Consider a steady flow of oil in a pipeline. | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The Reynolds number for a fluid flow is given by $R_e = \frac{2 v \rho r}{\eta}$, where $v$ is the velocity of flow, $\rho$ is the density of the

Vedclass · Accepted Answer

$\frac{1}{e}$

Vedclass · Answer

(A) The Reynolds number for a fluid flow is given by $R_e = \frac{2 v ho r}{\eta}$, where $v$ is the velocity of flow, $ho$ is the density of the fluid, $r$ is the radius of the tube, and $\eta$ is the viscosity of the fluid.From the equation of continuity, $A_1 v_1 = A_2 v_2$, where $A = \pi r^2$. Thus, $\pi r_1^2 v_1 = \pi r_2^2 v_2$, which implies $\frac{v_1}{v_2} = \frac{r_2^2}{r_1^2}$.The ratio of Reynolds numbers is $\frac{R_1}{R_2} = \frac{v_1 r_1}{v_2 r_2} = \left(\frac{r_2^2}{r_1^2}ight) 	imes \left(\frac{r_1}{r_2}ight) = \frac{r_2}{r_1}$.Given $r = r_0 e^{-\alpha x}$, we have $r_1 = r_0 e^{-\alpha x_1}$ and $r_2 = r_0 e^{-\alpha (x_1 + 3)}$.Substituting these values, $\frac{R_1}{R_2} = \frac{r_0 e^{-\alpha (x_1 + 3)}}{r_0 e^{-\alpha x_1}} = e^{-\alpha (x_1 + 3) + \alpha x_1} = e^{-3 \alpha}$.Given $\alpha = \frac{1}{3} 	ext{ m}^{-1}$, we get $\frac{R_1}{R_2} = e^{-3(1/3)} = e^{-1} = \frac{1}{e}$.

Similar Questions

What is Reynolds number?

What is critical velocity? Write its equation.

Consider an increase of $1 \%$ in each of the radius of the artery, the viscosity of blood, and the density of blood, respectively. The percentage change in the flow rate of blood in the artery is (in $\%$)

If $\rho$ is the density and $\eta$ is the coefficient of viscosity of a fluid which flows with a speed $v$ in a pipe of diameter $d$,the correct formula for Reynolds number $R_{e}$ is ..............

Water flows in a streamline manner through a capillary tube of radius $a$. The pressure difference is $P$ and the rate of flow is $Q$. If the radius is reduced to $\frac{a}{4}$ and the pressure is increased to $4P$,then the rate of flow becomes ................

Vedclass Test Series

Exam Paper Generator

Online Exam Module