Testing the "thin separation" assumption. Reconsider the pressure$-$driven flow problem between two

fixed plates problem that we explored in class. Now assume that the pressure varies in the $y-$direction due to

the hydrostatic pressure of the fluid shown below. The pressure gradient with respect to the $x-$direction can be

shown to varies $($by solving the y$-$balance$)$ as:

$\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{x}}=\frac{P}{x}=f(y)\frac{(P_0-P_1)}{L}+\frac{g}{L}(y-y_0)$

Thus, you can substitute the red dashed expression for $\frac{P}{}x$ in the $x-$balance.

a$.$ Solve for $v_x$ as a function of $y.$

b$.$ Plot $v_x$ with and without this additional pressure dependence given that $=8\mathrm{.}9{10}^{-4}k\frac{g}{m}-s,=1\frac{gm}{c}{m}^3,$

$y_0=0\mathrm{.}001m,L=1m,g=9\mathrm{.}8\frac{m}{s^2}$ and $(P_0-P_l)=-20Pa.$ Simplify the equation when $g=0$ and show this is

the same form when the hydrostatic pressure is not considered.

c$.$ Give the $y$ position for the maximum velocity with the additional hydrostatic pressure dependence.

d$.$ Give the $Re$ for the average velocity between the plates with the additional hydrostatic pressure dependence.

Is the assumption of laminar flow valid?