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\begin{enumerate}
\item Non-Newtonian polymer flow in a channel

  A polymer melt is forced through a flat channel of thickness $\delta$ between
  two fixed horizontal plates by a pressure difference $\Delta P$ (inlet
  pressure minus outlet pressure).  This channel length $L$ is much longer than
  its width $W$, and it is a lot longer and wider than it is thick ($\delta$),
  so you may assume fully-developed flow and neglect edge effects.  Take the
  $x$ direction to be the direction of flow, parallel to the plates, and the
  $y$ direction to be straight up, perpendicular to the plates.

  This polymer is a pseudoplastic non-Newtonian fluid, whose behavior can be
  modeled using a ``power law'':
  $$\tau_{yx} = -\mu_0 \left|\frac{\partial u_x}{\partial y}\right|^{n-1}
  \frac{\partial u_x}{\partial y},$$
  which is to say, shear stress is proportional to shear strain rate to the $n$
  power, with the extra $\partial u_x/\partial y$ there to get the sign right.
  The differential equation describing $x$-momentum equation for this laminar
  1-D flow (where $u_y=u_z=0$ and $d\vec{u}/dx=d\vec{u}/dz=0)$ can be written
  with the above power law substituted for $\tau_{yx}$:
  $$\rho\frac{\partial u_x}{\partial t} = -\frac{\partial P}{\partial x} +
  \mu_0\frac{\partial}{\partial y}\left(
    \left|\frac{\partial u_x}{\partial y}\right|^{n-1}
    \frac{\partial u_x}{\partial y}\right) + F_x.$$
  The general solution to this equation for steady-state flow driven only by
  pressure gradient is:
  $$u_x = -\frac{\mu_0L}{\Delta P}\frac{n}{n+1}
  \left(-\frac{\Delta P}{\mu_0L}y + C_1\right)^{\frac{n+1}{n}} + C_2.$$
  Note that this is only valid for negative $y$; the positive $y$ solution is
  symmetric.

  \begin{enumerate}
  \item If the polymer is pseudoplastic, is $n$ greater or less than one?

  \item Determine the specific solution which fits the above general solution
    to the no-slip boundary conditions at the two stationary plates, and/or the
    symmetry plane boundary condition halfway between them (you need just two
    of these three conditions).  Note: it might help to set $y=0$ halfway
    between the plates.

  \item Sketch this velocity profile for an appropriate pseudoplastic $n$ value
    of your choosing.

  \item For channel flow of this kind, a Newtonian fluid's average velocity
    will be 2/3 of its maximum velocity.  For this pseudoplastic fluid, will
    the average velocity be more or less than 2/3 of its maximum?
  \end{enumerate}
\end{enumerate}
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