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2-4spreadmud.tex
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\section{Spreading of a shallow mass on an incline}
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{\bf References:} \\
C. C. Mei, (1966), Nonlinear gravity waves in a thin sheet of viscous fluid, {\it J. Math. \& Phys.} 45, 482-496. \\
K. F. Liu \& C. C. Mei, 1989, Slow spreading of a sheet of a Bingham fluid on an inclined plane, {\it J. Fluid Mech. } 207, 505-529.\\
X. Huang \& M. H. Garcia,1997, Asymnptotic solution for Bingham debris flows, {\it Debris -flow Hazards and Mitigation,} ASCE Proc. pp.561-575.\\
The spreading of a finite mass of paint, paper pulp, mud or lava on an inclined plane is of interest to a variety of industrial and geological problems. For mud modeled as a Bingham-plastic non-Newtonian fluid, Liu \& Mei (1989) solved the equation similar to (2.3.22) numerically. An analytical solution for Herschel-Bulkley fluid was given later by Huang \& Garcia (1997). We modify their theories for the simpler case of Newtonion fluid here.
The free surface of a thin layer is expected to flatten in time
over most of the profile, but it should steepen near the downstream front where the spatial derivative is much more
important than elsewhere. Let us study the problem by dividing the total fluid extent into two: the far field not too
close to the steep
front and the near field around the front.
\subsection{Far field away from the front}
Let the total length of the thin layer be $L$ and the maximum depth be $H$, with $H/L\ll 1$. We introduce the normalized variables suitable for the far field.
\be h=Hh', \,\, x=Lx',\,\, t=Tt' \ee
where $T$ will be chosen to simplify the appearance of the final equation. Thus,
\[\f{H}{T} \f{\p h'}{\p t'} + \f{\rho g\sin \theta }{3 \mu}\f{H^3}{L} \f{\p}{\p x'} \lb \lp 1-\f{H}{L \tan\theta }\f{\p h'}{\p x'} \rp h'^3\rb= 0 \]
Let us choose the time time scale to be
\be T=\f{L}{ {\rho g H^2\sin \theta}/{3\mu}}\label{timescale}\ee
where the denominator is the average fluid speed across the depth. The dimensionless equation for the farfield reads,
\be \f{\p h'}{\p t'} + \f{\p}{\p x'} \lb \lp 1-\f{H}{L \tan\theta }\f{\p h'}{\p x'} \rp h'^3\rb= 0 \label{norm-basic}\ee
Assume \be \f{H}{L \tan\theta }\equiv \ep \ll 1\label{assumption}\ee
(\ref{norm-basic}) can then be approximated by the hyperbolic equation,
\be \f{\p h'}{\p t'} + \f{\p h'^3}{\p x'}
=\f{\p h'}{\p
t'} + 3 h'^2 \f{\p h' }{\p x'} =0
\label{far}\ee
It is of the class called kinematic wave equation in
flood hydrology. Solution can be obtained by the theory of characteristics.
In particular
(\ref{far}) can be rewritten in the form
\be \f{\p h}{\p
t}dt + \f{\p h }{\p x}dx =dh=0
\label{char-form}\ee
if
\be \f{dx'}{dt'} = 3h'^2 \label{char-curve}\ee
The last two equations imply that $h'$ remains constant for all $t'$ along the characteristic curve $x'(t')$ defined by the differential equation (\ref{char-curve}).
Moreover, if the initial profile is presecribed,
\be h'(x', 0) = h'_o(x'), \ee
then all characteristics are straight but of different slopes.
The characteristic originated from the initial point $x'=\xi'$ at $t'=0$ is the straight line
\be x(t,\xi) = 3h_o^2(\xi) t + \xi, \label{char}\ee
along which
\be h'(x',t') = h'_o(\xi') \label{formalsol}\ee
In principle we can solve for $\xi' $ from (\ref{char}) in term of $x',t'$ and substitute the result into (\ref{formalsol}) to get $h'(x',t')$. For any initial hump, the characteristics at the front must intersect one another, impling multivaluedness of solution, which is phycially unacceptable. The solution can still be proceeded if a discontinuity, {\it shock}, is allowed at $x'=x'_s(t')$, as long as mass is conserved.
As an example, consider a triangular initial profile:
\be h'_o (x') = \lcb \begin{array} {cc} x' & 01. \end{array} \right. \ee
The profile has a shock front to begin with, which is a mathematical idealization of course.
From (\ref{char}) we get
\be x' = \lcb\begin{array}{cc} 3\xi'^2 t' +
\xi', & 0<\xi' <1,\\
\xi' , & \xi'<0, \xi' >1\end{array} \right.\label{char-ex} \ee
Within $0<\xi'<1$, $\xi'$ can be solved in terms of $x',t'$,
\be \xi' = \f{1}{6t'} \lp -1+\sqrt{1+12 x't'} \rp\ee
For any $t'>0$,
\be h' = 0, \quad \mbox{ for}\quad x' <0, ~~\& ~~x'>x'_s(t'), \ee
and \be h'=\xi'=\f{1}{6t'} \lp -1+\sqrt{1+12 x't'}\rp, \quad \quad 0