The word 'fluid' is the generic title used to encompass the set of materials which cannot support a tangential or shear stress without flow. It therefore includes gases and liquids that may be both Newtonian and non-Newtonian. It follows from this definition that, for such a fluid material, it is possible to write a constitutive equation that relates the stress tensor τ to the rate of strain tensor E so that [Richardson (1989)]

where η itself may depend upon the rate of strain tensor E_{=} (see Flow of Fluids). In the case where η is a constant, this is the constitutive equation of a Newtonian fluid and η is the Viscosity. This class of fluids includes all gases and most liquids. However, non-Newtonian fluids are important in many circumstances and display a wide variety of different behavior since, for some fluids, the viscosity increases with increasing rate of strain (*dilutant fluids*) while for other materials, the viscosity decreases with increasing rate of strain (*pseudoplastic fluids*). Finally, there are materials where the behavior depends upon time such as *thixotropic fluids*, where the viscosity shows a limited decrease with time under a suddenly applied constant shear stress, and others (*viscoelastic*) fluids, where the material partially returns to its original form when the applied stress is removed.

The physical properties of fluids enter the equations of conservation of fluid mechanics through constitutive equations such as that given above for the viscosity and through Fourier's Law of Heat Conduction, and an Equation of State for the **Density** in terms of the Pressure and Temperature. These constitutive equations may sometimes be deduced from statistical mechanics, such as the equation of state for a perfect gas, but more often must be founded upon experimental observations. Furthermore, the physical properties of the fluids themselves are most usually determined empirically since statistical mechanical theory can provide no more than guidance to their evaluation. Direct measurement of the physical properties of all fluids of interest is not possible owing to the shear magnitude of the task, so that very often it is necessary to make use of predictive procedures to evaluate properties that have varying degrees of reliance on rigorous theory.

#### REFERENCES

Richardson, S. M. (1989) *Fluid Mechanics.* Hemisphere, New York.

Bird, R. B., Stewart, W. E., and Lightfoot, E. N. (1960) *Transport Phenomena*, Wiley, New York.

Reid, R. C., Prausnitz, J. M., and Sherwood, T. K. (1977) *The Properties of Gases and Liquids*, 3rd ed. McGraw-Hill, New York.

#### References

- Richardson, S. M. (1989)
*Fluid Mechanics.*Hemisphere, New York. - Bird, R. B., Stewart, W. E., and Lightfoot, E. N. (1960)
*Transport Phenomena*, Wiley, New York. - Reid, R. C., Prausnitz, J. M., and Sherwood, T. K. (1977)
*The Properties of Gases and Liquids*, 3rd ed. McGraw-Hill, New York.