Question

The rate of flow of a liquid through a capillary tube is:
A) Directly proportional to the length of the tube.
B) Inversely proportional to the difference of pressure between the ends of the tube.
C) Directly proportional to the 4th power of the radius of the tube.
D) Independent of the nature of the liquid.

Answer 1

Hint: Capillary action: when a tube of small diameter is held vertically on a water container the liquid rises on the water this is called capillary action. Adhesion of water to the walls of a vessel will cause an upward force on the liquid at the edges.
Adhesive forces: adhesive forces" refers to the attractive forces between unlike substance,
Cohesive forces: Cohesion refers to the attraction of molecules for other molecules of the same kind,
Surface Tension: Liquid surfaces tend to shrink into the minimum surface area possible. Surface tension allows insects (e.g. mosquitoes), to sit on the surface without drowning

Complete solution:
Capillary action: Tubes having very small diameters (narrow cylindrical tubes) are called capillary. If these narrow tubes are dipped in a liquid it is observed that liquid in the capillary either rises (or) falls relative to the surrounding liquid level. This phenomenon is called capillary action and such tubes are called capillary.
For easy understating, we will consider the capillary tube as a normal pipe.
The flow rate through a capillary is given by Poiseuille's equation
Poiseuille's equation
$Q = \dfrac{{\pi {r^4}\Delta P}}{{8\mu L}}$
Where,
Q = flow rate
r = radius of the capillary tube
\[\Delta P\] =pressure difference
L=length of pipe/tube
$\mu $=coefficient of viscosity
Flow rate is defined as discharge per unit of time or how much liquid is flowing through a pipe in a unit of time
Now consider a liquid flowing through a pipe and if we increase the pressure at one of the pipes the flow of liquid through the pipe will increase as there is an increase in pressure difference which will increase its kinetic energy.
So we can say that the flow rate depends upon the pressure
The same is evident from the above equation
I.e. Q \[ \propto \Delta P\]
Now, If water is flowing through a thin pipe and suddenly we increase the cross-sectional area of pipe now more amount of liquid will flow through the pipe per unit time, so from here we can conclude that flow rate also depends upon the cross-sectional area
The cross-sectional area increases or decreases as we change the radius of the pipe, so we can say that the flow rate depends upon the radius of the pipe.
It is also evident from the above equation i.e. Q$ \propto {r^4}$

Final answer is (C), Flow rate is directly proportional to the fourth power of the radius.

1. We can see from the equation that flow rate is inversely proportional to length so option a is ruled out.
2. Flow rate increases with an increase in pressure difference and decreases with a decrease in pressure difference so it is directly proportional to pressure difference so option b is also ruled out.
3. Now we have discussed that an increase in the cross-sectional area will increase the flow rate so we can say they are directly proportional to each other so option C is correct.
4. As we have discussed that the capillary rise occurs due to adhesive forces dominating over cohesive and adhesive forces are property of the material so flow rate depends upon the nature of liquid.

Note: 1. Adhesive and cohesive forces are the natural property of the liquid.
2. Flow rate also depends upon the viscosity of the fluid as more viscous liquid fills flow more slowly.
3. Meniscus are formed due to surface tension.
4. If adhesive forces are dominant then we will see the capillary rise and if cohesive force dominates then there will be capillary fall.
5. The flow rate depends upon Pressure difference, area of cross-section, length of pipe, and viscosity of the liquid.
6. Length and Viscosity will decrease the flow rate as they will offer resistance to flow.