Determination of the coefficient of viscosity of a liquid (water) by capillary flow method using Poiseuill’s formula

If $V$ is the volume of a liquid flowing in a time $t$ under a pressure $p$ through a capillary tube of length $l$ and radius $r$, then according to Poiseuille's formula,

$$\frac{V}{t} = \frac{p \pi r^4}{8 \eta l}$$

If the height between the liquid levels is $h$, the density of the liquid is $\rho$, and the acceleration due to gravity is $g$,

$$p = h \rho g$$

Hence,

$$\frac{V}{t} = \frac{h \rho g \pi r^4}{8 \eta l}$$

Wash the capillary tube first with sodium hydroxide solution, then with diluted hydrochloric acid, and finally with pure water. Connect the capillary tube to the constant-pressure apparatus using a rubber tube and fix it to the stand while leveling it with a spirit level.

Attach a piece of cotton thread near the open end of the tube, open the tap, and adjust the pressure so that the cotton thread vibrates slowly, indicating a constant pressure. Place a measuring cylinder under the open end of the tube and simultaneously start a stopwatch when the collected water reaches a specified level.

After a measured time interval (e.g., 1 minute), stop the watch and record the volume of water collected and the time taken. Measure the height of the liquid pressure head $h$ using a meter rule. Repeat the measurements for different values of $h$ by changing the pressure head and record the observations.

Measure the length of the capillary tube $l$ using a meter rule. Using a travelling microscope, measure the internal diameter of the tube in two mutually perpendicular directions.

Let:

• $t$ = time of flow of water
• $l$ = total length of the capillary tube
• $d_1$ = diameter of the capillary tube in one direction
• $d_2$ = diameter of the capillary tube in the perpendicular direction
• $V$ = volume of water collected
• $h$ = liquid pressure head

Flow rate:

$$V_t = \frac{V}{t}$$

Mean diameter:

$$d = \frac{d_1 + d_2}{2}$$

Radius of the capillary tube:

$$r = \frac{d}{2}$$

Using Poiseuille's equation,

$$\frac{V}{t} = \frac{h \rho g \pi r^4}{8 \eta l}$$

the coefficient of viscosity is

$$\eta = \frac{h \rho g \pi r^4 t}{8 V l}$$