The rate of flow of the liquid through the tube of length l and radius r, connected across a perssure haed h be V. If two tubes of the same length but of radius r and `r//2` are connected in series, across the same pressure head h, find the rate of flow of liquid through the combination. If both the tubes are connected inparallel to the same pressure head, then find the rate of flow of liquid through the combination.

Answer

[`V//17 ; 17 V//16`]

Explanation

For the pressure head h, the pressure difference at the two ends of tube is, `p=h rho g` The rate of flow of the liquid through tube A of radius r is
`v=(pi P r^(4))/(8eta l)=(pi g rho g r^(4))/(8eta l)` …..(i)
When tubes B and C of eadii r and `r//2` are connected inseries, let h' be the pressure head at the common point of the tubes B and C. The rate of flow of liquid V' through each of the tube B and C will the same. Thus
`V'=(pi(h-h')rho g r^(4))/(8 eta l)=(pi h' rho g (r//2)^(4))/(8 eta l)`
`:. h-h'=(h')/(16)` or `h=(17h')/(16)` or `h'=(16)/(17)h`
So, ` V'=(pi=(h-16h//17)rho g r^(4))/(8 eta l)`
`=(pi h rho g r^(4))/(8xx17 eta l)` ....(ii)
Dividing (ii) by (i), we get
`V'//V=1//7` or `V'=V//17`
When tubes are connected in parallel, then total rate of the flow of liquid througgh the tube is
`=V+V'=(pi p r^(4))/(8eta l)+(pi p(r//2)^(4))/(8 eta l)`
`=(17)/(16)((pi p r^(4))/(8eta l))=(17V)/(16)`