Problem $3.(5$ pts$.)$ An ideal liquid flows through a two$-$dimensional slot $($see figure below$)$ with a velocity distribution $($vector$)$ of vec$(V)=u(x,t)i=\frac{4q_{o}t}{b(x)t_o}i,$ where $t$ is time; the flow rate per width $q_0$ and time $t_o$ are reference values $($constants$).$ Velocity $u$ is uniform in the $y$ and $z$ directions. The width $b(x)$ linearly decreases with $x$ and is given as $b(x)=B-\frac{x}{6}($B is a constant width at $x=0,$ and the width is $\frac{B}{3}$ at $x=4$B$).$ Answer the following questions:

$(3\mathrm{.}1)$ Is the flow steady or unsteady?

$(3\mathrm{.}2)$ Is the flow one$-,$ two$-$ or three$-$dimensional flow?

$(3\mathrm{.}3).$ Is the flow uniform or non$-$uniform?

$(3\mathrm{.}4).$ Is the flow rotational or irrotational? Why? $($prove it$)$

$(3\mathrm{.}5).$ Can you apply the Bernoulli equation between $x=0$ and $x=4B?$ Why?

$(3\mathrm{.}6).$ What will be the local acceleration at $x=3B$ in terms of $B,t_o,$ and $q_o?$

$(3\mathrm{.}7).$ What will be the convective acceleration at $x=3B$ in terms of $B,t,t_o,$ and $q_o?$