$2.$ Consider the pressure$-$driven flow of a low density polyethylene $($LDPE$)$ melt at $200\backslash$deg C through a tube with a diameter of $2$ mm and length of $5$ cm$.$ Use the parameters in Table $3\mathrm{.}5.$ The pressure drop is $10$ MPa.

$($a$)$ Plot the velocity profile $($velocity vs$.$ r$).$

$($b$)$ Calculate the maximum shear rate of the melt. At what position is this shear stress found?

$($c$)$ Calculate maximum sheer stress.

$($d$)(4$ pts$.)$ Determine if flow is laminar or turbulent based on the maximum velocity $[$hint: assume lowest viscosity of the melt using the maximum shear rate from part $($b$)].$ The density of the polymer melt will be about $0\mathrm{.}8$ g$/$cm$^3(800$ kg$/$m$^3).$

TABLE $3\mathrm{.}5$ Pressure$-$Driven Flow Equations

Configuration

Pressure$-$driven flow

through a rectangular

channel $of$ width $W$

$v_x=\frac{P}{2L}(\frac{H^2}{4}-y^2),v_x=\frac{H}{2(s+1)}(\frac{HP}{2KL}{)}^{\text{'}}$

$Q=\frac{H^{3}WP}{12L}$

Non$-$Newtonian,

Newtonian

$[1-(\frac{2y}{H}{)}^{s+1}]$

power law liquid ${?}^{(}()()a)$

$Q=\frac{W{H}^2}{2(s+2)}(\frac{HP}{2KL}{)}^{\text{'}}$

Pressure$-$driven flow

through a tube of radius $R$

$v_x=\frac{P}{4L}(R^2-r^2),v_x=\frac{R}{s+1}(\frac{RP}{2KL}{)}^s[1-(\frac{r}{R}{)}^{s+1}]$

$Q=\frac{R^{4}P}{8L},Q=\frac{R^3}{s+3}(\frac{RP}{2KL}{)}^s$

$s_s=\frac{1}{n}$