According to Poiseuille’s formula, the rate of flow V of liquid through a capillary tube of length \( l \) and radius \( r \) is given by,
\( V=\frac{\pi{P}r^4}{8\eta{l}} \),
where, \( \eta \) is the coefficient of viscosity of the liquid and \( P \) is the pressure difference between the ends of the tube.
Or, \( \displaystyle{V=\frac{P}{\frac{8\eta{l}}{\pi{r^4}}}=\frac{P}{Z}} \)
or, \( P=VZ\tag{1} \)
According to Ohm’s law for electricity,
Potential Difference (E)= Resiatance(R) \( \times \) Current(I) \( \tag{2} \)
Comparing equations (1) & (2), we can say that the pressure difference \( P \) plays the same role of \( E \) and rate of flow \( V \) plays the same role of \( I \), So \( Z \) is the flow resistance.
(i) Series combination:
Let us consider a number of capillary tubes of length \( l_1 \), \( l_2 \), \( \cdots \) and redius \( r_1 \), \( r_2 \), \( \cdots \) respectively are joined in series combination then the rate of flow
\( V=\frac{\pi{P}}{8\eta}\frac{1}{ \left(\frac{l_1}{{r_1}^4}+ \frac{l_2}{{r_2}^4}+\cdots\right) } \)or, \( V=\displaystyle{\frac{P}{\frac{8\eta{l_1}}{\pi{r_1}^4}+\frac{8\eta{l_2}}{\pi{r_2}^4}+\cdots}}\\=\frac{P}{Z_1+Z_2+\cdots}\\=\frac{P}{Z_s} \)
where, \( Z_1=\frac{8\eta{l_1}}{\pi{r_1}^4} \), \( Z_2=\frac{8\eta{l_2}}{\pi{r_2}^4} \), \( \cdots \) re the flow resistances in the respective tubes and \( Z_s \) is the effective flow resistance in series combination.
(ii) Parallel combination:
For parallel combination
\( V=\frac{\pi{P}}{8\eta}\left(\frac{{r_1}^4}{l_1}+\frac{{r_2}^4}{l_2}+\cdots\right) \)or, \( V=P\left( \frac{1}{\frac{8\eta{l_1}}{\pi{r_1}^4}} + \frac{1}{\frac{8\eta{l_2}}{\pi{r_2}^4}} +\cdots \right) \)
or, \( V=P\left( \frac{1}{Z_1}+\frac{1}{Z_2}+\cdots \right) \)
If \( Z_p \) be the effective flow resistance in parallel combination then \( V=\frac{P}{Z_p} \).
Therefore, \( \frac{1}{Z_p}=\frac{1}{Z_1}+\frac{1}{Z_2}+\cdots \)