$(100$ points$)$ A $3$D velocity field is defined as vec$(V)=$uhat$(i)+$vhat$(j)+$what$(k)=(2x+y^2+z+t)hat(i)+$

$(x^2-3y+z^2-t)hat(j)+(x^2+y+z+t)hat(k).$

$($a$)$ Is the flow steady or unsteady $?(5)$

$($b$)$ Find he stagnation point of the flow, at time $t=0.(10)$

$($c$)$ Calculate the total acceleration $(\frac{D(vec(V))}{Dt})$ of the flow. Label temporal and spatial $($ad$-$

vection$)$ parts of the acceleration. $(25)$

$($d$)$ Is the flow rotational or irrotational $?$

Prove it by calculating the vorticity of the flow. $(20)$

$($e$)$ What is the linear strain rate of the flow in the three directions $({}_{},{}_{yy},{}_{zz})?$

Based on them, can you comment on the in incompressibility of the flow? $(20)$

$($f$)$ Write out the complete strain rate tensor $({}_{ij})$ for the flow. $(20)$

Answer: