The essential things you must remember in the section, ‘viscosity’ are given below:

**1. Viscous force between two layers of a fluid = ****ηAdv/dx **where η is the coefficient of viscosity, A is the common area of the layers and dv/dx is the velocity gradient.

**2. Poiseuille’s formula **for the volume (V) of a liquid flowing through a capillary tube of radius 'r’ in a time ‘t’ under a pressure difference ‘P’ between the ends of the tube is

**V = πPr ^{4}t / 8Lη**

where L is the length of the tube and η is the coefficient of viscosity of the liquid.

** **If the liquid flows through a horizontal capillary tube under a constant hydrostatic pressure produced by a height ‘h’ of liquid column, **V = πh****ρ****gr ^{4}t / 8Lη** where ρ is the density of the liquid.

** **It will be better to remember the rate of flow (which is the volume flowing per second) as

**Q = V/t = πPr ^{4}/ 8Lη**

If Q_{1} and Q_{2} are the rates of flow through two tubes (of radii r_{1}, r_{2} and lengths L_{1}, L_{2}) under a given pressure head P, then the rate of flow (Q_{series}) under the same pressure head P when the tubes are connected in *series* is given by the reciprocal relation,

1/Q_{series} = 1/Q_{1} +1/Q_{2}.

[You can easily prove this by combining the equations** **Q_{1} = πPr_{1}^{4}/ 8L_{1}η, Q_{2} = πPr_{2}^{4}/ 8L_{2}η, P = P_{1}+P_{2} (where P_{1} and P_{2} are the pressures between the ends of the two tubes when they are in series) and Q = πP_{1}r_{1}^{4}/ 8L_{1}η = πP_{2} r_{2}^{4}/ 8L_{2}η. Do this as an exercise]

If there are many tubes in series, the above relation gets modified as

1/Q_{series} = 1/Q_{1} +1/Q_{2} + +1/Q_{3} +1/Q_{4 }+…etc.

**3. Reynold’s number, R = v****ρr/**** ****η **where ‘v’ is the velocity of the liquid of density ρ and viscosity (coefficient) η through a tube of radius ‘r’.

Note that R is *dimensionless* and that the *flow will be streamlined only if R is less than 2000 (approximately). *It therefore follows that *stream lined flow is more likely in the case of liquids of small density and large viscosity.*

**4. Stokes formula** for the viscous force (F) on a sphere of radius ‘r’ moving with a velocity ‘v’ through a fluid having coefficient of viscosity η is

**F = 6πrηv**

If the sphere moves with terminal velocity v_{terminal} as is the case when it moves down under gravity through a column of viscous medium, we can equate the magnitude of viscous force to the apparent weight of the sphere so that

**6πrηv _{terminal} = (4/3)πr^{3}**(

**ρ**

**–**

**σ)g**where ρ

**is the**

**density of the material of the sphere and σ is the density of the viscous medium.**

Note that the *terminal velocity is directly proportional to the radius of the sphere.*

In the next post we will discus some typical multiple choice questions on viscosity.

Merry Christmas!