Question # $3($Competencies a$1,$ a$2,$ b$1,$ b$4,$ c$3)$

$(20$ Marks$)$

$(3-$a$)$ Consider steady, incompressible, laminar flow of a Newtonian fluid in an

infinitely long round pipe of diameter. Ignore the effects of gravity. A constant

pressure gradient $d\frac{P}{d}x$ is applied in the $x-$direction. Derive an expression for

the velocity field inside the pipe as function of $r$ and estimate the viscous shear

force per unit surface area acting on the pipe wall.