Potentially useful formulas

Bernoulli equation:

$p+\frac{1}{2}{u}^2+gz=$ constant along a streamline;

$p+{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{u^2}{R}dn+gz=$ constant across a streamline.

Continuity equation $($mass conservation$)$

$u_1{A}_1=u_2{A}_2$

Problem $1$: Flow in a converging pipe

Water $($assumed inviscid, incompressible$)$ flows steadily through a horizontal pipe of varying cross$-$sectional

area, shown in Figure $1.$ The centerline velocity is given by $V(x)=10(1+x)hat(i),$ where $x$ is in feet.

Determine the pressure gradient, del$\frac{p}{d}$elx $($as a function of $x),$ needed to produce this flow. $(1$ pt$)$

If the pressure at section $(1)$ is $50$ psi $,$ determine the pressure at $(2)$ by

$($i$)$ Integration of the pressure gradient obtained in Part $1$; $(1$ pt$)$

$($ii$)$ application of the Bernoulli equation. $(1$ pt$)$

Figure $1$: Flow in a converging pipe.