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(C) The flow rate $Q$ through a capillary tube is given by Poiseuille's law: $Q = \frac{\Delta P}{R}$,where $R = \frac{8 \eta l}{\pi r^4}$ is the flui

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$\frac{l_1 l_2}{l_1 + l_2}$

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(C) The flow rate $Q$ through a capillary tube is given by Poiseuille's law: $Q = \frac{\Delta P}{R}$,where $R = \frac{8 \eta l}{\pi r^4}$ is the fluid resistance.For tubes in parallel,the effective resistance $R_{eff}$ is given by $\frac{1}{R_{eff}} = \frac{1}{R_1} + \frac{1}{R_2}$.Substituting the expressions for resistance: $\frac{\pi r^4}{8 \eta l} = \frac{\pi r^4}{8 \eta l_1} + \frac{\pi r^4}{8 \eta l_2}$.Canceling the common terms $\frac{\pi r^4}{8 \eta}$,we get: $\frac{1}{l} = \frac{1}{l_1} + \frac{1}{l_2}$.Solving for $l$: $\frac{1}{l} = \frac{l_1 + l_2}{l_1 l_2}$,which implies $l = \frac{l_1 l_2}{l_1 + l_2}$.

Column - $I$	Column - $II$
$(a)$ $R_e > 2000$	$(i)$ Streamline
$(b)$ $1000 < R_e < 2000$	$(ii)$ Turbulent
	$(iii)$ Unsteady

Two capillary tubes of the same radius but lengths $l_1$ and $l_2$ are connected in parallel at the bottom of a vessel. What should be the length of a single capillary tube of the same radius that replaces both,such that the flow rate remains the same?

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$A$ liquid of density $10^3 \, kg/m^3$ and coefficient of viscosity $8 \times 10^{-2} \, \text{decapoise}$ is flowing in a tube of radius $2 \, cm$ with speed $2 \, m/s$. The Reynold's number is ..........

Range of Reynolds number are given in Column - $I$ and types of flow are given in Column - $II$. Match them appropriately.

Column - $I$ Column - $II$

$(a)$ $R_e > 2000$ $(i)$ Streamline

$(b)$ $1000 < R_e < 2000$ $(ii)$ Turbulent

$(iii)$ Unsteady

$A$ capillary tube is connected to the bottom of a container. If its radius is increased by $10\%$,what is the percentage change in the rate of flow of the liquid (in $\%$)?

Two capillary tubes of same length each of $50 \ cm$ but of different radii $4 \ mm$ and $2 \ mm$ are connected in series. When water flows,the pressure difference between the ends of the arrangement is $P$. Then the pressure difference between the ends of the first tube is

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