Consider a diffuser $($length $\frac{L}{2})$ attached downstream of a flow control valve as shown in Fig. $1.$ When valve is opened, flow rate increases such that the velocity along the centre line is given by $u=U(1-e^{-ct})(1-\frac{x}{L})hat(i).$ Here $Lc,$ and $U$ are constants.

Fig. $1.$

Determine the fluid acceleration $a=a(x,t)$ in the diffuser. Find the value of $c$ such that the acceleration is zero for any $x$ when $U=10\frac{m}{s}$; $L=5m$; at a time $t=1s.$ Explore the reason for the typical observation on acceleration.