2.3 Prediction of eroded path and erosion coefficient
Equation [12] suggests that hydraulic shear stress is the function of fluid properties and the radius. In the specific test or condition, the given fluid is assumed to be unaltered (i.e., \(\nu\), \(\rho\), and Q are viewed as constants), even though carrying more soil particles will change fluid density and viscosity slightly. Therefore, shear stress can be also expressed as
\begin{equation} \tau=a\frac{1}{R^{3}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [13]\nonumber \\ \end{equation}
Where \(a=\frac{4\nu Q\rho_{w}}{\pi}\). The radius of the eroded path is considered as the function of time (i.e.,\(R=R\left(t\right)\)).
Erosion rate can be defined by
\begin{equation} \dot{}=\frac{\text{dm}}{\text{Adt}}=\frac{\rho_{s}\text{dV}}{\text{Adt}}=\frac{\rho_{s}\bullet 2\pi R\bullet L\bullet dR}{2\pi RL\bullet dt}=\frac{\rho_{s}\text{dR}}{\text{dt}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [14]\nonumber \\ \end{equation}
Based on equation [1], erosion coefficient is calculated by the differential equation,
\begin{equation} K=\frac{d\dot{\varepsilon}}{\text{dτ}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [15]\nonumber \\ \end{equation}
It becomes as,
\begin{equation} \frac{d\dot{\varepsilon}}{\text{dt}}=K\frac{\text{dτ}}{\text{dt}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [16]\nonumber \\ \end{equation}
Replace equation [13] and [14] into equation [16]
\begin{equation} \frac{\rho_{s}\bullet d\left(\frac{\text{dR}}{\text{dt}}\right)}{\text{dt}}=Ka\frac{d\left(\frac{1}{R^{3}}\right)}{\text{dt}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [17]\nonumber \\ \end{equation}
Equation [17] is transformed into
\begin{equation} \rho_{s}\bullet R^{{}^{\prime\prime}}=Ka\frac{-3}{R^{4}}R^{{}^{\prime}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [18]\nonumber \\ \end{equation}
Organize,
\begin{equation} R^{{}^{\prime\prime}}+\frac{3Ka}{\rho_{s}}\bullet\frac{1}{R^{4}}R^{{}^{\prime}}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [19]\nonumber \\ \end{equation}
Solving the equation [19],
\begin{equation} R=\left(\frac{4Ka}{\rho_{s}}t+C\right)^{\frac{1}{4}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }[20]\nonumber \\ \end{equation}
Equation [20] is further simplified as,
\begin{equation} R=\lambda\left(t+C\right)^{\frac{1}{4}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [21]\nonumber \\ \end{equation}
Thus,
\begin{equation} R^{{}^{\prime}}=\frac{\lambda}{4}\left(t+C\right)^{-\frac{3}{4}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [22]\nonumber \\ \end{equation}\begin{equation} R^{{}^{\prime\prime}}=-\frac{3\lambda}{16}\left(t+C\right)^{-\frac{7}{4}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [23]\nonumber \\ \end{equation}
where,
\begin{equation} \lambda=\left(\frac{4Ka}{\rho_{s}}\right)^{\frac{1}{4}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [24]\nonumber \\ \end{equation}
It indicates that the temporal variation of the eroded path (expressed as Radius as function of time) follows the power function with a fixed power number of 0.25. The constant (\(C\)) depends on the boundary conditions (i.e., initial radius). The parameter of \(\lambda\) reflects the information about soil properties and hydraulic conditions.
Hence, erosion coefficient can be obtained as
\begin{equation} K=\frac{\rho_{s}\lambda^{4}}{4a}=\frac{\pi\rho_{s}\lambda^{4}}{16\rho_{w}\text{νQ}}\text{\ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [25]\nonumber \\ \end{equation}