## Fluids and continuum

**INTRODUCTION: Definition of a fluid: **A fluid is defined as a substance that deforms continuously under the action of a shear stress, however small magnitude present. It means that a fluid deforms under very small shear stress, but a solid may not deform under that magnitude of the shear stress.

By contrast a solid deforms when a constant shear stress is applied, but its deformation does not continue with increasing time. In Fig.1, deformation pattern of a solid and a fluid under the action of constant shear force is illustrated. We explain in detail here deformation behavior of a solid and a fluid under the action of a shear force.

**Introduction to Fluid Mechanics**: In Fig., a shear force F is applied to the upper plate to which the solid has been bonded, a shear stress resulted by the force equals to

Where A is the contact area of the upper plate. We know that in the case of the solid block the deformation is proportional to the shear stress t provided the elastic limit of the solid material is not exceeded.

When a fluid is placed between the plates, the deformation of the fluid element is illustrated in Fig. We can observe the fact that the deformation of the fluid element continues to increase as long as the force is applied. The fluid particles in direct contact with the plates move with the same speed of the plates. This can be interpreted that there is no slip at the boundary. This fluid behavior has been verified in numerous experiments with various kinds of fluid and boundary material.

In short, a fluid continues in motion under the application of a shear stress and cannot sustain any shear stress when at rest.

**Fluid as a continuum: **In the definition of the fluid the molecular structure of the fluid was not mentioned. As we know the fluids are composed of molecules in constant motions. For a liquid, molecules are closely spaced compared with that of a gas. In most engineering applications the average or macroscopic effects of a large number of molecules is considered. We thus do not concern about the behavior of individual molecules. The fluid is treated as an infinitely divisible substance, a continuum at which the properties of the fluid are considered as a continuous (smooth) function of the space variables and time.

To illustrate the concept of fluid as a continuum consider fluid density as a fluid property at a small region.(Fig.(a)). Density is defined as mass of the fluid molecules per unit volume. Thus the mean density within the small region C could be equal to mass of fluid molecules per unit volume. When the small region C occupies space which is larger than the cube of molecular spacing, the number of the molecules will remain constant. This is the limiting volume above which the effect of molecular variations on fluid properties is negligible. A plot of the mean density versus the size of unit volume is illustrated in Fig.(b).

**The density of the fluid is defined as: **

figure(a) figure(b)