Math 6410 • ODE

Consider the problem of a thin layer of paint of thickness \(h(x,t)\) and speed
\(u(x,y,t)\) flowing down a wall. The paint is assumed to be uniform in the \(z\)-direction.
The balance between gravity and viscosity (fluid friction) means that the velocity
satisfies the equation
\[
  \frac{\partial^2u}{\partial y^2}=-c,
\]
where \(c\) is a positive constant. This is supplemented by the boundary conditions
\[
  u(x,0,t)=0,\qquad
  \left.\frac{\partial u(x,y,t)}{\partial y}\right|_{y=h}=0.
\]
The density of paint per unit length in the \(x\)-direction is \(\rho_0h(x,t)\) where
\(\rho_0\) is a constant, and the corresponding flux is
\[
  q(x,t)=\rho_0\int_0^h u(x,y,t)\dd y.
\]
\begin{enumerate}
\item[(a)] Sketch a figure illustrating the physical problem.
\item[(b)] Using conservation of paint, and solving for \(u(x,y,t)\) in terms of
\(h(x,t)\) and \(y\), derive the following PDE for the thickness \(h\):
\[
  \frac{\partial h}{\partial t}+ch^2\frac{\partial h}{\partial x}=0.
\]
\item[(c)] Set \(c=1\). Show that the characteristics are straight lines and that the
Rankine-Hugoniot condition on a shock \(x=S(t)\) is
\[
  \frac{dS}{dt}=\frac{[h^3/3]^+_-}{[h]^+_-}.
\]
\item[(d)] A stripe of paint is applied at \(t=0\) so that
\[
  h(x,0)=
  \begin{cases}
    0, & x<0\text{ or }x>1,\\
    1, & 0<x<1.
  \end{cases}
\]
Show that, for small enough \(t\),
\[
  h=
  \begin{cases}
    0, & x<0,\\
    (x/t)^{1/2}, & 0<x<t,\\
    1, & t<x<S(t),\\
    0, & S(t)<x,
  \end{cases}
\]
where the shock is \(x=S(t)=1+t/3\). Explain why this solution changes at \(t=3/2\),
and show that thereafter
\[
  \frac{dS}{dt}=\frac{S}{3t}.
\]
\end{enumerate}