# Capillaries in series

The velocity of fluid of viscosity $\eta$ through a capillary of radius $r$ and length $l$ at a distance $x$ from the center of the capillary is given by; $v=\frac{P}{4l \eta }(r^2-x^2)$ (where $P$ is the pressure difference at the two ends of capillary). With the help of this I can find the rate of flow of fluid out of the capillary equal to $\frac {dV_{out}}{dt} = \frac{\pi Pr^4}{8l \eta }$.

But what happens when the capillaries are in series with different radius and length?

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$P_1+P_2 = P$
$V_1 = \frac{\pi P_1 r_1^4}{8 l_1\eta} = V_2 = \frac{\pi P_2 r_2^4}{8 l_2\eta}$
Solve this system for $P_1$ and $P_2$ then plug back in to find the flow rate in terms of $P, r_1, l_1, r_2, l_2$.