First slide
Fluid dynamics
Question

The rate of steady volume flow of water through a capillary tube of length 'l' and radius 'r' under a pressure difference of P is V. This tube is connected with another tube of the same length but half the radius in series. Then the rate of steady volume flow through them is (The pressure difference across the combination is P)

Moderate
Solution

P = P1 + P2
\large \frac Vt\left ( \frac {8\eta l}{\pi r^4} \right )=\frac {{V}'}{t}\left ( \frac {8\eta l}{\pi r_1^4} \right )+\frac {{V}'}{t}\left ( \frac {8\eta l}{\pi r_2^4} \right )
\large \Rightarrow \frac {V8\eta l}{\pi r^4}={V}'\frac {8\eta l}{\pi r^4}+{V}'\frac {8\eta l}{\pi (r/2)^4}
\large \Rightarrow \frac {V8\eta l}{\pi r^4}={V}'\frac {8\eta l}{\pi r^4}[1+16]\Rightarrow {V}'=\frac {V}{17}

   
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