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FLUID IN MOTION

HYDRODYNAMICS

Hydrodynamics is the branch of physics which deals with the study of properties of fluids in motion.

Viscosity of the fluid

Is the property of a moving fluid (liquid or gas) to oppose the relative motion between its layers.

Thus viscosity is that property of a fluid that indicates its internal friction.

The greater the viscosity of a fluid, the grater is the force required to cause one layer of fluid to slide past another.

For example:

The viscosity of honey is very large as compared to that of water. This means that for the same applied external force, the rate of flow of honey will be very small as compared to that of water.

The viscosity of a fluid not only retards its own motion but it also retards the motion of a solid through it.

The greater the viscosity of a fluid, the harder it is for a solid to move through it, imagine the difference between swimming in water and honey.

CAUSE OF VISCOSITY

Viscosity is the internal friction of a fluid which opposes the motion of one layer of fluid past another.

The forces of attraction between the molecules of a moving fluid determine the viscosity of the fluid.

Viscous force

Is the tangential force that tends to destroy the relative motion between different fluid layers.

Viscous Fluid

Is the fluid which offers a resistance to the motion through it of any solid body.

Non Viscous fluid

Is the fluid which does not offer a resistance to the motion through it of any solid.

Velocity Gradient

Is the change of velocity divided by the distance in a direction perpendicular to the velocity.

NEWTON’S LAW OF VISCOSITY

Newton’s law of viscosity states that “the frictional force F between the layers is directly proportional to area A of the layers in contact and to the velocity gradient”

is a constant of proportionality and is called coefficient of viscosity.

Note that the negative sign shows that the direction of viscous drag F is opposite to the direction of motion of the liquid.

From,

The coefficient of viscosity is the tangential force required to maintain a unit velocity gradient between two parallel layers each of unit area.

or

Is the tangential force per unit area of a layer, required to maintain unit velocity gradient normal to the direction of flow.

Coefficient of viscosity of the liquid is a measure of the degree to which the fluid exhibits viscous effects.

Units of

The SI units of F is 1N, The SI unit or A is 1 and that of velocity gradient is SI unit of is NSM-2. It also called.

The coefficient of viscosity of a liquid is if a tangential force of 1N is required to maintain a velocity gradient of between two parallel layers each of area .

is also called Dynamic viscosity or Absolute Viscosity The viscosity of an ideal liquid is zero.

Dimensional formula of

=

The coefficient of viscosity of a liquid decrease with the increases in temperature and vice versa. However, the coefficient of viscosity of gases increases with the increase in temperature.

Fluidity

Is a measure of re ability of a fluid of flow and is equal to the reciprocals of

Dimensional formula of fluidity is

NEWTONIAN AND NON NEWTONIAN FLUID

Newtonian fluid

Is the fluid with which the velocity gradient is proportional to the tangential stress.

These fluids obeys Newton’s law of viscosity

Non Newtonian fluid

Is the fluid with which the velocity gradient is not proportional to the tangential stress.

These fluids does not obey Newton‘s law of viscosity.

They don‘t have constant values of , Oil-paint is an example of a non-Newtonian liquid

VARIATION OF COEFFICIENT OF VISCOSITY WITH TEMPERATURE

- For liquids

In the case of liquids the viscosity is due to the attraction among molecules within the liquid and also between the molecules of the liquid and those of solids in contact.

With rise in temperature, the molecular attractions get weakened and hence viscosity decreases.

- For Gases

The molecules are much furt

her apart and the viscosity is due to the collisions between the fast moving (flowing) molecules and those flowing at lower velocities.

her apart and the viscosity is due to the collisions between the fast moving (flowing) molecules and those flowing at lower velocities.

During collisions the fast molecules give up momentum to the slow molecules and are retarded in their flow.

As temperature increases molecular activity increases and this led to the increase of viscosity with rise in temperature.

Differences between friction and viscosity.

Friction | Viscosity |

- Solid friction is independent of the temperature.
- In case of solid friction, heat is generated at the
| – The viscosity of a liquid decreases with the increase in temperature. |

surfaces between the solids. | – Heat is generated within the fluid and not at the interface of the solid and the fluid |

– Friction between two surfaces of solids is | – Viscosity depends upon the area of contact and the velocity gradient between the layers. |

velocity.

Similarities between friction and viscosity

- Both come into play wherever there is a relative motion
- Both oppose the relative motion (iii) Both arise from intermolecular forces (iv) Both depend on nature of surfaces.

STOKE’S LAW

Stoke’s law state that “for steady motion of a small spherical body, smooth and rigid moves slowly in a fluid of infinite extent, the viscous drag force experienced on the body is given by,

F – Viscous drag force

– Coefficient of viscosity of the fluid

r – Radius of spherical body

v – Terminal velocity

Derivation of the Formula

Consider a sphere of radius moving with velocity v through a fluid whose coefficient of viscosity.

It is desired to find the expression for the viscous force F of the sphere.

For this purpose, we shall use dimensional analysis.

Stokes observed that in case of a slowly moving small sphere, the viscous force F depends on

- The radius r of the spherical body
- The coefficient of viscosity of the fluid.
- The velocity v of the spherical body.
- Shape and size of the solid body.

Equating the indices of M, L, T

1 = y , 1 = x – y + z, -2 = y – z Solving gives y = 1 z = 1

x = 1

The value of k was found to be

Limitation of the Stoke’s law

- Strictly, the law applies to a fluid of infinite extent.
- The law does not hold good if the spherical body is moving so fast that conditions are not streamline.
- The spherical body must be smooth, small and rigid

Importance of Stoke’s law

- It accounts for the formation of goods
- It is used in Millikan‘s experiment for the measurement of charge on an electron
- It is used to find the size of small particles
- It explains why large rain drops hurt much more than small ones when falling on you. It is not just that they are heavier but they are actually falling faster.