$5($a$)$ Fluid flows through a pipe with a velocity of $7\mathrm{.}0f\frac{t}{s}$ and is being heated, so the fluid temperature $T$ at an axial position $x$ increases at a steady rate of $56\mathrm{.}0\frac{F}{m}in.$ In addition, the fluid temperature is increasing in the axial direction at the rate of $5\mathrm{.}0\frac{F}{f}t.$ Find the value of the material derivative DT$/$Dt at the position x $.$ Explain how you arrived at your answer.

$($b$)$ A layer of oil flows down a vertical plate as shown in the Figure. The flow velocity is given by $V=(\frac{V_0}{h^2})(2hx-x^2)hat(),$ where $V_0$ and $h$ are constants.

$($i$)$ Show that the fluid sticks to the plate and that the shear stress at the edge of the layer $(x=$ h$)$ is zero.

$($ii$)$ Determine the flowrate across surface $AB.$ Assume the width of the plate is $b.$

Note: The velocity profile for laminar flow in a pipe has a similar shape.