_{0s}^\ast (b) \dfrac{{\rm I}_1 \left( {\beta R} \right)}{{\rm I}_0 \left( {\beta R} \right)}} \right|\dot {\varepsilon }_z$ (19)

 $\left| {{Q}_{4}} \right|(\left| {{\tau }_{\text{w4}}} \right|)=1.5\left| {{Q}_{3}} \right|(\left| {{\tau }_{\text{w3}}} \right|)=4.9\left| {{Q}_{2}} \right|(\left| {{\tau }_{\text{w2}}} \right|)=32.6\left| {{Q}_{1}} \right|(\left| {{\tau }_{\text{w1}}} \right|)$
 图 5 应变率一定的情况下，频率对流量及切应力幅值的影响 Figure 5 The amplitudes of fluid flow rates (FFRA) and shear stress (FSSA)as a function of the loading frequency with the strain \rater kept onstant

Case Ⅲ和Ⅳ中的压力和流速场均高于中空的骨单元模型，这样含哈弗液体的骨单元模型内液体(骨小管中)的流量(流速)高于中空模型，从而导致产生的切应力较大.

3.3 时间

 图 6 应变率为$\dot {\varepsilon }_z = 0.001$ s$^{-1}$时，应轴向应变随时间的变化，\Re( ) 表示复数取虚部 Figure 6 History of strain loads at $\dot {\varepsilon }_z = 0.001$ s$^{-1}$, the operator Re( ) gives the real part of the complex number
 图 7 流量与切应力随时间的变化 Figure 7 Time responses of FFR and FSS
3.4 骨小管半径

 图 8 流量及切应力幅值与骨小管半径的关系 Figure 8 Evolutions of FFRA and FSSA with canalicular radius
3.5 渗透率

 图 9 流量及切应力幅值与渗透率的关系 Figure 9 Evolutions of FFRA and FSSA with permeability
4 讨论

5 结论

(1) 骨小管中的液体流量和切应力均跟骨单元受到的外部载荷幅值和频率呈线性关系.进一步研究表明，应变率是流量和切应力的生理载荷决定因素.

(2) 骨小管的半径越大，产生的流量和切应力越大.

(3) 在骨单元尺度,流量和切应力随着渗透率的增大而减小.

(4) 相对于中空的骨单元模型，哈弗液体能使骨小管产生较大的流量和切应力.