Question

Three capillaries of length $L,L/2$ and $L/3$ are connected in series. Their radii are $r,r/2$ and $r/3$ respectively. Then, if steam-line flow is to be maintained and the pressure across the first capillary is p, then the

• A. Pressure difference across the ends of second capillary is $8p$
• B. Pressure difference across the third capillary is $43p$
• C. Pressure difference across the ends of the second capillary is $16p$
• D. Pressure difference across the third capillary is $56p$

Answer 1

The correct option is A Pressure difference across the ends of second capillary is $8p$

In series, the rate of flow of liquid through capillaries will be same.
Rate of flow of liquid is given by
$\frac{dQ}{dt}=\frac{\pi p{r}^4}{8\eta L}$
As capillaries are joined in series, so $\left(\frac{dQ}{dt}\right)$ will be same for each capillary.
Hence, $\frac{\pi p{r}^4}{8\eta L}=\frac{\pi p^{\prime }(r/2{)}^4}{8\eta \left(L/2\right)}=\frac{\pi p^{\prime }(r/3{)}^4}{8\eta \left(L/3\right)}$
So, pressure difference across the ends of $2nd$ capillary
$p^{\prime }=8p$
and across the ends of $3rd$ capillary
$p^{\prime }=27p$
If two capillaries are joined in parallel, the pressure difference across them will be same, but rates of flow would be different.