Problem $3(40$ points$)$

Consider the flow of water vertically down due to gravity.

The density of water is $.$

The acceleration of gravity is $g.$

The separation between the two flat plates is $h.$

The viscosity of water is $$

Here is a list of assumptions valid for this problem:

$1-$ Steady state flow

$2-$ No pressure gradient along the channel

$3-2$D flow

$4-$ Incompressible flow

$5-$ Fully developed flow inside the channel

PART I $($do not use numerical values for $$;$g$;$$;$h)$

i$)$ Indicate a coordinate system to be used and its relative orientation to the channel.

ii$)$ Write the continuity equation and simplify it$,$ indicating the assumptions $1-5$ used in each case.

iii$)$ Write the component of the Navier$-$Stokes equation in the vertical direction and simplify it$,$ indicating assumptions used in each term that is eliminated.

iv$)$ Solve the resulting differential equation for the velocity along the channel with unknown constants of integrations.

v$)$ Write the boundary conditions for the velocity.

vi$)$ Find a final expression for the velocity along the channel.

vii$)$ Draw the velocity profile across the channel.

viii$)$ Calculate the flow rate per unit of width of the plates $($into the page$).$

ix$)$ Calculate the shear stress acting on each plate, indicating magnitude and direction.

x$)$ Calculate the shear stress acting across the fluid and plot it

xi$)($extra$)$ show that the Navier$-$Stokes equation is trivially satisfied in the horizontal direction.

PARTII

Calculate the numerical value of the flow rate per unit width and the shear stress using $=1000\frac{kg}{m^3}$;$g=9\mathrm{.}8\frac{m}{s^2}$;$=0\mathrm{.}001$ Pa s; and $h=1mm.$