Problem $4.$ Consider steady, incompressible, parallel, laminar flow of a film of oil falling slowly down an infinite vertical wall. The oil film thickness is h $,$ and gravity acts in the negative z $-$direction. There is no applied $($forced$)$ pressure driving the flow$-$the oil falls by gravity alone. Calculate the velocity and pressure fields in the oil film and sketch the normalized velocity profile. You may neglect changes in the hydrostatic pressure of the surrounding air. For a given geometry and set of boundary conditions, calculate the $($a$)$ velocity and $($b$)$ pressure fields and $($c$)$ plot the velocity profile

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$\backslash$mathrm$\{$W$\}^\{*\}=\backslash$mathrm$\{$f$\}\backslash$left$(\backslash$mathrm$\{$x$\}^\{*\}\backslash$right$)\backslash$text $\{,$ where $\}$

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Assumptions $1$ The wall is infinite in the $\backslash ($ y z $\backslash )-$plane $(\backslash ($ y $\backslash )$ is into the page for a right$-$handed coordinate system$).$

$2$ The flow is steady $($all partial derivatives with respect to time are zero$).$

$3$ The flow is parallel $($the $\backslash ($ x $\backslash )-$component of velocity, $\backslash ($ u $\backslash ),$ is zero everywhere$).$

$4$ The fluid is incompressible and Newtonian with constant properties, and the flow is laminar.

$5$ Pressure $\backslash ($ P$=$P$\_\{\backslash$mathrm$\{$atm$\}\}=\backslash )$ constant at the free surface. In other words, there is no applied pressure gradient pushing the flow; the flow establishes itself due to a balance between gravitational forces and viscous forces. In addition, since there is no gravity force in the horizontal direction, $\backslash ($ P$=$P$\_\{\backslash$text $\{$atm $\}\}\backslash )$ everywhere.

$6$ The velocity field is purely two$-$dimensional, which implies that velocity component n $50$ and all partial derivatives with respect to $\backslash ($ y $\backslash )$ are zero.

$7$ Gravity acts in the negative $\backslash ($ z $\backslash )-$direction. We express this mathematically as $\backslash (\backslash$vec$\{$g$\}=-$g $\backslash$vec$\{$k$\}\backslash ),$ or $\backslash ($ g$\_\{$x$\}=$g$\_\{$y$\}=\backslash$overline$\{0\}\backslash$quad g$\_\{$z$\}=-$g $\backslash ).$