Problem $1($b$)(15$ Points$)$ Use the following Navier Stoke's equation to derive the partial

derivative equation for steady state viscous flow of a fluid film down a vertical surface. The

surface on the left is stationary. Assume the flow is laminar, the fluid is incompressible and

viscosity is a constant. The flow is driven in one direction by gravity only. Write clearly all your

reasons to cancel any terms.

The continuity equation $(3\mathrm{.}6-24)$ for constant density is

$\frac{del{v}_x}{\text{d}\text{e}\text{l}\text{x}}+\frac{del{v}_y}{\text{d}\text{e}\text{l}\text{y}}+\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{z}}=0$

Equation of motion in rectangular coordinates. For Newtonian fluids for constant $$ and $$

for the $x$ component, $y$ component, and $z$ component we obtain, respectively,

$(\frac{del{v}_x}{\text{d}\text{e}\text{l}\text{t}}+v_x\frac{del{v}_x}{\text{d}\text{e}\text{l}\text{x}}+v_y\frac{del{v}_x}{\text{d}\text{e}\text{l}\text{y}}+v_z\frac{del{v}_x}{\text{d}\text{e}\text{l}\text{z}})=(\frac{de{l}^2{v}_x}{del{x}^2}+\frac{de{l}^2{v}_x}{del{y}^2}+\frac{de{l}^2{v}_x}{del{z}^2})-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{x}}+g_x$

$(\frac{del{v}_y}{\text{d}\text{e}\text{l}\text{t}}+v_x\frac{del{v}_y}{\text{d}\text{e}\text{l}\text{x}}+v_y\frac{del{v}_y}{\text{d}\text{e}\text{l}\text{y}}+v_z\frac{del{v}_y}{\text{d}\text{e}\text{l}\text{z}})=(\frac{de{l}^2{v}_y}{del{x}^2}+\frac{de{l}^2{v}_y}{del{y}^2}+\frac{de{l}^2{v}_y}{del{z}^2})-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{y}}+g_y$

$(\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{t}}+v_x\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{x}}+v_y\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{y}}+v_z\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{z}})=(\frac{de{l}^2{v}_z}{del{x}^2}+\frac{de{l}^2{v}_z}{del{y}^2}+\frac{de{l}^2{v}_z}{del{z}^2})-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{z}}+g_z$