$A$ dimensionally consistent relation for the volume $V$ of a liquid of coefficient of viscosity $\eta$ flowing per second through a tube of radius $r$ and length $l$ and having a pressure difference $p$ across its end,is

$A$ capillary tube of length $l$ and radius $r$ is connected to the bottom of a container. When the pressure difference across it is $P$,the volume of water flowing out per second is $V$. If another capillary tube of the same length but half the radius is connected in series with the first one,what will be the new volume of liquid flowing out per second? (The total pressure difference across the system is $P$.)

$(a)$ What is the largest average velocity of blood flow in an artery of radius $2 \times 10^{-3} \; m$ if the flow must remain laminar?
$(b)$ What is the corresponding flow rate? (Take viscosity of blood to be $2.084 \times 10^{-3} \; Pa \; s$).

The Reynolds number of a flow is the ratio of

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 streamlined manner through a capillary of radius $a$,the pressure difference being $p$ and the rate of flow $Q$. If the radius is reduced to $a/2$ and the pressure is increased to $4p$,the rate of flow becomes: