$(4)$ Fluid flowing in between two plates does not have a uniform velocity distribution but instead varies

across the plates as shown in the figure. If the flow is laminar, the velocity distribution $u(y)$ is

given by $u(y)=u_{\text{m}\text{a}\text{x}}[1-(\frac{y}{h}{)}^2]$ where u is the fluid velocity at any distance y from the

centerline; $u_{\text{m}\text{a}\text{x}}$ is the maximum velocity at the centerline, and $2$ h is the distance between the

plates. Assuming that water $(T=10C)$ is flowing between the plates at a distance of $2$ mm with

a centerline velocity of $18c\frac{m}{s}.$ Use the given coordinate system.

$($a$)$ Sketch the general shear stress distributions in the pipe for the given velocity profile.

$($b$)$ Estimate the magnitude and sign of the shear stress that the water exerts on the top and bottom

plates. Does your calculation match with the profiles drawn in part a$)?$

$($c$)$ Where is the magnitude of shear stress the largest and smallest between the plates?

$($d$)$ Estimate the shear force including the sign $($positive to the right$)$ acting on the top and bottom

plates if the area of each plate is $1\mathrm{.}2m^2.$

Ans: