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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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