Question $4$

MARKS

$($a$)$ Draw a $2$D infinitesimal fluid element and label it appropriately such that it can be used to derive an expression for the conservation of mass in fluid flows and then proceed to derive this equation.

$[10]$

$($b$)$ The following equation depicts the $x-$direction conservation of linear momentum in cylindrical coordinates such that it can be applied to pipe flow. The variable $\text{'}x\text{'}$ represents the axial dimension along the pipe length, $\text{'}r\text{'}$ represents the radial dimension, and $\text{'}\text{'}$ represents the circumferential dimension. The velocity is denoted $\text{'}u\text{'}$ with subscripts for each directional component and the pressure is denoted $\text{'}p\text{'}.$

$(\frac{d{u}_x}{dt}+u_r\frac{d{u}_x}{dr}+\frac{u_{}}{r}\frac{d{u}_x}{d}+u_x\frac{d{u}_x}{dx})=-\frac{dp}{dx}+[\frac{1}{r}\frac{d}{dr}(r\frac{d{u}_x}{dr})+\frac{1}{r^2}\frac{d_2{u}_x}{d{}^2}+\frac{d_2{u}_x}{d{x}^2}]$

$($i$)$ Simplify this equation for the problem of fully developed steady laminar pipe flow and justify the elimination of any terms from it$.$

$[8]$

$($ii$)$ Use the resulting simplified equation to derive an equation for the velocity profile across the pipe diameter, i$.$e$.u(r),$ i$.$e$.$ for fully developed, steady, and laminar pipe flow.

$15$

$[$Total $33]$