Problem $3(10$ pts$)$

Starting with the momentum equation,

$\frac{\text{d}\text{e}\text{l}{v}_i}{\text{d}\text{e}\text{l}\text{t}}=-\frac{\text{d}\text{e}\text{l}{v}_i{v}_j}{del{x}_j}-\frac{u_{ji}}{del{x}_j}-\frac{\text{d}\text{e}\text{l}\text{p}}{del{x}_i}+g_i$

obtain the equivalent of the momentum equation in Panton's table C$\mathrm{.}5$:

$(\frac{del{v}_i}{\text{d}\text{e}\text{l}\text{t}}+v_j\frac{del{v}_i}{del{x}_j})=\frac{de{l}^2{v}_i}{del{x}_j^2}-\frac{\text{d}\text{e}\text{l}\text{p}}{del{x}_i}+g_i$

Note that $\frac{de{l}^2}{del{x}_i^2}=\frac{\text{d}\text{e}\text{l}}{del{x}_i}\frac{\text{d}\text{e}\text{l}}{del{x}_i}$ is the Laplacian operator, $gra{d}^2,$ where there are implied repeated indices in $\frac{de{l}^2}{del{x}_i^2}.$

Problem $4(10$ pts$)$

In class we discussed the falling film problem. Let's extend our analysis a bit.

$(4$ pts$)$ Derive the expression for the mass flow rate.

$(2$ pts$)$ Plot the mass flow rate as a function of the film thickness for both water and glycerol at $20$C$.$ Use an angle of $20$

degrees for the incline.

$(4$ pts$)$ What force must be applied to the plate beneath the film to keep it stationary? Define this force in terms of its

individual components $F_x,F_y$ and $F_z.$