One of the first question we need to explore is –what is fluid? Or we might ask- what is the difference between a solid and a fluid? We have a general, vague idea of the difference. A solid is “hard” and not easily deformed; whereas a fluid is “soft” and is easily deformed (we can readily move through air). Although quite descriptive, these casual observations of the differences between solids and fluids are not very satisfactory from a scientific or engineering point of view. A more specific distinction is based on bow materials deform under the action of an external load. *A fluid is defined as a substance that deforms continuously when acted on by a shearing stress of any magnitude.* A shearing stress (force per unit area) is created whenever a tangential force acts on a surface as shown by the figure in the margin. When common solids such as steel or other metals are acted on by a shearing stress, they will initially deform (usually a very small deformation), but they will not continuously deform (flows). However, common fluids such as water, oil and air satisfy the definition of a fluid-that is, they will flow when acted on by a shearing stress. Some materials, such as slurries, tar, putty, toothpaste, and so on, are not easily classified since they will behave as a solid if the applied shearing stress is small, but if the stress exceeds some critical value, the substance will flow. The study of such materials is called Rheology and does not fall within the province of classical fluid mechanics. Although the molecular structure of fluids is important in distinguishing one fluid from another from another, because of the large number of molecules involved, it is not possible to study the behavior of individual molecules when trying to describe the behavior of fluids at rest or in motion. Rather, we characterize the behavior by considering the average, or macroscopic. Value of the quantity of interest, where the average is evaluated over a small volume containing a large number of molecules.

The equation based on the principle of conservation of mass is called continuity equation. Thus for a fluid flowing through the pipe at all the cross-section, the quantity of fluid per second is constant.

Consider two cross-section of a pipe as shown in the figure.

Let V_{1} = Average velocity at cross section 1-1

ρ_{1}= Density at section 1-1

A_{1}= Area of pipe at section 1-1

And V_{2}, ρ_{2}, A_{2} are corresponding valves at section, 2-2.

Then rate of flow at section 1-1 = ρ_{1}A_{1}V_{2}

According to law of conversation of mass

Rate of law of conservation of mass

Rate of flow at section 1-1 = Rate of flow at section 2-2

**Continuity Equation**. If the fluid is incompressible, then **ρ**_{1} = ρ_{2} and continuity equation (1) reduces to

**A**_{1}V_{1} = A_{2}V_{2}

Consider two cross-section of a pipe as shown in the figure.

Let V

ρ

A

And V

Then rate of flow at section 1-1 = ρ

According to law of conversation of mass

Rate of law of conservation of mass

Rate of flow at section 1-1 = Rate of flow at section 2-2

ρ_{1}A_{2}V_{2} = ρ_{2}A_{2}V_{2} .............................(1)

Equation (1) is applicable to the compressible as well as incompressible fluids and is called The flow of a fluid through a pipe can be divided into two general classes, streamline flow or turbulent flow, depending upon the type of path followed by the individual particles of the fluid. When the flow of all the fluid particles is essentially along lines parallel to the axis of the pipe, the flow is called streamline (also viscous or laminar). When the course followed by the individual particles of the fluid deviates greatly from a straight line so that vortices and eddies are formed in the fluid, the flow is called turbulent.

The distinction between streamline flow and turbulent flow can be shown clearly by means of a simple experiment. The experiment is carried out by injecting a small stream of colored liquid into a fluid flowing inside a glass tube. If the fluid is moving at a sufficiently low velocity, the colored liquid will flow through the system in a straight line. No appreciable mixing of the two fluids will occur, and the straight –line path of the colored liquid can be observed visually. Under these conditions, streamline flow exists. If the velocity of the main stream is increased steadily, a velocity will finally be reached where the colored liquid no longer flows in a straight line. It now starts to mix with the main body of the fluid, and eddies and whirls can actually be observed through the glass walls of the tube. As the main-stream velocity is further increased, the mixing effect becomes more noticeable until the colored liquid is finally dispersed at random throughout the entire body of the main fluid. Under these conditions, turbulent flow exists. The particles of the fluid are no longer moving in smooth, straight lines but are moving in irregular directions throughout the tube.

The distinction between streamline flow and turbulent flow can be shown clearly by means of a simple experiment. The experiment is carried out by injecting a small stream of colored liquid into a fluid flowing inside a glass tube. If the fluid is moving at a sufficiently low velocity, the colored liquid will flow through the system in a straight line. No appreciable mixing of the two fluids will occur, and the straight –line path of the colored liquid can be observed visually. Under these conditions, streamline flow exists. If the velocity of the main stream is increased steadily, a velocity will finally be reached where the colored liquid no longer flows in a straight line. It now starts to mix with the main body of the fluid, and eddies and whirls can actually be observed through the glass walls of the tube. As the main-stream velocity is further increased, the mixing effect becomes more noticeable until the colored liquid is finally dispersed at random throughout the entire body of the main fluid. Under these conditions, turbulent flow exists. The particles of the fluid are no longer moving in smooth, straight lines but are moving in irregular directions throughout the tube.

When a liquid flows, the velocity of liquid particles is not same at different points in the liquid. If the plane is considered normal to the directions of flow of liquid then on this plane also the velocity is not the same at all points. At low speed when the flow is steady, we can think of the flow as in different layers with different but definite velocities. All liquid particles in a particular layer are moving with the same velocity. Such a flow is called laminar flow and the whole liquid can be supposed to be made of such laminar. In a laminar flow the different layers of the liquid glide over one another at a slow and steady velocity without inter-mixing. It is also called streamline or viscous flow. **The property of fluid (liquid and gas) due to which it opposes the relative motion between its different layers is called viscosity (or fluid friction or internal friction).**

Newton’s formula and coefficient of Viscosity:

When a body, say a boat, is moving slowly on the surface of water in a calm river or canal, the top layer of water moves with the speed of the boat as it is dragged with it. The layer of water in contact with the bed of the river is at rest. Obviously, the layers from the bottom upwards will be moving with increasing velocity. Let v be the velocity of a layer at distance z from the bed and v + dv the velocity of a layer at distance z + dz from the bed. Thus, velocity differs by dv between two layers separated by a distance dz (perpendicular to the direction of v). The quantity $\frac{dv}{dz}$ is called the velocity gradient.

As a result of large number of experiments, Newton found that the viscous force F, acting on any layer of a fluid, is directly proportional to it’s a and to the velocity gradient at the layer.

F ∝ A and F ∝ $\frac{dv}{dz}$

Or F ∝ A$\frac{dv}{dz}$

When*η* is a constant called coefficient of viscosity or simply viscosity of the fluid. The negative sign is included as the force is frictional and opposes relative motion. The equation (1) is known as **Newton’s Law of Viscous Force.**

Newton’s formula and coefficient of Viscosity:

When a body, say a boat, is moving slowly on the surface of water in a calm river or canal, the top layer of water moves with the speed of the boat as it is dragged with it. The layer of water in contact with the bed of the river is at rest. Obviously, the layers from the bottom upwards will be moving with increasing velocity. Let v be the velocity of a layer at distance z from the bed and v + dv the velocity of a layer at distance z + dz from the bed. Thus, velocity differs by dv between two layers separated by a distance dz (perpendicular to the direction of v). The quantity $\frac{dv}{dz}$ is called the velocity gradient.

As a result of large number of experiments, Newton found that the viscous force F, acting on any layer of a fluid, is directly proportional to it’s a and to the velocity gradient at the layer.

F ∝ A and F ∝ $\frac{dv}{dz}$

Or F ∝ A$\frac{dv}{dz}$

F = -ηA$\frac{dv}{dz}$ .......................(1)

When

When a solid body moves through a fluid, the fluid in contact with the body is dragged with it. This creates layers of liquid with varying velocities and as a result, viscous forces come into play. The fluid layer adjacent to the solid experts viscous force on the body to oppose its motion. The magnitude of the viscous force depends on the shape and size of the solid body, its speed and the viscosity of the fluid. Stokes proved that when a small spherical body of radius a moves with velocity v through a fluid of viscosity n, the viscous force opposing the motion of the sphere is

This equation is called Stokes’ Law. It is based on the following assumptions that

F = 6πηav

This equation is called Stokes’ Law. It is based on the following assumptions that

- The spherical body is perfectly rigid and smooth

- The medium is homogeneous and infinite in extent,

- There is no slip between the body and medium,

- The body moves through the medium at slow speed, in any case not exceeding a certain value equal to 0.6 η/σa, where σ is the density of the medium.

- It is to be noted that the viscous force is proportional to the velocity of the moving body.

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