**1 Fluids and Continuum**

## Fluids and Continuum

### 1 Fluids and Continuum

INTRODUCTION

The flow of ideal non-viscous fluids was extensively studied and mathematical theories were developed during the last century. The field of study was called as ‘Hydrodynamics’. However the results of mathematical analysis could not be applied directly to the flow of real fluids. Experiments with water flow resulted in the formulation of empirical equations applicable to engineering designs. The field was called Hydraulics. Due to the development of industries there arose a need for the study of fluids other than water. Theories like boundary layer theory were developed which could be applied to all types of real fluids, under various conditions of flow. The combination of experiments, the mathematical analysis of hydrodynamics and the new theories is known as ‘Fluid Mechanics’. Fluid Mechanics encompasses the study of all types of fluids under static, kinematic and dynamic conditions.

The study of properties of fluids is basic for the understanding of flow or static condition of fluids. The important properties are density, viscosity, surface tension, bulk modulus and vapour pressure. Viscosity causes resistance to flow. Surface tension leads to capillary effects. Bulk modulus is involved in the propagation of disturbances like sound waves in fluids. Vapour pressure can cause flow disturbances due to evaporation at locations of low pressure. It plays an important role in cavitation studies in fluid machinery.

In this chapter various properties of fluids are discussed in detail, with stress on their effect on flow. Fairly elaborate treatment is attempted due to their importance in engineering applications. The basic laws used in the discussions are :

(i) Newton’s laws of motion,

(ii) Laws of conservation of mass and energy,

(iii) Laws of Thermodynamics, and

(iv) Newton’s law of viscosity.

A fluid is defined as a material which will continue to deform with the application of shear force however small the force may be.

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. In this text, only systems satisfying continuum requirements are discussed.

The flow of ideal non-viscous fluids was extensively studied and mathematical theories were developed during the last century. The field of study was called as ‘Hydrodynamics’. However the results of mathematical analysis could not be applied directly to the flow of real fluids. Experiments with water flow resulted in the formulation of empirical equations applicable to engineering designs. The field was called Hydraulics. Due to the development of industries there arose a need for the study of fluids other than water. Theories like boundary layer theory were developed which could be applied to all types of real fluids, under various conditions of flow. The combination of experiments, the mathematical analysis of hydrodynamics and the new theories is known as ‘Fluid Mechanics’. Fluid Mechanics encompasses the study of all types of fluids under static, kinematic and dynamic conditions.

The study of properties of fluids is basic for the understanding of flow or static condition of fluids. The important properties are density, viscosity, surface tension, bulk modulus and vapour pressure. Viscosity causes resistance to flow. Surface tension leads to capillary effects. Bulk modulus is involved in the propagation of disturbances like sound waves in fluids. Vapour pressure can cause flow disturbances due to evaporation at locations of low pressure. It plays an important role in cavitation studies in fluid machinery.

In this chapter various properties of fluids are discussed in detail, with stress on their effect on flow. Fairly elaborate treatment is attempted due to their importance in engineering applications. The basic laws used in the discussions are :

(i) Newton’s laws of motion,

(ii) Laws of conservation of mass and energy,

(iii) Laws of Thermodynamics, and

(iv) Newton’s law of viscosity.

A fluid is defined as a material which will continue to deform with the application of shear force however small the force may be.

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. In this text, only systems satisfying continuum requirements are discussed.