Problem $2$: Venturi tube as a flowmeter 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$)$

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

A typical Venturi tube has a converging$-$diverging contour, as shown in figure $2$a$,$ where the cross$-$sectional

areas at stations $(1)$ and $(2)$ are prescribed respectively as $A_1$ and $A_2.$ When a Venturi tube is used as a

flowmeter, we take pressure measurements at stations $(1)$ and $(2)$ and use Bernoulli's equation to find the

mass flow rate $m^{}=A_1{u}_1=A_2{u}_2,$ where $$ is the fluid density and $u_1$ and $u_2$ are flow speeds at stations

$(1)$ and $(2).$ Assuming constant density and no change in elevation, determine the mass flow rate $m^{}$ as a

function of the known $,p_1,p_2,A_1$ and $A_2.(3$ pts$)$

Figure $2$: $($a$)$ A Venturi tube; $($b$)$ Elegant fluid mechanics taking place inside a carburetor: carburetors use a Venturi tube

to provide air$-$fuel mixture to a spark$-$ignition $($SI$)$ engine at a controlled mixing ratio by adjusting $A_{\text{j}\text{e}\text{t}}$ via a jet needle.