## 4 Analysis
Since the slipping plane is very close to the compact layer, we can further deduce that the zeta \(\zeta(x)\) is also proportional to \(\sigma(x)\); that is, \(\zeta(x)=D\sigma(x)\), and \(D\) is a proportional factor. Thus,  
\[\zeta(x)=D\left\{\left[\bigg{(}x-l_{0}-\frac{l_{1}}{2}\bigg{)}A\right]^{2}+B \right\}\bigg{[}1-cos\bigg{(}\frac{x-l_{0}-l_{1}}{l_{1}}2\pi\bigg{)}\bigg{]}C +D\zeta_{0}\,\ x\in[l_{0},l_{0}+l_{1}] \tag{6}\]  
where \(\zeta_{0}\), corresponding to \(\sigma_{0}\), is a constant. The Equations (3) and (4) with the boundary conditions (5) and (6) can be solved using the similar approach as described by [PERSON] [32]. The fluid flow at stead state is governed by the Navier-Stokes equation  
\[\mu\frac{1}{r}\frac{d}{dr}\bigg{(}r\frac{dv}{dr}\bigg{)}=\frac{dp}{dx}-E_{x} \rho_{e}, \tag{7}\]  
where \(v\) is the fluid velocity along the canal axis in direction \(x\), \(\mu\) is the dynamic viscosity of the fluid, \(E_{x}\) is the flow induced electric field along the canal, and \(E_{x}\)\(\rho_{e}\) represents the flow induced electro-viscous resistance. The boundary conditions of Equation (7) are  
\[v=0,r=R;\ \frac{\partial v}{\partial r}=0,r=0; \tag{8}\]  
If the \(\psi\) is less than several millivols, that is \(\frac{ze\psi(x,r)}{k_{B}T}\ll 1\), the \(\sinh\bigg{(}\frac{ze\psi(x,r)}{k_{B}T}\bigg{)}\approx\frac{ze\psi(x,r)}{k_{B}T}\). In fact, the measured data in Figure 2 show that \(\ \psi\) is in the order of millivols (streaming potential is the difference of \(\psi\) between two points along the canal). Thus, the Poisson-Boltzmann equation can be linearized as:  
\[\frac{1}{r}\frac{d}{dr}\bigg{(}r\frac{\partial\psi(r)}{\partial r}\bigg{)}=\kappa^ {2}\psi(x,r), \tag{9}\]  
where \(\kappa^{-1}=\sqrt{\frac{\varepsilon k_{B}T}{2n_{0}e^{2}\kappa^{2}}}\) is the Debye length in the double layer [34]. Equation (9) is a zero order Bessel equation, and with the boundary conditions (5) its solution is given by  
\[\psi(x,r)=\zeta(x)\frac{I_{0}(\kappa r)}{I_{0}(\kappa R)}\, \tag{10}\]  
where \(I_{0}\) is the modified zero order of Bessel function. Substituting the solution (10) into Equation (4) yields  
Figure 5:0  
Substituting \(\rho_{e}(x,r)\) into Equation (7), fluid velocity can be obtained by integrating Equation (7) with the boundary conditions (8)  
Figure 5:1  
The flowing fluid in a microcanal carries electric charges along the flowing direction, which forms the streaming current \(I_{s}\) as  
Figure 5:2  
Substituting Equation (11) into Equation (13) and integrating it yields  
Figure 5:3  
In addition, the electric field \(E_{x}\) produces a conduction current \(I_{c}\) to the opposite direction of the streaming current, and  
Figure 5:4  
where \(\lambda_{b}\) and \(\lambda_{s}\) denote the conductivity of the bulk fluid and the solid bone respectively. At steady state, the total current \(I_{s}+I_{c}=0\). Generally, the conductivity of fluid is far greater than that of solid bone \(\Big{(}\frac{2\lambda_{s}}{R}\ll\lambda_{b}\Big{)}\), and combining Equations (14) and (15) yields  
Figure 5:5  
The streaming potential between the two ends of the capillary is  
Figure 5:6  
Substituting \(\zeta(x)=D\zeta_{0}\) with Equation (6) into Equation (17), it results in  
Figure 5:7If the \((x)=D\zeta_{0}\,x\in[0,L]\), which means that \(l_{1}=0\), then the second term becomes zero. And in this case, according to Equation (6), \(\zeta(x)\) becomes \(\zeta_{0}\) along \(L\) or \(D=1\). Thus,  
Figure 5:8  
which is a typical formula of streaming potential [32] by considering the action of the electro-viscous resistance. When the radius R is large enough \((I_{2}\,(\kappa R))/(I_{0}\,(\kappa R)\,)\) approaching one, the Equation (19) leads to \(\phi=\varepsilon\zeta_{0}P/(\mu\lambda_{b})\), the classical formula of streaming potential. The curve in Figure 6A shows the influence of charge density \(\sigma_{1}\) on the streaming potentials. The horizontal coordinate represents the relative length \(l_{1}\)/\(L\) and the vertical coordinate is the streaming potential.