Three capillaries of length L, L/2 and L/3 ar | vedclass

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(A) According to Poiseuille's equation,the rate of flow $Q$ is given by $Q = \frac{\pi P r^4}{8 \eta L}$.Since the capillaries are connected in series

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The pressure difference across the ends of the second capillary is $8 P$.

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(A) According to Poiseuille's equation,the rate of flow $Q$ is given by $Q = \frac{\pi P r^4}{8 \eta L}$.Since the capillaries are connected in series,the rate of flow $Q$ is the same for all three capillaries.Therefore,the pressure difference $\Delta P$ across a capillary is given by $\Delta P = Q \cdot \frac{8 \eta L}{\pi r^4} \propto \frac{L}{r^4}$.Let $P_1, P_2, P_3$ be the pressure differences across the first,second,and third capillaries respectively.Given $P_1 = P$,$L_1 = L, r_1 = r$.For the second capillary: $L_2 = L/2, r_2 = r/2$.$P_2 = P_1 \cdot \frac{L_2}{L_1} \cdot \left(\frac{r_1}{r_2}\right)^4 = P \cdot \frac{L/2}{L} \cdot \left(\frac{r}{r/2}\right)^4 = P \cdot \frac{1}{2} \cdot (2)^4 = P \cdot \frac{16}{2} = 8P$.For the third capillary: $L_3 = L/3, r_3 = r/3$.$P_3 = P_1 \cdot \frac{L_3}{L_1} \cdot \left(\frac{r_1}{r_3}\right)^4 = P \cdot \frac{L/3}{L} \cdot \left(\frac{r}{r/3}\right)^4 = P \cdot \frac{1}{3} \cdot (3)^4 = P \cdot \frac{81}{3} = 27P$.Thus,the pressure difference across the second capillary is $8P$.

Three capillaries of length $L, L/2$ and $L/3$ are connected in series. Their radii are $r, r/2$ and $r/3$ respectively. If stream-line flow is to be maintained and the pressure difference across the first capillary is $P$,then:

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