Problem $1$

Consider the viscous flow of air over a flat plate. At a given station in the flow direction the

variation of flow velocity, $u,$ in the direction perpendicular to the plate, $y$ direction is given by:

$u=1582(1-e^{-\frac{y}{L}})$

where $L$ is characteristic length $=1$ inch and units of $u$ are in $f\frac{t}{sec}.$ The viscosity coefficient

$=3\mathrm{.}7373{10}^{-7}$ slug$/($ft$.$sec$)$ find shear stress acting on wall ${}_w=($du$.$dy$)$ using $y=0,1$ inch

and :

a$.$ First order forward difference methods,

b$.$ First$-$order backward methods

c$.$ Second$-$order central difference methods

d$.$ Find exact values and calculate the percentage of errors for each of the above methods

of formulations.