Apply Navier$-$Stokes equation to find an expression for velocity distribution for laminar flow of Newtonian fluid in thin film along an inclined flat plate. The equations of motion for Cartesian co$-$ordinate $(x,y,z)$ are given below.

$(\frac{de{l}_x}{\text{d}\text{e}\text{l}\text{x}}+v_x\frac{del{x}_x}{\text{d}\text{e}\text{l}\text{x}}+v_y\frac{del{v}_x}{\text{d}\text{e}\text{l}\text{y}}+v_z\frac{del{v}_x}{\text{d}\text{e}\text{l}\text{z}})=-\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}+[\frac{de{l}^2{v}_x}{del{x}^2}+\frac{de{l}^2{v}_x}{del{y}^2}+\frac{de{l}^2{v}_x}{del{z}^2}]+D{g}_x$

$p(\frac{del{v}_y}{\text{d}\text{e}\text{l}}+v_x\frac{del{v}_y}{\text{d}\text{e}\text{l}\text{x}}+v_y\frac{del{v}_y}{\text{d}\text{e}\text{l}\text{y}}+v_z\frac{del{v}_y}{\text{d}\text{e}\text{l}\text{z}})=\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{y}}+[\frac{de{l}^2{v}_y}{del{x}^2}+\frac{de{l}^2{v}_y}{del{y}^2}+\frac{de{l}^2{v}_y}{de{l}^2}]+pg,$

$(\frac{del{v}_z}{\text{d}\text{e}\text{l}}+v_x\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{x}}+v_y\frac{del{v}_x}{\text{d}\text{e}\text{l}\text{y}}+v_z\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{z}})=\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{z}}+[\frac{de{l}^2{v}_z}{del{x}^2}+\frac{de{l}^2{v}_z}{del{y}^2}+\frac{de{l}^2{v}_z}{z^2}]+p{g}_3$