Q$1)$ A $2-$D flow is described by the following velocity components:

$u=5x^3$ and $v=-15x^{2}y.$ Evaluate the $($i$)$ stream function, $($ii$)$ velocity, and $($iii$)$ acceleration at point $A$

and $y=2m.$

Q$2)$ Check whether the following functions satisfy the continuity equation and are valid potential

functions $($A is a constant$)$:

$($a$)\frac{O}{?}=(\frac{A}{2})(x^2-y^2)$

$(30\%)$

$($b$)\frac{O}{?}=(A)(cosx+siny)$

$($c$)\frac{O}{?}=(A)ln(xy)$

Q$3)$ A viscous incompressible fluid flows between two infinite, vertical, parallel plates. Determine,

using the Navier Stokes equations, an expression for the pressure gradient in the direction of

flow. Express your answer in terms of the mean velocity. Assume that the flow is laminar,

steady, and uniform.

$(40\%)$