Chapter 10, Problem 82GP

To determine

To Derive: The equation of diameter of the stream.

Answer to Problem 82GP

The equation of diameter of the stream as a function of distance $y$ is $d_1=d_0\times {\left(\frac{v_0^2}{v_0^2+2gy}\right)}^{\frac{1}{4}}$ .

Explanation of Solution

Given info:

  $0$ is used for faucet and $1$ is used for the lower position.

  $z_1=0\text{ m, }{z}_0=y\text{ m}$

Pressure at faucet $P_0$ = Pressure at lower position $P_1$

Formula used:

Continuity equation is used to find the velocity at lower position.

  $\begin{array}{l}{A}_0\times v_0=A_1\times v_1\\ v_1=\frac{A_0\times v_0}{A_1}\end{array}$

Here,

  $v$ is velocity of fluid,

  $A$ is the area through which fluid is flowing.

Bernoulli's Equation will be used to find the equation of diameter of a stream as a function of distance.

  $\frac{P_0}{\rho g}+\frac{v_0^2}{2g}+Z_0=\frac{P_1}{\rho g}+\frac{v_1^2}{2g}+Z_1$

Here.

  $P$ is the pressure,

  $v$ is velocity of fluid,

  $Z$ is the distance from the reference.

Calculation:

Substitute the given values, velocity at lower position is found by continuity equation and equation for diameter as a function of distance by Bernoulli's equation.

From the continuity equation,

  $\begin{array}{l}{A}_0\times v_0=A_1\times v_1\\ v_1=\frac{A_0\times v_0}{A_1}\\ A_0=\frac{\pi }{4}\times d_0^2\\ A_1=\frac{\pi }{4}\times d_1^2\\ v_1=\frac{\frac{\pi }{4}\times d_0^2\times v_0}{\frac{\pi }{4}\times d_1^2}=v_0\times \frac{d_0^2}{d_1^2}\end{array}$

From the Bernoulli’s equation,

  $\begin{array}{l}\frac{P_0}{\rho g}+\frac{v_0^2}{2g}+Z_0=\frac{P_1}{\rho g}+\frac{v_1^2}{2g}+Z_1\\ \frac{v_0^2}{2\times g}+y=\frac{v_1^2}{2\times g}\\ v_0^2+2\times g\times y=v_1^2\\ v_0^2+2\times g\times y=v_0^2\times {\left(\frac{d_0}{d_1}\right)}^4\\ {\left(\frac{d_0}{d_1}\right)}^4=\frac{v_0^2+2\times g\times y}{v_0^2}\\ \frac{d_1}{d_0}={\left(\frac{v_0^2}{v_0^2+2\times g\times y}\right)}^{\frac{1}{4}}\\ d_1=d_0\times {\left(\frac{v_0^2}{v_0^2+2\times g\times y}\right)}^{\frac{1}{4}}\end{array}$

Conclusion:

Thus, the equation of diameter of the stream as a function of distance y is $d_1=d_0\times {\left(\frac{v_0^2}{v_0^2+2gy}\right)}^{\frac{1}{4}}$ .