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2-2lubri.tex\\
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\section{Lubrication approximation for flow in a thin layer}
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An essential first step of any analytical approximtation is the art of scaling, which we shall emphasize repeatedly throughout this course.
Let $H$ be the characteristic depth and $L$ the characteristic
length in the direction of the flow, and assume a shallow layer, i.e.,
\be H/L\ll 1\ee
Let $U$ be the scale of $u$, then by continuity, the scale of $v$ must be $U\f{H}{L}$ in order not to violate mass conservation. Leaving the velocity and pressure scales $U,P$ undermined for the time being, we
introduce the following scales and normalized variables, denoted by primes,
\be t=Tt', \,\, x=Lx', y=Hy', \,\, u=Uu', \, \, v=U\f{H}{L} v', \quad p=Pp', \ee
The normalized continuity equation is
\be \f{\p u'}{\p x'} + \f{\p v'}{\p y'} = 0 \label{cont-norm}\ee
Both terms are equally important, reflecting the holyness of the law of mass concentration. The longitudinal momentum equation is normalized to
\be \f{U}{T}\f{\p u'}{\p t'} + \f{U^2}{L}\lp u'\f{\p u'}{\p x'} + v'\f{\p u'}{\p y'}\rp = g \sin \theta -\f{P}{\rho L} \f{\p p'}{\p x'} +\f{ \nu U}{H^2} \lp\f{H^2}{L^2} \f{\p^2 u'}{\p x'^2} + \f{\p^2 v'}{\p y'^2 } \rp \ee
Dividing by $\f{\nu U}{H^2}$, we get
\be \f{H^2}{\nu T}\f{\p u}{\p t} + \f{UH}{\nu}\f{H}{L}\lp u'\f{\p u'}{\p x'} + v'\f{\p u'}{\p y'}\rp = \f{g \sin \theta H^2}{\nu U}-\f{PH^2}{\rho L\nu U} \f{\p p'}{\p x'} + \lp\f{H^2}{L^2} \f{\p^2 u'}{\p x'^2} + \f{\p^2 u'}{\p y'^2 } \rp \ee
For a shallow layer ($H/L\ll 1$) we assume in addition,
\be \f{UH}{\nu} =O(1)\ee
and \be \f{H^2}{\nu T} \ll 1\ee
Omitting terms of the order $H/L$ and smaller,
the above equation can be approximated to the leading order by
\be 0= \f{g \sin \theta H^2}{\nu U}-\f{PH^2}{\rho L\nu U} \f{\p p'}{\p x'} + \f{\p^2 u'}{\p y'^2 } \label{lub-x'}\ee
or in dimensional form,
\be 0 = g\sin \theta -\f{1}{\rho}\f{\p p}{\p x} + \nu \f{\p^2 u}{\p y^2} \label{lub-x}\ee
All inertia terms are inconsequential; the most important balance is among gravity, the pressure gredient and the dominant viscous stress. This balance
also implies a pressure scale,
\be P=\f{\rho L\nu U}{H^2}\label{pressure-h}\ee
From the transverse momentum equation,
\be \f{H}{L}\lb\f{U}{T}\f{\p v'}{\p t'} + \f{U^2}{L}\lp u'\f{\p v'}{\p x'} + v'\f{\p v'}{\p y'}\rp\rb = - g \cos \theta - \f{P}{\rho H} \f{\p p'}{\p y'} + \f{H}{L}\f{\nu U}{H^2}\lp \f{\p^2 v'}{\p x'^2} + \f{\p^2 v'}{\p y'^2} \rp \ee
or
\begin{eqnarray}\lefteqn{
\f{H^2}{L^2}\lcb \f{H^2}{\nu T}\f{\p v'}{\p t'} +\f{UH}{\nu}\f{H}{L}\lp u'\f{\p v'}{\p x'} + v'\f{\p v'}{\p y'}\rp\rcb =}\nonumber \\
&& - \f{g \sin \theta H^2}{\nu U}\f{H}{L\tan\theta}-\f{PH^2}{\rho L\nu U} \f{\p p'}{\p y' }
+ \f{H^2}{L^2}\lp\f{H^2}{L^2} \f{\p^2 v'}{\p x'^2} + \f{\p^2 v'}{\p y'^2 } \rp
\end{eqnarray}
Either for finite bed slope or for small slope but \be O\lp \f{H}{L}\rp = \tan \theta \ll 1\ee
the left hand side above is negligible with an error of $O(H/L)^3$. In physical variables the approximate result is
\be 0 = -g\cos \theta -\f{1}{\rho}\f{\p p}{\p y}\label{lub-y} \ee
Not only the inertia terms are insignificant, the pressure is hydrostatic. This balance also implies the pressure scale
\be P=\rho g H\cos\theta \label{pressure-v}
\ee
Note that (\ref{pressure-h}) and (\ref{pressure-v}) together implies the velocity scale
\be U=\f{H}{L}\f{gH\cos \theta}{\nu}\ee
The distinguishing feature of negligible inertia is shared by the slow flow through thin gaps of bearings in the theory of lubrication. Hence (\ref{lub-x}) and (\ref{lub-y}) can be called the lubrication approximation.
We leave it as an exercise to show by similar normalization, that
the dynamic boundary conditions on $y=h$ can be approximated to the leading order by
\be \f{\p u}{\p y} = 0 \ee for the tangential stress,
and
\be p = 0 \ee
for the normal stress. It follows by integrating (\ref{lub-y}))
that
\be p(x,y,t) = \rho g\cos \theta[h(x,t)-y] \ee
The longitudinal momentum equation can also be readily integrated,
\be u=- \f{\rho g}{\mu} \lp \sin \theta- \cos\theta \f{\p h}{\p x} \rp \lp \f{y^2}{2} - hy\rp \ee
The
total discharge is \be Q=\overline u h=\int_0^hu\,dy = \f{\rho g h^3}{3\mu} \lp \sin\theta - \cos\theta\f{\p h}{\p x}\rp \ee
which can be inserted in (\ref{int-mass}) to give
\be\fbox{$\displaystyle
\f{\p h}{\p t} + \f{\rho g}{3\mu} \f{\p}{\p x}\lb h^3\lp\sin\theta - \cos \theta\f{\p h}{\p x}\rp\rb = 0$} \label{basic}\ee
This is a nonlinear diffusion equation governing the evolution of the fluid depth.
In the special limit of a uniform flow, $\p /\p x\equiv 0$. The velocity profile is then
\be u=\f{\rho gh^2}{\mu} \sin \theta \lp \f{y}{h}-\f{y^2}{2h^2} \rp \ee
with $h$ being a pure constant. The coresponding discharge is
\be Q= \f{\rho g h^3}{3\mu} \sin\theta \ee
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