## 4 Analysis
\tag{2}\]  
where, \(A\), \(B\) and \(C\) are undetermined parameters. The distribution curve of \(\sigma(x)\) (red line) is shown in Figure 5. \(\sigma(x)\) is symmetrical with respect to the central section (point \(\beta\)) of the capillary and has three extreme points, \(\alpha\), \(\beta\) and \(\gamma\) (Figure 5). Let \(x_{\alpha}\), \(x_{\beta}\) and \(x_{\gamma}\) be the coordinates of the three extreme points, respectively. If relatively small differences are selected \(\sigma_{1}-\sigma(x_{\alpha})\), or \(\sigma_{1}-\sigma(x_{\gamma})\) and \(\sigma_{1}-\sigma(x_{\beta})\) and a relatively larger distance \((x_{\gamma}-x_{\alpha})\) exist between the two maximums, the \(\sigma(x)\) can provide a good approximation of \(\sigma_{1}\). Once the three differences are selected, three equations are obtained and the three parameters of \(A\), \(B\) and \(C\) can be found. If the \(\sigma_{1}\;-\sigma(x_{\alpha})\), \(\sigma_{1}-\sigma(x_{\beta})\) and \(x_{\gamma}-x_{\alpha}\) are assigned values of 0.05 \(\sigma_{1}\), 0.025 \(\sigma_{1}\) and 0.7 \(l_{1}\), and when \(l_{1}=0.4\;L\;and\;\;\sigma_{1}=2\;\sigma_{0}\), the \(A\), \(B\) and \(C\) are obtained as 8.0, 0.7 and 0.72 respectively. Then the specific distribution curve of the charge density in Figure 5 has been determined. In the classical analysis of streaming potentials in a round capillary, the potentials in the electric double layer \(\psi\) and the net charge density in the fluid \(\rho_{e}\) are the functions of radius \(r\) of the capillary only. In the current conditions, the \(\psi\) and \(\rho_{e}\) are also functions of \(x\), owing to the function \(\sigma(x)\). That is, \(\psi=\psi(x,r)\) and \(\rho_{e}=\rho_{e}(x,r)\). Take the classical analysis approach for the streaming potentials in a  
Figure 5: Geometric model with different electric charge density distributions. round capillary [32; 33]: The electrical potential \(\psi(x,r)\) and the net charge density \(\rho_{e}(x,r)\) in bone fluid are described by the Poisson-Boltzmann equation  
\[\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}r\frac{\partial\psi(x,r)}{ \partial r}\bigg{)}=\frac{2n_{0}ze}{\varepsilon}\sinh\bigg{(}\frac{ze\psi(x,r) }{k_{B}T}\bigg{)} \tag{3}\]  
\[\rho_{e}(x,r)=-\varepsilon\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}r \frac{\partial\psi(x,r)}{\partial r}\bigg{)} \tag{4}\]  
where \(n_{0}\), \(z\) and \(e\) are the ionic concentrations in the bulk bone fluid, the valence of ions and the elementary charge respectively; \(\varepsilon\) denotes the dielectric constant of the fluid, \(k_{B}\) the Boltzmann constant and \(T\) the absolute temperature. The boundary conditions are  
\[\psi(x,r)=\zeta(x),\ r=R\ ;\ \frac{\partial\psi(x,r)}{\partial r}=0,r=0 \tag{5}\]  
where \(\zeta(x)\) is called Zeta potential, being the value of \(\psi\) at the slipping plane, which refers to the plane where the fluid velocity decreases to zero. Because the diameters of the microcanals in bone are very small relative to their lengths, it can be considered that the charge density \(\sigma(x)\) on the solid only affects the \(\psi(x,r)\) and \(\rho_{e}(x,r)\) in fluid at the cross section with the same \(x\)-coordinate. Accordingly, \(x\) can be regard as a parameter instead of a variable, and then \(\psi(x,r)\) and \(\rho_{e}(x,r)\) become unary functions \(\psi(r)\) and \(\rho_{e}(r)\). Because of the electrostatic attraction of the charges on the solid surface, charges with different signs in the fluid are absorbed to the surface and form an immobile layer called the compact layer. The charges on the solid and the unlike charges in the compact layer are similar to a capacitor. One electrode is the solid surface with charge density \(\sigma(x)\) and the compact layer is approximately equal to the other electrode. Therefore, it can be speculated the charge density on the compact layer is proportional to \(\sigma(x)\).