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}