Fluid Mechanics: 

Definition: Fluid Mechanics encompasses the study of all types of fluids under static, kinematic and dynamic conditions. The combination of experiments, the mathematical analysis of hydrodynamics and the new theories is known as ‘Fluid Mechanics’.

Fluid: A fluid is defined as a material which will continue to deform with the application of shear force however small the force may be. Generally matter exists in three phases namely (i) Solid (ii) Liquid and (iii) Gas (includes vapour). The last two together are also called by the common term fluids. 

(i) Solids: In solids atoms/molecules are closely spaced and the attractive (cohesive) forces between atoms/molecules is high. The shape is maintained by the cohesive forces binding the atoms.When an external force is applied on a solid component, slight rearrangement in atomic positions balances the force. Depending upon the nature of force the solid may elongate or shorten or bend. When the applied force is removed the atoms move back to the original position and the former shape is regained. Only when the forces exceed a certain value (yield), a small deformation called plastic deformation will be retained as the atoms are unable to move to their original positions. When the force exceeds a still higher value (ultimate), the cohesive forces are not adequate to resist the applied force and the component will break.

(ii) Liquids: In liquids the inter molecular distances are longer and the cohesive forces are of smaller in  magnitude. The molecules are not bound rigidly as in solids and can move randomly. However, the cohesive forces are large enough to hold the molecules together below a free surface that forms in the container. Liquids will continue to deform when a shear or tangential force is applied. The deformation continues as long as the force exists. In fluids the rate of deformation controls the force (not deformation as in solids). More popularly it is stated that a fluid (liquid) cannot withstand applied shear force and will continue to deform. When at rest liquids will assume the shape of the container forming a free surface at the top.

(iii) Gases: In gases the distance between molecules is much larger compared to atomic dimensions and the cohesive force between atoms/molecules is low. So gas molecules move freely and fill the full volume of the container. If the container is open the molecules will diffuse to the outside. Gases also cannot withstand shear. The rate of deformation  is proportional to the applied force as in the case of liquids.

Vapour is gaseous state near the evaporation temperature. The state in which a material exists depends on the pressure and temperature. For example, steel at atmospheric temperature exists in the solid state. At higher temperatures it can be liquefied. At still higher temperatures it will exist as a vapour. A fourth state of matter is its existence as charged particles or ions known as plasma. This is encountered in MHD power generation.

COMPRESSIBLE AND INCOMPRESSIBLE FLUIDS:

If the density of a fluid varies significantly due to moderate changes in pressure or temperature, then the fluid is called compressible fluid. Generally gases and vapours under normal conditions can be classified as compressible fluids. In these phases the distance between atoms or molecules is large and cohesive forces are small. So increase in pressure or temperature will change the density by a significant value.

If the change in density of a fluid is small due to changes in temperature and or pressure, then the fluid is called incompressible fluid. All liquids are classified under this category.

When the change in pressure and temperature is small, gases and vapours are treated as incompressible fluids. For certain applications like propagation of pressure disturbances, liquids should be considered as compressible.
CONTINUUM: 

As gas molecules are far apart from each other and as there is empty space between molecules doubt arises as to whether a gas volume can be considered as a continuous matter like a solid for situations similar to application of forces. 

Under normal pressure and temperature levels, gases are considered as a continuum (i.e., as if no empty spaces exist between atoms). The test for continuum is to measure properties like density by sampling at different locations and also reducing the sampling volume to low levels. If the property is constant irrespective of the location and size of sample volume, then the gas body can be considered as a continuum for purposes of mechanics (application of force, consideration of acceleration, velocity etc.) and for the gas volume to be considered as a single body or entity. This is a very important test for the application of all laws of mechanics to a gas volume as a whole. When the pressure is extremely low, and when there are only few molecules in a cubic metre of volume, then the laws of mechanics should be applied to the molecules as entities and not to the gas body as a whole.