Water flows in a streamlined manner through a | vedclass

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(C) According to Poiseuille's law,the rate of flow $Q$ through a capillary of radius $r$ and length $\ell$ under a pressure difference $p$ is given by

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$Q/4$

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(C) According to Poiseuille's law,the rate of flow $Q$ through a capillary of radius $r$ and length $\ell$ under a pressure difference $p$ is given by:$Q = \frac{\pi p r^4}{8 \eta \ell}$Here,$\eta$ is the coefficient of viscosity.From the formula,we see that $Q \propto p r^4$.Given the initial state: $Q \propto p a^4$.Given the final state: $Q' \propto p' (r')^4$,where $p' = 4p$ and $r' = a/2$.Substituting these values:$Q' \propto (4p) \left(\frac{a}{2}\right)^4 = 4p \cdot \frac{a^4}{16} = \frac{p a^4}{4}$.Comparing the two:$\frac{Q'}{Q} = \frac{p a^4 / 4}{p a^4} = \frac{1}{4}$.Therefore,$Q' = Q/4$.

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:

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